Composite Likelihood Ratio Tests

Question 1

Define uniformly most powerful.

Answer 1

A simple null hypothesis H₀ against a composite alternative hypothesis H₁ is uniformly most powerful (UMP) at significance level α if the test is most powerful at significane level α against every simple hypothesis in H₁.

Question 2

Define a montone likelihood ratio.

Answer 2

Question 3

What is the method for monotone likelihood ratio test?

Answer 3

Question 4

What sided tests are monotone likelihood ratio tests?

Answer 4

One-sided

Question 5

What is the name given to the two-sided composite likelihood ratio tests?

Answer 5

Generalised Likelihood Ratio Tests

Question 6

Define a generalised likelihood ratio test.

Answer 6

Question 7

Wbat is the generalised likelihod ratio test statistic?

Answer 7

Question 8

What is the test for the generalised ratio test statistic?

Answer 8

Question 9

What is the symbol for the MLE if θ ∈ ω₀ and if θ ∈ Ω?

Answer 9

1. θ^{^}₀
2. θ^{^}

Question 10

What does dim(ω₀) and dim(Ω) stand for?

Answer 10

The number of free parameters in ω₀, Ω respectively

Question 11

Under suitable smoothness conditions the null distribution of -2log(⋀) for large n is aprroximtely equal to what?

Answer 11

A chi-square distribution with degrees of freedom equal to dim(Ω) - dim(ω₀)

Question 12

What does ⋀ equal?

Answer 12

Question 13

Prove the following therom

Answer 13

Need to do.

Question 14

What is the maximum likelihood estimates for each p_i from a multinomial sample x₁, …. x_m of size n?

Answer 14

Question 15

Prove the following theorem

Answer 15

Need to do.

Question 16

What is the generalsied likelihood ratio test for a multinormial sample?

Answer 16

Question 17

For large n, what is the approximate distribution for -2log(⋀)?

Answer 17

Chi-squared distribution with m - 1 - d degrees of freedom, where d is the number of fitted parameters under H₀.

Question 18

For large n what is the generalized ratio test asymptotically equal to?

Answer 18

The chi-squared goodness of fit test

Question 19

Proove that the chi-square goodness of fit test is asymptotically equal to the generalised ratio test which uses -2log(⋀).

Answer 19

Need to do - Last page of notes.