Maximum likelihood estimates possess the property of functional invariance, which means that if is the MLE of θ, and h is any function of θ, then h (θ) is the MLE of h(θ).

a. Let .X ~ Bin(n, p) where n is known and p is unknown. Find the MLE of the odds ratio p/(1- p).

b. Use the result of Exercise 5 to find the MLE of the odds ratio p/(1 - p) if X ~ Geom(p).

c. If X ~ Poisson(λ), then P(X = 0) = e-λ. Use the result of Exercise 6 to find the MLE of P(X = 0) if X1,…,Xn is a random sample from a population with the-Poisson(λ) distribution.

Answer:

Step 1 of 3:

Let, be the maximum likelihood estimator for and h() be the MLE of h().

Let X follows Binomial distribution with probability mass function

f(x;p) = (1-p

The claim is to find the MLE of the odds ratio

The likelihood function is

L(x:p) = (1-p

Take log on both sides

Log L(x:p) = Log()

= [ log(n!) - log(x!) - log(n-x)! + x log p + (n - x) log (1-p) ]

Differentiate with respect to p and equate it to zero.

(log(n!) - log(x!) - log(n-x)! + x log p + (n - x) log (1-p) ) = 0

= (x log p + (n - x) log (1-p) )

Where, (x log p) = and (n - x) log (1-p) =

Substitute these values to (x log p + (n - x) log (1-p) )

Therefore, (x log p + (n - x) log (1-p) ) = -

Then, =

For =

=

=

Hence, MLE of the odds ratio is

Step 2 of 3:

b) Let X follows Geometric distribution with probability mass function with parameter p.

f(x ; p) = p ( 1 - p

The claim is to find the MLE of the odds ratio

the Maximum likelihood function (MLE) of p is

Then, =

=

=

Hence, MLE of the odds ratio is