Elena Di Bernardino
email: elena.di-bernardino@univ-lyon1.fr

Thomas Laloë
email: thomas.laloe@unice.fr

Estimating level sets of a distribution function using a plug-in method: a multidimensional extension

Keywords: Level sets, multidimensional distribution function, plug-in estimation, Hausdorff distance

. Finally we investigate the effects of scaling data on our consistency results.

Introduction

In this present paper, we consider the problem of estimating the level sets of a d-variate distribution function. To this aim, we generalize the results obtain in a previous paper (Di [START_REF] Di Bernardino | Plug-in estimation of level sets in a non-compact setting with applications in multivariable risk theory[END_REF].

As yet remarked in Di Bernardino et al. (2011), considering the level sets of a distribution function, the commonly assumed property of compactness for these sets is no more reasonable. Then, differing from the classical literature [START_REF] Cavalier | Nonparametric estimation of regression level sets[END_REF][START_REF] Cuevas | A plug-in approach to support estimation[END_REF][START_REF] Baíllo | Convergence rates in nonparametric estimation of level sets[END_REF][START_REF] Baíllo | Total error in a plug-in estimator of level sets[END_REF][START_REF] Cuevas | Plug-in estimation of general level sets[END_REF][START_REF] Biau | A graph-based estimator of the number of clusters[END_REF][START_REF] Laloë | Sur Quelques Problèmes d'Apprentissage Supervisé et Non Supervisé[END_REF], we need to work in a non-compact setting and this requires special attention in the statement of our problem.

We follow the same general approach than in Di Bernardino et al. (2011), and we will keep as much as possible the same notation. Considering a consistent estimator F n of the distribution function F , we propose a plug-in approach (e.g. see Baíllo et al., 2011;[START_REF] Rigollet | Optimal rates for plug-in estimators of density level sets[END_REF][START_REF] Cuevas | Plug-in estimation of general level sets[END_REF]) to estimate the level set

L(c) = {x ∈ R d + : F (x) ≥ c}, for c ∈ (0, 1), by L n (c) = {x ∈ R d + : F n (x) ≥ c}, for c ∈ (0, 1).
The regularity properties of F and F n as well as the consistency properties of F n will be specified in the statements of our theorems.

As in Di Bernardino et al. (2011) our consistency results are stated with respect to two criteria of "physical proximity" between sets: the Hausdorff distance and volume of the symmetric difference.

If the consistency in term of the Hausdorff distance is a trivial extension of Theorem 2.1 in Di Bernardino et al. (2011) (see Theorem 2.1 below), things are a little more complex for the volume of the symmetric difference. In particular, in this latter case the convergence rate suffers from the well-known curse of dimensionality (see Theorem 3.1).

A second aim of this paper is to analyze the effects of scaling data on our consistency results (see Theorem 4.1).

The paper is organized as follows. We introduce some notation, tools and technical assumptions in Section 1. Consistency and asymptotic properties of our estimator of L(c) are given in Sections 2 and 3. Section 4 is devoted to investigate the effects of scaling data on our consistency results. Finally, proofs are postponed to Section 5.

Notation and preliminaries

In this section we introduce some notation and tools which will be useful later.

Let Define, for c ∈ (0, 1), the upper c-level set of F ∈ F and its plug-in estimator

N * = N \ {0}, R * + = R + \ {0} and R d + * = R d + \ {0}. Let F be the set of continuous distribution functions R d + → [0, 1] and X := (X
L(c) = {x ∈ R d + : F (x) ≥ c}, L n (c) = {x ∈ R d + : F n (x) ≥ c},
and

{F = c} = {x ∈ R d + : F (x) = c}.
In addition, given T > 0, we set

L(c) T = {x ∈ [0, T ] d : F (x) ≥ c}, L n (c) T = {x ∈ [0, T ] d : F n (x) ≥ c}, {F = c} T = {x ∈ [0, T ] d : F (x) = c}.
Given a set A ⊂ R d + we denote by ∂A its boundary, and by β A the scaled set {β x, with x ∈ A}.

Note that, in the presence of a plateau at level c, {F = c} can be a portion of quadrant R d + instead of a set of Lebesgue measure null in R d + . In the following we introduce suitable conditions in order to avoid this situation.

We denote by B(x, ρ) the closed ball centered on x ∈ R d + and with positive radius ρ. Let B(S, ρ) = x∈S B(x, ρ), with S a closed set of R d + . For r > 0 and ζ > 0, define

E = B({x ∈ R d + : | F (x) -c |≤ r}, ζ),
and, for a twice differentiable function F ,

m ▽ = inf x∈E (∇F ) x , M H = sup x∈E (HF ) x ,
where (∇F ) x is the gradient vector of F evaluated at x and (∇F ) x its Euclidean norm, (HF ) x the Hessian matrix evaluated in x and (HF ) x its matrix norm induced by the Euclidean norm.

For sake of completeness, we recall that if A 1 and A 2 are compacts sets in R d + , the Hausdorff distance between A 1 and A 2 is defined by

d H (A 1 , A 2 ) = max sup x∈A 1 d(x, A 2 ), sup x∈A 2 d(x, A 1 ) , where d(x, A 2 ) = inf y∈A 2 x -y .
The above expression is well defined even when A 1 and A 2 are just closed (not necessarily compacts) sets but, in this case, the value d H (A 1 , A 2 ) could be infinity. Then in our setting, in order to avoid these situations, we introduce the following assumption.

H: There exist γ > 0 and

A > 0 such that, if | t -c | ≤ γ then ∀ T > 0 such that {F = c} T = ∅ and {F = t} T = ∅, d H ({F = c} T , {F = t} T ) ≤ A | t -c | .
For

Proposition 1.1 Let c ∈ (0, 1). Let F ∈ F be twice differentiable on R d * + . Assume there exist r > 0, ζ > 0 such that m ▽ > 0 and M H < ∞. Then F satisfies Assumption H, with A = 2 m ▽ .

Consistency in terms of the Hausdorff distance

In this section we study the consistency properties of L n (c) T with respect to the Hausdorff distance between ∂L n (c) T and ∂L(c) T .

From now on we note, for n ∈ N * ,

F -F n ∞ = sup x ∈ R d + | F (x) -F n (x) |,
and for T > 0 F -F n T ∞ = sup x ∈ [0,T ] d | F (x) -F n (x) | .
The following result can be considered a trivially adapted version of Theorem 2.1 in Di Bernardino et al. (2011).

Theorem 2.1 Let c ∈ (0, 1). Let F ∈ F be twice differentiable on R d * + . Assume that there exist r > 0, ζ > 0 such that m ▽ > 0 and M H < ∞. Let T 1 > 0 such that for all t : | t -c | ≤ r, ∂L(t) T 1 = ∅. Let (T n ) n∈N * be
an increasing sequence of positive values. Assume that, for each n and for almost all samples of size n, F n is a continuous function and that

F -F n ∞ → 0, a.s.
Then, for n large enough,

d H (∂L(c) Tn , ∂L n (c) Tn ) ≤ 6 A F -F n Tn ∞ a.s., where A = 2 m ▽ . Therefore we have d H (∂L(c) Tn , ∂L n (c) Tn ) = O( F -F n ∞ ) a.s.
Under assumptions of Theorem 2.1, d H (∂L(c) Tn , ∂L n (c) Tn ) converges to zero and the quality of our plug-in estimator is obviously related to the quality of the estimator F n . For comments and discussions about this result we refer the interested reader to Remark 2 in Di Bernardino et al.

(2011).

L 1 consistency

The previous section was devoted to the consistency of L n (c) in terms of the Hausdorff distance.

We consider now another consistency criterion: the consistency of the volume (in the Lebesgue measure sense) of the symmetric difference between L(c) Tn and L n (c) Tn . This means that we define the distance between two subsets A 1 and

A 2 of R + d by d λ (A 1 , A 2 ) = λ(A 1 △ A 2 ),
where λ stands for the Lebesgue measure on R d and △ for the symmetric difference.

Let us introduce the following assumption:

A1 There exist positive increasing sequences

(v n ) n∈N * and (T n ) n∈N * such that v n [0,Tn] d | F -F n | p λ(dx) P → n→∞ 0,
for some 1 ≤ p < ∞.

We now establish our consistency result with convergence rate, in terms of the volume of the symmetric difference. We can interpret the following theorem as an extension of Theorem 3. we remark that a sequence T n , whose divergence rate is large, implies a convergence rate p n quite slow. Moreover, this phenomenon is emphasized by the dimension d of the data, and we face here the well-known curse of dimensionality. In the following we will illustrate this aspect by giving convergence rate in the case of the empirical distribution function (see Example 1). Firstly, from Theorem 3.1 we can derive the following result. .

This result comes trivially from Theorem 3.1 and the fact that that

v n F -F n ∞ P → n→∞ 0 implies ∀ p ≥ 1, w n [0,Tn] d | F -F n | p λ(dx) P → n→∞ 0, with w n = v p n T d n .
Let us now present a more practical example.

Example 1 (The empirical distribution function case)

Let F n the d-variate empirical distribution function. Then, it holds that v n F -F n ∞ P → n→∞ 0, with v n = o( √ n).
From Theorem 3.1, with p = 2, we obtain for instance:

p n = o n 1/3 T n 7/3 , for d = 3; p n = o n 1/3 T n 10/3 , for d = 4.
The next section is dedicated to study the effects of scaling data.

About the effects of scaling data

Suppose now to scale our data using a scale parameter a ∈ R * + . In our case, the scaled random vector will be (a X 1 , a X 2 , . . . , a X d ) := a X. From now on we denote F a X (resp. F X ) the distribution function associated to a X (resp. to X). Using notation of Section 1, let

L a (c) = {x ∈ R d + : F a X (x) ≥ c}.
It is easy to prove (see for instance Section 3 in Tibiletti, 1993) that

L a (c) = a L(c),
and

E a = B({x ∈ R d + : | F a X (x) -c |≤ r}, ζ) = a E. Define now m ▽ a = inf x∈Ea ∇F a X (x) .
First, we can obtain the following result whose proof is postponed to Section 5.

Lemma 4.1 It holds that

m ▽ a = 1 a m ▽ , ∀ a ∈ R * + .
Furthermore, if

M H = sup x∈E (HF X ) x < +∞ then M H,a = sup x∈a E (HF a X ) x < +∞, with a ∈ R * + .
We can now consider the effects of scaling data on Theorem 2.1 and 3.1.

Theorem 4.1

1. Under same notation and assumptions of Theorem 2.1, for n large enough, it holds that We propose a rate of convergence for this second criterion. Moreover, we analyze the impact of scaling data on our results. As a future work, a complete simulation study and an R-package are in preparation.

d H (∂L a (c) a Tn , ∂L n, a (c) a Tn ) ≤ 6 A a F -F n Tn ∞ , a.s.

Proofs

Proof of Theorem 3.1 Under assumptions of Theorem 3.1, we can always take T 1 > 0 such that for all t : | tc | ≤ r, ∂L(t) T 1 = ∅. Then for each n, for all t : | tc | ≤ r, ∂L(t) Tn is a non-empty (and compact) set on R d + .

We consider a positive sequence ε n such that ε n → n→∞ 0. For each n ≥ 1 the random sets

L(c) Tn △ L n (c) Tn , Q εn = {x ∈ [0, T n ] d : | F -F n |≤ ε n } and Q εn = {x ∈ [0, T n ] d : | F -F n |> ε n } are measurable and λ(L(c) Tn △ L n (c) Tn ) = λ(L(c) Tn △ L n (c) Tn ∩ Q εn ) + λ(L(c) Tn △ L n (c) Tn ∩ Q εn ). Since L(c) Tn △ L n (c) Tn ∩ Q εn ⊂ {x ∈ [0, T n ] d : c -ε n ≤ F < c + ε n } we obtain λ(L(c) Tn △ L n (c) Tn ) ≤ λ({x ∈ [0, T n ] d : c -ε n ≤ F < c + ε n }) + λ( Q εn ).
From Assumption H (Section 1) and Proposition

1.1, if 2 ε n ≤ γ then d H (∂L(c + ε n ) Tn , ∂L(c -ε n ) Tn ) ≤ 2 ε n A.
From assumptions on first derivatives of F (see Assumption H and Proposition 1.1) and Propriety 1 in Imlahi et al. (1999), we can write

λ({x ∈ [0, T n ] d : c -ε n ≤ F < c + ε n }) ≤ (2 ε n A) d T d-1 n .
If we now choose

ε n = o 1 p n T d-1 n , (1)
we obtain that, for n large enough, 2 ε n ≤ γ and

p n λ({x ∈ [0, T n ] d : c -ε n ≤ F < c + ε n }) → n→∞ 0.
Let us now prove that p n λ( Q εn ) P → n→∞ 0. To this end, we write

p n λ( Q εn ) = p n 1 {x∈[0,Tn] d : |F -Fn|>εn} λ(dx) ≤ p n ε p n [0,Tn] d | F -F n | p λ(dx).
Take ε n such that First, we remark that F a X (x) = F X x a .

ε n = p n v n 1 p . ( 2
Then, we obtain The second part of this demonstration is equal to proof of Theorem 3.1. Then we take ε n such that Hence the result.

m ▽ a = inf x∈a E ∂ ∂x 1 F X x a , . . . , ∂ ∂x d F X x a , = inf x∈a E
ε n = p n, a v n 1 p . ( 4

Remark 1

 1 Under assumptions of Proposition 1.1, {F = t} is a set of Lebesgue measure null in R d + . Furthermore we obtain ∂L(c) T = {F = c} T = {F = c} ∩ [0, T ] d (we refer for details to Remark 1 in Di Bernardino et al., 2011 and Theorem 3.2 in Rodríguez-Casal, 2003).

Theorem 3 . 1 .

 31 1 in Di Bernardino et al. (2011), in the case of a d-variate distribution function F . Let c ∈ (0, 1). Let F ∈ F be a twice differentiable distribution function on R d * + . Assume that there exist r > 0, ζ > 0 such that m ▽ > 0 and M H < ∞. Assume that for each n, with probability one, F n is measurable. Let (v n ) n∈N * and (T n ) n∈N * positive increasing sequences such that Assumption A1 is satisfied and that for all t : | tc | ≤ r, ∂L(t) T 1 = ∅. Then, it holds that p n d λ (L(c) Tn , L n (c) Tn ) with p n an increasing positive sequence such that p n = o v The proof is postponed to Section 5. This demonstration is basically based on the proof of Theorem 3.1 in Di Bernardino et al. (2011). Theorem 3.1 provides a convergence rate, which is closely related to the choice of the sequence T n . Note that, as in Theorem 3 in Cuevas et al. (2006), Theorem 3.1 above does not require any continuity assumption on F n . Furthermore, as in Theorem 3.1 in Di Bernardino et al. (2011),

Corollary 3 . 1

 31 Under the assumptions and notations of Theorem 3.1. Assume that there exists a positive increasing sequence (v n ) n∈N * such that v n F -F n ∞ P → n→∞ 0. Then, it holds that p n d λ (L(c) Tn , L n (c) Tn ) with p n an increasing positive sequence such that p n = o v n p p+1 /T n d+(d-1) p p+1

2 . 1 . 2 ..

 212 Under same notation and assumptions of Theorem 3.1 it holds that p n, a d λ (L a (c) a Tn , L n, a (c) a Tn ) p n, a an increasing positive sequence such that p n, a = o v The first result of Theorem 4.1 states that a change of scale of the data implies the same change of scale for the Hausdorff distance. The second result states that a change of scale of the data implies a rate in So, we see logically that the scale factor a impacts the volume in R d with an exponent d. Conclusion Starting from previous results obtained in Di Bernardino et al. (2011), we propose in this paper a generalization to the estimation of level sets in the case of a d-variate distribution function. The consistency results are stated in term of Hausdorff distance and volume of the symmetric difference.

  )So, from Assumption A1 in Section 3, we obtain p n λ( Q εn ) ε n that satisfies (1) and (2). Hence the result.Proof of Lemma 4.1

1 Proof of 1 .. 1 n ( 3 )

 1113 Second part of Lemma 4.1 comes down from trivial calculus. Hence the result.Proof of Theorem 4.Following the proof of Theorem 2.1, it holds thatd H (∂L a X (c) a Tn , ∂L n, a (c) a Tn ) ≤ 6 2 m ▽ a sup x∈[0,a T n] d | F X x a -F n x a | .Using Lemma 4.1 and the fact thatsup x∈[0,a T n] d | F X x a -F n x a |= sup x∈[0,T n] d | F X (x) -F n (x) |,we get the result.Proof of 2. As in the proof of Theorem 3.1 and using same notation, we can writeλ({x ∈ [0, a T n ] d : cε n ≤ F a X < c + ε n }) ≤ (2 ε n A a) d a d-1 T d-1 nIf we now chooseε n = o 1 p n, a a d T d-we obtain that for n large enough 2 ε n ≤ γ andp n, a λ({x ∈ [0, a T n ] d : cε n ≤ F a X < c + ε n }) → n→∞ 0.

  )Then, from Assumption A1, in Section 3, we obtain p n, a λ({x∈ [0, a T n ] d : | F a X -F a, n |> ε n }) ε n that satisfies (3) and (4).

  1 , X 2 , . . . , X d ) a random vector with distribution function F ∈ F. Given an i.i.d sample {X i } n i=1 in R d + with distribution function F ∈ F, we denote by F n an estimator of F based on this finite sample.

  further details about this assumption the interest reader is referred to Di Bernardino et al.

(2011), Cuevas et al. (2006), Tsybakov (1997). Remark that a sufficient condition for Assumption H can be obtained in terms of the differentiability properties of F . Proposition 1.1 below is a trivial extension in d-variate setting of Proposition 1.1 in Di Bernardino et al. (2011).

Acknowledgements: This work has been partially supported by the French research national agency (ANR) under the reference ANR-08BLAN-0314-01. The authors thank Yannick Baraud, Christine Tuleau-Malot and Patricia Reynaud-Bouret for fruitful discussions.

References: