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    Likelihood Ratio Test

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    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.

    $2.19

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