Solution Found!
Maximum likelihood estimates possess the property of
Chapter 4, Problem 7E(choose chapter or problem)
Maximum likelihood estimates possess the property of functional invariance, which means that if \(\widehat{\theta}\) is the MLE of , and h\(\(\widehat{\theta}\)\) is any function of
, then h\(\(\widehat{\theta}\)\) is the MLE of h\(\(\widehat{\theta}\)\).
a. Let where
is known and
is unknown. Find the MLE of the odds ratio
.
b. Use the result of Exercise 5 to find the MLE of the odds ratio if
.
If , then \(P(X=0)=e^{-\lambda}\) Use the result of Exercise 6 to find the MLE of
if \(X_{1}, \ldots, X_{n}\) is a random sample from a population with the Poisson \((\lambda)\) distribution.
Equation transcription:
Text transcription:
\widehat{\theta}
P(X=0)=e^{-\lambda}
X_{1}, \ldots, X_{n}
(\lambda)
Questions & Answers
QUESTION:
Maximum likelihood estimates possess the property of functional invariance, which means that if \(\widehat{\theta}\) is the MLE of , and h\(\(\widehat{\theta}\)\) is any function of
, then h\(\(\widehat{\theta}\)\) is the MLE of h\(\(\widehat{\theta}\)\).
a. Let where
is known and
is unknown. Find the MLE of the odds ratio
.
b. Use the result of Exercise 5 to find the MLE of the odds ratio if
.
If , then \(P(X=0)=e^{-\lambda}\) Use the result of Exercise 6 to find the MLE of
if \(X_{1}, \ldots, X_{n}\) is a random sample from a population with the Poisson \((\lambda)\) distribution.
Equation transcription:
Text transcription:
\widehat{\theta}
P(X=0)=e^{-\lambda}
X_{1}, \ldots, X_{n}
(\lambda)
ANSWER:
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