# Likelihood Ratio Test

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#### Solution Preview

Please see the attachment for solution.

Solution (a)

We have the joint p.d.f. of order statistics is

=

Define

Then , , ,...,

Now we can write

, , ,...,

Now the joint p.d.f. of is given by,

,

where

= = =

Thus the joint p.d.f. of is given by,

,

where

This shows that are independently distributed with are i.i.d. as exponential distribution with parameter 1/σ.

If , then are i.i.d. as exponential distribution with parameter 1/σ.

Solution (b)

We have the joint p.d.f. of order statistics is

=

Define and

Then we can write

, , ,...,

Now the joint p.d.f. of is given by,

,

where

= = =1

Thus the joint p.d.f. of is given by,

,

where

This shows that are independently distributed with are i.i.d. as ...

#### Solution Summary

The Likelihood Ratio Test and construction of confidence interval for the parameters of an extreme value distribution are discussed in the solution.