Question: MLE and Score function Suppose that X 1 ,..., X n is a random sample from a mixture distribution with cdf F ( x )

MLE and Score function

  • Suppose thatX1,...,Xnis a random sample from a mixture distribution with cdf

F(x)=G(x)+(1)H(x), where 0<<1 is the unknown parameter, and G and H are

cdf of two known continuous univariate distributions with respective densities g and h.

a) Find the score function and simplify it.

b) Set up the likelihood equation whose solution gives the MLE of .

Even though the MLE is not available in an explicit form

and must be computed numerically,

argue that the likelihood equation has at most one solution.

c) Show that the Fisher information about in a single observation is

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