Stochastic Orderings - University of Michigan



5 - Partial Orderings for Random Variables

Given two random variables, we often want to compare them. In some cases we might want to claim that one is less than the other. For example, we have two different types of disc drives and we are comparing how long it takes them each to break. Because of the element of chance, which one breaks first may differ in different trials. Here we look at different types of conditions under which we say “one lasts longer than the other”. In other words, we look at different types of conditions under which we say one random variable is less than another. Much of this is based on the paper of Keilson and Sumita [5].

Section 5.1 discusses the "Ordinary Ordering" of random variables as functions on the sample space while Section 5.2 considers the "Stochastic Ordering". Section 5.3 and 5.4 look at the "Hazard Rate Ordering" while Section 5.5 discusses the "Likelihood Ratio Ordering".

5.1 Ordinary Ordering

The simplest way of saying that one random variable R is less than or equal another T is to say that it is almost always less than or equal to the other. We will use the symbol ( for this.

Definition 1. If R and T are random variables then R is less than or equal to T if R(ω) ( T(ω) for almost all ω. We will write R ( T if this is the case.

This type of ( posseses the usual properties of ( for numbers.

Proposition 1. (a) For any U one has R ( T ( R + U ( T + U

(b) R1 ( T1 and R2 ( T2 ( R1 + R2 ( T1 + T2

(c) R ( 0 ( T ( T + R

(d) R ( T and U ( 0 ( UR ( UT

(e) R ( 1 and T ( 0 ( T ( RT

Proof. R ( T ( R(ω) ( T(ω) for almost all ω ( R(ω) + U(ω) ( T(ω)+ U(ω) for almost all ω ( R + U ( T + U. R + U ( T + U ( R ( T follows from R ( T ( R + U ( T + U by replacing R, T and U by R + U, T + U and -U. The proof of (d) is similar. (b) follows from (a), (c) follows from (b) and (e) follows from (d). //

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