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 .
d. Use the second moment E(Y) to find a method of moment estimator for .
e. Find the MLE of . Denote this as .
f. What is the asymptotic variance of ?
g. Show whether is an unbiased estimator for .
h. Show whether is a consistent estimator for .
i. Show whether is sufficient for .
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.