Uniformly Most Powerful Test - Monotonic likelihood Ratio

<Uniformly Most Powerful Test>

When critical region of a test is uniformly more powerful than any other critical region having same significance lvela(α), the test is called as ' Uniformly Most Powerful Test.' Also the critical region of the test is called as 'Uniformly Most Powerful Critical Region.' We use Uniformly most powerful test when testing a simple hypothesis versus composite hypothesis ,or testing composite hypothesis versus composite hypothesis. The notation θ ∈{ } is that the parameter is included in a certain groups which is not a specified value. 

<Monotonic Likelihood Ratio>

Having monotonic likelihood ratio is an important condition for being uniformly most powerful test, because the test must 'uniformly' powerful than any other tests having same significance level. 

<Monotonic Likelihood Ratio of exponential family>

Exponential family is a specific probability distribution of a certain form. In statistics, there are some properties that distributions of exponential family have in common. One of the properties is to have monotonic likelihood ratio. Lots of familiar distributions are exponential family. One parameter exponential family has monotone likelihood ratio for the sufficient statistics T(X). Exponential distribution, Geometric distribution is one parameter distribution so they can have monotonic likelihood ratio. However for more than one parameter exponential families such as normal distribution, one parameter should be known or rejected. 

[Statistics] - Exponential family

<Uniformly Most Powerful Test using Monotonic likelihood ratio> 

When critical region of a test is uniformly more powerful than any other critical region having same significance level(α), the test is called as ' Uniformly Most Powerful Test.' The term ' uniformly' is proved by monotonic likelihood ratio. 

If a distribution has monotonic likelihood ratio in a sufficient statistic T(X), for any two values of the parameters, and have the most powerful rejection region of test under a significance level α, the test is called 'Uniformly Most Powerful Test.'

<exponential distribution>

<poisson distribution>

<Normal distribution>

<Sample Size for a certain power for a test> 

If there is a condition about power, rejection region or type 2 error, we can calculate sample size which is not given.