a.
Find the MLE of θ.
a.
Answer to Problem 103SE
TheMLE of θ is,
Explanation of Solution
The density
From Exercise 9.82, the MLE of θ for Weibull’s distribution is given as
It is known that the Rayleigh’s distribution is a special case of Weibull’s distribution with
Therefore, by substituting
Hence, the MLE of θ for Rayleigh’sdistribution is obtained as follows:
Thus, the MLE of θ is,
b.
Find the asymptotic variance of the MLEof θ.
b.
Answer to Problem 103SE
The asymptotic variance of the MLE is
Explanation of Solution
Under the regulatory conditions, the formula for asymptotic variance is given as follows:
The log-likelihood function of θ is obtained as follows:
By obtaining the first- and second-order partial derivations of the log-likelihood function with respect to θ, the asymptotic variance is found as follows:
Consider,
By substituting Equation
Therefore, the asymptotic variance of the MLE is
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Chapter 9 Solutions
Mathematical Statistics with Applications
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage