Bivariate Weighted Residual and Past Entropies

Abstract

The weighted entropy introduced by Belis and Guiasu (1968) is viewed as a measure of uncertainty. Di Crescenzo and Longobardi (2006) proposed dynamic form of these measure namely weighted residual (WRE) and past entropies (WPE). In this paper, we extend the definition of weighted residual and past entropies to bivariate setup and obtain some of its properties. Several properties, including monotonicity and bounds of BWRE and BWRP are obtained. We also look into the problem of extending WRE and WPE for conditionally specified models. Several properties, including bounds of CWRE and CWPE are obtained for conditional distributions. It is shown that the proposed measure uniquely determines the distribution function.