Question

Finding the MLE under the null hypothesis:

Answer 1

\((X_1, X_2, X_3) \) follow a multinomial\((n,\theta_1,\theta_2,\theta_3)\) distribution. I need to find the likelihood ratio statistic for testing \(H_0: \theta_1=\theta^2, \theta_2=2\theta(1-\theta)\) against all alternatives. I'm having trouble finding the MLE under the null Ho.

The pmf of the multinomial:
\[ \frac{n!}{x_1!x_2! (x_3)!}\theta_1^{x_1}\theta_2^{x_2}(\theta_3)^{x^3}\] with constraints \( x_1+x_2+x_3=n\) and \(\theta_1+\theta_2+\theta_3=1\) thus giving the following pmf:
\[\frac{n!}{x_1!x_2! (n-x_1-x_2)!}\theta_1^{x_1}\theta_2^{x_2}(1-\theta_1-\theta_2)^{n-x_1-x_2}\]

Answer 2

This might just be a shot in the dark, but I think you'll be able to maximize the likelihood function using Lagrange multipliers.

Answer 3

Ohhh, hmm. I will try that!

Answer 4

You were given some constraints, and if I recall correctly, Lagrange multipliers work with exactly that. Hope it helps.

Answer 5

Yeah you're right. I kind of forgot about Lagrange multipliers... haven't used them in a while..