Suppose is exponentially distributed with density Consider where Our goal is to estimate
(a) Find the density function of
(b) Let be an i.i.d. sample from the density of Find the MLE of denoted by
(c) Find the asymptotic variance of
Hello,
Use the Jacobian transformation to get the pdf of Z.
You should get
The likelihood function is then the pdf of the n-tuple, which gives :
(where all the zi are >0, and where )
So the log likelihood function is
The function is not important, since we will differentiate the log likelihood function with respect to h.
Now find its maximum and you'll get
Because in the pdf, it's 1/z, so you're summing 1/z_i. You just have to know that e^x e^y = e^(x+y)
You can, but you have to find the cdf of X first.Also, notable shortcut: instead of taking the Jacobian. Consider the cdf,
What is z ??? It's nothing near from a parameter...~
You should read again some basics...