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    Distribution Theory - Estimation

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    1. A density function sometimes used by engineers to model length of life of electronic components is the Rayleigh density, given by if 0 < y1 <infinity, 0 otherwise. Assume Y1, Y2, ...Yn is a random sample from a Rayleigh distribution.
    a. If Y has the Rayleigh density, find the probability density for U = Y2.
    b. Use part a to find E(Y) and V(Y).
    c. Use the first moment E(Y) to find a method of moment estimator for &#61553;.
    d. Use the second moment E(Y) to find a method of moment estimator for &#61553;.
    e. Find the MLE of &#61553;. Denote this as .
    f. What is the asymptotic variance of ?
    g. Show whether is an unbiased estimator for &#61553;.
    h. Show whether is a consistent estimator for &#61553;.
    i. Show whether is sufficient for &#61553;.

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    Solution Summary

    The moment and maximum likelihood estimation of the parameters of a Rayleigh density are discussed in the solution. The examination of unbiasedness, consistency, sufficiency of the proposed estimators are also discussed.