Most powerful test - Neyman-Pearson Theorem /proof/ examples/

<Some important definitions in Test>

Test statistic

Test statistics: Statistics from random samples in hypothesis test. We can use statistics from complete sufficient statistics, maximum likelihood estimator, estimator from method of moment, minimum variance unbiased estimator and so on. 

Null hypothesis is a general state(people have accepted naturally) that we want to reject. 

Alternative hypothesis is a state that we want to accept. (new state) 

Critical region is the rejection region, that is, Interval of test statistic that leads to rejection of null hypothesis, acceptance of alternative hypothesis. 

Size of critical region is the probability to obtain a value in critical region when the null hypothesis is true , which is P(reject the null hypothesis when it is true). Type 1 error, which is level of significance level, is the size of critical region. 

There are two types of errors in hypothesis test. Type 1 error is the probability to reject null hypothesis even though that is true. Type 2 error is the probability not to reject null hypothesis even though the null hypothesis is not true. When we reject the null hypothesis, it is the same meaning of accept alternative hypothesis. Type 1 error which can be also called as significance level,is regarded as more serious error than Type 2 error. 

Power function is probability to reject null hypothesis. So if null hypothesis is true, then the less power function , the better. 

In contrast, when alternative hypothesis is true(null hypothesis is false), then the larger power function, the better. 

<Most Powerful Test> 

Among unbiased estimator, we want to find the best estimator from collections of unbiased estimators, which is Minimum Variance Unbiased Estimator. Like this, we want to find the best one for test. That is, in collection of tests with same significance level, test that has the biggest power function is should the best test. We can find the test by using Neyman-Pearson test. 

Neyman-Pearson method uses likelihood ratio. When test statistics is included in the rejection region, It would be the better test if the probability of getting null hypothesis is smaller than probability of getting alternative hypothesis. So expressing with likelihood ratio, when the test statistics is included in the rejection region, the smaller  L(null, alternative), the better the test. 

So the test is the best when the likelihood ratio is smallest under same significance level when sample statistics be included in rejection region. 

When test statistics is not included in the rejection region, It would be the better test if the probability of getting alternative hypothesis is smaller than probability of getting null hypothesis. So expressing with likelihood ratio, when the test statistics is not included in rejection region, the larger L(null, alternative), the better the test.

So the test is the best when the likelihood ratio is largest under same significance level when sample statistics be not included in rejection region. 

<Neyman-Pearson Theorem - A way to find best test >