Fisher's Approximation of a Binomial Distribution

[gajohnson]

Homework Statement

Suppose that X is the number of successes in a Binomial experiment with n trials and
probability of success θ/(1+θ), where 0 ≤ θ < ∞. (a) Find the MLE of θ. (b) Use Fisher’s
Theorem to find the approximate distribution of the MLE when n is large.

Homework Equations

Fisher's Approximation in its full form can be see here:

http://grab.by/s1bA

The Attempt at a Solution

The MLE is easy enough to find, but the question I have is one about Fisher's approximation. In the formula for calculating the 1/(tau)^2 portion of the variance, note that you must use the distribution of only one of X1, X2, etc. This makes plenty of sense if you have a collection of some other kinds of i.i.d. variables (normal, exponential, etc.), but what about if you have one binomial experiment made up of n trials?

What does this imply in the binomial case? Do I calculate 1/(tau)^2 using just the entire binomial distribution given in the problem, or do I use something else (one bernoulli trial, maybe)? Thanks!

 

[mighty2000]

In the binomial case, the MLE of θ is given by ^θ = X/n, where X is the number of successes in n trials. To find the approximate distribution of the MLE using Fisher's theorem, we can use the formula:

1/(tau)^2 = nI(θ), where I(θ) is the Fisher information.

In the binomial case, the Fisher information is given by I(θ) = nθ/(1+θ)^2. Therefore, 1/(tau)^2 = nθ/(1+θ)^2. This implies that the approximate distribution of the MLE ^θ is approximately normal with mean θ and variance θ/(1+θ)^2, when n is large.

To clarify, in this case, we use the entire binomial distribution given in the problem to calculate the Fisher information and subsequently the variance of the MLE.

Hope this helps!