Kybernetika

The aim of the paper is to present a test of goodness of fit with weigths in the classes based on weighted $\left( h,\phi \right) $-divergences. This family of divergences generalizes in some sense the previous weighted divergences studied by Frank et al \cite{frank} and Kapur \cite{kapur}. The weighted $\left( h,\phi \right)$-divergence between an empirical distribution and a fixed distribution is here investigated for large simple random samples, and the asymptotic distributions are shown to be either normal or equal to the distribution of a linear combination of independent chi-square variables. Some approximations to the linear combination of independent chi-square variables are presented.