Let X1, , Xn be an i.i.d. sample from a Rayleigh distribution with parameter > 0:

a Find the method of moments estimate of .

b Find the mle of .

c Find the asymptotic variance of the mie.

The double exponential distribution is

For an i.i.d. sample of size n =2m ± 1, show that the mie of  is the median of the sample (the observation such that half of the rest of the observations are smaller and half are larger). [Hint: The function g(x) =  is not differentiable. Draw a picture for a small value of n to try to understand what is going on.]

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