Discuss the below:
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 <∞, 0 otherwise. Assume Y1, Y2, ...Yn is a random sample from a Rayleigh distribution.
f(y)=(2y e-y2/θ)/θ
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 θ.
d. Use the second moment E(Y) to find a method of moment estimator for θ.
e. Find the MLE of θ. Denote this as θ^ .
f. What is the asymptotic variance of ?
g. Show whether θ^ is an unbiased estimator for θ.
h. Show whether θ^ is a consistent estimator for θ.
i. Show whether θ^ is sufficient for θ.