Uniform strong estimation under α-mixing, with rates

Abstract

Let s{; X n s};, n ⩾ 1, be a stationary α-mixing sequence of real-valued r.v.'s with distribution function (d.f.) F, probability density function (p.d.f.) f and mixing coefficient α( n). The d.f. F is estimated by the empirical d.f. F n , based on the segment X 1,…, X n . By means of a mixingale argument, it is shown that F n ( x) converges almost surely to F( x) uniformly in x∈ R . An alternative approach, utilizing a Kiefer process approximation, establishes the law of the iterated logarithm for sups{;vb; F n ( x)− F( xvb;; x∈ R . The d.f. F is also estimated by a smooth estimate F n , which is shown to converge almost surely (a.s.) to F, and the rate of convergence of sups{;vb; F n ( x) − F( x)vb;;|; x∈ R s}; is of the order of O((log log n/n) 1 2 ). The p.d.f. f is estimated by the usual kernel estimate f n , which is shown to converge a.s. to f uniformly in x∈ R , and the rate of this convergence is of the order of O((log log n/nh 2 n ) 1 2 ), where h n is the bandwidth used in f n . As an application, the hazard rate r is estimated either by r n or r n , depending on whether F n or F n is employed, and it is shown that r n ( x) and r n ( x) converge a.s. to r( x), uniformly over certain compact subsets of R , and the rate of convergence is again of the order of O((log log n/nh 2 n ) 1 2 ). Finally, the rth order derivative of f, f ( r) , is estimated by f ( r) n , and is shown that f ( r) n ( x) converges a.s. to f ( r) ( x) uniformly in x∈ R .The rate of this convergence is of the order of O((log log n/nh 2( r+1) n ) 1 2 ).