Maximum Likelihood Estimator + Prior

1. Nov 5, 2012

Scootertaj

1.Suppose that X~B(1,∏). We sample n times and find n1 ones and n2=n-n1zeros
a) What is ML estimator of ∏?
b) What is the ML estimator of ∏ given 1/2≤∏≤1?
c) What is the probability ∏ is greater than 1/2?
d) Find the Bayesian estimator of ∏ under quadratic loss with this prior

2. The attempt at a solution
a) $$L=\pi^n_1 *(1-\pi)^{n_2}$$
Do $$\frac{d(logL)}{d\pi} = \frac{n_1}{\pi} - \frac{n_2}{1-\pi}$$ → $$\pi_{ML} = \frac{n_1}{n}$$
b) not sure how to go about
c) not sure
d) I think I know how.

Last edited: Nov 5, 2012
2. Nov 5, 2012

Ray Vickson

In (b) you are asked to maximize L (or log L) subject to 1/2 ≤ π ≤ 1. Your solution to )a)_ may, or may not work in this case. When does it work? When does it fail? If it fails, what then must be the solution?

RGV

3. Nov 5, 2012

Scootertaj

What do you mean fail?
Intuitively, $$\pi_{ML}=\frac{n_1}{n}$$ would "fail" in the case that it is $$\frac{n_1}{n} < 1/2$$
But, I'm not sure what our solution must be then if it fails.

4. Nov 5, 2012

Ray Vickson

"Fail" = does not succeed = is wrong = does not work. When that is the case, something must have happened; what was that? What does that tell you about the behaviour of L(π)? (Hint: draw a hypothetical graph.)

RGV

5. Nov 5, 2012

Scootertaj

Well, based off the graph of $$\pi^{n_1}(1-\pi)^{n_2}$$ with several different n1 and n2 values plugged in that the best choice would be $$\pi=n1/n$$ when $$1/2≤n1/n≤1$$, else we choose $$\pi=1/2$$ since we usually look at the corner points (1/2 and 1)