Generalization of Varma and Tsallis Entropies and their Properties

Abstract

The objective of this thesis entitled Generalization of Varma and Tsallis Entropies and Their Properties is to study the monotonic behaviour convolution results characterizations and reliability properties of residual life time of Generalized Shannon information measures. newline In chapter 1 we present the literature survey related to the Shannon information measure Generalized information measure the reliability properties of residual life time and well known life time distributions. In addition to this the basic fundamental background is also provided. The entropy measures are the uncertainty about the outcomes of a random experiment. In case the outcome is captured in an interval which is contracting the measure of entropy should be decreasing. In the chapter 2 Varma entropy of order alpha and type beta has been studied for this monotonic behaviour. For an absolutely continuous type random variable necessary and sufficient conditions on the distribution function have been provided so that the conditional Varma entropy is a monotonic on an interval. Further the results on the convolutions of Varma entropy have also been provided. newlineIn chapter 3 we propose a generalized cumulative residual information measure based on Tsallis entropy and its dynamic version. We study the characterizations of the proposed information measure and define new classes of life distributions based on this measure. Some applications are provided in relation to weighted and equilibrium probability models. Finally the empirical cumulative newlineTsallis entropy is proposed to estimate the new information measure. newlineIn chapter 4 we extend the definition of dynamic cumulative residual Tsallis entropy DCRTE into the Bivariate setup and study its properties in the context of reliability theory. Earlier Sati and Gupta 2015 proposed two measures of uncertainty based on non-extensive entropy called the dynamic cumulative residual Tsallis entropy DCRTE and the empirical cumulative Tsallis entropy.