INFORMATION THEORETIC MEASURES BASED ON RECORD STATISTICS

Abstract

The various information theoretic measures for example Shannon entropy measure [104] and its various additive generalizations like Renyi entropy [99], and Varma entropy [122], and non-additive generalizations like Havrda and Charvat entropy measure [56] have applications in different fields like statistics, physics, electronics etc. All these are based on single random variable. So by taking idea from this, we try to explore the behaviour of various information theoretic measures when these are applied on the sequence of record random variables and onthesequenceofk-recordrandomvariables. Inthisthesiswestudygeneralized Varma entropy measure, Kerridge inaccuracy measure, Kullback-Leibler discrimination measure, cumulative residual inaccuracy measure and entropy measure for past lifetime for the sequence of record values. We introduce these measures for record values and study some characterization results based on them. This thesis includes seven chapters including the first chapter which is on introduction and literature survey. The thesis is organized as follows: In Chapter 2, we have considered a measure of past entropy based on Shannon [104] entropy measure for nth upper k-record value. A characterization result for the measure under consideration has given. We have discussed some basic properties of the proposed measure. Also we have constructed some bounds to the proposed past entropy measure for nth k-records. The work reported in this chapter has been published in the paper entitled, Measure of Entropy for Past Lifetime and k-Record Statistics in Physica A, 2018, 503, 623-631. In Chapter 3, we have introduced a measure of inaccuracy between distributions of the nth record value and parent random variable and discussed some xiii properties of it. It has also been shown that the proposed inaccuracy measure characterizes the distribution of parent random variable uniquely. Measure of inaccuracy for some specific distributions has also been studied. F α distributions are equally important, so keeping this in mind we have also studied inaccuracy measure for F α distributions. The part of the work reported in this chapter has been published in the paper entitled, Kerridge Measure of Inaccuracy for Record Statistics in Journal of Information and Optimization Sciences, 2018, 39(5), 1149-1161 and some work has been presented in the International Conference on interdisciplinary Mathematics, Statistics and Computational Techniques held at Manipal University, Jaipur, Dec 22-24, 2016. In Chapter 4, we have studied a measure of inaccuracy between nth upper k-record value and mth upper k-record value. A simplified expression for the proposed inaccuracy measure has also been derived to find the inaccuracy measure for some specific probability distributions. We have also shown that the proposed inaccuracy measure characterizes the underlying distribution function uniquely. Furtherwehaveconsideredresidualmeasureofinaccuracybetweentwok-record values and given a characterization result for that. The results reported in this chapter have been published in the paper entitled, Measure of Inaccuracy and k-Record Statistics in Bulletin of Calcutta Mathematical Society, 2018, 110 (2), 151-166 and some work has been presented in National Seminar on Recent Developments in Mathematical Sciences held at MDU, Rohtak, Mar 07-08, 2017. In Chapter 5, we have provided an extension of cumulative residual inaccuracy as suggested by Taneja and Kumar [116] to k-record values. We have studied some properties of this measure. Also we have studied some stochastic ordering and have found the proposed measure for some of the distributions which occur oftenin manyrealistic situations and haveapplications in variousfields of science and engineering. The work reported in this chapter is communicated under the title, Cumulative Residual Inaccuracy Measure for k-Record Values and some work has been presented in International Conference on Recent Advances in Pure and Applied Mathematics held at Delhi Technological University, Delhi, Oct 23-25, 2018. xiv In Chapter 6, we have provided an extension of Kullback-Leibler [66] information measure to k-record values. The distance between two k-record distributions of residual lifetime has been found. We have found the measure of discrepancy between nth k-record value and mth k-record value. Also keeping the record times fixed, we have derived the distance between k-record value and l-record value. Wehavealsostudiedsomepropertiesofthemeasureproposedandacharacterizationresultforthat. Theworkreportedinthischapteriscommunicatedunderthe title, A Measure of Discrimination Between Two Residual Lifetime Distributions For k-Record Values and some work has been presented in International ResearchSymposiumonEngineeringandTechnologyheldatSingapore,August 28-30, 2018. In Chapter 7, we have considered and studied a generalized two parameters entropy based on Varma’s entropy [122] function for k-record statistic. A general expression for this entropy measure of k-record value has been derived. Furthermore based on this, we have proposed a generalized residual entropy measure for k-record value and proved a characterization result that the proposed measure determines the distribution function uniquely. Also, an upper bound to the dynamic generalized entropy measure has been derived. The part of the work reported in this chapter has been communicated under the title, On Generalized Information Measure of Order ( α , β ) and k-Record Statistics. In the last we have presented the conclusion of the work reported in this thesis and further scope of work.