Consider the following partial ordering between randomvariables X 1 and X 2 . The randomvariable X…


Consider the following partial ordering between randomvariables X1 and X2. The randomvariable X1 is said to be smaller than the random variable X2 in the completion rate sense if the completion rate of X1 at time t, say λ1(t), is larger than or equal to the completion rate of X2, say λ2(t), for any t.

(a) Show that this ordering is equivalent to the ratio (1−F1(t))/(1−F2(t)) being monotone decreasing in t. (b) Show that

 monotone likelihood ratio ordering ⇒ completion rate ordering ⇒ stochastic ordering.