Transformed nonparametric functions estimation

Abstract

The estimation of an unknown probability density functions of a random variable or its distribution function or a function related to it using standard kernel density estimate is the most popular technique among many density estimation methods. This is due to its favorable features such as it does not assume any functional form, data guide the underlying density and it accurately detects any multimodality present in the target density. Often, the standard kernel chosen has its support on whole Euclidean space. However, in many situations such as in survival, reliability, social and ecological analyses, the random variables have support only on positive half of the real line or on a compact interval and using standard kernel to estimate the density of these random variables assigns positive probabilities outside the support of the target density. Ignoring the probability mass outside the support of random variables will result in erroneous bias. To circumvent this problem, transformed kernel density and distribution functions estimates are proposed. A similar approach is used to estimate the density and distribution functions of data from weighted distribution. These estimates are used to estimate failure rate and regression functions. The asymptotic properties of these estimators are studied including the most crucial bandwidth selection. These new estimators have the same support as the data and preserve the fundamental properties of the random variables. Simulation studies and some real data examples are presented.