Properties and applications of pdf-related information measures and distributions

Abstract

The study of the information measures gives rise to di erent measures according to the contests in which it is applied. In the contest of reliability theory and survival analysis, an ever-growing interest is given by the entropy applied to continuous random variables.This quantity gives the expectation of the information content and is known as di erential entropy. Another quantity, the di erential varentropy is the variance of the information con- tent. Di erential entropy and di erential varentropy are mainly applied to the study of brand-new item. Other measures of interests in reliability contests are the dynamical mea- sures of information. In this thesis a particular attention is devoted to resid- ual entropy and residual varentropy, that are the expectation and the variance of the information content of a residual lifetime distribution. They can be very useful in the cases in which the item has a nite age. In particular, the residual varentropy is a largely unexplored subject and a focus on this quantity constitute the central part of the thesis. Stimulated by the need of describing useful notions related to the quan- tity described above, we introduce the `pdf-related distributions'. These are de ned in terms of transformation of absolutely continuous random variables through their own probability density functions. We investigate their main characteristics, with reference to the general form of the distribution, the quantiles, and some related notions of reliability theory. This allows us to obtain a characterization of the uniform distribution based on pdf-related dis- tributions of exponential, Laplace and power type as well. We also face the problem of stochastic comparing the pdf-related distributions by resorting to suitable stochastic orders. Finally, the given results are used to analyse properties and to compare some useful information measures, such as the di erential entropy and the varentropy. This work of thesis covers di erent arguments in the contest of infor- mation for continuous random variables. Firstly, mathematical properties of residual varentropy are discussed, such as conditions for which it is con- stant or monotonic and the determination of the upper and the lower bound. Another theoretical aspect that will be discussed concerns the properties of entropy and varentropy for stochastically ordered distributions. In addition, some applications of residual varentropy are proposed. The rst, the propor- tional hazards model gives an example of application of the varentropy in the context of reliability theory and survival analysis. The second is an applica- tion to stochastic process. More speci cally, residual varentropy is applied to the rst passage-time of an Ornstein{Uhlenbeck jump-di usion process. A kernel-type estimation of the residual varentropy is nally proposed, as a further application to real data. ... [edited by Author]