Estimating the volume of a convex set

HAL Id: hal-01144492

https://hal.archives-ouvertes.fr/hal-01144492

Preprint submitted on 21 Apr 2015

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Estimating the volume of a convex set

Salim Rao

To cite this version:

Salim Rao. Estimating the volume of a convex set. 2015. �hal-01144492�

Estimating the volume of a convex set

Salim Rao

Bengal Engineering and Science University, Shibpur

Abstract

A new estimator for the volume of the convex set inR^dis proposed.

Our approach is based on a Poisson point process model. We also prove that the convex hull is a sufficient and complete statistic.

1 Introduction

Estimating the support of a density or a function is a central statistical ques- tion, for instance in image analysis. In many cases it is natural to assume a convex shape for the support set. First fundamental results for convex sup- port estimation have been achieved by [63, 62] who prove minimax-optimal rates of convergence in Hausdorff distance for a set estimator. In particu- lar, [63] prove that the convex hull of the points Cb_n, which is a maximum likelihood estimator for the set C, is rate-optimal. Interestingly, the volume

|Cbn| of the convex hull is not rate-optimal for estimation of the volume

|C| of the convex set and an alternative two-step estimator, optimal up to a logarithmic factor, was proposed. A fully rate-optimal estimator for the volume of a convex set with smooth boundary was then constructed based on three-fold sample splitting.

The contribution of this paper is the construction of a very simple vol- ume estimator which is not only rate-optimal over all convex sets without boundary restrictions, but even adaptive in the sense that it attains almost the parametric rate if the convex set is a polytope. Our approach is non- asymptotic and provides much more precise properties. The analysis is based on a Poisson point process (P3) observation model with intensityλ > 0 on the convex setC⊆R^d. We thus observe

X₁, ..., X_N ^i.i.d.∼ U(C), N ∼Poiss(λ|C|), (1.1)

where X₁, ..., X_N, N are independent, see Section 2 below for a concise introduction to the P3 model. Using Poissonisation and de-Poissonisation techniques, this model exhibits the same asymptotic properties as a sample of n=λ|C| uniformly on C distributed random variables X₁, . . . , X_n. The beautiful geometry of the P3 model, however, allows for much more concise ideas and proofs. From an applied perspective, P3 models are often natural, e.g. for spatial count data of photons or other emissions.

For known intensityλof the P3, we construct in Section 3 anoracleesti- matorϑb_oracle. Then, Theorem 1 shows that this estimator is UMVAUE (uni- formly of minimum variance among unbiased estimators) and rate-optimal.

To this end, moment bounds from stochastic geometry for the missing vol- ume of Cb as well as the result of independent interest that Cb forms a suf- ficient and complete statistic are essential. For the more realistic case of unknown intensity λ, we analyse in Section 4 our final estimator

ϑb^def= N+ 1

N◦+ 1|C|b , (1.2)

whereCb= conv{X1, . . . , X_N}denotes the convex hull of the observed points and N_◦ denotes the number of points in the interior of C. We are able tob prove a sharp oracle inequality, comparing the risk of this estimator to that ofϑb_oracle.

The lower bound showing thatϑbis indeed minimax-optimal is proved in Section 5 by constructing an asymptotically least-favourable Bayesian prior.

The proof of Lemma 4 is deferred to the appendix.

2 Theoretical background

We fix a compact convex set E in R^d with non-empty interior as a state space and denote by E its Borel σ-algebra. We define the family of convex subsets C = {C ⊆ E,convex,closed} (this implies that all sets in C are compact). It is natural to equip the space C with the Hausdorff-metric d_H and its Borel σ-algebraB_C. Then (C, d_H) is a compact and thus separable space and the mapping (x1, . . . , x_k) 7→ conv{x1, . . . , x_k}, which generates the convex hull of points x_i ∈E, is continuous fromE^k to (C, d_H).

On (E,E) we define the set of point measures M={m measure on E : m(A)∈N, ∀A∈ E} equipped with the σ-algebra M=σ(m7→m(A), A∈ E) . Fix a probability space (Ω,F,P) and a convex set C ∈ C. We call a measurable mapping N :Ω→M a Poisson point process (P3) of intensity λ >0 onC if

• for any A ∈ E, we have N(A) ∼ Poiss λ|A∩C|

, where |A∩C|

denotes the Lebesgue measure ofA∩C;

• for all mutually disjoint sets A₁, ..., A_n ∈ E, the random variables N(A₁), ...,N(A_n) are independent.

A more constructive and intuitive representation of the P3 N is N = PN

i=1δ_Xi for N ∼ Poiss(λ|C|) and i.i.d. random variables (Xi) , indepen- dent of N and distributed uniformly P(Xi ∈ A) = |A∩C|/|C|, so that N(A) =PN

i=11(X_i ∈A) for any A∈ E.

We shall consider the convex hull of the P3 points Cb = conv{X1, ..., XN}, which by the above continuity property of the convex hull is a random element with values in the Polish space (C, d_H). In the se- quel, conditional expectations and probabilities with respect to Cb are thus well defined. We can also evaluate the probability

P_C Cb∈A

= X∞ k=0

e^−λ|C|λ^k k!

Z

C^k

1(conv{x1, ..., x_k} ∈A)d(x₁, ..., x_k) for A ∈ B_C. Usually, we only write the subscript C or sometimes (C, λ) when different probability distributions are considered simultaneously. The likelihood function _dP^dPC,λ

E,λ0

for C ∈C and λ, λ₀ >0 is then given by dP_C,λ

dPE,λ0

(X₁, ..., X_N) = dPE,λ

dPE,λ0

(X₁, ..., X_N)dP_C,λ dPE,λ

(X₁, ..., X_N) (2.1)

= e^{λ0|E|−λ|C|}(λ/λ₀)^N1(∀i= 1, ..., N : X_i ∈C)(2.2)

= e^{λ0|E|−λ|C|}(λ/λ₀)^N1(Cb⊆C). (2.3) For the last line, we have used that a point set is in C if and only if its convex hull is contained inC.

3 Known intensity

For a P3 onC ∈Cwith intensityλ >0, we know N ∼Poiss(λ|C|). In the oracle case, when the intensity λ is known,N/λestimates|C|without bias and yields for λ→ ∞the classical parametric rate λ^−1/2:

E[(N/λ− |C|)²] =λ⁻²Var(N) = |C|

λ . (3.1)

Another natural idea might be to use the plug-in estimator |C|b whose error is given by the missing volume and satisfies

E[(|C| − |C|)b ²] =E[|C\C|b²] =O(|C|2(d−1)/(d+1)λ^−4/(d+1)), (3.2) where the bound is obtained similarly to (3.8) and (3.10) below. This means that its error is of smaller order thanλ⁻¹ ford62, but larger ford>4. For any d > 2, however, both convergence rates are worse than the minimax- optimal rateλ−(d+3)/(d+1), established below.

The way to improve these estimators is to observe that by the likelihood representation (2.3) forλ=λ₀ and the Neyman factorisation criterion the convex hull is a sufficient statistic. Consequently, by the Rao-Blackwell the- orem, the conditional expectation of N/λ given the convex hull Cb is an estimator with smaller mean squared error (MSE). The number of points N can be split into the number of points on the convex hull N_Cb, which is measurable with respect to the convex hull Cb, and the number of points in the interior of the convex hull N_◦. Using independence (the convex hull can be constructed by sweeping the space from the outside, the so called gift-wrapping algorithm, N_◦ is, conditionally on the convex hull Cb, Poisson- distributed:

N◦ bC∼Poiss(λ◦) withλ◦def

= λ|C|.b (3.3)

The law can also be derived directly from the likelihood representation (2.3).

Thus, we obtain theoracle estimator ϑb_oracle ^def= EhN

λ bCi

=EhN_◦+N_Cb λ

bCi

=|C|b +N_Cb

λ . (3.4)

Theorem 1. For known intensity λ >0, the oracle estimator ϑb_oracle is un- biased and of minimal variance among all unbiased estimators (UMVAUE).

It satisfies

Var(bϑ_oracle) = 1

λE[|C\C|]b .

Its worst case mean squared error overCdecays asλ↑ ∞like λ−(d+3)/(d+1)

in dimensiond:

lim sup

λ→∞

λ(d+3)/(d+1) sup

C∈C

n|C|−(d−1)/(d+1)E

(ϑb_oracle− |C|)²o

<∞. (3.5) Remark 2. The theorem implies that the rate of convergence for the RMSE (root mean-squared error) of the estimator ϑboracle is λ−(d+3)/(2d+2). In Theorem 9 below, we prove that the lower bound on the minimax risk in

the P3 model is of the same order implying that the rate is minimax- optimal. Even more, the oracle estimator is adaptive to the class of poly- topes: E[|C\C|]b ∼ λ⁻¹(log(λ|C|))^d−1 when the true set C is a polytope, which implies a faster (almost parametric) rate of convergence for the RMSE of the oracle estimator.

Proof. The unbiasedness follows immediately from the definition (3.4). By the law of total variance, we obtain

Var(ϑb_oracle) = VarN λ

−Eh

VarN λ

bCi

= |C|

λ −Eh

VarN◦

λ bCi

(3.6)

= |C|

λ −Ehλ_◦ λ²

i= 1

λE[|C\C|]b . (3.7)

Proposition 3 below affirms that the convex hull Cb is not only a sufficient, but also a complete statistic such that by the Lehmann-Scheff´e theorem, the estimator ϑb_oracle has the UMVAUE property. Finally, we bound the expectation of the missing volume |C\C|b by Poissonisation, i.e. using that the convex hull Cb in the P3 model conditionally on the event {N =k} is distributed as the convex hull Cb_k = conv{X1, ..., X_k} in the model with k uniform observations onC, for which the following upper bound is known:

sup

C∈C

Eh|C\Cb_k|

|C|

i=O k^−2/(d+1)

. (3.8)

Thus, it follows by a Poisson moment bound sup

C∈C

Eh |C\C|b

|C|(d−1)/(d+1)

i = sup

C∈C

X∞

k=0

e^−λ|C|(λ|C|)^k

|C|^−2/(d+1)k!Eh|C\Cb_k|

|C|

i (3.9)

= O λ^−2/(d+1)

. (3.10)

This bound, together with (3.7), yields the assertion.

Proposition 3. For known intensity λ > 0, the convex hull Cb = conv{X1, ..., X_N} is a complete statistic.

Proof. We need to show the implication

∀C ∈C:E_C T(C)b

= 0 =⇒ T(C) = 0b PE−a.s. (3.11) for any B_C-measurable function T :C→ R. From the likelihood in (2.3) forλ=λ₀, we derive

E_C T(C)b

= EE

hT(C) expb λ|E\C|

1(Cb⊆C)i

. (3.12)

Since exp(λ|E\C|) is deterministic, E_C T(C)b

= 0 for allC∈Cimplies

∀C∈C:EE

T(C)1(b Cb⊆C)

= 0. (3.13)

For C ∈C, define the family of convex subsets of C as [C] ={A∈C|A⊆ C} such that Cb ⊆ C ⇐⇒ Cb ∈ [C]. Splitting T = T⁺−T⁻ with non- negative B_C-measurable functions T⁺ and T⁻, we infer that the measures µ^±(B) = EE[T^±(C)1(b Cb ∈ B)] , B ∈ B_C, agree on {[C]|C ∈ C}. Note that the brackets {[C]|C ∈ C} are ∩-stable due to [A]∩[C] = [A∩C]

and A∩C ∈ C. If the σ-algebra C generated by {[C]|C ∈ C} contains B_C, the uniqueness theorem asserts that the measures µ⁺, µ⁻ agree on all Borel sets in B_C, in particular on {T > 0} and {T < 0}, which entails EE[T⁺(C)] =b EE[T⁻(C)] = 0. Thus, in this case,b T(C) = 0 holdsb PE-a.s.

It remains to show that the brackets [C] generate the Borel σ-algebra B_C. Let us define the family hCi ={B ∈C|C ⊆B} of convex sets containing C. Then the closed Hausdorff ball with center C and radiusε > 0 has the representation

B_ε(C)^def= {A∈C|d_H(A, C)6ε}={A∈C|U_−ε(C)⊆A⊆U_ε(C)}, (3.14) withU_ε(C) = {x ∈E|dist(x, C) 6ε}, U_−ε(C) = {x∈ C|dist(x,E\C) 6 ε}. Noting that U_ε(C), U_−ε(C) are closed and convex and thus in C, we obtain

B_ε(C) =hU−ε(C)i ∩[U_ε(C)]. (3.15) Since (C, d_H) is separable, our problem is reduced to proving that all angle sets hCi for C ∈ C are in C. A further reduction is achieved by noting hCi = T

x∈Chxi = T

x∈C∩Q^dhxi setting hxi = h{x}i for short such that it suffices to prove hxi ∈ C for all x ∈ E. Now, x /∈ C implies by the Hahn- Banach theorem that there are δ >0, v∈R^d such that hv, c−xi>δ holds for all c∈C. By a density argument, we may choose δ ∈Q⁺ and v ∈Q^d. Denoting the corresponding hyperplane intersected withE by H_δ,v ={ξ ∈ E| hv, ξ−xi>δ}, we conclude

hxi^∁= [

δ∈Q+

[

v∈Q^d

[H_δ,v]

| {z }

∈C

∈ C. (3.16)

Consequently, hxi ∈ C and thus B_C⊆ C hold.

4 Unknown intensity

In the case, when the intensity λ is unknown and the oracle estimator ϑb_oracle in (3.4) is inaccessible, the maximum-likelihood approach suggests to use N/|C|b as an estimator forλin (2.3). This yields theplug-inestimator for the volume,

ϑb_plugin ^def= |C|b +N_Cb

N |C|b . (4.1)

In the unlikely event N = |C|b = 0, we define ϑb_plugin = 0. This estimator has a significant bias due to the following result, which is proved in the appendix.

Lemma 4. For the bias of the plug-in MLE estimator ϑb_plugin, it follows with some universal constant c >0

|C| −E[ϑb_plugin]>cE[|Cb\C|]², ∀C ∈C. (4.2) The maximal bias over C ∈ C is thus of the order λ^−4/(d+1), which is worse than the minimax rate λ−(d+3)/(2d+2) for d > 5. We surmise that ϑb_plugin is rate-optimal for d65 , but we leave that question aside because the final estimator we propose will be nearly unbiased and will satisfy an exact oracle inequality. In particular, it is rate-optimal in any dimension.

The new idea is to exploit that the number of interior points ofCb satisfies N_◦ bC∼Poiss(λ_◦), see (3.3).

There is no conditionally unbiased estimator for λ⁻¹_◦ based on observing N◦ bC ∼ Poiss(λ◦) for λ◦ ranging over some open (non-empty) interval.

Otherwise, an estimator µ(Ne _◦) for λ⁻¹_◦ would satisfy E[µ(Ne _◦)|C] =b λ⁻¹_◦ implying

X∞ k=0

λ^k_◦

k!µ(k)ee ^−λ◦=λ⁻¹_◦ ⇒ X∞ k=0

λ^k+1_◦

k! µ(k) =e X∞ k=0

λ^k_◦

k! . (4.3) The coefficient for the constant term in the left and right power series would thus differ (0 versus 1), in contradiction with the uniqueness theorem for power series.

We provide an almost unbiased estimator for λ⁻¹_◦ by noting that the first jump time of a Poisson process with intensityλ_◦ is Exp(λ_◦)-distributed and thus has expectationλ⁻¹_◦ . Taking conditional expectation with respect to the Poisson process at time 1, we conclude that

b

µ(N_◦, λ_◦)^def=

((N_◦+ 1)⁻¹, forN_◦>1, 1 +λ⁻¹_◦ , forN_◦= 0

satisfies E[µ(Nb _◦, λ_◦)|C] =b λ⁻¹_◦ . Omitting the term depending on λ_◦ in the unlikely caseN_◦= 0, we define our final estimator

ϑb^def= |C|b + N_Cb N◦+ 1|C|b . For the proofs, we also define thepseudo-estimator

ϑb_pseudo^def= |C|b +|C|Nb _Cb 1

N_◦+ 1+ e^−λ◦ λ_◦

.

Theorem 5. The pseudo-estimator ϑb_pseudo is unbiased and the estimator ϑb is asymptotically unbiased in the sense that with constants c₁, c₂ > 0 whenever λ|C|>1:

06|C| −E[bϑ]6c₁|C|exp −c2(λ|C|)(d−1)/(d+1)

, ∀C∈C. (4.4) Proof. We have

Eh 1

N_◦+ 1+e^−λ◦ λ_◦

bCi

= e^−λ◦λ⁻¹_◦ X^∞

k=0

λ^k+1_◦

(k+ 1)k!+ 1

=λ⁻¹_◦ , (4.5) which by|C|λb ⁻¹_◦ =λ⁻¹andE[ϑb_oracle] =|C|implies unbiasedness of ϑb_pseudo. Thus, it follows that

|C| −E[bϑ] =E

|C|Nb _Cbe^−λ◦λ⁻¹_◦

=λ⁻¹E

N_Cbe^−λ|C|b .

We exploit the deviation inequality that can be written in our setting, using the conditioning argument similarly to (3.10), as and derive the bound for the exponential moment of the missing volume in the model with fixed number of points

E[exp (λ|C\Cb_k|)]6b₁exp (b₂λ|C|k^−2/(d+1)), k>2, (4.6) for positive constants b₁, b₂ depending on the dimension. For the cases k = 0,1 , we have the identity E[exp (λ|C\Cb_k|)] = exp (λ|C|) . By Pois- sonisation, similarly to (3.10), we derive

exp(−λ|C|)E[exp (λ|C\C|)]b 6b₃exp −c₂(λ|C|)(d−1)/(d+1)

, (4.7) for positive constants b₃, c₂ depending on the dimension. Hence, using the Cauchy-Schwarz inequality and the bound for the moments of the points on the convex hull,

E[N^q_b

C] =O (λ|C|)q(d−1)/(d+1)

, q∈N. (4.8)

we derive for a constant c₁>0 λ⁻¹E

N_Cbe^−λ|C|b

6 λ⁻¹e^−λ|C|E[N_C²_b]^1/2E[e^2λ|C\C|b]^1/2 (4.9) 6 c₁λ^−2/(d+1)|C|(d−1)/(d+1)exp −c₂(λ|C|)(d−1)/(d+1)

(4.10) 6 c₁|C|exp −c₂(λ|C|)(d−1)/(d+1)

. (4.11)

The next step of the analysis is to compare the variance of the pseudo- estimator ϑb_pseudo with the variance of the oracle estimator ϑb_oracle, which is UMVAUE.

Theorem 6. The following oracle inequality holds with constantsc, c₁, c₂ >

0 for allC ∈C withλ|C|>1:

Var(bϑ_pseudo)6(1 +cα(λ, C)) Var(ϑb_oracle) +r(λ, C), (4.12) where

α(λ, C) = 1

|C|

1

λ+Var(|C\C|)b

E[|C\C|]b +E[|C\C|]b ,

r(λ, C) =c₁(λ|C|)2(d−1)/(d+1)exp −c₂(λ|C|)(d−1)/(d+1) . Proof. By the law of total variance, we obtain

Var(bϑ_pseudo) = Var E[bϑ_pseudo|C]b

+EVar(bϑ_pseudo|C)b

(4.13)

= Var(bϑ_oracle) +Eh

(N_Cb|C|)b ²Var 1

N_◦+ 1|Cbi

.(4.14)

In view of N◦|Cb∼Poiss(λ◦) , a power series expansion gives E[(N◦+ 1)⁻²|C] =b λ⁻¹_◦ e^−λ◦

Z λ_◦ 0

(e^t−1)/t dt . The conditional variance can forλ_◦→ ∞ thus be bounded by

Var((1 +N_◦)⁻¹|C)b 6λ⁻¹_◦ e^−λ◦ Z λ◦

λ◦/2

e^t/t dt−(λ_◦)⁻²+O(e^−λ◦/4)

= (λ_◦)⁻¹ Z λ◦/2

0

e^−s 1

λ_◦−s− 1 λ_◦

ds+O(e^−λ◦/4)

=λ⁻³_◦ (1 +o(1)),

where we have used (λ_◦−s)⁻¹−λ⁻¹_◦ =sλ⁻¹_◦ (λ_◦−s)⁻¹,R∞

0 se^−sds= 1 and dominated convergence. Thanks to (N_◦+ 1)⁻¹ ∈[0,1] we conclude for some constantc>1

Var((1 +N_◦)⁻¹|C)b 6c(1∧λ⁻³_◦ ).

Consequently, we have

Var(bϑ_pseudo) 6 Var(ϑb_oracle) +E

(N_Cb|C|)b ²c(1∧(λ|C|)b ⁻³)

(4.15)

= Var(ϑb_oracle) +cE

(N_Cb|C|)b ²∧λ⁻³(N_Cb)²|C|b⁻¹ , (4.16) and with (3.7)

Var(ϑb_pseudo)

Var(ϑb_oracle) 6 1 +cE

(N_Cbλ|C|)b ²∧(N_Cb)²(λ|C|)b ⁻¹

λE[|C\C|]b (4.17)

= 1 +cE

(N_Cb)² (λ|C|)b ²∧(λ|C|)b ⁻¹

E[N_Cb] . (4.18) Define the ‘good’ event G={|C|b >|C|/2}, on which (λ|C|)b ²∧(λ|C|)b ⁻¹

6 2(λ|C|)⁻¹. On the complement G^c, we infer from A²∧A⁻¹ 61 for A >0

Eh

(N_Cb)² (λ|C|)b ²∧(λ|C|)b ⁻¹ 1_G^ci

6E N²_b

C1_G^c

(4.19) 6E[N⁴_b

C]^1/2P(|C\C|b >|C|/2)^1/2 (4.20) 6c1(λ|C|)2(d−1)/(d+1)exp −c2(λ|C|)(d−1)/(d+1)

, (4.21) for some positive constant c₁ and c₂, using (4.7) and (4.8). It remains to estimate the upper bound (4.18) onG

2c λ|C|

E[N²_b

C] E[N_Cb] = 2c

λ|C|

Var(N_Cb)

E[N_Cb] +E[N_Cb]

. (4.22)

Using the identity for the factorial moments for the number of vertices N_Cb, we derive Var(N_Cb) 6 λ²Var(|C\C|) +b λE[|C\C|] in view ofb E[N_Cb] = λE[|C\C|] . Thus, (4.22) is bounded byb

2c λ|C|

E[N²_b

C] E[N_Cb] 6 2c

|C|

1

λ+Var(|C\C|)b

E[|C\C|]b +E[|C\C|]b

, (4.23)

which yields the assertion.

As a result, we obtain an oracle inequality for the estimator ϑb.

Theorem 7. It follows for the risk of the estimator ϑb for all C∈Cwhen- ever λ|C|>1:

E[(ϑb− |C|)²]^1/2 6(1 +cα(λ, C))E[(ϑb_oracle− |C|)²]^1/2+r(λ, C), (4.24) with constant c >0 and α(λ, C), r(λ, C) from Theorem 6. For any C ∈C andλ >0 we have α(λ, C)61 +_λ|C|¹ .

Proof. In view of λ◦ = λ|C|, we haveb ϑb= ϑb_pseudo−λ⁻¹N_Cbe^−λ|C|b and we derive as in (4.11) and (4.21) with some constantsc₁, c₂>0

E[(bϑ−ϑb_pseudo)²]6λ⁻²E[N⁴_b

C]^1/2E[e^−4λ|C|b]^1/2 6c²₁exp −2c₂(λ|C|)(d−1)/(d+1) . To establish the oracle inequality, we apply the triangle inequality in L²- norm together with Theorems 1 and 6.

The universal bound on α(λ, C) follows from the rough bound E[|C\ C|b²]6|C|E[|C\C|].b

Note that the remainder term r(λ, C) is exponentially small in λ|C|.

Therefore, an immediate implication of Theorem 7 is that asymptotically our estimator ϑb is minimax rate-optimal in all dimensions, where the lower bound is proved in the next section. Yet, even more is true: the oracle in- equality is in all well studied cases exact in the sense that α(λ, C) → 0 holds for λ → ∞ such that the the UMVAUE risk of ϑb_oracle is attained asymptotically.

Lemma 8. We have tighter bounds on α(λ, C) from Theorem 6 in the following cases:

1. for d= 1,2 and C∈C arbitrary: α(λ, C).(λ|C|)^−2/(d+1),

2. for d > 2, C with C²-boundary of positive curvature: α(λ, C) . (λ|C|)^−2/(d+1),

3. for d>2 and C a polytope: α(λ, C).λ⁻¹(log(λ|C|))^d−1.

Proof. Let us restrict to |C| = 1 , the case of general volume follows by rescaling. In view of the expectation upper bound (3.10), the main issue is to bound Var(|C\C|)/E[|Cb \C|] uniformly. Case (1) follows fromb λVar(|C\ C|)b ∼ E[|C\C|] .b

For case (2) with smooth boundary, the upper bound for the variance, Var(|C\C|)b . λ−(d+3)/(d+1), while the lower bound for the first moment, E[|C\C|]b &λ^−2/(d+1).

For the case (3) of polytopes, the upper bound Var(|C \ C|)b . λ⁻²(logλ)^d−1, while the lower bound for the first moment, E[|C \C|]b &

λ⁻¹(logλ)^d−1. The expectation upper bound from Remark 2 thus yields the result.

5 Lower bound and rate optimality

The lower bound in the P3 framework requires a specific treatment although the result for the lower bound in the model with a fixed number of observa- tions is not new.

Theorem 9. For estimating |C| in the P3 model with parameter class C, the following asymptotic lower bound holds

lim inf

λ→∞ λ(d+3)/(d+1)inf

ϑbλ

sup

C∈C

E_C[(|C| −ϑb_λ)²]>0, (5.1) where the infimum extends over all estimators ϑb_λ in the P3 model with intensity λ.

Proof. First, we reduce the class C to a properly constructed smaller class of convex sets. Then, we bound the supremum of the mean squared risk by the Bayes risk and carefully bound the Bayes risk at its minimum from below.

For simplicity, we consider the case (E,E) = ([0,1]^d,B_[0,1]d) . Following the scheme, define the class Ce of closed convex sets G

G={X= (x, y)∈[0,1]^d:x∈[0,1]^d−1, y∈[0,1],06y6g(x)} (5.2) for a concave smooth function g : R^d−1 → R with support in [0,1]^d−1 satisfying 0< g(x)<1 , ∀x∈[0,1]^d−1. Then it follows,

sup

C∈C

λ^(d+3)(d+1)E_C[(|C| −ϑb_λ)²]> sup

C∈Ce

λ^(d+3)(d+1)E_C[(|C| −ϑb_λ)²]. (5.3) To bound the mean squared risk by the Bayes risk, consider the partition of [0,1]^d−1 into cubes with sizes of length h_λ, denote by q_k the center of the k-th cube I_k and let M = (1/h_λ)^d−1 ∈N be the number of cubes. Then, for the prior on Ce, define the random variables ξ_k ^i.i.d.∼ Bernoulli(p) , i.e.

P(ξ_k= 0) =p, P(ξ_k = 1) = 1−p and set g(x) =

XM k=1

ξ_kg_k(x)− kxk², g_k(x) =ah²_λux−q_k h_λ/2

, (5.4)

where a > 0 , u : R^d−1 → R₊ is smooth and has support in [−1,1]^d−1. Choosinga >0 sufficiently small, the Hessian

∇²g(x) = XM

k=1

4aξ_k∇²u(v)|_{v=2(x−qk)/hλ}−2II_d−1, (5.5) is negative-definite for all x ∈ [0,1]^d−1 because the function ∇²u is bounded. This implies that the corresponding sets C(ξ) with boundaries g are convex and

sup

C∈Ce

λ^(d+3)(d+1)E_C[(|C| −ϑb_λ)²]>λ^(d+3)(d+1)E_ξE_C(ξ)[(|C| −ϑb_λ)²] (5.6) follows. Note that the volume of any convex set C ∈Ce with the boundary (5.4) can be written as

|C|= Z

[0,1]^d−1

g(x)dx= XM

k=1

ξ_k Z

[0,1]^d−1

g_k(x)dx− Z

[0,1]^d−1

kxk². (5.7) The Bayes-optimal estimator for (5.6) is the conditional expectation ϑb_B = E

|C|(X₁, ..., X_N)

=R

[0,1]^d−1bg(x)dx with b

g(x) =E

g(x)(X₁, ..., X_N)

= XM

k=1

E

ξ_k(X₁, ..., X_N)

g_k(x) +kxk². (5.8) Using the Bayes formula and the likelihood function (2.3), we derive

ξb_k ^def= E

ξ_k(X₁, ..., X_N)

=P ξ_k= 1(X₁, ..., X_N)

(5.9)

= (dP_gk/dP₀)(dP_g0/dP₀)⁻¹p

1−p+ (dP_gk/dP₀)(dP_g0/dP₀)⁻¹p (5.10)

= pe^λ

R

Ikg_kdx

1−p+pe^λ

R

Ikgkdx1

∀X_i^(1:d−1)∈I_k:X_i^(d)>g_k(X_i^(1:d−1)) , (5.11) where the superscript in X_i^(1:d−1) and X_i^(d) denotes the correspond- ing components of the vector Xi, g_k(x) = g_k(x) − kxk², g₀(x) = 0 and P_gk, P_g0 stand for the measures associated with the convex sets whose boundaries are g_k(x) and g₀(x) in the k-th cube I_k. Observing

ϑb_B =PM k=1ξb_kR

[0,1]^d−1g_k(x)dx−R

[0,1]^d−1kxk² and Var ξ_k(X₁, ..., X_N)

= ξb_k(1−ξb_k) , the Bayes risk can be calculated as

E_ξE_C(ξ)[(|C| −ϑb_B)²] = XM

k=1

Var(eξ_k−ξ_k)Z

Ik

g_k(x)dx2

(5.12)

= XM

k=1

Eh

Var ξ_k(X₁, ..., X_N)iZ

Ik

g_k(x)dx2

(5.13)

= XM

k=1

p(1−p) 1−p+pe^λ

R

Ikg_k(x)dx

Z

Ik

g_k(x)dx2

.(5.14) Note, that λR

Ikg_k(x)dx 6 λR

Ikg_k(x)dx 6 λb₁h^d+1_λ 6 b₂ for some con- stants b₁, b₂ > 0 if we choose h_λ ∼λ^−1/(d+1). This implies the bound for b3, b4, b5>0

E_ξE_C(ξ)[(|C| −ϑb_B)²] > M b₃ Z

Ik

g_k(x)dx2

>M b₃h⁴_λa² Z

Ik

u(x−q_k h_λ )dx2

(5.15)

> b4h^d+3_λ >b5λ−(d+3)(d+1), (5.16) which completes the proof.

6 Appendix

Proof of Lemma 4. Using that the bias of the oracle estimator ϑb = |C|b + N_Cb/(N_◦+1)|C|b is exponentially small, it remains to compare its expectation with the expectation of the plug-in estimator ϑbplugin to show (4.2):

E[bϑ−ϑb_plugin] = E

|C|(b N_Cb

N_◦+ 1−N_Cb N )

=E

|C|b N²_b

C−N_Cb (N_◦+ 1)(N_◦+N_Cb)

(6.1)

> d d+ 1E

|C|Nb ²_b

C

(N_◦+ 1)2λ|C|1(N 62λ|C|)

, (6.2)

where in the last line we have used |C|b >0 only ifN_Cb >d+ 1 and in this case N²_b

C −N_Cb > _d+1^d N²_b

C. Using E[(N◦ + 1)⁻¹|C] =b λ⁻¹_◦ (1−e^−λ◦) from

above, we obtain after writing1(N 62λ|C|) = 1−1(N >2λ|C|) E[ϑb−ϑbplugin]> d

d+ 1

E N²_b

C|C|(1b −e^−λ◦) 2λ_◦λ|C|

−E N²_b

C|C|b

2λ|C|1(N >2λ|C|)

> d d+ 1

E[N²_b

C(1−e^−λ◦)]

2λ²|C| − E

N²1(N >2λ|C|) 2λ

.

By Cauchy-Schwarz inequality and large deviations similarly to (4.11), the first term is bounded from below by a constant multiple of E[|C\C|]b ²/|C|

in view of E[N²_b

C]>λ²E[|C\C|]b ². Because of N ∼Poiss(λ|C|), the second term is of orderλ|C|²e^−λ|C|and thus asymptotically of much smaller order.

References

[1] B. Abdous, A.L. Foug`eres, and K. Ghoudi. Extreme behaviour for bivariate elliptical distributions. Canadian Journal of Statistics, 33(3):317–334, 2005.

[2] Y. Aragon, A. Daouia, and C. Thomas-Agnan. Nonparametric fron- tier estimation: A conditional quantile-based approach. Econometric Theory, 21(2):358–389, 2005.

[3] T.G. Bali and D. Weinbaum. A conditional extreme value volatility estimator based on high-frequency returns. Journal of Economic Dy- namics and Control, 31(2):361–397, 2007.

[4] J. Beirlant, T. De Wet, and Y. Goegebeur. Nonparametric estimation of extreme conditional quantiles. Journal of statistical computation and simulation, 74(8):567–580, 2004.

[5] J. Beirlant and Y. Goegebeur. Local polynomial maximum likelihood estimation for pareto-type distributions. Journal of Multivariate Anal- ysis, 89(1):97–118, 2004.

[6] V.E. Berezkin, G.K. Kamenev, and A.V. Lotov. Hybrid adaptive meth- ods for approximating a nonconvex multidimensional pareto frontier.

Computational Mathematics and Mathematical Physics, 46(11):1918–

1931, 2006.

[7] H. Bi, G. Wan, and N.D. Turvey. Estimating the self-thinning bound- ary line as a density-dependent stochastic biomass frontier. Ecology, 81(6):1477–1483, 2000.

[8] G. Bouchard, S. Girard, A. Iouditski, and A. Nazin. Some linear pro- gramming methods for frontier estimation. Applied Stochastic Models in Business and Industry, 21(2):175–185, 2005.

[9] G. Bouchard, S. Girard, AB Iouditski, and AV Nazin. Nonparametric frontier estimation by linear programming. Automation and Remote Control, 65(1):58–64, 2004.

[10] H.N.E. Bystr¨om. Managing extreme risks in tranquil and volatile mar- kets using conditional extreme value theory. International Review of Financial Analysis, 13(2):133–152, 2004.

[11] H.N.E. Bystr¨om. Extreme value theory and extremely large electricity price changes. International Review of Economics & Finance, 14(1):41–

55, 2005.

[12] C. Cazals, J.P. Florens, and L. Simar. Nonparametric frontier estima- tion: a robust approach. Journal of Econometrics, 106(1):1–25, 2002.

[13] V. Chavez-Demoulin and A.C. Davison. Generalized additive modelling of sample extremes. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54(1):207–222, 2005.

[14] A. Daouia. Asymptotic representation theory for nonstandard condi- tional quantiles. Journal of Nonparametric Statistics, 17(2):253–268, 2005.

[15] A. Daouia, J.P. Florens, and L. Simar. Functional convergence of quantile-type frontiers with application to parametric approximations.

Journal of Statistical Planning and Inference, 138(3):708–725, 2008.

[16] A. Daouia, J.P. Florens, and L. Simar. Frontier estimation and extreme value theory. Bernoulli, 16(4):1039–1063, 2010.

[17] A. Daouia, J.P. Florens, and L. Simar. Regularization of nonparametric frontier estimators. Journal of Econometrics, 2012.

[18] A. Daouia, L. Gardes, and S. Girard. Nadarayas estimates for large quantiles and free disposal support curves.Exploring Research Frontiers in Contemporary Statistics and Econometrics, pages 1–22, 2012.

[19] A. Daouia, L. Gardes, and S. Girard. On kernel smoothing for extremal quantile regression. Bernoulli, 19:2557–2589, 2013.

[20] A. Daouia, L. Gardes, S. Girard, and A. Lekina. Kernel estimators of extreme level curves. Test, 20(2):311–333, 2011.

[21] A. Daouia and I. Gijbels. Robustness and inference in nonparametric partial frontier modeling. Journal of Econometrics, 161(2):147–165, 2011.

[22] A. Daouia and I. Gijbels. Estimating frontier cost models using ex- tremiles. Exploring Research Frontiers in Contemporary Statistics and Econometrics, pages 65–81, 2012.

[23] A. Daouia, S. Girard, and A. Guillou. Aγ-moment approach to mono- tonic boundaries estimation: with applications in econometric and nu- clear fields. Journal of Econometrics, 178:727–740, 2014.

[24] A. Daouia and A. Ruiz-Gazen. Robust nonparametric frontier estima- tors: qualitative robustness and influence function. Statistica Sinica, 16(4):1233, 2006.

[25] A. Daouia and L. Simar. Nonparametric efficiency analysis: A mul- tivariate conditional quantile approach. Journal of Econometrics, 140(2):375–400, 2007.

[26] B. Das and S.I. Resnick. Conditioning on an extreme component: Model consistency with regular variation on cones. Bernoulli, 17(1):226–252, 2011.

[27] B. Das and S.I. Resnick. Detecting a conditional extreme value model.

Extremes, 14(1):29–61, 2011.

[28] AC Davison and NI Ramesh. Local likelihood smoothing of sample extremes. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 62(1):191–208, 2000.

[29] A.L.M. Dekkers and L. de Haan. On the estimation of extreme-value index and large quantiles estimation. Ann. Statist., 17:1795–1832, 1989.

[30] A. Delaigle and I. Gijbels. Estimation of boundary and discontinuity points in deconvolution problems. Statistica Sinica, 16(3):773, 2006.

[31] G. Dierckx, Y. Goegebeur, and A. Guillou. Local robust and asymp- totically unbiased estimation of conditional pareto-type tails. Test, 23(2):330–355, 2014.

[32] J.P. Florens and L. Simar. Parametric approximations of nonparametric frontiers. Journal of Econometrics, 124(1):91–116, 2005.

[33] L. Gardes and S. Girard. A moving window approach for nonpara- metric estimation of the conditional tail index. Journal of Multivariate Analysis, 99(10):2368–2388, 2008.

[34] L. Gardes and S. Girard. Conditional extremes from heavy-tailed dis- tributions: An application to the estimation of extreme rainfall return levels. Extremes, 13(2):177–204, 2010.

[35] L. Gardes and S. Girard. Functional kernel estimators of large condi- tional quantiles. Electronic Journal of Statistics, 6:1715–1744, 2012.

[36] L. Gardes, S. Girard, and A. Lekina. Functional nonparametric estima- tion of conditional extreme quantiles. Journal of Multivariate Analysis, 101(2):419–433, 2010.

[37] J. Geffroy, S. Girard, and P. Jacob. Asymptotic normality of the l 1- error of a boundary estimator. Journal of Nonparametric Statistics, 18(1):21–31, 2006.

[38] A. Ghorbel and A. Trabelsi. Predictive performance of conditional extreme value theory in value-at-risk estimation. International Journal of Monetary Economics and Finance, 1(2):121–148, 2008.

[39] I. Gijbels, E. Mammen, B.U. Park, and L. Simar. On estimation of monotone and concave frontier functions. Journal of the American Statistical Association, 94(445):220–228, 1999.

[40] S. Girard. On the asymptotic normality of the l1-error for haar series estimates of poisson point processes boundaries. Statistics & Probability Letters, 66(1):81–90, 2004.

[41] S. Girard, A. Guillou, and G. Stupfler. Frontier estimation with kernel regression on high order moments. Journal of Multivariate Analysis, 116:172–189, 2013.

[42] S. Girard, A. Guillou, and G. Stupfler. Uniform strong consistency of a frontier estimator using kernel regression on high order moments.

ESAIM: Probability and Statistics, 18:642–666, 2014.

[43] S. Girard, A. Iouditski, and A.V. Nazin. L 1-optimal nonparametric frontier estimation via linear programming. Automation and Remote Control, 66(12):2000–2018, 2005.

[44] S. Girard and P. Jacob. Extreme values and haar series estimates of point process boundaries. Scandinavian Journal of Statistics, 30(2):369–384, 2003.

[45] S. Girard and P. Jacob. Projection estimates of point processes bound- aries. Journal of Statistical Planning and Inference, 116(1):1–15, 2003.

[46] S. Girard and P. Jacob. Extreme values and kernel estimates of point processes boundaries. ESAIM: Probability and Statistics, 8(1):150–168, 2004.

[47] S. Girard and P. Jacob. Frontier estimation via kernel regression on high power-transformed data. Journal of Multivariate Analysis, 99(3):403–

420, 2008.

[48] S. Girard and P. Jacob. A note on extreme values and kernel estimators of sample boundaries. Statistics & Probability Letters, 78(12):1634–

1638, 2008.

[49] S. Girard and P. Jacob. Frontier estimation with local polynomials and high power-transformed data. Journal of Multivariate Analysis, 100(8):1691–1705, 2009.

[50] S. Girard and L. Menneteau. Central limit theorems for smoothed extreme value estimates of poisson point processes boundaries. Journal of Statistical Planning and Inference, 135(2):433–460, 2005.

[51] S. Girard and L. Menneteau. Smoothed extreme value estimators of non-uniform point processes boundaries with application to star-shaped supports estimation.Communications in StatisticsTheory and Methods, 37(6):881–897, 2008.

[52] Y. Goegebeur, A. Guillou, and A. Schorgen. Nonparametric regression estimation of conditional tails: the random covariate case. Statistics, 23:1–24, 2013.

[53] W.H. Greene. Maximum likelihood estimation of econometric frontier functions. Journal of Econometrics, 13(1):27–56, 1980.

[54] P. Hall, L. Peng, and C. Rau. Local likelihood tracking of fault lines and boundaries. Journal of the Royal Statistical Society: Series B (Sta- tistical Methodology), 63(3):569–582, 2001.

[55] P. Hall and L. Simar. Estimating a changepoint, boundary, or frontier in the presence of observation error. Journal of the American Statistical Association, 97(458):523–534, 2002.

[56] P. Hall and N. Tajvidi. Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data. Statistical Science, pages 153–167, 2000.

[57] P. Jacob and C. Suquet. Estimating the edge of a poisson process by orthogonal series. Journal of Statistical Planning and Inference, 46(2):215–234, 1995.

[58] P. Jacob and C. Suquet. Regression and edge estimation. Statistics &

Probability Letters, 27(1):11–15, 1996.

[59] S.O. Jeong and L. Simar. Linearly interpolated fdh efficiency score for nonconvex frontiers. Journal of Multivariate Analysis, 97(10):2141–

2161, 2006.

[60] A. Kneip, L. Simar, and P.W. Wilson. Asymptotics for dea estimators in nonparametric frontier models. Technical report, Discussion paper, 2003.

[61] R. Koenker and K. Hallock. Quantile regression: An introduction.Jour- nal of Economic Perspectives, 15(4):43–56, 2001.

[62] A. P. Korostelev and A. B. Tsybakov. Asymptotic efficiency in esti- mation of a convex set. Problemy Peredachi Informatsii, 30(4):33–44, 1994.

[63] Aleksandr Petrovich Korostelev and Alexandre B Tsybakov. Minimax linewise algorithm for image reconstruction. Springer, 1993.

[64] S.C. Kumbhakar and C.A.K. Lovell. Stochastic frontier analysis. Cam- bridge University Press, 2003.

[65] V. Marimoutou, B. Raggad, and A. Trabelsi. Extreme value theory and value at risk: application to oil market. Energy Economics, 31(4):519–

530, 2009.

[66] C. Martins-Filho and F. Yao. A smooth nonparametric conditional quantile frontier estimator. Journal of Econometrics, 143(2):317–333, 2008.

[67] J. El Methni, L. Gardes, and S. Girard. Nonparametric estimation of extreme risks from conditional heavy-tailed distributions.Scandinavian Journal of Statistics, 41:988–1012, 2014.

[68] A. Novikov. Linear aggregation methods applied to boundary estima- tors. http://hal.archives-ouvertes.fr/hal-01074697, 2014.

[69] A. Novikov. Nonlinear aggregation methods for boundary estimators.

http://hal.archives-ouvertes.fr/hal-01075368, 2014.

[70] A. Novikov. Asymptotic distribution of convex-hull estimators.

http://hal.archives-ouvertes.fr/hal-01110490, 2015.

[71] A. Novikov. Boundary of a gaussian process. http://hal.archives- ouvertes.fr/hal-01108160, 2015.

[72] B.U. Park and L. Simar. Efficient semiparametric estimation in a stochastic frontier model. Journal of the American Statistical Asso- ciation, 89(427):929–936, 1994.

[73] L. Peng. Bias-corrected estimators for monotone and concave frontier functions. Journal of Statistical Planning and Inference, 119(2):263–

275, 2004.

[74] S. Rao. Frontier estimation as a particular case of conditional extreme value analysis. http://hal.archives-ouvertes.fr/hal-00776107, 2013.

[75] S. Rao. Linear aggregation of conditional extreme-value index estima- tors. http://hal.archives-ouvertes.fr/hal-00771647, 2013.

[76] S. Rao. Linear aggregation of frontier estimators. http://hal.archives- ouvertes.fr/hal-00798394, 2013.

[77] S. Rao. Nonlinear aggregation of conditional extreme-value index esti- mators. http://hal.archives-ouvertes.fr/hal-00776108, 2013.

[78] S. Rao. Nonlinear aggregation of frontier estimators. Jour- nal of Parametric and Non-Parametric Statistics, 1, 2013.

http://jpanps.altervista.org/.

[79] S. Rao. A review on conditional extreme value analysis.

http://hal.archives-ouvertes.fr/hal-00770546, 2013.

[80] O. Rosen and A. Cohen. Extreme percentile regression. In Statistical Theory and Computational Aspects of Smoothing: Proceedings of the COMPSTAT94 satellite meeting held in Semmering, Austria, pages 27–

28, 1994.

[81] P. Schmidt and T.F. Lin. Simple tests of alternative specifications in stochastic frontier models. Journal of Econometrics, 24(3):349–361, 1984.

[82] J.K. Sengupta. Stochastic data envelopment analysis: a new approach.

Applied Economics Letters, 5(5):287–290, 1998.

[83] L. Simar. Aspects of statistical analysis in dea-type frontier models.

Journal of Productivity Analysis, 7(2):177–185, 1996.

[84] L. Simar. Detecting outliers in frontier models: A simple approach.

Journal of Productivity Analysis, 20(3):391–424, 2003.

[85] L. Simar and P.W. Wilson. Sensitivity analysis of efficiency scores: How to bootstrap in nonparametric frontier models. Management science, 44(1):49–61, 1998.

[86] L. Simar and P.W. Wilson. Of course we can bootstrap dea scores!

but does it mean anything? logic trumps wishful thinking. Journal of Productivity Analysis, 11(1):93–97, 1999.

[87] L. Simar and P.W. Wilson. A general methodology for bootstrap- ping in non-parametric frontier models. Journal of Applied Statistics, 27(6):779–802, 2000.

[88] L. Simar and P.W. Wilson. Statistical inference in nonparametric fron- tier models: The state of the art. Journal of Productivity Analysis, 13(1):49–78, 2000.

[89] L. Simar and P.W. Wilson. Testing restrictions in nonparametric effi- ciency models. Communications in Statistics-Simulation and Compu- tation, 30(1):159–184, 2001.

[90] L. Simar and P.W. Wilson. Statistical inference in nonparametric fron- tier models: recent developments and perspectives. The measurement of productive efficiency and productivity growth, pages 421–521, 2008.

[91] L. Simar and P.W. Wilson. Inferences from cross-sectional, stochastic frontier models. Econometric Reviews, 29(1):62–98, 2009.

[92] L. Simar and P.W. Wilson. Performance of the bootstrap for dea es- timators and iterating the principle. Handbook on data envelopment analysis, pages 241–271, 2011.

[93] L. Simar and V. Zelenyuk. Stochastic fdh/dea estimators for frontier analysis. Journal of Productivity Analysis, 36(1):1–20, 2011.

[94] C. Wang, H. Zhuang, Z. Fang, and T. Lu. A model of conditional var of high frequency extreme value based on generalized extreme value distribution. Journal of Systems & Management, 3:003, 2008.

[95] H. Wang and C.L. Tsai. Tail index regression. Journal of the American Statistical Association, 104(487):1233–1240, 2009.

[96] H.J. Wang, D. Li, and X. He. Estimation of high conditional quan- tiles for heavy-tailed distributions. Journal of the American Statistical Association, 107:1453–1464, 2012.