text: given by the hardware capabilities of that worker, that is, if the worker is able to run the speciﬁed service. This restriction on the edges is speciﬁed via a binary matrix H, where R is a ⊆ × ∈ binary relation R W Z that holds iff. a worker w W ∈ i has the hardware capab(cid:40)ilties to run a service zj Z. As a consequence, the entries of H are deﬁned by ∈ 1 (w , z ) R H = i j ∈ (4) i,j 0 (w , z ) / R. i j The corresponding transformation function is a mapping × × → κ : Z W H E. The overall costs a for running a uv service on a worker is given by the workload of the worker and the pre-deﬁned costs for a single container. The overall workload of the worker is composed of the CPU load denoted by (cid:15), the VRAM load denoted by η, the SWAP load denoted Fig. 5. Contour Plots of the Workload Cost Functions - The several x-axes by ζ and the bandwith of the worker denoted by θ, all represent the relative workload, the y-axes represent the pre-deﬁned costs multiplied by a weighting factor δ: per image and the color of the contour plot indicates the resulting cost. · · · · a = (cid:15) δ + η δ + ζ δ + θ δ , (5) uv uv (cid:15) uv η uv ζ uv θ such that the sum of all weights δ equal 1. The single cost than the execution on a single machine, the set of service of each load is given by their related cost functions: vertices is the set of the single services united with the set ∪ ∪ → · of the pooled services such that τ : (W Z Φ) V . We (cid:15)uv = α β4 (CPU load) (6a) assume that k lies in the interval [1, n]. The dependencies of · × η = α β4 (VRAM load) (6b) services are represented via a binary matrix Ξ of size n n, ζuuvv = α · β (SWAP load) (6c) where D is∈a binary relation D ⊆ Z × Z that ho∈lds iff. a · − service z Z depends on a another service z Z. The θ = α (1 β)4 (Bandwidth), (6d) k (cid:40) j uv entries of Ξ are given by ∈ { ∈ R| ≤ ≤ } where α x 0 x 100 is the predeﬁned ∈ { ∈ R| ≤ ≤ } ∈ costs for the service and β x 0 x 1 is 1 (z , z ) D the currently measured relative workload. Fig. 5 illustrates Ξk,j = 0 (zk, zj) ∈/ D. (7) the cost functions as contour plots. It shows that (cid:15) and η k j are progressive cost functions, where the exponent is chosen ∈ ∈ such that the cost increase with a strong degree of progres- The cost of a pooled service φ Φ for a worker w W is given by the sum of cost of their respective containers, sivity in the second half of the workload scale to relieve high-loaded workers. Because of the assumption that a high multiplied by a discount factor δ. The corresponding cell swap indicates a very high workload of some worker, the of ∈the hardware capability ∈matrix H of a pooled container exponent of the cost function ζ is chosen such that the cost φ Φ for a worker w W is the logical conjuction rise earlier than (cid:15) and η. In the case of θ, the cost function is of all entries of H for that service. The entries of the chosen such that the cost decrease along with the available capacity matrix C stays the same as for single services. The problem is solved using the cost-scaling push-relabel (CSPR) bandwith of the worker. If the bandwith tends to zero, the algorithm by Goldberg and Kennedy [15] implemented in the cost increases. The cost calculation algorithm is designed as Google Optimization Toolbox [16]. a distributed algorithm that allows uses parallelization of the V. EVALUATION cost calculation. We assume that m is the total number of agents, n is the total number of services, i lies in the interval The following section discusses the results of the eval- [1, m] and j lies in the interval [1, n]. The cost per worker uation. To reveal possible differences between system ar- and×per service are represented as a cost matrix A of size chitectures, the experiments were performed using an x64- m n, where the entries of A are deﬁned by Ai,j = auv. based system consisting of an Intel Core i5-6200U with Each worker can run exactly one container, which implies 2.30 GHz, 4096 MB LPDDR3 and IEEE 802.11ac-WLAN that each capacity is given by cuv = 1. T×he overall capacities connection and an ARM-based system consisting of an ARM are represented as a matrix C of size m n, where the entries Cortex-A7 with 900 MHz, 1024MB DDR2-SDRAM and a of C are deﬁned by Ci,j = cuv. IEEE 802.3-10/100-Mbit/s connection. The evaluation of the In order to enable that two or more services can be service allocation algorithm as well as the overall system executed on one worker, the system supports pooled services ⊂ P evaluation were performed using the previously described Φ, where Φ (Z). To take into account that in some cases ARM-based system composed as a 12-node cluster. (e.g. when all machines have a high workload) the execution A. Container Performance of a pooled service on different machines is more favorable ##LTFigure##

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text: The

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text: V.

text: EVALUATION

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text: m]

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text: interval

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text: The

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text: To

text: reveal

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text: and×per

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text: experiments

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text: computed with the widely available GMM package. We propose two data- based methods for selection of the number of moments. A small scale sim- ulation study reveals that the proposed estimation indeed out-performs the existing alternatives in ﬁnite samples. Keywords: Nonignorable nonresponse; Generalized method of moments; Semi- parametric eﬃciency. 1 Introduction Missing data is common in many ﬁelds of applications. One way to deal with the missing data problem is to delete observations containing missing data. In doing so we may produce biased estimates and erroneous conclusions, depend- ing on the data missing mechanism. If data are missing completely at random, standard estimation and inference procedures are still consistent when the missing data observations are ignored, see Heitjan and Basu (1996), Little (1988) among others. If data are missing at random (MAR) in the sense that the propensity of missingness depends only on the observed covariates, consistent estimation can still be obtained through covariate balancing, see Rubin (1976a,b), Little and Ru- bin (1989), Robins and Rotnitzky (1995), Robins et al. (1995), Bang and Robins (2005), Qin and Zhang (2007), Chen et al. (2008), Tan (2010), Rotnitzky et al. (2012), Little and Rubin (2014) among others. In many applications, data are missing not at random (MNAR). For example, the income question in sample sur- veys is often not answered by people at the top end of the distribution, that is, their response frequency depends on an outcome variable that is often the key focus. An investigator is examining the eﬀect of sleep on pain by calling subjects daily to ask them about last night’s sleep and their pain today. Patients who are experiencing severe pain are more likely to not come to the phone leaving the data missing for that particular day; again this would violate the MAR assump- tion. From political science, roll-call votes, which measure legislatures ideological positions, are subject to non-ignorable nonresponse because, unsurprisingly, politi- cians behave strategically. In the MNAR case, the parameter of interest may not even be identiﬁed (e.g., Robins et al. (1997)), let alone be consistently estimated. ∈ { } To be more speciﬁc, let T 0, 1 denote the binary random variable indicating the missing status of the outcome variable Y : Y is observed if T takes the value one and Y is not observed if T takes the value zero. Let X denote a vector of P | explanatory variables, let π(x, y) = (T = 1 X = x, Y = y) denote the propen- | sity score function and let f | (y x) denote the conditional density function of Y Y X given X. Robins et al. (1997) shows that if both the propensity score function and the conditional density function are completely unknown, the joint distribution of (T, Y ) given X is not point identiﬁable. In this case, a necessary identiﬁcation 2

text: computed

label: abstract

text: with

text: the

text: widely

text: available

text: GMM

text: package.

text: We

text: propose

text: two

text: data-

text: based

text: methods

text: for

text: selection

text: of

text: the

text: number

text: of

text: moments.

text: A

text: small

text: scale

text: sim-

text: ulation

text: study

text: reveals

text: that

text: the

text: proposed

text: estimation

text: indeed

text: out-performs

text: the

text: existing

text: alternatives

text: in

text: ﬁnite

text: samples.

text: Keywords:

label: paragraph

text: Nonignorable

text: nonresponse;

text: Generalized

text: method

text: of

text: moments;

text: Semi-

text: parametric

text: eﬃciency.

label: section

text: Introduction

text: Missing

text: data

text: is

text: common

text: in

text: many

text: ﬁelds

text: of

text: applications.

text: One

text: way

text: to

text: deal

text: with

text: the

text: missing

text: data

text: problem

text: is

text: to

text: delete

text: observations

text: containing

text: missing

text: data.

text: In

text: doing

text: so

text: we

text: may

text: produce

text: biased

text: estimates

text: and

text: erroneous

text: conclusions,

text: depend-

text: ing

text: on

text: the

text: data

text: missing

text: mechanism.

text: If

text: data

text: are

text: missing

text: completely

text: at

text: random,

text: standard

text: estimation

text: and

text: inference

text: procedures

text: are

text: still

text: consistent

text: when

text: the

text: missing

text: data

text: observations

text: are

text: ignored,

text: see

text: Heitjan

text: and

text: Basu

text: Little

text: among

text: others.

text: If

text: data

text: are

text: missing

text: at

text: random

text: (MAR)

text: in

text: the

text: sense

text: that

text: the

text: propensity

text: of

text: missingness

text: depends

text: only

text: on

text: the

text: observed

text: covariates,

text: consistent

text: estimation

text: can

text: still

text: be

text: obtained

text: through

text: covariate

text: balancing,

text: see

text: Rubin

text: Little

text: and

text: Ru-

text: bin

text: Robins

text: and

text: Rotnitzky

text: Robins

text: et

text: al.

text: Bang

text: and

text: Robins

text: Qin

text: and

text: Zhang

text: Chen

text: et

text: al.

text: Tan

text: Rotnitzky

text: et

text: al.

text: Little

text: and

text: Rubin

text: among

text: others.

text: In

text: many

text: applications,

text: data

text: are

text: missing

text: not

text: at

text: random

text: (MNAR).

text: For

text: example,

text: the

text: income

text: question

text: in

text: sample

text: sur-

text: veys

text: is

text: often

text: not

text: answered

text: by

text: people

text: at

text: the

text: top

text: end

text: of

text: the

text: distribution,

text: that

text: is,

text: their

text: response

text: frequency

text: depends

text: on

text: an

text: outcome

text: variable

text: that

text: is

text: often

text: the

text: key

text: focus.

text: An

text: investigator

text: is

text: examining

text: the

text: eﬀect

text: of

text: sleep

text: on

text: pain

text: by

text: calling

text: subjects

text: daily

text: to

text: ask

text: them

text: about

text: last

text: night’s

text: sleep

text: and

text: their

text: pain

text: today.

text: Patients

text: who

text: are

text: experiencing

text: severe

text: pain

text: are

text: more

text: likely

text: to

text: not

text: come

text: to

text: the

text: phone

text: leaving

text: the

text: data

text: missing

text: for

text: that

text: particular

text: day;

text: again

text: this

text: would

text: violate

text: the

text: MAR

text: assump-

text: tion.

text: From

text: political

text: science,

text: roll-call

text: votes,

text: which

text: measure

text: legislatures

text: ideological

text: positions,

text: are

text: subject

text: to

text: non-ignorable

text: nonresponse

text: because,

text: unsurprisingly,

text: politi-

text: cians

text: behave

text: strategically.

text: In

text: the

text: MNAR

text: case,

text: the

text: parameter

text: of

text: interest

text: may

text: not

text: even

text: be

text: identiﬁed

text: (e.g.,

text: Robins

text: et

text: al.

text: let

text: alone

text: be

text: consistently

text: estimated.

text: To

text: be

text: more

text: speciﬁc,

text: let

text: T

text: denote

text: the

text: binary

text: random

text: variable

text: indicating

text: the

text: missing

text: status

text: of

text: the

text: outcome

text: variable

text: Y

text: Y

text: is

text: observed

text: if

text: T

text: takes

text: the

text: value

text: one

text: and

text: Y

text: is

text: not

text: observed

text: if

text: T

text: takes

text: the

text: value

text: zero.

text: Let

text: X

text: denote

text: a

text: vector

text: of

text: P

text: explanatory

text: variables,

text: let

text: π(x,

text: y)

text: (T

text: X

text: x,

text: Y

text: y)

text: denote

text: the

text: propen-

text: sity

text: score

text: function

text: and

text: let

text: f

text: (y

text: x)

text: denote

text: the

text: conditional

text: density

text: function

text: of

text: Y

text: Y

text: X

text: given

text: X.

text: Robins

text: et

text: al.

text: shows

text: that

text: if

text: both

text: the

text: propensity

text: score

text: function

text: and

text: the

text: conditional

text: density

text: function

text: are

text: completely

text: unknown,

text: the

text: joint

text: distribution

text: of

text: (T,

text: Y

text: given

text: X

text: is

text: not

text: point

text: identiﬁable.

text: In

text: this

text: case,

text: a

text: necessary

text: identiﬁcation


text: • Scenario IV. The design is similar to that in Kang and Schafer (2007). Z = (Z , Z ) is generated from the standard bivariate normal distribution, 1 2 and Y is generated from the normal distribution with mean 2 + Z and unit 1 variance. The missing mechanism is modeled as P | − (T = 1 Y, X , X ) = [1 + exp(α Z + β Y )] 1 1 2 0 1 0 − with (α , β ) = (1, 1). The true value of the parameter of interest is θ = 0 0 0 E [Y ] = 2. Instead of directly observing covariates Z, we observe a non-linear transformation of Z: X = exp(Z /2) and X = Z /(1 + exp(Z )). 1 1 2 2 1 In all scenarios, we generate J = 500 random samples, and for each sample, we compute the following three estimators: 1. Naive estimator. We compute t(cid:88)he missing at random estimator (α˜ , β˜ , θ˜ ) MAR MAR MAR as N 1 T ˜ i θ = Y , MAR N (cid:104)π(X ; α˜ , β˜ ) i (cid:105) i=1 i MAR MAR ˜ where π(X ; α˜ , β ) is an estimated response model. In Scenarios I, i MAR MAR (cid:104) − (cid:105) 1 ˜ ˜ II & III, π(X ; α˜ , β ) = 1 + exp(α˜ + β X ) and in Sce- i MAR MAR MAR MAR i − 1 ˜ ˜ nario IV π(X ; α˜ , β ) = 1 + exp(α˜ Z + β X ) , where i MAR MAR MAR 1i MAR 2i ˜ (α˜ , β ) are estimated by GMM. MAR MAR ˆ ˆ 2. MK2 estimator. We compute (αˆ , β , θ ) using the approach of Morikawa (cid:88) (cid:16) ˆ MKˆ MK MK (cid:17) and Kim (2016), i.e. (αˆ , β , θ ) is the solution of MK MK MK N (cid:62) (cid:62) ˆ ˆ S (T , Z ; α, β) , S (T , Z ; α, β, θ) = 0 , 1 i i 2 i i i=1 (cid:18) (cid:19) (cid:20) (cid:12) (cid:21) (cid:12) (cid:12) where (cid:12) ∇ (cid:18) (cid:19) − − T E π(Z; α, β) Sˆ (T, Z; α, β) = 1 (cid:63) −γ X , 1 π(Z; α, β) 1 π(Z; α, β) − T − − T E | Sˆ (T, Z; α, β, θ) = U(Z) + θ 1 (cid:63) [U(Z) X] , 2 π(Z; α, β) π(Z; α, β) (cid:80) E | and for any function g(Z) the quantity (cid:63)[g(Z) X] is deﬁned by (cid:80) − − E(cid:63)[g(Z)|X = x] := Nj=1 TjKh(x X−j)Tjπ(Zj; α, β) 1O−(x, Yj; α, β)g(x, Yj) ; N K (x X )T π(Z ; α, β) 1O(x, Y ; α, β) − j=1 h j j j j 1 π(z; α, β) O(z; α, β) = , π(z; α, β) 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: Scenario

text: IV.

text: The

text: design

text: is

text: similar

text: to

text: that

text: in

text: Kang

text: and

text: Schafer

text: Z

text: (Z

text: Z

text: is

text: generated

text: from

text: the

text: standard

text: bivariate

text: normal

text: distribution,

text: and

text: Y

text: is

text: generated

text: from

text: the

text: normal

text: distribution

text: with

text: mean

text: Z

text: and

text: unit

text: variance.

text: The

text: missing

text: mechanism

text: is

text: modeled

text: as

text: P

text: (T

text: Y,

text: X

text: X

text: exp(α

text: Z

text: β

text: Y

text: with

text: (α

text: β

text: The

text: true

text: value

text: of

text: the

text: parameter

text: of

text: interest

text: is

text: θ

text: E

text: [Y

text: Instead

text: of

text: directly

text: observing

text: covariates

text: Z,

text: we

text: observe

text: a

text: non-linear

text: transformation

text: of

text: Z:

text: X

text: exp(Z

text: and

text: X

text: Z

text: exp(Z

text: In

text: all

text: scenarios,

text: we

text: generate

text: J

text: random

text: samples,

text: and

text: for

text: each

text: sample,

text: we

text: compute

text: the

text: following

text: three

text: estimators:

text: Naive

text: estimator.

text: We

text: compute

text: missing

text: at

text: random

text: estimator

text: (α˜

text: β˜

text: θ˜

text: MAR

text: MAR

text: MAR

text: as

text: N

text: T

text: i

text: θ

text: Y

text: MAR

text: N

text: α˜

text: β˜

text: i

text: i

text: MAR

text: MAR

text: where

text: π(X

text: α˜

text: β

text: is

text: an

text: estimated

text: response

text: model.

text: In

text: Scenarios

text: I,

text: i

text: MAR

text: MAR

text: II

text: III,

text: π(X

text: α˜

text: β

text: exp(α˜

text: β

text: X

text: and

text: in

text: Sce-

text: i

text: MAR

text: MAR

text: MAR

text: MAR

text: i

text: nario

text: IV

text: π(X

text: α˜

text: β

text: exp(α˜

text: Z

text: β

text: X

text: where

text: i

text: MAR

text: MAR

text: MAR

text: MAR

text: (α˜

text: β

text: are

text: estimated

text: by

text: GMM.

text: MAR

text: MAR

text: ˆ

text: ˆ

text: estimator.

text: We

text: compute

text: (αˆ

text: β

text: θ

text: using

text: the

text: approach

text: of

text: Morikawa

text: ˆ

text: MKˆ

text: MK

text: MK

text: and

text: Kim

text: i.e.

text: (αˆ

text: β

text: θ

text: is

text: the

text: solution

text: of

text: MK

text: MK

text: MK

text: N

text: ˆ

text: ˆ

text: S

text: (T

text: Z

text: α,

text: β)

text: S

text: (T

text: Z

text: α,

text: β,

text: θ)

text: i

text: i

text: i

text: i

text: where

text: T

text: E

text: π(Z;

text: α,

text: β)

text: Sˆ

text: (T,

text: Z;

text: α,

text: β)

text: −γ

text: X

text: π(Z;

text: α,

text: β)

text: π(Z;

text: α,

text: β)

text: T

text: T

text: E

text: Sˆ

text: (T,

text: Z;

text: α,

text: β,

text: θ)

text: U(Z)

text: θ

text: [U(Z)

text: X]

text: π(Z;

text: α,

text: β)

text: π(Z;

text: α,

text: β)

text: E

text: and

text: for

text: any

text: function

text: g(Z)

text: the

text: quantity

text: X]

text: is

text: deﬁned

text: by

text: x]

text: TjKh(x

text: X−j)Tjπ(Zj;

text: α,

text: β)

text: Yj;

text: α,

text: β)g(x,

text: Yj)

text: N

text: K

text: (x

text: X

text: )T

text: π(Z

text: α,

text: β)

text: Y

text: α,

text: β)

text: h

text: j

text: j

text: j

text: j

text: π(z;

text: α,

text: β)

text: O(z;

text: α,

text: β)

text: π(z;

text: α,

text: β)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:16)(cid:104) (cid:105) (cid:17) (cid:62) − (cid:62) − where g (T, Z; γ, θ) := 1 T u (X) , θ T U(Z) . The GMM K π(Z;γ) K π(Z;γ) estimator of γ and θ is deﬁned as 0 0 · · (γˇ, θˇ) = arg min G (γ, θ)T W G (γ, θ) ∈ ∈ K K γ Γ,θ Θ × ≥ where W is a (K+1) (K+1) symmetric weighting matrix. For every ﬁxed K p, Hansen (1982) shows that, under some regularity conditions, the estimator − − − (γˇ γ , θˇ θ ) = O (N 1/2) (5) 0 0 p is asymptotically normally distributed, but generally not the best unless the best (cid:2) (cid:3) weighting matrix is used. The best weighting matrix is the inverse of E (cid:62) D × := g (T, Z; γ , θ )g (T, Z; γ , θ ) . (K+1) (K+1) K 0 0 K 0 0 The best estimator (within the class deﬁned by the speciﬁc unconditional mo- ments) is deﬁned as · − · (γ, θ) = arg min G (γ, θ)T D 1 × G (γ, θ). ∈ ∈ K (K+1) (K+1) K γ Γ,θ Θ Suppose that the propensity score function is (cid:104)diﬀerentiable with(cid:105) respect to γ. Denote (cid:20) (cid:21)    E (cid:104) ∇ (cid:62)(cid:105)  u (X) γπ(Z;γ0) , 0 × ∇ E 1 K π(Z;γ0) K 1 B × = G (γ , θ ) = (K+1) (p+1) γ,θ N K 0 0 E ∇ (cid:62) U(Z) γπ(Z;γ0) , 1 π(Z;γ0) (cid:110) (cid:111) (cid:0) (cid:1) (cid:0) (cid:1) and − (cid:62) − 1 V = B × D 1 × B × . K (K+1) (p+1) (K+1) (K+1) (K+1) (p+1) (cid:18) (cid:19) ≥ Hansen (1982) shows that, for every ﬁx(cid:0)ed K p, (cid:1) √ − − √N(γ γ ) → V 1/2 − 0 N 0, I × in distribution. (6) K N(θ θ ) (p+1) (p+1) 0 Since the best weighting matrix depends on the unknown parameter value, the best estimator (γ, θ) is infeasible. Hansen (1982) suggests the following two-step procedure: 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: g

text: (T,

text: Z;

text: γ,

text: θ)

text: T

text: u

text: (X)

text: θ

text: T

text: U(Z)

text: The

text: GMM

text: K

text: π(Z;γ)

text: K

text: π(Z;γ)

text: estimator

text: of

text: γ

text: and

text: θ

text: is

text: deﬁned

text: as

text: (γˇ,

text: θˇ)

text: arg

text: min

text: G

text: (γ,

text: θ)T

text: W

text: G

text: (γ,

text: θ)

text: K

text: K

text: γ

text: Γ,θ

text: Θ

text: where

text: W

text: is

text: a

text: symmetric

text: weighting

text: matrix.

text: For

text: every

text: ﬁxed

text: K

text: p,

text: Hansen

text: shows

text: that,

text: under

text: some

text: regularity

text: conditions,

text: the

text: estimator

label: equation

text: (γˇ

text: γ

text: θˇ

text: θ

text: O

text: (N

text: p

text: is

text: asymptotically

text: normally

text: distributed,

text: but

text: generally

text: not

text: the

text: best

text: unless

text: the

text: best

text: weighting

text: matrix

text: is

text: used.

text: The

text: best

text: weighting

text: matrix

text: is

text: the

text: inverse

text: of

text: E

text: D

text: g

text: (T,

text: Z;

text: γ

text: θ

text: )g

text: (T,

text: Z;

text: γ

text: θ

text: K

text: K

text: The

text: best

text: estimator

text: (within

text: the

text: class

text: deﬁned

text: by

text: the

text: speciﬁc

text: unconditional

text: mo-

text: ments)

text: is

text: deﬁned

text: as

text: (γ,

text: θ)

text: arg

text: min

text: G

text: (γ,

text: θ)T

text: D

text: G

text: (γ,

text: θ).

text: K

text: K

text: γ

text: Γ,θ

text: Θ

text: Suppose

text: that

text: the

text: propensity

text: score

text: function

text: is

text: respect

text: to

text: γ.

text: Denote

text: E

text: u

text: (X)

text: E

text: K

text: K

text: B

text: G

text: (γ

text: θ

text: γ,θ

text: N

text: K

text: E

text: U(Z)

text: and

text: V

text: B

text: D

text: B

text: K

text: Hansen

text: shows

text: that,

text: for

text: every

text: K

text: p,

text: √N(γ

text: γ

text: V

text: N

text: I

text: in

text: distribution.

text: K

text: N(θ

text: θ

text: Since

text: the

text: best

text: weighting

text: matrix

text: depends

text: on

text: the

text: unknown

text: parameter

text: value,

text: the

text: best

text: estimator

text: (γ,

text: θ)

text: is

text: infeasible.

text: Hansen

text: suggests

text: the

text: following

text: two-step

text: procedure:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: √ (cid:32) (cid:80) (cid:33) Step I. Compute the initial N-consistent estimator (cid:99) (cid:62) 1 N u (X )u (X ) 0 × W 0 := N i=1 K(cid:62) i K i (cid:99) K 1 , 0 × 1 K 1 · − · (γˇ, θˇ) = arg min G (γ, θ)T W 1 G (γ, θ). ∈ × K 0 K (γ,θ) Γ Θ Step II. Compute the bes(cid:88)t weighting matrix and the best estimator N 1 (cid:62) Dˆ × := g (T , Z ; γˇ, θˇ)g (T , Z ; γˇ, θˇ) , (K+1) (K+1) K i i K i i (cid:98) N (cid:98) (cid:98) i=1 · − · 1 (γ, θ) = arg min G (γ, θ)T D × G (γ, θ). ∈ ∈ K (K+1) (K+1) K γ Γ,θ Θ (cid:18) (cid:19) ≥ Hansen (1982) establ(cid:98)ishes that, for (cid:0)every ﬁxed K (cid:1) p, √ (cid:98) − − √N(γ γ ) → V 1/2 − 0 N 0, I × in distribution. (7) K N(θ θ ) (p+1) (p+1) 0  (cid:80) Moreover, denote  (cid:98)  (cid:98)  − (cid:80) ∇ (cid:98) (cid:62)  N 1 N u (X ) γπ(Zi;γ) , 0 × i=1 K i π(Zi;(cid:98)γ) K 1 B × := (K+1) (p+1) − ∇ (cid:98) (cid:62) N 1 N U(Z ) γπ(Zi;γ) , 1 i=1 i π(Zi;γ) (cid:26) (cid:27) (cid:16) (cid:17) (cid:16) (cid:17) and (cid:98) (cid:98) (cid:98) (cid:98) − (cid:62) − 1 1 V := B × D × B × . K (K+1) (p+1) (K+1) (K+1) (K+1) (p+1) ≥ Hansen (1982) proves that, f(cid:98)or every ﬁxed K p, → V V in probability. (8) K K The best estimator (within the class deﬁned by the speciﬁc unconditional mo- ments) is generally not semiparametrically eﬃcient. To obtain the eﬃcient estima- tor, we shall allow K to increase with the sample size at the rate o(N1/3) so that { } u (X) span the space of measureable functions (see also Geman and Hwang K (1982) and Newey (1997)). In the next two sections, we shall establish that results in (5)-(8) still hold with expanding K = o(N1/3). 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Step

text: I.

text: Compute

text: the

text: initial

text: N-consistent

text: estimator

text: N

text: u

text: (X

text: )u

text: (X

text: W

text: N

text: i

text: K

text: i

text: K

text: K

text: (γˇ,

text: θˇ)

text: arg

text: min

text: G

text: (γ,

text: θ)T

text: W

text: G

text: (γ,

text: θ).

text: K

text: K

text: (γ,θ)

text: Γ

text: Θ

text: Step

text: II.

text: Compute

text: the

text: weighting

text: matrix

text: and

text: the

text: best

text: estimator

text: N

text: Dˆ

text: g

text: (T

text: Z

text: γˇ,

text: θˇ)g

text: (T

text: Z

text: γˇ,

text: θˇ)

text: K

text: i

text: i

text: K

text: i

text: i

text: N

text: (γ,

text: θ)

text: arg

text: min

text: G

text: (γ,

text: θ)T

text: D

text: G

text: (γ,

text: θ).

text: K

text: K

text: γ

text: Γ,θ

text: Θ

text: Hansen

text: that,

text: for

text: ﬁxed

text: K

text: p,

text: √N(γ

text: γ

text: V

text: N

text: I

text: in

text: distribution.

text: K

text: N(θ

text: θ

text: 0

text: Moreover,

text: denote

text: N

text: N

text: u

text: (X

text: γπ(Zi;γ)

text: K

text: i

text: K

text: B

text: N

text: N

text: U(Z

text: γπ(Zi;γ)

text: i

text: π(Zi;γ)

text: and

text: V

text: B

text: D

text: B

text: K

text: Hansen

text: proves

text: that,

text: every

text: ﬁxed

text: K

text: p,

text: V

text: V

text: in

text: probability.

text: K

text: K

text: The

text: best

text: estimator

text: (within

text: the

text: class

text: deﬁned

text: by

text: the

text: speciﬁc

text: unconditional

text: mo-

text: ments)

text: is

text: generally

text: not

text: semiparametrically

text: eﬃcient.

text: To

text: obtain

text: the

text: eﬃcient

text: estima-

text: tor,

text: we

text: shall

text: allow

text: K

text: to

text: increase

text: with

text: the

text: sample

text: size

text: at

text: the

text: rate

text: so

text: that

text: u

text: (X)

text: span

text: the

text: space

text: of

text: measureable

text: functions

text: (see

text: also

text: Geman

text: and

text: Hwang

text: K

text: and

text: Newey

text: In

text: the

text: next

text: two

text: sections,

text: we

text: shall

text: establish

text: that

text: results

text: in

text: still

text: hold

text: with

text: expanding

text: K

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Determining the projection constants of univariate polynomial spaces (cid:40)(cid:12) (cid:12) (cid:12) (cid:12)(cid:41) (cid:88) (cid:12)(cid:88) (cid:12) (cid:12)(cid:88) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) U ≤ × K Me Mo (55) λ( ) ρ Ae/o∈mRMine/o×K (cid:96)∈m(cid:74)1a:xL(cid:75) k=1 max (cid:88)m=1 Aem(cid:101),kuem(w(cid:96)+) , m=1 Aom,kuom(w(cid:96)+) (cid:74) (cid:75) K (cid:48) ∈ s.to Ae/o ue/(cid:48)o(v ) = δ (cid:48), m, m 1:M , m,k m k m,m e/o k=1 − where ρ = cos ((πd)/(4L)) 1. To transform the latter into a linear program, we introduce the ∈ R × ∈ R × collocation mat(cid:101)rices V e/o K Me/o and W e/o L Me/o deﬁned by (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) ∈ ∈ ∈ (56) V e/o = ue/o(v ) and W e/o = ue/o(w+), k 1:K , (cid:96) 1:L , m 1:M , k,m m k (cid:96),m m (cid:96) e/o ∈ R × ∈ R a(cid:39)s well as slack variables through B L K and c . All in all, we obtain the following(cid:36) implementable form of the upper bound. Upper bound for the projection constant — symmetry exploited  U ≤ Inputs: basis for a sym(cid:18)metr(cid:19)ic space of polynomials of degree d, parameters S > d and L. − Ae/oV e/o = I , U ≤ π d 1 × − ≤ Me/o≤ (57) λ( ) cos min c s.to B W e/oAe/o B, 4 L Ae/o∈∈RRM×e/o×K B1 ≤ c1, B ∈LR K (cid:38) c (cid:37) where the matrices V e/o and W e/o are deﬁned in (56). 4.2 Implication for the lower bound As in Section , the minimization program (46) is ﬁrst transformed to make it amenable to the trigonometric moment problem. The reformulation will involve a maximum over [0, π/2], which is ∈ lower bounded by the maximum over a grid θ+, . . . , θ+. With w+ := cos(θ+) [0, 1], we obtain 1 L (cid:96) (cid:96) (cid:40)(cid:12) (cid:12) (cid:12) (cid:12)(cid:41) (cid:90) (cid:12)(cid:88) (cid:12) (cid:12)(cid:88) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:101) (cid:12) (cid:12) (cid:101) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) U ≥ (cid:101) (cid:101) π Me Mo (58) λ( ) µµ(cid:101)o1e1,,..i..n..,,µµf(cid:101)oMeMoe (cid:96)∈m(cid:74)1a:xL(cid:75) 0 max (cid:90) m(cid:101)=1 uem(w(cid:96)+)dµ(cid:101)em , m=1 uom(w(cid:96)+)dµom (cid:74) (cid:75) π (cid:48) ∈ s.to ue/(cid:48)o(cos(θ))dµe/o(θ) = δ (cid:48), m, m 1:M , m m m,m e/o 0 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Determining

text: the

text: projection

text: constants

text: of

text: univariate

text: polynomial

text: spaces

text: U

text: K

text: Me

text: Mo

text: λ(

text: ρ

text: Ae/o∈mRMine/o×K

text: max

text: K

text: s.to

text: Ae/o

text: δ

text: m,

text: m

text: m,k

text: m

text: k

text: m,m

text: e/o

text: where

text: ρ

text: cos

text: To

text: transform

text: the

text: latter

text: into

text: a

text: linear

text: program,

text: we

text: introduce

text: the

text: R

text: R

text: collocation

text: V

text: e/o

text: K

text: Me/o

text: and

text: W

text: e/o

text: L

text: Me/o

text: deﬁned

text: by

text: V

text: e/o

text: ue/o(v

text: and

text: W

text: e/o

text: ue/o(w+),

text: k

text: m

text: k,m

text: m

text: k

text: m

text: e/o

text: R

text: R

text: well

text: as

text: slack

text: variables

text: through

text: B

text: L

text: K

text: and

text: c

text: All

text: in

text: all,

text: we

text: obtain

text: the

text: implementable

text: form

text: of

text: the

text: upper

text: bound.

text: Upper

text: bound

text: for

text: the

text: projection

text: constant

text: symmetry

text: exploited

text: U

text: Inputs:

text: basis

text: for

text: a

text: space

text: of

text: polynomials

text: of

text: degree

text: d,

text: parameters

text: S

text: d

text: and

text: L.

text: Ae/oV

text: e/o

text: I

text: U

text: π

text: d

text: Me/o≤

text: λ(

text: cos

text: min

text: c

text: s.to

text: B

text: W

text: e/oAe/o

text: B,

text: L

text: Ae/o∈∈RRM×e/o×K

text: B

text: ∈LR

text: K

text: c

text: where

text: the

text: matrices

text: V

text: e/o

text: and

text: W

text: e/o

text: are

text: deﬁned

text: in

text: Implication

text: for

text: the

text: lower

text: bound

text: As

text: in

text: Section

text: the

text: minimization

text: program

text: is

text: ﬁrst

text: transformed

text: to

text: make

text: it

text: amenable

text: to

text: the

text: trigonometric

text: moment

text: problem.

text: The

text: reformulation

text: will

text: involve

text: a

text: maximum

text: over

text: which

text: is

text: lower

text: bounded

text: by

text: the

text: maximum

text: over

text: a

text: grid

text: θ+,

text: θ+.

text: With

text: w+

text: cos(θ+)

text: we

text: obtain

text: L

text: U

text: π

text: Me

text: Mo

text: λ(

text: max

text: π

text: s.to

text: δ

text: m,

text: m

text: m

text: m

text: m,m

text: e/o

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Determining the projection constants of univariate polynomial spaces lower bounds to match up to the desired accuracy. Five digits of accuracy can typically be achieved for three-dimensional spaces of moderate degree. The following table summarizes the values of projection constants obtained for several diﬀerent spaces.8 V { } { } { } { } { } { } spanned by 1, x, x3 T , T , T U , U , U x, x2, x3 T , T , T U , U , U 0 2 3 0 2 3 1 2 3 1 2 3 V ≈ ≈ ≈ ≈ ≈ ≈ λ( ) 1.4723 1.4460 1.1522 1.3325 1.4065 1.2354 Table 1: Projection constants of three-dimensional spaces spanned by monomials and Chebyshev polynomials of the ﬁrst and second kind. 5.3 Polynomial spaces of higher dimensions Looking at (57) and (63), we see that the dimension M = M + M is far from being a factor e o inﬂuencing the computational cost of the optimization programs, so our code can easily be executed for polynomial spaces of dimension higher than three. For instance, we can deal with the spaces of cubic, quartic, and quintic polynomials and compute their projection constants with four digits of accuracy as P ≈ P ≈ P ≈ (70) λ( ) 1.365, λ( ) 1.459, λ( ) 1.538. 3 4 5 Obtaining the same accuracy necessitates larger parameters K, L, S when the dimension increases. For degree d > 5, with our modest computational investment, we could locate the projection P constants of the spaces of polynomials of degree at most d in the ranges presented in Table 2 d below. The improvement with respect to the ranges found in [10] is particularly noticeable for the lower bounds (recall that this is where the method of moments came into the picture). Let us now come to a close by examining the approximate minimal projection onto the cubics obtained by solving (57). For the measures µ , µ , µ , µ associated with the functionals dual to − 1 2 3 4 1, x, x2, x3, we detected atoms at 1 and 1 (but none at 0) and Figure 2 indicates that their − continuous parts seem to possess a common support strictly included in [ 1, 1], as was the case for quadratics. But a disparity with the quadratics now occurs in terms of shape preservation. It was P conjectured in [13], and proved for d = 2, that (one of the) minimal projections onto preserve C − P d d-convexity. This means that if P is a minimal projection from [ 1, 1] onto , then, for any ∈ C − d f d[ 1, 1], (cid:80) ≥ − ⇒ ≥ − (71) f(d) 0 on [ 1, 1] =? (P (f))(d) 0 on [ 1, 1], − or equivalently, writing P (f)(x) = d+1 η (f)xm 1, m=1 m ≥ − ⇒ ≥ (72) f(d) 0 on [ 1, 1] =? η (f) 0. d+1 { } 8For the space span T , T , T , it was more eﬀective to compute the lower bound using a ‘symmetrization’ of (42) 1 2 3 more direct than (63). Its implementation is also included in the reproducible. 18 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Determining

text: the

text: projection

text: constants

text: of

text: univariate

text: polynomial

text: spaces

text: lower

text: bounds

text: to

text: match

text: up

text: to

text: the

text: desired

text: accuracy.

text: Five

text: digits

text: of

text: accuracy

text: can

text: typically

text: be

text: achieved

text: for

text: three-dimensional

text: spaces

text: of

text: moderate

text: degree.

text: The

text: following

text: table

text: summarizes

text: the

text: values

text: of

text: projection

text: constants

text: obtained

text: for

text: several

text: diﬀerent

text: V

label: table

text: spanned

text: by

text: x,

text: T

text: T

text: T

text: U

text: U

text: U

text: x,

text: T

text: T

text: T

text: U

text: U

text: U

text: V

text: λ(

text: Table

text: Projection

text: constants

text: of

text: three-dimensional

text: spaces

text: spanned

text: by

text: monomials

text: and

text: Chebyshev

text: polynomials

text: of

text: the

text: ﬁrst

text: and

text: second

text: kind.

text: Polynomial

text: spaces

text: of

text: higher

text: dimensions

text: Looking

text: at

text: and

text: we

text: see

text: that

text: the

text: dimension

text: M

text: M

text: M

text: is

text: far

text: from

text: being

text: a

text: factor

text: e

text: o

text: inﬂuencing

text: the

text: computational

text: cost

text: of

text: the

text: optimization

text: programs,

text: so

text: our

text: code

text: can

text: easily

text: be

text: executed

text: for

text: polynomial

text: spaces

text: of

text: dimension

text: higher

text: than

text: three.

text: For

text: instance,

text: we

text: can

text: deal

text: with

text: the

text: spaces

text: of

text: cubic,

text: quartic,

text: and

text: quintic

text: polynomials

text: and

text: compute

text: their

text: projection

text: constants

text: with

text: four

text: digits

text: of

text: accuracy

text: as

text: P

text: P

text: P

text: λ(

text: λ(

text: λ(

text: Obtaining

text: the

text: same

text: accuracy

text: necessitates

text: larger

text: parameters

text: K,

text: L,

text: S

text: when

text: the

text: dimension

text: increases.

text: For

text: degree

text: d

text: with

text: our

text: modest

text: computational

text: investment,

text: we

text: could

text: locate

text: the

text: projection

text: P

text: constants

text: of

text: the

text: spaces

text: of

text: polynomials

text: of

text: degree

text: at

text: most

text: d

text: in

text: the

text: ranges

text: presented

text: in

text: Table

text: d

text: below.

text: The

text: improvement

text: with

text: respect

text: to

text: the

text: ranges

text: found

text: in

text: is

text: particularly

text: noticeable

text: for

text: the

text: lower

text: bounds

text: (recall

text: that

text: this

text: is

text: where

text: the

text: method

text: of

text: moments

text: came

text: into

text: the

text: picture).

text: Let

text: us

text: now

text: come

text: to

text: a

text: close

text: by

text: examining

text: the

text: approximate

text: minimal

text: projection

text: onto

text: the

text: cubics

text: obtained

text: by

text: solving

text: For

text: the

text: measures

text: µ

text: µ

text: µ

text: µ

text: associated

text: with

text: the

text: functionals

text: dual

text: to

text: x,

text: we

text: detected

text: atoms

text: at

text: and

text: (but

text: at

text: and

text: Figure

text: indicates

text: that

text: their

text: continuous

text: parts

text: seem

text: to

text: possess

text: a

text: common

text: support

text: strictly

text: included

text: in

text: as

text: was

text: the

text: case

text: for

text: quadratics.

text: But

text: a

text: disparity

text: with

text: the

text: quadratics

text: now

text: occurs

text: in

text: terms

text: of

text: shape

text: preservation.

text: It

text: was

text: P

text: conjectured

text: in

text: and

text: proved

text: for

text: d

text: that

text: (one

text: of

text: the)

text: minimal

text: projections

text: onto

text: preserve

text: C

text: P

text: d

text: d-convexity.

text: This

text: means

text: that

text: if

text: P

text: is

text: a

text: minimal

text: projection

text: from

text: onto

text: then,

text: for

text: any

text: C

text: d

text: f

text: d[

text: f(d)

text: on

text: (P

text: (f))(d)

text: on

text: or

text: equivalently,

text: writing

text: P

text: (f)(x)

text: η

text: (f)xm

text: m

text: f(d)

text: on

text: η

text: (f)

text: the

text: space

text: span

text: T

text: T

text: T

text: it

text: was

text: more

text: eﬀective

text: to

text: compute

text: the

text: lower

text: bound

text: using

text: a

text: ‘symmetrization’

text: of

text: more

text: direct

text: than

text: Its

text: implementation

text: is

text: also

text: included

text: in

text: the

text: reproducible.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: S. Foucart, J. B. Lasserre 261–289. Academic Press, 1970. [4] E. Cheney, C. Hobby, P. Morris, F. Schurer, and D. Wulbert. On the minimal property of the Fourier projection. Trans. Amer. Math. Soc., 143, 249–258, 1969. [5] D. Coppersmith and T. Rivlin. The growth of polynomials bounded at equally spaced points. SIAM J. Math. Anal., 23, 970–983, 1992. [6] T. Driscoll, N. Hale, and L. Trefethen, editors. Chebfun Guide. Pafnuty Publications, Oxford, 2014. [7] H. Ehlich and K. Zeller. Schwankung von Polynomen zwischen Gitterpunkten. Mathematische Zeitschrift, 86(1), 41–44, 1964. [8] S. Foucart. Computation of minimal projections and extensions. Numerical Functional Analysis and Optimization, 37(2), 159–185, 2016. [9] S. Foucart and V. Powers. Basc: constrained approximation by semideﬁnite programming. IMA Journal of Numerical Analysis, 37(2), 1066–1085, 2017. [10] C. Helzel and K. Petras. Numerical estimation of projection constants. Numerical Functional Analysis and Optimization, 18, 555–566, 1997. [11] J. Lasserre. Moments, Positive Polynomials and Their Applications, Vol. 1. Imperial College Press, 2009. [12] S. Lozinski. On a class of linear operators. Dokl. Akad. Nauk SSSR, 61, 193–196, 1948. [13] M. Prophet, B. Chalmers and F. Metcalf. Minimal shape-preserving projections onto Π . n Numerical Functional Analysis and Optimization, 18, 507–520, 1997. [14] E. Rakhmanov. Bounds for polynomials with a unit discrete norm. Annals of Mathematics, 165, 55–88, 2007. [15] K. Schmu¨dgen. The Moment Problem. Springer, Graduate Texts in Mathematics 277, 2017. [16] B. Shekhtman and L. Skrzypek. Geometric aspects of minimal projections onto planes. In: Constructive Theory of Functions, Varna 2005 (B. Bojanov, editor), 267–277, Marin Drinov Academic Publishing House, 2006. 21 ##LTLine##

text: S.

text: Foucart,

text: J.

text: B.

text: Lasserre

label: reference

text: Academic

text: Press,

text: E.

text: Cheney,

text: C.

text: Hobby,

text: P.

text: Morris,

text: F.

text: Schurer,

text: and

text: D.

text: Wulbert.

text: On

text: the

text: minimal

text: property

text: of

text: the

text: Fourier

text: projection.

text: Trans.

text: Amer.

text: Math.

text: Soc.,

text: D.

text: Coppersmith

text: and

text: T.

text: Rivlin.

text: The

text: growth

text: of

text: polynomials

text: bounded

text: at

text: equally

text: spaced

text: points.

text: SIAM

text: J.

text: Math.

text: Anal.,

text: T.

text: Driscoll,

text: N.

text: Hale,

text: and

text: L.

text: Trefethen,

text: editors.

text: Chebfun

text: Guide.

text: Pafnuty

text: Publications,

text: Oxford,

text: H.

text: Ehlich

text: and

text: K.

text: Zeller.

text: Schwankung

text: von

text: Polynomen

text: zwischen

text: Gitterpunkten.

text: Mathematische

text: Zeitschrift,

text: S.

text: Foucart.

text: Computation

text: of

text: minimal

text: projections

text: and

text: extensions.

text: Numerical

text: Functional

text: Analysis

text: and

text: Optimization,

text: S.

text: Foucart

text: and

text: V.

text: Powers.

text: Basc:

text: constrained

text: approximation

text: by

text: semideﬁnite

text: programming.

text: IMA

text: Journal

text: of

text: Numerical

text: Analysis,

text: C.

text: Helzel

text: and

text: K.

text: Petras.

text: Numerical

text: estimation

text: of

text: projection

text: constants.

text: Numerical

text: Functional

text: Analysis

text: and

text: Optimization,

text: J.

text: Lasserre.

text: Moments,

text: Positive

text: Polynomials

text: and

text: Their

text: Applications,

text: Vol.

text: Imperial

text: College

text: Press,

text: S.

text: Lozinski.

text: On

text: a

text: class

text: of

text: linear

text: operators.

text: Dokl.

text: Akad.

text: Nauk

text: SSSR,

text: M.

text: Prophet,

text: B.

text: Chalmers

text: and

text: F.

text: Metcalf.

text: Minimal

text: shape-preserving

text: projections

text: onto

text: Π

text: n

text: Numerical

text: Functional

text: Analysis

text: and

text: Optimization,

text: E.

text: Rakhmanov.

text: Bounds

text: for

text: polynomials

text: with

text: a

text: unit

text: discrete

text: norm.

text: Annals

text: of

text: Mathematics,

text: K.

text: Schmu¨dgen.

text: The

text: Moment

text: Problem.

text: Springer,

text: Graduate

text: Texts

text: in

text: Mathematics

text: B.

text: Shekhtman

text: and

text: L.

text: Skrzypek.

text: Geometric

text: aspects

text: of

text: minimal

text: projections

text: onto

text: planes.

text: In:

text: Constructive

text: Theory

text: of

text: Functions,

text: Varna

text: (B.

text: Bojanov,

text: editor),

text: Marin

text: Drinov

text: Academic

text: Publishing

text: House,

text: ##LTLine##


text: ∼ Figure 4: Distribution of A (left) and m (right), normalised to the total number of parameter points of 8000. t t˜1 corrections which are indispensable to shift the SM-like Higgs boson mass to 125 GeV. These corrections are dominated by the (s)top loop contributions. The distributions of the trilinear soft SUSY breaking stop-sector coupling A and of the t lightest stop mass m are shown in Fig. 4. Due to the additional contribution proportional to λ t˜1 to the tree-level mass, cf. Eq. (4.24), less important radiative corrections ∆m2 are necessary to h shift m to 125 GeV. Contrary to the MSSM, therefore zero mixing in the stop sector leads to h allowed scenarios in the NMSSM. For the same reason, the lightest stop mass can still be rather light, down to about 270 GeV. The upper bound is limited by our scan range. Experiments set a low mass bound in searches for stop quarks decaying into a charm quark and the lightest − neutralino, χ˜0 [51,76–79].7 This is the dominating decay for mass diﬀerences m m < M , 1 t˜1 χ˜01 W where M denotes the charged W boson mass, [83, 84]. The latest results by ATLAS [51] W exclude top squark masses up to about 240 GeV at 95% C.L. for arbitrary neutralino masses, within the kinematic boundaries. ∼ ± The distribution for A peaks at large values of 2 TeV. Large A values entail large t t radiative corrections to the SM-like Higgs boson mass, so that the tree-level mass value can be shifted to the required 125 GeV. At the same time this results in a large splitting of the stop mass values, so that the lightest stop mass m distribution reaches its maximum around t˜1 500 GeV. Another accumulation is at the upper m range as a result of large radiative mass t˜1 corrections induced by heavy stop masses. 4.2 Mass Distributions In Fig. 5 we show the mass distribution for the non-SM-like H , which is either H or i 1 H , versus the lightest pseudoscalar mass values M . For mass values below about 115 GeV th2e non-SM-like Higgs boson is H , while for M A>∼1 170 GeV it is H . There is a gap for 115 GeV <∼ M <∼ 170 GeV and a1nother one forH1i15 GeV <∼ M <∼ 1230 GeV. Both are due to the LHC excHliusion limits. There are only a few points for MA1, M <∼ 62 GeV. Here the Hi A1 decay of the SM-like 125 GeV Higgs boson into very light scalars or pseudoscalars would be kinematically possible, inducing reduced branching ratios in the SM decays and hence reduced 7Investigating the stop four-body decays into a neutralino, bottom quark and fermion pair [80, 81], similar bounds can be derived in the searches for monojets [51,82]. 10 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

label: caption

text: Figure

text: Distribution

text: of

text: A

text: (left)

text: and

text: m

text: (right),

text: normalised

text: to

text: the

text: total

text: number

text: of

text: parameter

text: points

text: of

text: t

text: corrections

text: which

text: are

text: indispensable

text: to

text: shift

text: the

text: SM-like

text: Higgs

text: boson

text: mass

text: to

text: GeV.

text: These

text: corrections

text: are

text: dominated

text: by

text: the

text: (s)top

text: loop

text: contributions.

text: The

text: distributions

text: of

text: the

text: trilinear

text: soft

text: SUSY

text: breaking

text: stop-sector

text: coupling

text: A

text: and

text: of

text: the

text: t

text: lightest

text: stop

text: mass

text: m

text: are

text: shown

text: in

text: Fig.

text: Due

text: to

text: the

text: additional

text: contribution

text: proportional

text: to

text: λ

text: to

text: the

text: tree-level

text: mass,

text: cf.

text: Eq.

text: less

text: important

text: radiative

text: corrections

text: are

text: necessary

text: to

text: h

text: shift

text: m

text: to

text: GeV.

text: Contrary

text: to

text: the

text: MSSM,

text: therefore

text: zero

text: mixing

text: in

text: the

text: stop

text: sector

text: leads

text: to

text: h

text: allowed

text: scenarios

text: in

text: the

text: NMSSM.

text: For

text: the

text: same

text: reason,

text: the

text: lightest

text: stop

text: mass

text: can

text: still

text: be

text: rather

text: light,

text: down

text: to

text: about

text: GeV.

text: The

text: upper

text: bound

text: is

text: limited

text: by

text: our

text: scan

text: range.

text: Experiments

text: set

text: a

text: low

text: mass

text: bound

text: in

text: searches

text: for

text: stop

text: quarks

text: decaying

text: into

text: a

text: charm

text: quark

text: and

text: the

text: lightest

text: neutralino,

text: χ˜0

text: This

text: is

text: the

text: dominating

text: decay

text: for

text: mass

text: diﬀerences

text: m

text: m

text: M

text: χ˜01

text: W

text: where

text: M

text: denotes

text: the

text: charged

text: W

text: boson

text: mass,

text: The

text: latest

text: results

text: by

text: ATLAS

text: W

text: exclude

text: top

text: squark

text: masses

text: up

text: to

text: about

text: GeV

text: at

text: C.L.

text: for

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text: neutralino

text: masses,

text: within

text: the

text: kinematic

text: boundaries.

text: The

text: distribution

text: for

text: A

text: peaks

text: at

text: large

text: values

text: of

text: TeV.

text: Large

text: A

text: values

text: entail

text: large

text: t

text: t

text: radiative

text: corrections

text: to

text: the

text: SM-like

text: Higgs

text: boson

text: mass,

text: so

text: that

text: the

text: tree-level

text: mass

text: value

text: can

text: be

text: shifted

text: to

text: the

text: required

text: GeV.

text: At

text: the

text: same

text: time

text: this

text: results

text: in

text: a

text: large

text: splitting

text: of

text: the

text: stop

text: mass

text: values,

text: so

text: that

text: the

text: lightest

text: stop

text: mass

text: m

text: distribution

text: reaches

text: its

text: maximum

text: around

text: GeV.

text: Another

text: accumulation

text: is

text: at

text: the

text: upper

text: m

text: range

text: as

text: a

text: result

text: of

text: large

text: radiative

text: mass

text: corrections

text: induced

text: by

text: heavy

text: stop

text: masses.

text: Mass

text: Distributions

text: In

text: Fig.

text: we

text: show

text: the

text: mass

text: distribution

text: for

text: the

text: non-SM-like

text: H

text: which

text: is

text: either

text: H

text: or

text: i

text: H

text: versus

text: the

text: lightest

text: pseudoscalar

text: mass

text: values

text: M

text: For

text: mass

text: values

text: below

text: about

text: GeV

text: non-SM-like

text: Higgs

text: boson

text: is

text: H

text: while

text: for

text: M

text: GeV

text: it

text: is

text: H

text: There

text: is

text: a

text: gap

text: for

text: GeV

text: M

text: GeV

text: and

text: one

text: GeV

text: M

text: GeV.

text: Both

text: are

text: due

text: to

text: the

text: LHC

text: excHliusion

text: limits.

text: There

text: are

text: only

text: a

text: few

text: points

text: for

text: M

text: GeV.

text: Here

text: the

text: Hi

text: decay

text: of

text: the

text: SM-like

text: GeV

text: Higgs

text: boson

text: into

text: very

text: light

text: scalars

text: or

text: pseudoscalars

text: would

text: be

text: kinematically

text: possible,

text: inducing

text: reduced

text: branching

text: ratios

text: in

text: the

text: SM

text: decays

text: and

text: hence

text: reduced

text: the

text: stop

text: four-body

text: decays

text: into

text: a

text: neutralino,

text: bottom

text: quark

text: and

text: fermion

text: pair

text: similar

text: bounds

text: can

text: be

text: derived

text: in

text: the

text: searches

text: for

text: monojets

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: Figure 7: Production rates in pb for the non-SM-like CP-even Higgs boson H (i = 1, 2) into the γγ (upper left), i the b¯b (upper right), the ZZ (lower left√) and tt¯(lower right) ﬁnal states for tan β < 5 (blue) and tan β > 5 (pink), as a function of M at a c.m. energy s = 13 TeV. The full black line shows the production rate for a SM Higgs Hi boson with same mass. The ττ ﬁnal states are not shown in Fig. 7, but exhibit the same pattern as the b¯b ﬁnal state ± with an additional suppression factor of 10. The Z and W boson ﬁnal states reach maximum ± signal rates of about 0.1 pb for the former and 1 pb for the latter. Again the W ﬁnal states, not depicted in Fig. 7, show the same behaviour as the Z boson ﬁnal states. They are enhanced by a factor 10 compared to these, however with missing energy in the ﬁnal state, so that only transverse Higgs masses can be reconstructed. b) Signal rates for H : We now turn to possible discovery channels for H being the non- i=2 2 SM-like Higgs boson, hence investigate the mass regions above 170 GeV in Fig. 7. The next- ≡ to-lightest Higgs boson H with H h is mostly singlet-like for small tan β values and mostly ≥ 2 1 doublet-like for tan β 5. The SM-like Higgs H needs to be doublet-like with a large H 1 u component, in order to have a substantial coupling to top quarks and hence a gluon fusion cross section large enough to lead to signal rates compatible with the LHC data. Due to the unitarity R of the mixing matrix S rotating the current to the mass eigenstates, H is hence doublet- 2 like but with a large H component, so that its couplings to down-type fermions are enhanced d ± compared to the SM. The coupling G to massive vector bosons V = Z, W on the other HiV V hand is suppressed. This can be understood by looking at the coupling, which normalised to 14 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: Figure

text: Production

text: rates

text: in

text: pb

text: for

text: the

text: non-SM-like

text: CP-even

text: Higgs

text: boson

text: H

text: (i

text: into

text: the

text: γγ

text: (upper

text: left),

text: i

text: the

text: b¯b

text: (upper

text: right),

text: the

text: ZZ

text: (lower

text: left√)

text: and

text: tt¯(lower

text: right)

text: ﬁnal

text: states

text: for

text: tan

text: β

text: (blue)

text: and

text: tan

text: β

text: (pink),

text: as

text: a

text: function

text: of

text: M

text: at

text: a

text: c.m.

text: energy

text: s

text: TeV.

text: The

text: full

text: black

text: line

text: shows

text: the

text: production

text: rate

text: for

text: a

text: SM

text: Higgs

text: Hi

text: boson

text: with

text: same

text: mass.

text: The

text: ττ

text: ﬁnal

text: states

text: are

text: not

text: shown

text: in

text: Fig.

text: but

text: exhibit

text: the

text: same

text: pattern

text: as

text: the

text: b¯b

text: ﬁnal

text: state

text: with

text: an

text: additional

text: suppression

text: factor

text: of

text: The

text: Z

text: and

text: W

text: boson

text: ﬁnal

text: states

text: reach

text: maximum

text: signal

text: rates

text: of

text: about

text: pb

text: for

text: the

text: former

text: and

text: pb

text: for

text: the

text: latter.

text: Again

text: the

text: W

text: ﬁnal

text: states,

text: not

text: depicted

text: in

text: Fig.

text: show

text: the

text: same

text: behaviour

text: as

text: the

text: Z

text: boson

text: ﬁnal

text: states.

text: They

text: are

text: enhanced

text: by

text: a

text: factor

text: compared

text: to

text: these,

text: however

text: with

text: missing

text: energy

text: in

text: the

text: ﬁnal

text: state,

text: so

text: that

text: only

text: transverse

text: Higgs

text: masses

text: can

text: be

text: reconstructed.

text: b)

text: Signal

text: rates

text: for

text: H

text: We

text: now

text: turn

text: to

text: possible

text: discovery

text: channels

text: for

text: H

text: being

text: the

text: non-

text: SM-like

text: Higgs

text: boson,

text: hence

text: investigate

text: the

text: mass

text: regions

text: above

text: GeV

text: in

text: Fig.

text: The

text: next-

text: to-lightest

text: Higgs

text: boson

text: H

text: with

text: H

text: h

text: is

text: mostly

text: singlet-like

text: for

text: small

text: tan

text: β

text: values

text: and

text: mostly

text: doublet-like

text: for

text: tan

text: β

text: The

text: SM-like

text: Higgs

text: H

text: needs

text: to

text: be

text: doublet-like

text: with

text: a

text: large

text: H

text: u

text: component,

text: in

text: order

text: to

text: have

text: a

text: substantial

text: coupling

text: to

text: top

text: quarks

text: and

text: hence

text: a

text: gluon

text: fusion

text: cross

text: section

text: large

text: enough

text: to

text: lead

text: to

text: signal

text: rates

text: compatible

text: with

text: the

text: LHC

text: data.

text: Due

text: to

text: the

text: unitarity

text: R

text: of

text: the

text: mixing

text: matrix

text: S

text: rotating

text: the

text: current

text: to

text: the

text: mass

text: eigenstates,

text: H

text: is

text: hence

text: doublet-

text: like

text: but

text: with

text: a

text: large

text: H

text: component,

text: so

text: that

text: its

text: couplings

text: to

text: down-type

text: fermions

text: are

text: enhanced

text: d

text: compared

text: to

text: the

text: SM.

text: The

text: coupling

text: G

text: to

text: massive

text: vector

text: bosons

text: V

text: Z,

text: W

text: on

text: the

text: other

text: HiV

text: V

text: hand

text: is

text: suppressed.

text: This

text: can

text: be

text: understood

text: by

text: looking

text: at

text: the

text: coupling,

text: which

text: normalised

text: to

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: the SM coupling is given by G ≡ R R HiV V g = ( S cos β + S sin β) , (4.26) g HiV V i1 i2 HSMV V R R where S ( S ) quantiﬁes the H (H ) component of H . The H component can be substantial i1 i2 d u i d for large tan β but its contribution in the coupling is suppressed by the factor cos β. The up-type part of the coupling comes with sin β and also for small tan β values would be small, as the H ≡ u component is taken by H h. As can be inferred from the plot, for Higgs mass values above ∼ 1 400 GeV the rates into b¯b are enhanced for large tan β. This is due to an enhanced branching ratio into bottom pairs. The total decay width in this mass range is dominated by the decay into V = W, Z, and the H branching ratio into b¯b can be approximated by 2 g2 ΓSM → BRNM→SSM = H2dd¯ HSM b¯b H2 b¯b g2 ΓSM → + g2 ΓSM → + ... H2V V HSM WW H2V V HSM ZZ ≈ gH2 2dd¯ BRSM → , (4.27) g2 HSM b¯b H2V V with M = M and where g denotes the H coupling to down-type fermions in terms H2 HSM H2dd¯ 2 of the SM coupling. The enhanced coupling to down-type quarks and the suppressed coupling to massive gauge bosons explain the observed behaviour. For large values of tan β and mass values above 400 GeV, the heavier H may therefore be 2 discovered in its b¯b and ττ ﬁnal state, which exhibits the same behaviour as the b¯b ﬁnal state, the latter suﬀering from large backgrounds though, so that the τ production rate, suppressed by roughly a factor 10 compared to the former, may be more promising for discovery. In particular, the production in association with a b-quark pair (not shown here) increases the signal rates by ∼ another factor 10 compared to production in gluon fusion. The gauge boson and top quark ﬁnal states are more challenging on the other hand, even in associated production, due to the (above discussed) suppression of the couplings to gauge bosons and top quarks. For tan β <∼ 5 the H searches have to rely on a combined search in the ττ ﬁnal states (most interesting for 2 ∼ H masses up to 250 GeV) and the vector-boson and top-pair channels. The associated 2 production with b-quarks is not eﬀective for these tan β values. Production rates of A : Figure 8 displays the signal rates for the lightest CP-odd Higgs boson 1 A produced in association with a b-quark pair in the b¯b ﬁnal state (left) and in gluon fusion 1 in the top quark decay (right). The rates into ττ and µµ ﬁnal states show the same pattern ∼ as the ones into b¯b but are suppressed by a factor of 10 and 104 each. Massive gauge boson ﬁnal states are forbidden due to the CP-odd nature of A . The decay rates into photons both − 1 in associated and in inclusive production are below 10 4 already in the low mass range. In this mass range A may be discovered in its decays into τ’s, as the b-quark ﬁnal states are notoriously 1 diﬃcult due to large backgrounds. This discovery channel is also interesting for large M at tan β > 5. For M >∼ 400 GeV the pseudoscalar, which in most scenarios is singlet-likeA, 1can A1 also be MSSM-like with a large a component, in particular for tan β > 5, cf. Fig. 6. Associated d production with a b-quark pair here leads to cross sections of up to 4 pb. Otherwise the top quark ﬁnal state has to be exploited for heavy A production, though challenging with cross 1 sections of at most a few pb and a complicated ﬁnal state. We note, that for small tan β values, there is a step in Fig. 8 in the areas covered by the scatter points around M = 350 GeV. This A1 is due to the opening of the decays into the top pair ﬁnal state, which steeply increase at the 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: SM

text: coupling

text: is

text: given

text: by

text: G

text: R

text: R

text: HiV

text: V

text: g

text: S

text: cos

text: β

text: S

text: sin

text: β)

text: g

text: HiV

text: V

text: HSMV

text: V

text: R

text: R

text: where

text: S

text: S

text: quantiﬁes

text: the

text: H

text: (H

text: component

text: of

text: H

text: The

text: H

text: component

text: can

text: be

text: substantial

text: d

text: u

text: i

text: d

text: for

text: large

text: tan

text: β

text: but

text: its

text: contribution

text: in

text: the

text: coupling

text: is

text: suppressed

text: by

text: the

text: factor

text: cos

text: β.

text: The

text: up-type

text: part

text: of

text: the

text: coupling

text: comes

text: with

text: sin

text: β

text: and

text: also

text: for

text: small

text: tan

text: β

text: values

text: would

text: be

text: small,

text: as

text: the

text: H

text: u

text: component

text: is

text: taken

text: by

text: H

text: h.

text: As

text: can

text: be

text: inferred

text: from

text: the

text: plot,

text: for

text: Higgs

text: mass

text: values

text: above

text: GeV

text: the

text: rates

text: into

text: b¯b

text: are

text: enhanced

text: for

text: large

text: tan

text: β.

text: This

text: is

text: due

text: to

text: an

text: enhanced

text: branching

text: ratio

text: into

text: bottom

text: pairs.

text: The

text: total

text: decay

text: width

text: in

text: this

text: mass

text: range

text: is

text: dominated

text: by

text: the

text: decay

text: into

text: V

text: W,

text: Z,

text: and

text: the

text: H

text: branching

text: ratio

text: into

text: b¯b

text: can

text: be

text: approximated

text: by

text: ΓSM

text: BRNM→SSM

text: HSM

text: b¯b

text: b¯b

text: ΓSM

text: ΓSM

text: V

text: HSM

text: WW

text: V

text: HSM

text: ZZ

text: BRSM

text: HSM

text: b¯b

text: V

text: with

text: M

text: M

text: and

text: where

text: g

text: denotes

text: the

text: H

text: coupling

text: to

text: down-type

text: fermions

text: in

text: terms

text: HSM

text: of

text: the

text: SM

text: coupling.

text: The

text: enhanced

text: coupling

text: to

text: down-type

text: quarks

text: and

text: the

text: suppressed

text: coupling

text: to

text: massive

text: gauge

text: bosons

text: explain

text: the

text: observed

text: behaviour.

text: For

text: large

text: values

text: of

text: tan

text: β

text: and

text: mass

text: values

text: above

text: GeV,

text: the

text: heavier

text: H

text: may

text: therefore

text: be

text: discovered

text: in

text: its

text: b¯b

text: and

text: ττ

text: ﬁnal

text: state,

text: which

text: exhibits

text: the

text: same

text: behaviour

text: as

text: the

text: b¯b

text: ﬁnal

text: state,

text: the

text: latter

text: suﬀering

text: from

text: large

text: backgrounds

text: though,

text: so

text: that

text: the

text: τ

text: production

text: rate,

text: suppressed

text: by

text: roughly

text: a

text: factor

text: compared

text: to

text: the

text: former,

text: may

text: be

text: more

text: promising

text: for

text: discovery.

text: In

text: particular,

text: the

text: production

text: in

text: association

text: with

text: a

text: b-quark

text: pair

text: (not

text: shown

text: here)

text: increases

text: the

text: signal

text: rates

text: by

text: another

text: factor

text: compared

text: to

text: production

text: in

text: gluon

text: fusion.

text: The

text: gauge

text: boson

text: and

text: top

text: quark

text: ﬁnal

text: states

text: are

text: more

text: challenging

text: on

text: the

text: other

text: hand,

text: even

text: in

text: associated

text: production,

text: due

text: to

text: the

text: (above

text: discussed)

text: suppression

text: of

text: the

text: couplings

text: to

text: gauge

text: bosons

text: and

text: top

text: quarks.

text: For

text: tan

text: β

text: the

text: H

text: searches

text: have

text: to

text: rely

text: on

text: a

text: combined

text: search

text: in

text: the

text: ττ

text: ﬁnal

text: states

text: (most

text: interesting

text: for

text: H

text: masses

text: up

text: to

text: GeV)

text: and

text: the

text: vector-boson

text: and

text: top-pair

text: channels.

text: The

text: associated

text: production

text: with

text: b-quarks

text: is

text: not

text: eﬀective

text: for

text: these

text: tan

text: β

text: values.

text: Production

text: rates

text: of

text: A

text: Figure

text: displays

text: the

text: signal

text: rates

text: for

text: the

text: lightest

text: CP-odd

text: Higgs

text: boson

text: A

text: produced

text: in

text: association

text: with

text: a

text: b-quark

text: pair

text: in

text: the

text: b¯b

text: ﬁnal

text: state

text: (left)

text: and

text: in

text: gluon

text: fusion

text: in

text: the

text: top

text: quark

text: decay

text: (right).

text: The

text: rates

text: into

text: ττ

text: and

text: µµ

text: ﬁnal

text: states

text: show

text: the

text: same

text: pattern

text: as

text: the

text: ones

text: into

text: b¯b

text: but

text: are

text: suppressed

text: by

text: a

text: factor

text: of

text: and

text: each.

text: Massive

text: gauge

text: boson

text: ﬁnal

text: states

text: are

text: forbidden

text: due

text: to

text: the

text: CP-odd

text: nature

text: of

text: A

text: The

text: decay

text: rates

text: into

text: photons

text: both

text: in

text: associated

text: and

text: in

text: inclusive

text: production

text: are

text: below

text: already

text: in

text: the

text: low

text: mass

text: range.

text: In

text: this

text: mass

text: range

text: A

text: may

text: be

text: discovered

text: in

text: its

text: decays

text: into

text: τ’s,

text: as

text: the

text: b-quark

text: ﬁnal

text: states

text: are

text: notoriously

text: diﬃcult

text: due

text: to

text: large

text: backgrounds.

text: This

text: discovery

text: channel

text: is

text: also

text: interesting

text: for

text: large

text: M

text: at

text: tan

text: β

text: For

text: M

text: GeV

text: the

text: pseudoscalar,

text: which

text: in

text: most

text: scenarios

text: is

text: singlet-likeA,

text: also

text: be

text: MSSM-like

text: with

text: a

text: large

text: a

text: component,

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text: Fig.

text: Associated

text: d

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text: are compatible with all constraints discussed above and that arise from our scan of the parameter space deﬁned in Eqs. (3.9)-(3.15), i.e. not restricted to the Natural NMSSM. The presented scenarios not only lead to suﬃciently large cross sections for double Higgs production from heavier Higgs decays but some of them also entail exotic multi-photon and/or multi-fermion ﬁnal states. The possibility of such ﬁnal states should be taken into consideration by the experiments in order not to miss possible Higgs signatures, that can appear in multi-Higgs models like the NMSSM without particular ﬁne-tuning of the parameters. We also include scenarios featuring a light CP-even Higgs boson with a mass below 10 GeV. The presented scenarios highlight speciﬁc features of NMSSM Higgs decay chains and can be viewed as benchmark scenarios for future LHC searches. In the following we will again use the convenient notation introduced in section 5. Note, however, that the SM-like Higgs boson h can now be H or H . The more doublet-like scalar 1 2 H can be given by H or H , depending on the scenario. The pseudoscalar doublet state A, 2 3 as before, is given by A , and the singlet like CP-odd A by A . The singlet-like scalar H , 2 s 1 s however, can be H , H or even H . 1 2 3 The benchmark scenarios that we present can be classiﬁed as follows: A) H = h, H = H , tan β small, light spectrum <∼ 350 GeV: 2 1 s The scenarios A are characterized by the second lightest Higgs being SM-like, the lightest Higgs boson singlet-like and small tan β. The overall spectrum is rather light with maximal Higgs mass values of M = 341 GeV and M = 347 GeV. H A A.1: The CP-odd singlet is close in mass to the SM-like Higgs with M = 119 GeV. The As heavy Higgs bosons have masses around 341 and 347 GeV. A.2: The CP-odd singlet is light with M = 79 GeV, the heavy Higgs bosons with masses As around 326 GeV are somewhat lighter than in scenario A.1. B) H = h, H = H , tan β small: 1 2 s The scenarios B are characterized by the lightest Higgs being SM-like, the second lightest Higgs boson singlet-like and small tan β. B.1: The CP-odd singlet is light with M = 73 GeV and the overall spectrum as well As with the heavy H mass of 323 GeV and M = 312 GeV. A B.2: The pseudoscalar singlet with M = 107 GeV is still lighter than the SM-like Higgs, As but the overall spectrum is heavier with M = 556 GeV and M = 578 GeV. H A B.3: The singlet mass M = 133 GeV is now heavier than M . The maximum mass is As h given by M = 463 GeV, closely followed by M = 457 GeV. H A C) tan β > 5: Here we explicitly chose a scenario where tan β = 17 is large. The lightest Higgs boson is SM-like H = h. The spectrum is peculiar in the sense that the heavy scalar with a ∼ 1 mass of 3.5 TeV is very heavy. Additionally, H is singlet-like. The heavy pseudoscalar 3 O and second lightest scalar are doublet-like with masses of (600 GeV). The pseudoscalar singlet is much heavier than h, M = 312 GeV. As D) The SM-like h, given by H , can decay into Higgs pairs: 2 The scenarios D allow for decays of the SM-like Higgs h into lighter singlet Higgs pairs. 25

text: are

text: compatible

text: with

text: all

text: constraints

text: discussed

text: above

text: and

text: that

text: arise

text: from

text: our

text: scan

text: of

text: the

text: parameter

text: space

text: deﬁned

text: in

text: Eqs.

text: i.e.

text: not

text: restricted

text: to

text: the

text: Natural

text: NMSSM.

text: The

text: presented

text: scenarios

text: not

text: only

text: lead

text: to

text: suﬃciently

text: large

text: cross

text: sections

text: for

text: double

text: Higgs

text: production

text: from

text: heavier

text: Higgs

text: decays

text: but

text: some

text: of

text: them

text: also

text: entail

text: exotic

text: multi-photon

text: and/or

text: multi-fermion

text: ﬁnal

text: states.

text: The

text: possibility

text: of

text: such

text: ﬁnal

text: states

text: should

text: be

text: taken

text: into

text: consideration

text: by

text: the

text: experiments

text: in

text: order

text: not

text: to

text: miss

text: possible

text: Higgs

text: signatures,

text: that

text: can

text: appear

text: in

text: multi-Higgs

text: models

text: like

text: the

text: NMSSM

text: without

text: particular

text: ﬁne-tuning

text: of

text: the

text: parameters.

text: We

text: also

text: include

text: scenarios

text: featuring

text: a

text: light

text: CP-even

text: Higgs

text: boson

text: with

text: a

text: mass

text: below

text: GeV.

text: The

text: presented

text: scenarios

text: highlight

text: speciﬁc

text: features

text: of

text: NMSSM

text: Higgs

text: decay

text: chains

text: and

text: can

text: be

text: viewed

text: as

text: benchmark

text: scenarios

text: for

text: future

text: LHC

text: searches.

text: In

text: the

text: following

text: we

text: will

text: again

text: use

text: the

text: convenient

text: notation

text: introduced

text: in

text: section

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text: that

text: the

text: SM-like

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text: more

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text: depending

text: on

text: the

text: scenario.

text: The

text: pseudoscalar

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text: singlet-like

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text: H

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text: benchmark

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text: classiﬁed

text: as

text: follows:

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text: H

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text: GeV:

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text: scenarios

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text: second

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text: we

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text: scenarios

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text: lighter

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text: pairs.