Tests of hypotheses: most powerful and uniformly most powerful tests, likelihood ratio tests | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

A. We first review the general definition of a hypothesis test. 
A hypothesis test is like a lawsuit:
H₀ : the defendant is innocent  versus
H_a : the defendant is guilty 

The significance level and the power of the test. 
 
α = P(Type I error) = P(Reject H₀|H₀) ← signiicance level  
β = P(Type II error) = P(Fail to reject H₀ |H_a ) 

B. Now we derive the likelihood ratio test for  the usual two sided hypotheses for a population mean.  (It has a simple null hypothesis and a composite alternative hypothesis.) 

Example 1. Please derive the likelihood ratio test for H₀: μ = μ₀ versus Ha: μ ≠ μ₀, when the population is normal and population variance σ² is known.  
 Solution:  For a 2-sided test of H₀: μ = μ₀ versus H_a: μ ≠ μ₀, when the population is normal and population variance σ² is known, we have: 

The likelihoods are: 

There is no free parameter in

There is only one free parameter μ in   L(Ω) . Now we shall find the value of μ that maximizes the log likelihood 

It is easy to verify that  indeed maximizes the loglikelihood, and thus the likelihood function. 
 
Therefore the likelihood ratio is:

Therefore, the likelihood ratio test that will reject H₀ when *   is equivalent to the z-test that will reject H₀ when cZ 0 , where c can be determined by the significance level α as /2 c z   . 
 

C.  MP Test, UMP Test, and the Neyman-Pearson Lemma 
 
Now considering some one-sided tests where we have : 
                         
Given the composite null hypothesis for the second one-sided 
test, we need to expand our definition of the significance level as  
follows: