UNIFORM STRONG ESTIMATION UNDER ALPHA-MIXING, WITH RATES

Let {X(n)}, n greater-than-or-equal-to 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. F(n), based on the segment X1,..., X(n). By means of a mixingale argument, it is shown that F(n)(x) converges almost surely to F(x) uniformly in x is-an-element-of R. An alternative approach, utilizing a Kiefer process approximation, establishes the law of the iterated logarithm for sup{\F(n)(x)-F(x)\; x is-an-element-of 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 sup{\F(n)(x)-F(x)\; x is-an-element-of R} 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 is-an-element-of R, and the rate of this convergence is of the order of O((log log n/nh(n)2)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(n)2)1/2 Finally, the rth order derivative of f, f(r), is estimated by f(n)(r), and is shown that f(n)(r)(x) converges a.s. to f(r)(x) uniformly in x is-an-element-of R. The rate of this convergence is of the order of O((log log n/nh(n)2(r+1))1/2).