A nonparametric estimate of a multivariate density function

摘要: Let $x_1, \cdots, x_n$ be independent observations on a $p$-dimensional random variable $X = (X_1, X_p)$ with absolutely continuous distribution function $F(x_1, x_p)$. An observation $x_i$ $X$ is $x_i (x_{1i}, x_{pi})$. The problem considered here the estimation of probability density $f(x_1, x_p)$ at point $z (z_1, z_p)$ where $f$ positive and continuous. estimator proposed consistency shown. estimating has only recently begun to receive attention in literature. Several authors [Rosenblatt (1956), Whittle (1958), Parzen (1962), Watson Leadbetter (1963)] have univariate function. In addition, Fix Hodges (1951) were concerned connection nonparametric discrimination. Cacoullos (1964) generalized Parzen's work multivariate case. this paper arose out discrimination problem.