Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Delhi)


Kochar, Subhash C. (TSMU-Delhi; ISI)

Abstract (Summary of the Work)

The simplest and the most common way of comparing two random variables is through their means and variances. It may happen that in some cases the median of X is larger than the median of Y, while the mean of X is smaller than the mean of Y. However, this confusion will not arise if the random variables are stochastically ordered. Similarly, the same may happen if one would like to compare the variability of X with that of Y based only on numerical measures of variability. Besides, these characteristics of distributions might not exist in some cases. In most cases one can express various forms of knowledge about the underlying distributions in terms of their survival functions, hazard rate functions, mean residual functions, quantile functions and other suitable functions of probability distributions. These methods are much more informative than those based on comparing only few numerical characteristics of distributions. Comparisons of random variables based on such functions usually establish partial orders among them. We call them as stochastic ordersStochastic models are usually sufficiently complex in various fields of statistics, particularly in reliability theory. Obtaining bounds and approximations for their characteristics is of practical importance. That is, the approximation of a stochastic model either by a simpler model or by a model with simple constituent components might lead to convenient bounds and approximations for some particular and desired characteristics of the model. The study of changes in the properties of a model, as the constituent components vary, is also of great interest. Accordingly, since the stochastic components of models involve random variables, the topic of stochastic orders and dependence among random variables plays an important role in these areas.Now we introduce the notation and give some definitions of various types of stochastic orders and dependence among random variables. Throughout this thesis increasing means nondecreasing and decreasing means nonincreasing. We assume that expectations are well defined and multiple integrals can be evaluated irrespective of order. Let X and Y be univariate random variables with distribution functions F and G, survival functions F and G, density functions f and g; and hazard rates rF (= f/F) and ra (- 9/G), respectively.Stochastic orderingsDefinition 1.1.1X is said to be stochastically smaller than Y (denoted by X Sa Y) F(z) S G(2) for all x.This is equivalent to saying that Eg(x) < Eg(Y) for any increasing function g.Definition 1.1.2 X is said to be smaller than Y in hazard rate ordering (de- noted by X Shr Y) if G(1)/F(x) is increasing in x.It is worth noting that X Sar Y is equivalent to the inequalitiesP[X – t> a|X > t) S P[Y - t > zY > t), for all z 20 and t.In other words, the conditional distributions, given that the random variables are at least of a certain size, are all stochastically ordered (in the standard sense) in the same direction. Thus, if X and Y represent the survival times of different models of an appliance that satisfy this ordering, one model is better (in the sense of stochastic ordering) when the appliances are new, the same appliance is better when both are one month old, and in fact is better no matter how much time has elapsed. It is clearly useful to know when this strong type of stochastic ordering holds since quantities judgements are then easy to make. In case the hazard rates exist, it is easy to see that X Shr Y, if and only if, ra(x) SrF(x) for every z. The hazard rate ordering is also known as uniform stochastic ordering in the literature.


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Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


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