Statistical Theory and Inference by David J. Olive

Author:David J. Olive
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

The LHS =

So reject Ho if

or if ∑(x i −μ)2 > k where

Under Ho, so where

b) In the above argument,

but

for any . Hence the UMP test is the same as in a).

Alternatively, use the fact that this is an exponential family where is an increasing function of σ 2 with . Hence the test in a) is UMP for a) and b) by Theorem 7.3.

Example 7.6.

Let Y 1, . . . . , Y n be iid with pdf

where ν > 0, θ is known and t(y) is a function such that

a) Find the UMP level α test for H 0: ν = 1 versus H 1: ν = 1. 19.

b) Suppose n = 12 and α = 0. 05. Find the power β(1. 19) when ν = 1. 19.

Solution: a) This is an exponential family. Note that 2ν 2 is increasing, so 1∕(2ν 2) is decreasing and is increasing. Thus the UMP test rejects H 0 if where .

b) Use α to find k and then find the power. If H 0 is true so ν = 1, then Thus using a chi-square table. If ν = 1. 19, then So using a chi-square table where .

Example 7.7.

Let be independent identically distributed random variables with pdf

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