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

where Θ is a C ∞ function satisfying Θp, 0 = 1 for all p ∈ ∂ S f . See Appendix B in Biau, Cadre, and Pelletier, 2008. Denote by D 2 e p the directional differentiation operator of order 2 on V ∂ S f , ρ in the direction e p . 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 exists ρ 0 such that, for all p ∈ ∂ S f , the map u 7→ f p + ue p is of class C 2 on [0, ρ]. b There exists ρ 0 such that sup p ∈∂ S f sup ≤u≤ρ D 2 e p f p + ue p ∞. c There exists ρ 0 such that inf p ∈∂ S f inf ≤u≤ρ D 2 e p f p + ue p 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 is continuous on R d . 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 maps u 7→ f p + ue p , with p ∈ ∂ S f , are strictly convex see the Appendix.

2.2 Main result

Let σ 2 f = 2 d Z ∂ S f Z ∞ Z B0,1 Φp, t, ududt v σ dp, 2.3 with Φp, t, u = exp −ω d D 2 e p f pt 2 – exp ‚ βuD 2 e p f p t 2 2 Œ − 1 ™ , ω d denoting the volume of B0, 1 and βu = λ B0, 1 ∩ B2u, 1 . 2621 Remark 2.1. Let Γ be the Gamma function. We note that βu has the closed expression Hall, 1988, p. 23 βu =    2 π d−12 Γ € 1 2 + d 2 Š Z 1 |u| 1 − y 2 d−12 d y, if 0 ≤ |u| ≤ 1 0, if |u| 1, which, in particular, gives β 0 = ω d = π d 2 Γ € 1 + d 2 Š . 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 r n → 0, r.ii nr d n ln n 4 3 → ∞ and r.iii nr d+1 n → 0, then ‚ n r d n Œ 1 4 λ € S n △S f Š − Eλ € S n △S f Š D → N 0, σ 2 f , where σ 2 f 0 is 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 r n = Oln nn 1 d gives the fastest convergence rate of S n towards S f for the Hausdorff metric, that is Oln n n 1 d . For such a radius choice, the concentration speed of λS n ∆S f around its expectation as given by Theorem 2.1 is O p n ln n 1 4 , close to the parametric rate. Theorem 2.1 assumes d ≥ 2 Assumption 1-a. We restrict ourselves to the case d ≥ 2 for the sake of technical simplicity. However, the case d = 1 can be derived with minor adaptations, assuming r n → 0, nr n ln n 4 3 → ∞, and nr 3 2 n → 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 λS 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 with S f in the integrals does not effect things too much and, in fact, the asymptotic distributional behaviors of λS n △S f and V 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 r d n Œ 1 4 V n − EV n D → N 0, σ 2 f , where σ 2 f 0 is as in 2.3. 2622 Surprisingly, the limiting variance σ 2 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 d n → ∞ remained open. 3 Proof of Theorem 2.1 Our proof of Theorem 2.1 will borrow elements from Mason and Polonik 2009. Set ǫ n = 1 nr d 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 n p nr d n → ∞. For future reference we note that from r.i and r.iii, we get that r n ǫ n → 0. 3.2 Set E n = {x ∈ R d : f x ≤ ǫ n }. Furthermore, let L n ǫ n = Z E n 1{ f n x 0} − 1{ f x 0} d x and L n ǫ n = Z E c n 1{ f n x 0} − 1{ f x 0} d x . Noting that, under Assumption Set 1, λS n ∆S f = L n ǫ n + L n ǫ n , our plan is to show that ‚ n r d n Œ 1 4 L n ǫ n − EL n ǫ n D → N 0, σ 2 f 3.3 and ‚ n r d n Œ 1 4 L n ǫ n − EL n ǫ n P → 0, 3.4 which together imply the statement of Theorem 2.1. To prove a central limit theorem for the random variable L n ǫ n , it turns out to be more convenient to first establish one for the Poissonized version of it formed by replacing f n x with π n x = N n X i=1 1 Bx,r n X i , where N n is a mean n Poisson random variable independent of the sample X 1 , . . . , X n . By convention, we set π n x = 0 whenever N n = 0. The Poissonized version of L n ǫ n is then defined by Π n ǫ n = Z E n 1{π n x 0} − 1{ f x 0} d x . 2623 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 ǫ n . Then Subsection 3.2, we prove a central limit theorem for Π 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 Poissonizationde-Poissonization methodology goes back to at least Beirlant, Györfi, and Lugosi 1994.

3.1 Exact asymptotic behavior of VarΠ