Verify that (1.1) holds with probability one for any sequence of in Example 10. (a) Consider a decision problem and decision rule 8 for which Imagine a sequence of repetitions of the decision...

Verify that (1.1) holds with probability one for any sequence of

in Example 10.

(a) Consider a decision problem and decision rule 8 for which

Imagine a sequence of repetitions of the decision problem, where the parameters 8j can change and the data X(j) (from the density

are independent. Show that, if

for all
, then

with probability one for any sequence

(b) Construct an example of a decision problem where

and yet (1.2) still holds with probability one for any sequence

Verify, in Example 11, that the proportion of rejections that are in error (i.e., the proportion of the rejections for which Ho is true) will tend to! with probability one.