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Paul Deheuvels, Paris VI Print
Friday, 04 May 2012, 14:30 - 15:30

Paul Deheuvels, Université Paris VI


Non-Uniform Spacings Processes

Abstract: We provide a joint strong approximation of the uniform spacings empirical process and of the uniform quantile process by sequences of independent Gaussian processes.This allows us to obtain an explicit description of the limiting Gaussian process generated by the sample spacings from a non-uniform distribution. It is of the form B(t)+(1−σ_F ){(1− 4t) log(1 − t)}\int_0^1{B(s)/(1 − s)}ds, for 0 ≤ t ≤ 1, where {B(t) : 0 ≤ t ≤ 1} denotes a Brownian bridge, and where σ_F^2= Var(log f (X)) is a factor depending upon the underlying distribution function F(·) = P(X ≤ x) through its density f (x) = d/dxF(x). We provide a strong approximation of the non-uniform spacings processes by replicæ of this Gaussian process, with limiting sup-norm rate OP(n^{−1/8}(log n)^{1/2}). The limiting process reduces to a Brownian bridge if and only if σ_F^2= 1, which is the case when the sample observations are exponential. For uniform spacings, we get σ_F^2= 0, which is in agreement with the results of Beirlant (In: Limit theorems in probability and statistics, Proc Coll Math Soc J Bolyai, vol 13 36, Akadémiai Kiadó, Budapest, pp 77–80, 1984), and Aly et al. (Z Wahrsch Verw Gebiete

14 66:461–484, 1984).

Location: plaine, building NO, 9th floor, salle des professeurs
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