Kernel estimates under association: Strong uniform consistency (Q1186645)

A strictly stationary process \(X_ 1,X_ 2,\dots\) is called associated if for any functions \(g_ 1,g_ 2\), increasing in each variable, the r.v. \(g_ 1(X_{i_ 1},\dots,X_{i_ n})\), \(g_ 2(X_{i_ 1},\dots,X_{i_ n})\) are positively correlated for any choice of \(X_{i_ 1},\dots,X_{i_ n}\). For the \(r\) th order derivative \(f^{(r)}\) of the density \(f\) of \(X_ 1\) the strong consistency of a kernel estimator is established under some regularity conditions on the kernel functions and the covariance function. The rate of convergence is discussed and applications are given to the estimation of the hazard rate.