Mixture distribution. Suppose that the joint distribution of the two random variables x and y is

a. Find the maximum likelihood estimators of β and θ and their asymptotic joint distribution.

b. Find the maximum likelihood estimator of θ/(β + θ) and its asymptotic distribution.

c. Prove that f (x) is of the form

f (x) = γ (1 ?? γ )x, x = 0, 1, 2, . . . ,

and find the maximum likelihood estimator of γ and its asymptotic distribution.

d. Prove that f (y | x) is of the form

Prove that f (y | x) integrates to 1. Find the maximum likelihood estimator of λ and its asymptotic distribution.

e. Prove that

f (y) = θe^??θy, y ?? 0, θ>0.

Find the maximum likelihood estimator of θ and its asymptotic variance.

f. Prove that

Based on this distribution, what is the maximum likelihood estimator of β?

Transcribed Image Text:

e-B+0)y (By)* f(x, y) = В,0 > 0, у 2 0, х%3D0, 1,2, .... x!