Uniform strong estimation under [alpha]-mixing, with rates

Let s{;Xns};, n [greater-or-equal, slanted] 1, be a stationary [alpha]-mixing sequence of real-valued r.v.'s with distribution function (d.f.) F, probability density function (p.d.f.) f and mixing coefficient [alpha](n). The d.f. F is estimated by the empirical d.f. Fn, based on the segment X1,..., Xn. By means of a mixingale argument, it is shown that Fn(x) converges almost surely to F(x) uniformly in x[set membership, variant]. An alternative approach, utilizing a Kiefer process approximation, establishes the law of the iterated logarithm for sups{;vb;Fn(x)-F(xvb;; x[set membership, variant]. The d.f. F is also estimated by a smooth estimate n, which is shown to converge almost surely (a.s.) to F, and the rate of convergence of sups{;vb;n(x) - F(x)vb;;; x[set membership, variant]s}; is of the order of O((log log n/n)). The p.d.f. f is estimated by the usual kernel estimate fn, which is shown to converge a.s. to f uniformly in x[set membership, variant], and the rate of this convergence is of the order of O((log log n/nh2n)), where hn is the bandwidth used in fn. As an application, the hazard rate r is estimated either by rn or n, depending on whether Fn or n is employed, and it is shown that rn(x) and n(x) converge a.s. to r(x), uniformly over certain compact subsets of , and the rate of convergence is again of the order of O((log log n/nh2n)). 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[set membership, variant].The rate of this convergence is of the order of O((log log n/nh2(r+1)n)).