A single observation of a random variable having a geometric

A single observation of a random variable having a geometric distribution is to be used to test the null hypothesis that its parameter equals (theta0) \"theta subzero\" against the alternative that it equals (theta1) > [(theta0)]. Use the Nayman-Perason Lemma to find the best critical region of size alpha.

Solution

geometric distribution is :(1-P)^(X-1)P

Consider the test of the simple null hypothesis H₀: ? = ?₀ against the simple alternative hypothesis H_A: ? = ?_a. Let C and D be critical regions of size ?, that is, let:

?=P(C;? 0 ) and ?=P(D;? 0 )

Then, C is a best critical region of size ? if the power of the test at ? = ?_a is the largest among all possible hypothesis tests. More formally, C is the best critical region of size ? if, for every other critical region D of size ?, we have:

P(C;? ? )?P(D;? ? )