Stochastic Comparison of Random Entities

Ghosh, Sugata (2024) Stochastic Comparison of Random Entities. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Sugata Ghosh (18RS017)) 18RS017.pdf - Submitted Version Restricted to Repository staff only Download (3MB)

Abstract

The ordering of two real numbers is trivial as there is a well-defined order imposed on the set of real numbers. However, ordering of random entities, which are measurable functions or various kinds of classes of measurable functions, is far less trivial. One of the primary goals of probability and statistics is to develop methods to compare various random entities, which is a practical problem in several areas of studies. In the classical theory of comparison of random variables, stochastic orders provide useful measuring sticks to capture the notion of one random variable being larger or smaller than another. Many such orders extensively studied in the literature, with areas of applications in probability and statistics, actuarial science, risk management, economics, operations research, wireless communications and other related fields. In this thesis, we introduce the concept of asymptotic stochastic comparison of stochastic processes. We define and analyze several asymptotic stochastic orders, exploring their properties and interrelationships. Sufficient conditions for these orders to hold for certain stochastic processes, evolving from some statistical entities of interest, are derived. On a different direction, we observe that most of the stochastic orders for comparing random variables, considered in the literature, lack the connex property and do not consider dependence between random variables. These drawbacks can be overcome at the cost of transitivity with the stochastic precedence order, which may seem to be a good choice, in particular, when only two random variables are under consideration. We show that even under such favorable conditions, stochastic precedence order may direct to misleading conclusion in certain situations. Variations of the order are developed to address the phenomenon.

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