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# 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.

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