Studies On Statistical Properties Of Entropy And Extropy

Much attention have been paid to measure the uncertainty contained in a random variable. The Shannon entropy introduced by Shannon (1948) is one of the most impor-tant measures. Recently, Lad et al. (2015) proposed a new measure termed by extropy.The extropy of a random variable can be viewed as the complement dual of the entropy of this random variable. Duing to their intriguing properties and fruitful applications,we also focus our interests on entropy and extropy in this dissertation.The residual quantile entropy was studied firstly. It is shown that the decreasing mean residual life class implies the decreasing residual quantile entropy class, and the decreasing residual quantile entropy class is not closed under the formation of mixture.The less quantile entropy order is proved to be closed under the accelerated life models and the generalized order statistics models. Meanwhile, bounds of the entropy and the residual quantile entropy of some ageing classes are established.Secondly, we investigate the properties of extropy of order statisitics and record values. It is shown that the dispersive order of two random variables determines the ex-tropy order of their order statistics and record values. As a consequence, upper or lower bounds of the extropy of some aging classes are built. It is also shown that the equality of the extropy of order statistics and record values can determine uniquely their parent distributions. Moreover, it is proved that the extropy of order statistics is decreasing(increasing) in the sample size under the assumption that the parent distribution is DFR(IRHR). For upper record values, the extropy is decreasing in the sample size if the density-quantile function of underlying distributions is increasing. Finally, the lower bounds of the extropy of order statistics and record values are discussed. When the probability density function of the underlying random variables is symmetric about the mean, some symmetric properties of the extropy of order statistics are provided.Thirdly, we consider the probelem of estimating the extropy of an absolutely con-tinuous random variables with known supports. Two estimators are proposed and their behaviors are studied by means of real data and simulations. We propose goodness-of-fit test for uniformity based on the second extropy estimator, and compare its powers with that of other tests for uniformity. The results from simulations show that our proposed extropy-based test performs well in comparison with other tests of uniformity.Finally, we show that the residual extropy of an absolutely continuous random variable is determined uniquely by its failure rate function. Based on this point, sev-eral distributions are characterized in terms of its residual extropy. We also build the monotone properties of the residual extropy of the first order statistic, and show that the underlying distributions can be characterized by the residual extropy of order statis-tics. Moreover, two lifetime distributions and a stochastic order based on the notion of residual extropy are defined. Some properties of these new concepts are explored.