Maximum likelihood estimates possess the property of functional invariance, which means that if 
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
