getdoca8fd. 198KB Jun 04 2011 12:04:49 AM

(1)

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y}

Vol. 14 (2009), Paper no. 91, pages 2617–2635. Journal URL

http://www.math.washington.edu/~ejpecp/

Asymptotic Normality in Density Support Estimation

Gérard Biau

∗

Benoît Cadre

†

David M. Mason

‡

Bruno Pelletier

§

Abstract

Let X1, . . . ,Xnbe nindependent observations drawn from a multivariate probability density f with compact supportSf. This paper is devoted to the study of the estimatorSˆnofSf defined as the union of balls centered at theXiand with common radiusrn. Using tools from Riemannian geometry, and under mild assumptions on f and the sequence (rn), we prove a central limit theorem forλ(Sn∆Sf), whereλdenotes the Lebesgue measure onRd and∆the symmetric dif-ference operation.

Key words:Support estimation, Nonparametric statistics, Central limit theorem, Tubular neigh-borhood.

AMS 2000 Subject Classification:Primary 62G05, 62G20.

Submitted to EJP on April 29, 2009, final version accepted December 5, 2009.

∗_{LSTA & LPMA, Université Pierre et Marie Curie – Paris VI, Boîte 158, 175 rue du Chevaleret, 75013 Paris, France,}

[email protected]

†_{IRMAR, ENS Cachan Bretagne, CNRS, UEB, Campus de Ker Lann, Avenue Robert Schuman, 35170 Bruz, France,}

[email protected]

‡_{University of Delaware, Food and Resource Economics, 206 Townsend Hall, Newark, DE 19717, USA,}

[email protected]

§_{IRMAR, Université Rennes 2, CNRS, UEB, Campus Villejean, Place du Recteur Henri Le Moal, CS 24307, 35043}

(2)

1

Introduction

Let X1, . . . ,Xn be independent and identically distributed observations drawn from an unknown

probability density f defined onRd. It is assumed thatd≥2 throughout this paper. We investigate the problem of estimating the support of f, i.e., the closed set

S_f =_{x_∈Rd :f(x)>0_},

based on the sample X₁, . . . ,X_n. Here and elsewhere,Adenotes the closure of a Borel set A. This problem is of interest due to the broad scope of its practical applications in applied statistics. These include medical diagnosis, machine condition monitoring, marketing and econometrics. For a review and a large list of references, we refer the reader to Molchanov (1998), Baíllo, Cuevas, and Justel (2000), Biau, Cadre, and Pelletier (2008) and Mason and Polonik (2009).

Devroye and Wise (1980) introduced the following very simple and intuitive estimator ofS_f. It is defined as

S_n=

n

[

i=1

B(X_i,r_n), (1.1)

where_B(x,r)denotes the closed Euclidean ball centered at x and of radiusr>0, and where(rn)

is an appropriately chosen sequence of positive smoothing parameters. For x∈Rd, let

fn(x) = n

X

i=1

1_B(x,rn)(Xi)

be the (unnormalized) kernel density estimator of f. We see that

Sn={x ∈Rd: fn(x)>0}.

In other words, S_n = S_f

n, i.e., it is just a plug-in-type kernel estimator with kernel having a

ball-shaped support. Baíllo, Cuevas, and Justel (2000) argue that this estimator is a good generalist when no a priori information is available about S_f. Moreover, from a practical perspective, the relative simplicity of the estimation strategy (1.1) is a major advantage over competing multidi-mensional set estimation techniques, which are often faced with a heavy computational burden.

Biau, Cadre, and Pelletier (2008) proved, under mild regularity assumptions on f and the sequence

(r_n), that for an explicit constantc>0,

Æ

nr_ndE_λ(Sn∆Sf)→c,

where_△denotes the symmetric difference operation andλis the Lebesgue measure onRd. In the present paper, we go one step further and establish the asymptotic normality ofλ(S_n_△S_f). Precisely, our main Theorem 2.1 states, under appropriate regularity conditions on f and(rn), that

n

r_nd

1/4

(3)

for some explicit positive varianceσ2_f.

Denoting by∂Sf the boundary ofSf, it turns out that, under our conditions,λ(∂Sf) =0 and f >0

on the interior ofS_f. Therefore, we have the equality

λSf△

¦

x∈Rd : f(x)>0©=0.

Thus,λ(S_n_△S_f)may be expressed more conveniently as

λ(S_n_△S_f) = Z

Rd

1{fn(x)>0} −1{f(x)>0} dx.

This quantity is related to the so-called vacancy Vn left by randomly distributed spheres (see Hall

1985, 1988), which in this notation is

V_n=λS_f ₋S_n= Z

Sf

1{fn(x)>0} −1{f(x)>0}

dx. (1.2)

Hall (1985) has proved a number of central limit theorems for V_n. One of them, his Theorem 1, states that if f has support in[0, 1]d and is continuous then, as long asnr_nd_→awhere 0<a<∞, for some 0< σ2

a<∞,

p

n V_n₋EV_n D

→ N(0,σ2_a).

As pointed out in Hall’s paper, and to the best of our knowledge, the case when nr_nd → ∞has not been examined, except for some restricted cases in dimension 1. It turns out that, by adapting our arguments to the vacancy problem, we are also able to prove a general central limit theorem for

V_n when nr_nd _{→ ∞}, thereby extending Hall’s results. For more about large sample properties of vacancy and their applications consult Chapter 3 of Hall (1988).

Another result closely related to ours is the following special case of the main theorem in Mason and Polonik (2009). For any 0 < c < sup_{f(x) : x _∈Rd_}, let C(c) = _{x _∈Rd : f(x) > c_} and

ˆ

Cn(c) ={x ∈Rd:fˆn(x)>c}, where fˆn denotes a kernel estimator of f. Then

λCˆ_n(c)_△C(c)= Z

Rd

1{fˆn(x)>c} −1{f(x)>c} dx.

Mason and Polonik (2009) prove, subject to regularity conditions on f, as long as pnr_nd+2 →γ, with 0≤γ <∞andnr_nd/logn→ ∞, whereγ=0 in the cased=1, that for some 0< σ2

c<∞,

n

r_nd

1/4

λCˆ_n(c)_△C(c)₋E_λCˆ_n(c)_△C(c)_{→ N}D (0,σ2_c).

(4)

2

Asymptotic normality of

λ(S

_n

∆S

_f

)

2.1

Notation and assumptions

Throughout the paper, we shall impose the following set of assumptions related to the support of f. To state some of them we shall require a few concepts and terms from Riemannian geometry. For a good introduction to the subject we refer the reader, for instance, to the book by Gallot, Hulin and

Lafontaine (2004). We make use of the notationS◦_f to denote the interior ofS_f.

Assumption Set 1

(a) The supportS_f of f is compact inRd, with d_≥2.

(b) f is of class_C1onRd, and of class_C2 on_{T ∩}S◦_f, where_T is a tubular neighborhood ofS_f.

(c) The boundary∂S_f ofS_f is a smooth submanifold ofRd of codimension 1.

(d) The set_{x ∈Rd:f(x)>0_}is connected.

(e) f >0 onS◦_f.

Roughly speaking, under Assumption 1-(c), the boundary∂S_f is a subset of dimension(d₋1) of the ambient spaceRd_{. For instance, consider a density supported on the unit ball of}Rd_{. In this case,} the boundary is the unit sphere of dimension(d₋1). Note also that one can relax Assumption 1-(d) to the case where the set_{x _∈Rd: f(x)>0_}has multiple connected components, see Remark 3.3 in Biau, Cadre, and Pelletier (2008).

More precisely, under Assumption 1-(c),∂S_f is a smooth Riemannian submanifold with Riemannian metric, denoted byσ, induced by the canonical embedding of ∂S_f inRd. The volume measure on

(∂Sf,σ)will be denoted by vσ. Furthermore,(∂Sf,σ) is compact and without boundary. Then by

the tubular neighborhood theorem (see e.g., Gray, 1990; Bredon, 1993, p. 93),∂Sf admits a tubular

neighborhood of radiusρ >0,

V(∂S_f,ρ) =¦x ∈Rd: dist(x,∂S_f)< ρ©,

i.e., each point x ∈ V(∂Sf,ρ) projects uniquely onto ∂Sf. Let {ep; p ∈ ∂Sf} be the unit-norm

section of the normal bundle T∂S⊥_f that is pointing inwards, i.e., for all p _∈ ∂S_f, e_p is the unit normal vector to∂Sf directed towards the interior ofSf. Then each point x ∈ V(∂Sf,ρ) may be

expressed as

x=p+ve_p, (2.1)

where p∈∂Sf, and where v∈R satisfies|v| ≤ρ. Moreover, given a Lebesgue integrable function

ϕonV(∂Sf,ρ), we may write

Z

V(∂Sf,ρ)

ϕ(x)dx = Z

∂Sf

Z ρ

−ρ

(5)

whereΘ is a_C∞ function satisfyingΘ(p, 0) =1 for all p_∈∂S_f. (See Appendix B in Biau, Cadre, and Pelletier, 2008.)

Denote byD2_e

p the directional differentiation operator of order 2 onV(∂Sf,ρ)in the directionep. It

will be seen in the proofs in Section 3 that the variance in our central limit theorem is determined by the second order behavior of f near the boundary of its support. Therefore to derive this variance we shall need the following set of second order smoothness assumptions on f.

Assumption Set 2

(a) There existsρ >0 such that, for allp_∈∂S_f, the mapu_7→ f(p+ue_p)is of class_C2 on[0,ρ].

(b) There existsρ >0 such that

0< sup

p∈∂Sf

sup

0≤u≤ρ

D2_e

pf(p+uep)<∞.

(c) There existsρ >0 such that

inf

p∈∂Sf

inf

0≤u≤ρD 2

epf(p+uep)>0.

For similar smoothness assumptions see Section 2.4 of Mason and Polonik (2009). The imposition of such conditions appears to be unavoidable to derive a central limit theorem. Note also that Assumption Sets 1 and 2 are the same as the ones used in Biau, Cadre, and Pelletier (2008). In particular, we assume throughout that the density f iscontinuouson Rd. Thus, we are in the case of a non-sharp boundary, i.e., f decreases continuously to zero at the boundary of its support. The case where f has sharp boundary requires a different approach (see for example Härdle, Park, and Tsybakov, 1995). The analytical assumptions on f (Assumption Set 2) are stipulations on the local behavior of f at the boundary of the support. In particular, the restrictions on f imply that inside the support and close to the boundary the mapsu_7→ f(p+ue_p), with p_∈∂S_f, are strictly convex (see the Appendix).

2.2

Main result

Let

σ2_f =2d

Z

∂Sf

Z ∞

0 Z

B(0,1)

Φ(p,t,u)dudt vσ(dp), (2.3)

with

Φ(p,t,u) =exp−ω_dD_e2

pf(p)t

2

exp

β(u)D2_e

pf(p)

t2

2

−1

,

ωd denoting the volume ofB(0, 1)and

(6)

Remark 2.1. LetΓbe the Gamma function. We note thatβ(u)has the closed expression (Hall, 1988, p. 23)

β(u) = 





2π(d−1)/2

Γ1 2+

d

2

Z 1

|u|

(1₋ y2)(d−1)/2dy, if0_{≤ |}u_{| ≤}1

0, if |u|>1,

which, in particular, gives

β(0) =ω_d= π

d/2

Γ1+ d₂.

We are now ready to state our main result.

Theorem 2.1. Suppose that both Assumption Sets 1 and 2 are satisfied. If (r.i) rn → 0, (r.ii)

nr_nd/(lnn)4/3_{→ ∞}_{and (r.iii) nr}d+1

n →0, then

n

r_nd

1/4

λS_n_△S_f₋E_λS_n_△S_f _{→ N}D (0,σ2_f),

whereσ2_f >0is as in (2.3).

Remark 2.2. A referee pointed out that the methods in the paper may be applicable to obtain a central limit theorem for the histogram-based support estimator studied in Baíllo and Cuevas (2006). The reference Cuevas, Fraiman, and Rodríguez-Casal (2007) should be a starting point for such an investi-gation.

It is known from Cuevas and Rodríguez-Casal (2004) that the choice rn=O((lnn/n)1/d)gives the

fastest convergence rate ofS_n towardsS_f for the Hausdorff metric, that is O((lnn/n)1/d). For such a radius choice, the concentration speed ofλ(S_n∆S_f)around its expectation as given by Theorem 2.1 is O(pn/(lnn)1/4), close to the parametric rate.

Theorem 2.1 assumesd_≥2 (Assumption 1-(a)). We restrict ourselves to the cased_≥2 for the sake of technical simplicity. However, the cased =1 can be derived with minor adaptations, assuming

rn →0, nrn/(lnn)4/3 → ∞, andnrn3/2 →0. In fact, the one-dimensional setting has already been

explored in the related context of vacancy estimation (Hall, 1984).

As we mentioned in the introduction, the quantity λ(S_n_△S_f) is closely related to the vacancy V_n

(Hall 1985, 1988), which is defined as in (1.2). A close inspection of the proof of Theorem 2.1 reveals that taking intersection withSf in the integrals does not effect things too much and, in fact,

the asymptotic distributional behaviors ofλ(S_n_△S_f)andV_n are nearly identical. As a consequence, we obtain the following result:

Theorem 2.2. Suppose that both Assumption Sets 1 and 2 are satisfied. If (r.i), (r.ii) and (r.iii) hold, then

n

rd n

1/4

(V_n₋EV_n)_{→ N}D (0,σ2_f),

(7)

Surprisingly, the limiting varianceσ2_f remains as in (2.3). Theorem 2.2 was motivated by a remark by Hall (1985), who pointed out that a central limit theorem for vacancy in the case nr_nd → ∞

remained open.

3

Proof of Theorem 2.1

Our proof of Theorem 2.1 will borrow elements from Mason and Polonik (2009).

Set

ǫn=

1

(nrd n)1/4

. (3.1)

Observe that, from (r.ii) and (r.iii), the sequence(ǫ_n)satisfies (e.i)ǫ_n→0 and (e.ii)ǫ_npnrd n → ∞.

For future reference we note that from (r.i) and (r.iii), we get that

r_n

ǫn

→0. (3.2)

Set

En={x ∈Rd: f(x)≤ǫn}.

Furthermore, let

L_n(ǫn) =

Z

En

1{fn(x)>0} −1{f(x)>0} dx

and

L_n(ǫn) =

Z

Ec n

1{fn(x)>0} −1{f(x)>0} dx.

Noting that, under Assumption Set 1,λ(Sn∆Sf) =Ln(ǫn) +Ln(ǫn), our plan is to show that

n

rd n

1/4

L_n(ǫn)−ELn(ǫn)

_D

→ N(0,σ2_f) (3.3)

and

n

rd n

1/4

L_n(ǫn)−ELn(ǫn)

P

→0, (3.4)

which together imply the statement of Theorem 2.1. To prove a central limit theorem for the random variableL_n ǫn

, it turns out to be more convenient to first establish one for the Poissonized version of it formed by replacing f_n(x)with

πn(x) = Nn

X

i=1

1_B₍_x_,_r n)(Xi),

whereNnis a meannPoisson random variable independent of the sampleX1, . . . ,Xn. By convention,

we setπ_n(x) =0 wheneverN_n=0. The Poissonized version ofL_n ǫ_n

is then defined by

Π_n(ǫn) =

Z

En

(8)

The proof of Theorem 2.1 is organized as follows. First (Subsection 3.1), we determine the exact asymptotic behavior of the variance of Π_n ǫn

. Then (Subsection 3.2), we prove a central limit theorem forΠ_n ǫ_n

. By means of a de-Poissonization result (Subsection 3.3), we then infer (3.3). In a final step (Subsection 3.4) we prove (3.4), which completes the proof of Theorem 2.1. This Poissonization/de-Poissonization methodology goes back to at least Beirlant, Györfi, and Lugosi (1994).

3.1

Exact asymptotic behavior of Var

(Π

_n

(

ǫ

_n

))

Let

∆_n(x) =

1{πn(x)>0} −1{f(x)>0} .

In the sequel, the letterC will denote a positive constant, the value of which may vary from line to line.

Let(ǫn)be the sequence of positive real numbers defined in (3.1). In this subsection, we intend to

prove that, under the conditions of Theorem 2.1,

lim

n→∞ r_n

r_ndVar Πn(ǫn)

=σ2_f, (3.5)

whereσ2

f is as in (2.3).

Towards this goal, observe first that

Π_n(ǫn) =

Z

˜ En

1{πn(x)>0} −1{f(x)>0} dx,

where we set

˜

En=En∩S rn

f ,

with

Srn

f =

¦

x∈Rd : dist(x,Sf)≤rn

©

.

Clearly,

Var(Π_n ǫn

) =

Z

˜ En

Z

˜ En

C ∆_n(x),∆_n(y)

dxdy,

where here and elsewhereCdenotes ‘covariance’. Since ∆_n(x)and∆_n(y)are independent when-ever_kx₋ y_k>2r_n, we may write

Var Π_n(ǫ_n)₌ Z

˜ En

Z

˜ En

1

kx−yk ≤2r_n C ∆_n(x),∆_n(y)

dxdy.

Using the change of variable y =x+2rnu, we obtain

Var(Π_n ǫn

(9)

=2dr_nd

Z

Rd

Z

Rd

1_E˜_n(x)1_E˜_n(x+2rnu)1B(0,1)(u)C ∆n(x),∆n(x+2rnu)dxdu.

By construction, whenevernis large enough, ˜_E_n is included in the tubular neighborhood_V(∂S_f,ρ)

of∂Sf of radiusρ >0. In this case, each x ∈E˜n may be written as x = p+vep as described in

(2.1). Hence, for all large enoughn, we obtain

Var(Π_n ǫn

)

=2dr_nd

Z

∂Sf

Z ρ

−rn

Z

B(0,1)

1_E˜_n(p+vep)1_E˜_n(p+vep+2rnu)

×Θ(p,v)C∆_n(p+vep),∆n(p+vep+2rnu)

dudv vσ(dp).

For allp_∈∂S_f, letκp(ǫn)be the distance between pand the point x of the set{x ∈Rd:f(x) =ǫn}

such that the vectorx₋pis orthogonal to∂S_f. Using the change of variable v=t/pnrd

n, we may

write

Var Π_n(ǫn)

= 2

d_rd n

p

nrd n

Z

∂Sf

Z p

nrd nκp(ǫn)

−pnrd+2

n

Z

B(0,1)

1_E˜_n 

p+

t

p

nrd n

e_p+2r_nu



Θ 

p,

t p nrd n   ×C  ∆_n 

p+

t

p

nr_nd ep



,∆_n 

p+

t

p

nr_nd

ep+2rnu



 

dudt v_σ(dp).

For a justification of this change of variable, refer to equation (2.2) and equation (4.2) in the Ap-pendix. By conditions (r.i) and (r.iii),nr_nd+2_→0. Consequently,

r _n

rd n

Var Π_n(ǫ_n)

=o(1)

+2d

Z

∂Sf

Z p

nrd nκp(ǫn)

0

Z

B(0,1)

1_E˜_n 

p+

t

p

nr_nd

sep+2rnu



Θ 

p,

t

p

nr_nd

  ×C  ∆_n 

p+

t p nrd n e_p 

,∆_n 

p+

t

p

nrd n

e_p+2r_nu



 

dudt v_σ(dp). (3.6)

To get the limit asn→ ∞of the above integral, we will need the following lemma, whose proof is deferred to the end of the subsection.

Lemma 3.1. Let p∈∂Sf, t>0and u∈ B(0, 1)be fixed. Suppose that the conditions of Theorem 2.1

hold. Then

lim

n→∞C 

∆_n 

p+

t

p

nr_nd ep



,∆_n 

p+

t

p

nr_nd

ep+2rnu



 

= Φ(p,t,u),

(10)

Returning to the proof of (3.5), we notice that by (4.4) in the Appendix and (e.ii) we have

p

nr_ndκp(ǫn) → ∞ as n → ∞ and Θ(p, 0) = 1. Therefore, using Lemma 3.1 and the fact that

for allt>0 andu_{∈ B}(0, 1)

1_E˜_n 

p+

t

p

nrd n

+2r_nu



→1 asn→ ∞,

we conclude that the function inside the integral in (3.6) converges pointwise toΦ(p,t,u)asn_{→ ∞}.

We now proceed to sufficiently bound the function inside the integral in (3.6) to be able to apply the Lebesgue dominated convergence theorem. Towards this goal, fix p ∈ ∂Sf, u∈ B(0, 1) and

0< t ≤ pnr_ndκp(ǫn). Since∆n(x) ≤ 1 for all x ∈Rd, using the inequality |C(Y1,Y2)| ≤ 2E|Y1|

whenever|Y2| ≤1, we have

C  ∆_n 

p+

t

p

nr_nd ep



,∆_n 

p+

t

p

nr_nd

ep+2rnu

   

≤2E∆_n 

p+

. (3.7)

By the bound in (4.3) in the Appendix, we see that

sup

p∈∂Sf

κ_p(ǫ_n)_→0 as n→ ∞. (3.8)

Then, since e_p is a normal vector to ∂S_f at p which is directed towards the interior ofS_f, there exists an integerN₀ independent of p, t andusuch that, for alln_≥N₀, the pointp+ (t/pnrd

n)ep

belongs to the interior ofS_f. Therefore, f(p+ (t/pnrd

n)ep)>0 and, letting

ϕn(x) =P X ∈ B(x,rn)

,

we obtain

E∆_n 

p+

=P 

π_n 

p+

=0   =E  P 

∀i≤N_n : X_i∈ B/ 

p+

t

p

nrd n

e_p,r_n

  N_n     =E 

1−ϕ_n 

p+

t

p

nr_nd ep     Nn =exp 

−nϕ_n 

p+

t p nrd n e_p   

, (3.9)

where we used the fact thatNn is a meannPoisson distributed random variable independent of the

(11)

under Assumption 1-(b), one deduces that for all x ∈ V(∂S_f,ρ)_∩S◦_f, there exists a quantityK_n(x)

such that

ϕn(x) =rndωdf(x) +rnd+2Kn(x) and sup n

sup

x∈V(∂Sf,ρ)∩ ◦

Sf

|Kn(x)|<∞. (3.10)

For all x in_V(∂S_f,ρ)written as x= p+ue_pwithp_∈∂S_f and 0_≤u_≤ρ, a Taylor expansion of f

atpgives the expression

f(x) = 1

2D

2

epf(p+ξep)u

2_,

for some 0_≤ξ≤usince, by Assumption 1-(b), D_e_pf(p) =0. Thus, in our context, expanding f at

p, we may write

nϕn



p+

t

p

nrd n

e_p



=ω_dD_e2

pf(p+ξep)

t2

2 +nr

d+2

n Rn(p,t),

for some 0_≤ξ≤κp(ǫn), and whereRn(p,t)satisfies

sup

n

sup

§

R_n(p,t)

:p∈∂S_f and 0≤t ≤ Æ

nr_ndκp(ǫn)

ª

<∞.

Furthermore, by (3.8), each pointp+ξe_p falls in the tubular neighborhood_V(∂S_f,ρ)for all large enough n. Consequently, by Assumption 2-(c) there existsα > 0 independent of nand N1 ≥ N0

independent ofp, t andusuch that, for alln_≥N₁,

inf

p∈∂Sf

D2_e

pf(p+ξep)>2α.

This, together with identity (3.9) and (r.i), (r.iii), which implynr_nd+2_→0, leads to

E∆_n 

p+

t

p

nrd n

e_p



≤Cexp(−ω_dαt2) (3.11)

forn≥N1 and all

0≤t≤Ænr_nd sup

p∈∂Sf

κp(ǫn).

Thus, using inequality (3.11), we deduce that the function on the left hand side of (3.7) is dominated by an integrable function of(p,t,u), which is independent ofnprovidedn_≥N₁. Finally, we are in a position to apply the Lebesgue dominated convergence theorem, to conclude that

lim

n→∞ r _n

r_nd Var Πn(ǫn)

=2d

Z

∂Sf

Z ∞

0 Z

B(0,1)

Φ(p,t,u)dudt vσ(dp) =σ2_f.

To be complete, it remains to prove Lemma 3.1.

Proof of Lemma 3.1 Let x_n = p+ (t/pnrd

n)ep. Since nrnd → ∞ and nr d+2

n →0, both xn and

xn+2rnulie in the interior ofSf for all large enough n. As a consequence, f(xn)>0 and f(xn+

2r_nu)>0 for all large enoughn. Thus,

(12)

=C 1_{πn(xn) =0},1{πn(xn+2rnu) =0}

=P _π_n(x_n) =0,πn(xn+2rnu) =0

−P _π_n(x_n) =0

P _π_n(x_n+2r_nu) =0 =P _∀i≤N_n : X_i∈ B/ (x_n,r_n)_{∪ B}(x_n+2r_nu,r_n)

−P _∀_i_≤_N_n _: _X_i_{∈ B}_/ (xn,rn)

P _∀_i_≤_N_n _: _X_i _{∈ B}_/ (xn+2rnu,rn)

=exp

−nµ B(x_n,r_n)_{∪ B}(x_n+2r_nu,r_n) −exp

−nµ(_B(x_n,r_n))₋nµ(_B(x_n+2r_nu,r_n))

,

whereµdenotes the distribution ofX. LetB_n=_B(x_n,r_n)_{∩ B}(x_n+2r_nu,r_n). Using the equality

µ B(xn,rn)∪ B(xn+2rnu,rn)=ϕn(xn) +ϕn(xn+2rnu)−µ(Bn),

we obtain

C ∆_n(x_n),∆_n(x_n+2r_nu)

(3.12)

=exp

−n ϕn(xn) +ϕn(xn+2rnu) exp nµ(Bn)

−1

.

Now,µ(B_n)may be expressed as

µ(B_n) = f(x_n)λ(B_n) + Z

Bn

f(v)₋f(x_n)

dv.

Since f is of class_C1onRd, by developingf at x_n in the above integral, we obtain

Z

Bn

f(v)₋f(xn)

dv=r_nd+1Rn,

whereRnsatisfies

R_n

≤Csup

K k

gradfk,

and K is some compact subset of Rd containing ∂S_f and of nonempty interior. Next, note that λ(B_n) =r_ndβ(u), where

β(u) =λ(_B(0, 1)_{∩ B}(2u, 1)).

Therefore, expanding f atpin the directione_p, we obtain

µ(B_n) =β(u)t 2

2nD 2

epf

p+ξ_pt

nr_nd

e_p+r_nd+1R_n,

whereξ∈(0, 1). Hence by (r.iii),

lim

n→∞nµ(Bn) =β(u)D 2

epf(p)

t2

2.

(13)

3.2

Central limit theorem for

Π

_n

(

ǫ

n

)

In this subsection we establish a central limit theorem forΠ_n(ǫn). Set

S_n(ǫ_n) = an Πn(ǫn)−EΠn(ǫn)

σn

,

wherea_n= (n/r_nd)1/4 and

σ2_n=Varan Πn(ǫn)−EΠn(ǫn)

.

We shall verify that asn→ ∞

S_n(ǫn)

D

→ N(0, 1). (3.13)

To show this we require the following special case of Theorem 1 of Shergin (1990).

Fact 3.1. Let(X_i_,_n :i_∈Zd) denote a triangular array of mean zero m-dependent random fields, and letJn⊂Zd be such that

(i) VarP_i_∈J

nXi,n

→1as n→ ∞, and

(ii) For some2<s<3,P_i_∈J

nE|Xi,n|

s

→0as n_{→ ∞}.

Then

X

i∈Jn

Xi,n

D

→ N(0, 1).

We use Shergin’s result as follows. Recall the definition ofǫnin (3.1) and also that

Var(Π_n ǫn

) =

Z

˜ En

Z

˜ En

C ∆_n(x),∆_n(y)

dxdy,

with

˜

En=En∩S rn

f .

Next, consider the regular grid given by

A_i= (x_i

1,xi1+1]×. . .×(xid,xid+1],

where i=(i₁, . . . ,i_d),i₁, . . . ,i_d_∈Zandx_i=i r_n fori_∈Z. Define

Ri=Ai∩E˜n.

With_J_n=_{i_∈Zd:Ai∩E˜n6=; }we see that{Ri: i∈ Jn}constitutes a partition of ˜En. Note that for

eachi∈ Jn,

λ R_i

≤r_nd.

We claim that for all largen

(14)

To see this, we use the fact that, according to (4.3), there exists ¯ρ > 0 such that for all large n, ˜

En⊂ V

∂Sf, ¯ρpǫn

. Thus, sincern/pǫn→0 by (3.2),

[

i∈Jn

A_i_{⊂ V} ∂S_f,(ρ¯+2)pǫn

and, consequently,

r_ndCard(_J_n)_≤λV ∂S_f,(ρ¯+2)pǫn

≤Cpǫn.

Keeping in mind the fact that for any disjoint setsB₁, . . . ,B_kinRd such that, for 1_≤i₆= j_≤k,

then

Z

Bi

∆_n(x)dx,i=1, . . . ,k, are independent,

we can easily infer that

Xi,n=

a_n

Z

Ri

∆_n(x)₋E∆_n(x)

dx

σn

, i∈ Jn,

constitutes a 1-dependent random field onZd.

Recalling thatan= (n/rnd)

1/4 _and_σ2

n→σ

2

f asn→ ∞by (3.5) we get, for alli∈ Jn,

X_i_,_n

≤

an

σn

λ(R_i)_≤C(nr_n3d)1/4.

Hence, by (3.14),

X

i∈Jn

E_|X_i_,_n_|5/2_≤C Card(_J_n)

(nr_n3d)5/8_≤C(nr_n3d/2)1/2.

Clearly this bound when combined with (r.iii) andd≥2, gives asn→ ∞,

X

i∈Jn

E_|X_i_,_n_|5/2_→0,

which by the Shergin Fact 3.1 (withs=5/2) yields

S_n ǫn

= X

i∈Jn

X_i_,_n_{→ N}D (0, 1).

(15)

3.3

Central limit theorem for

L

_n

(

ǫ

n

)

Now we shall de-Poissonize the central limit forΠ_n(ǫn)to obtain one forLn ǫn

. Observe that

(S_n(ǫn)|Nn=n)

D

= an Ln(ǫn)−EΠn(ǫn)

σn

. (3.15)

Our next goal is to apply the following version of a theorem in Beirlant and Mason (1995) (see also Polonik and Mason, 2009) to infer from (3.13) that

a_n L_n(ǫn)−EΠn(ǫn)

σn

D

→ N(0, 1). (3.16)

Fact 3.2. Let N1,n and N2,n be independent Poisson random variables with N1,n being Poisson(nβn)

and N2,n being Poisson(n(1−βn))whereβn∈(0, 1). Denote Nn=N1,n+N2,n and set

Un=

N_1,_n₋nβn

p_n and Vn=

N_2,_n₋n(1₋βn)

p_n .

Let(S_n)be a sequence of real-valued random variables such that

(i) For each n≥1, the random vector(Sn,Un)is independent of Vn.

(ii) For someσ2_<_∞_{, S}

n

D

→σZ as n→ ∞.

(iii) βn→0as n→ ∞.

Then, for all x,

P(S_n_≤x _|N_n=n)_→P(σZ_≤x).

Let

Dn={x ∈Rd: f(x)≤2ǫn}.

We shall apply Fact 3.2 toSn(ǫn)with

N_1,_n=

Nn

X

i=1

1_{X_i_{∈ D}_n_}, N_2,_n=

Nn

X

i=1

1_{X_i∈ D/ n}

andβn=P(X ∈ Dn). Let

M= sup

x∈Rd

d

X

i=1

∂f (x)

∂x_i

.

We see that for all large enoughn, wheneverx ∈ Enandy ∈ B x,rn, by the mean value theorem,

f(y)_≤ f(x) +M rn≤ǫn

1+ M rn

ǫn

.

This combined with (3.2) implies for all largen



 

[

x∈En

B(x,rn)



 ∩ D

(16)

Therefore for all large enough n, the random variables S_n(ǫn)and N2,n are independent. Thus by

(3.15) andβn→0, we can apply Fact 3.2 to conclude that (3.16) holds.

Next we proceed just as in Mason and Polonik (2009) to apply a moment bound given in Lemma 2.1 of Giné, Mason, and Zaitsev (2003) to show that

E a_n L_n(ǫn)−EΠn(ǫn)

2

≤2σ2_n.

Therefore, since by (3.5),

σ2_n→σ2_f <∞,

the sequence(a_n(L_n(ǫn)−EΠn(ǫn)))is uniformly integrable. Hence we get using (3.16) that

a_n EL_n(ǫ_n)₋EΠ_n(ǫ_n) →0.

Thus, still by (3.16),

a_n L_n(ǫ_n)₋EL_n(ǫ_n)

σ_n

D

→ N(0, 1).

This in turn implies (3.3).

3.4

Completion of the proof of Theorem 2.1

It remains to verify (3.4). Observe that

L_n(ǫn) =

Z

Ec n

1{fn(x)>0} −1{f(x)>0} dx =

Z

Ec n

1_{f_n(x) =0_}dx.

To begin with note that for allx _{∈ E}_nc,

P(f_n(x) =0) = 1−ϕ_n(x)n

≤exp −nϕ_n(x)

,

where we recall that

ϕn(x) =P X ∈ B(x,rn)

.

Since f is of class_C1, we have, for allt_{∈ B}(x,r_n),

|f(t)₋f(x)_{| ≤}κ1rn,

where κ1 > 0 is independent of x. Therefore, using the properties f(x) ≥ ǫn and rn/ǫn →0 by

(3.2), we obtain

ϕn(x) =P X∈ B(x,rn)

= f(x)ωdrnd+

Z

B(x,rn)

f(t)₋f(x)

dt

≥ǫnrnd

ωd−κ1 rn

ǫ_n

(17)

whereκ2>0 is independent of x. Thus,

P(f_n(x) =0)_≤exp(₋κ2ǫnnrnd).

Consequently,

anELn(ǫn)≤

n

rd n

1/4

exp(₋κ2ǫnnrnd).

By (r.ii), for allnlarge enough, r_n−d≤n. Hence,

n

r_nd ≤n 2_.

Besides, providednis large enough, we get by (r.ii) thatnr_nd _≥(lnn/κ2)4/3. Consequently,

ǫnnrnd= (nr d n)

3/4 ≥ 1

κ2

lnn.

This gives the bound holding for all largen,

n

rd n

1/4

exp−κ2ǫnnrnd

≤pnexp(₋lnn) =n−1/2,

which goes to 0 as n→ ∞. This implies that both anELn(ǫn) → 0 and anLn(ǫn)

P

→0, and thus establishes (3.4). The proof of Theorem 2.1 now follows from (3.3) and (3.4).

4

Appendix: Properties of

x

: 0

<

f

(

x

)

≤

ǫ

Under Assumption Sets 1 and 2, we know that there exists a tubular neighborhood_V(∂S_f,ρ) of ∂Sf of radiusρsuch that first,

0< inf

p∈∂Sf

inf

0≤u≤ρD 2

epf(p+uep)≤ sup

p∈∂Sf

sup

0≤u≤ρ

D2_e

pf(p+uep)<∞, (4.1)

and second,

inf¦f(x) : x ∈Sf\V(∂Sf,ρ)

©

=sup¦f(x) : x ∈ V(∂Sf,ρ)

©

:=ǫ0>0.

Consequently, for all 0< ǫ < ǫ0, we have

{x_∈Rd : 0< f(x)_≤ǫ} ⊂ V(∂S_f,ρ).

Moreover, (4.1), together with the fact that f =0 on ∂S_f, entails that for all p _∈∂S_f, the maps

u_7→ f(p+ue_p)are strictly convex and strictly increasing on [0,ρ]. Therefore, for all 0< ǫ < ǫ0,

and for allp∈∂Sf there exists a unique real numberκp(ǫ)such that

(18)

Note that we also have the relation

[

p∈∂Sf

¦

p+ue_p: 0_≤u_≤κp(ǫ)

© =

x: 0< f (x)_≤ǫ ∪∂S_f, (4.2)

for all 0< ǫ < ǫ0.

Since D_e_pf(p) =0 for all p ∈∂S_f by assumption, by using a second order expansion of f at p in combination with (4.1), we have

sup

p∈∂Sf

κp(ǫ)≤

1 2p∈inf∂Sf

inf

0≤u≤ρD 2

epf(p+uep)

−1₂ p

ǫ. (4.3)

The same argument gives

inf

p∈∂Sf

κp(ǫ)≥





1 2_psup_∈∂Sf

sup

0≤u≤ρ

D2_ǫ

pf

p+ue_p



 −1/2

p

ǫ. (4.4)

Acknowledgments. We thank two referees for valuable comments and insightful suggestions.

References

[1] Baíllo, A. and Cuevas, A. (2006). Image estimators based on marked bins, Statistics,Vol. 40, pp. 277–288. MR2263620

[2] Baíllo, A., Cuevas, A., and Justel, A. (2000). Set estimation and nonparametric detection,

Canadian Journal of Statistics,Vol. 28, pp. 765-782. MR1821433

[3] Beirlant, J., Györfi, L., and Lugosi, G. (1994). On the asymptotic normality of the L₁- and L₂ -errors in histogram density estimation, Canadian Journal of Statistics, Vol. 22, pp. 309-318. MR1309318

[4] Beirlant, J. and Mason, D.M. (1995). On the asymptotic normality of L_p-norms of empirical functionals,Mathematical Methods of Statistics,Vol. 4, pp. 1-19. MR1324687

[5] Biau, G., Cadre, B., and Pelletier, B. (2008). Exact rates in density support estimation,Journal of Multivariate Analysis,Vol. 99, pp. 2185-2207. MR2463383

[6] Bredon, G.E. (1993). Topology and Geometry, Volume 139 of Graduate Texts in Mathematics, Springer-Verlag, New York. MR1224675

[7] Cuevas, A., Fraiman, R., and Rodríguez-Casal, A. (2007). Nonparametric approach to the estimation of lengths and surface areas, The Annals of Statistics, Vol. 35, pp. 1031–1051. MR2341697

(19)

[9] Devroye, L. and Wise, G. (1980). Detection of abnormal behavior via nonparametric estima-tion of the support,SIAM Journal on Applied Mathematics,Vol. 38, pp. 480-488. MR0579432

[10] Gallot, S., Hulin, D. and Lafontaine, J. (2004).Riemannian Geometry, 3rd Ed., Springer-Verlag, New York. MR2088027

[11] Giné, E., Mason, D.M., and Zaitsev, A. (2003). The L₁-norm density estimator process, The Annals of Probability,Vol. 31, pp. 719-768. MR1964947

[12] Gray, A. (1990).Tubes, Addison-Wesley, Redwood City, California. MR1044996

[13] Hall, P. (1984). Random, nonuniform distribution of line segments on a circle,Stochastic Pro-cesses and their Applications,Vol. 18, pp. 239-261. MR0770192

[14] Hall, P. (1985). Three limit theorems for vacancy in multivariate coverage problems, Journal of Multivariate Analysis,Vol. 16, pp. 211-236. MR0790601

[15] Hall, P. (1988).Introduction to the Theory of Coverage Processes, John Wiley & Sons, New York. MR0973404

[16] Härdle, W., Park, B.U., and Tsybakov, A.B. (1995). Estimation of non-sharp support boundaries,

Journal of Multivariate Analysis,Vol. 55, pp. 205–218. MR1370400

[17] Mason, D.M. and Polonik, W. (2009). Asymptotic normality of plug-in level set estimates,The Annals of Applied Probability,Vol. 19, pp. 1108–1142 . MR2537201

[18] Molchanov, I.S. (1998). A limit theorem for solutions of inequalities,Scandinavian Journal of Statistics,Vol. 25, pp. 235–242. MR1614288

[19] Shergin, V.V. (1990). The central limit theorem for finitely dependent random variables. In: