Question

1) Define the following statistic tests

      a)   A log likelihood ratio test

      b)   Wald's test

2) How is a log likelihood ratio test is constructed to assess theadequacy of a given model

3) Describe features of the covariance correlation matrix

Answer 1

1) A log likelihood ratio test.

Now consider a general family of distributions:

The null hypothesis H0 will state that the parameters belong to some subspace of the parameter space .

Let x1,x2,...,xn be a random sample of size n>1 from a population with p.d.f f(x,)

where , the parameter space is the totality of all points that can assume.

We want to test the null hypothesis :

Against all alternative hypothesis of the type:

The likelihood function ofthe sample observations is given by

(i)

According to the pronciple of maximum likelihood , the likelihood equation for estimating any parameter is given by

(ii)

We can obtain the maximum likelihood estimates for the parameters as they are allowed to vary over the parameter space and the subspace . Substituting these estimates in above equation (i) we obtain themaximum values of the likelihhod function for variation of the parameters in and respectively. Then the criterion for the likelihood ratio test is defined as the quotient of these two maxima and is given by

where

and are the maxima of the likelihood function (i) with respect to parameters in the region and respectively.

ii) Wald's test: The Wald test is a way to find out if explanatory variables in a given model are significant.

Test statistic is

W=

where is Maximum Likelihood estimator

= Expected fisher information.

NOTE: I have done the first question please repost rest of the questions. Thank you .