Composite Likelihood Ratio Tests Flashcards
Define uniformly most powerful.
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.
Define a montone likelihood ratio.
What is the method for monotone likelihood ratio test?
What sided tests are monotone likelihood ratio tests?
One-sided
What is the name given to the two-sided composite likelihood ratio tests?
Generalised Likelihood Ratio Tests
Define a generalised likelihood ratio test.
Wbat is the generalised likelihod ratio test statistic?
What is the test for the generalised ratio test statistic?
What is the symbol for the MLE if θ ∈ ω0 and if θ ∈ Ω?
- θ^0
- θ^
What does dim(ω0) and dim(Ω) stand for?
The number of free parameters in ω0, Ω respectively
Under suitable smoothness conditions the null distribution of -2log(⋀) for large n is aprroximtely equal to what?
A chi-square distribution with degrees of freedom equal to dim(Ω) - dim(ω0)
What does ⋀ equal?
Prove the following therom
Need to do.
What is the maximum likelihood estimates for each pi from a multinomial sample x1, …. xm of size n?
Prove the following theorem
Need to do.
What is the generalsied likelihood ratio test for a multinormial sample?
For large n, what is the approximate distribution for -2log(⋀)?
Chi-squared distribution with m - 1 - d degrees of freedom, where d is the number of fitted parameters under H0.
For large n what is the generalized ratio test asymptotically equal to?
The chi-squared goodness of fit test
Proove that the chi-square goodness of fit test is asymptotically equal to the generalised ratio test which uses -2log(⋀).
Need to do - Last page of notes.
