Universal Kernel Estimators in a Heteroscedastic Nonparametric Regression Model

Abstract We propose new kernel estimators for the conditional variance function in a heteroscedastic nonparametric regression model and prove the consistency of these estimators under a more general condition on the regressors than those previously known in this problem. With respect to the regressors, which are considered in an array scheme, it is only required that these variables would asymptotically densely (with increasing sample) fill the domain of the conditional variance function with high probability. This assumption, which is essentially necessary for recovering the function, encompasses both random and fixed regressors, but without the requirement of their weak dependence or regularity. As an auxiliary result, which is also of independent interest, the uniform consistency of universal locally constant estimators for the regression function in the heteroscedastic model under consideration is proved. A special feature of the kernel estimators studied in this paper is the construction of multiple Riemann integral sums in the structure of these estimators, from which we deliver universal conditions regarding the nature of the regressors that do not use the character of their dependence.