Stochastic Orderings - University of Michigan
5.2 The Stochastic Ordering
The ordering considered in the previous section (ordinary less than or equal to ) is too strong to be of use in most situations where probability is involved. A more useful one is "stochastically less than or equal to". One random variable R is stochastically less than or equal another T if for each t the probability that R exceeds t is no more than the probability T exceeds t.
Definition 1. If R and T are random variables then R is stochastically less than or equal to T if for each t one has P{R > t} ( P{T > t}. We will write R ( T st if this is the case.
Propostion 1.
(a) R ( st T ( SR(t) ( ST(t) for all t ( P{T ( t} ( P{R ( t} for all t ( FT(t) ( FR(t) for all t.
(b) R ( st T ( P{R ( t} ( P{T ( t} for all t ( P{T < t} ( P{R < t} for all t.
Proof. The first ( of (a) follows from the fact that SR(t) = P{R > t} and ST(t) = P{T > t}. The second ( of (a) follows from the fact that Pr{R ( t} = 1 - P{R > t} and Pr{T ( t} = 1 - P{T > t}. The third ( of (a) follows from the fact that FR(t) = P{R ( t} and FT(t) = P{T ( t}. This completes the proof of (a). The second ( of (b) follows from the fact that Pr{R ( t} = 1 - P{R > t} and Pr{T ( t} = 1 - P{T > t}. To show R ( st T ( P{R ( t} ( P{T ( t} for all t note that {R ( t} = {R > t - } and {R > t - } decreases with n. So P{R ( t} = P{R > t - } and similarly for P{T ( t}. The fact that R ( st T ( P{R ( t} ( P{T ( t} follows from this. To show that P{R ( t} ( P{T ( t} for all t ( R ( st T note that {R > t} = {R ( t - } and {R ( t - } increases with n. So P{R > t} = P{R ( t - } and similarly for P{T > t}. The fact that P{R ( t} ( P{T ( t} for all t ( R ( st T follows from this. This completes the proof of the first ( of (b) and completes the proof of part (b). (
For exponential random variables one can simply compare the decay rates.
Example 1. Let Tλ and Tμ be exponential random variables with breakdown rates λ and μ. Then λ ≤ μ ( Tμ ( st Tλ. This follows from the fact that for t ≥ 0 one has STμ(t) = e-μt ( e-λt = STλ(t).
The ordinary ordering of the previous section is stronger than the stochastic ordering. However, when one of the random variable is a constant the two orderings are equivalent.
Proposition 2. (a) If R ( T then R ( st T.
(b) Let c be a number. Let c also denote the random variable that associates the number c to each ( in (. If c ( st T then c ( T. If then T ( c.
Proof. {(: R(() > t} ( {(: T(() > t} ( P{(: R(() > t} ( P{(: T(() > t} ( SR(t) ( ST(t). This proves (a). To prove (b) note that by Propostion 1 (b) the relation c ( st T implies P{c ( c} ( P{T ( c}. Since P{c ( c} = 1 it follows that P{T ( c} = 1. So c ( T(() for almost all (. So c ( T. The proof in the case T ( st c is similar. //
( st posseses some of the usual properties of ( for numbers.
Proposition 3. (a) R ( 0 ( T ( st T + R.
(b) For any c one has R ( st T ( R + c ( st T + c
(c) R ( st T and U is independent of R and T ( R + U ( st T + U
(d) R1 ( st T1, …, Rn ( st Tn and R1, …, Rn, T1, …, Tn all independent ( R1 + … + Rn ( st T1 + … + Tn
(e) R ( st T ( sup(R) ( sup(T)
Proof. For each t one has {(: T(() > t} ( {(: T(() > t}, so P{T > t} ( P{T + R > t} which implies (a). R ( st T ( SR(t) ( ST(t) for all t ( SR(t-c) ( ST(t-c) for all t ( SR+c(t) ( ST+c(t) for all t ( R + c ( st T + c which proves (b). R ( st T ( SR(t) ( ST(t) for all t ( ( for all s ( SR+U(t) ( ST+U(t) for all s by Proposition 2 in section 1.5 ( R + U ( st T + U which proves (c). (d) follows from (c). To prove (e) note that R ( st T implies SR(sup(T)) ( ST(sup(T)) = 0. So SR(sup(T)) = 0 which implies sup(R) ( sup(T). //
However, some properties of ( for numbers do not carry through for ( st.
Example 2. Part (c) of Proposition 3 is not necessarily true if U is not independent of R and T. Let ( = {0, 1} with Pr{i} = ½ for each i. Let R(i) = i for each i and T = U = 1-R. Then R ( st T and R + U = 1 and T + U = 2T. It is not true that 1 ( st 2T.
Recall from section 3.4 that a hypoexponential random variable is a sum of independent exponential random variables. If (1, …, (n are positive numbers then T(1,…,(n denotes the hypoexponential random variable T(1,…,(n = T(1 + ( + T(n where T(1, …, T(n are independent and each T(j is an exponential random variable with decay rate (j. For hypoexponential random variables we have the following.
Proposition 4. Let (1, …, (n are positive numbers and Fn(t;(1,…,(n) be the cummulative distribution function of T(1,…,(n.
(a) If m ( n then
T(1,…,(m ( st T(1,…,(n
Fn(t;(1,…,(n) ( Fm(t;(1,…,(m) for all t
(b) If (1, …, (n are postive numbers with (j ( (j for all j then
T(1,…,(n ( st T(1,…,(n
Fn(t;(1,…,(n) ( Fn(t;(1,…,(n) for all t
Proof. T(1,…,(n = T(1,…,(m + T(m+1,…,(n and T(m+1,…,(n ( 0. By Proposition 2 (a) we get the first part of (a). The second part follows from Proposition 1. To prove (b) note that (j ( (j implies T(j ( st T(j so the first part of (b) follows from Proposition 3 (d). The second part follows from Propositon 1. //
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