# Constructing likelihood ratio test

Let $y_1$ and $y_2$ be independent Poissons with means $\lambda_1$ and $\lambda_2$. Find the likelihood ratio statistic for the test $H_0: \lambda_1 = \lambda_2$. Specify the asymptotic null distribution, and describe the condition under which the asymptotics apply.

I have the answer. It's $2\sum_i y_i\log\left(\frac{y_i}{ \bar{y}}\right)$. The asymptotic distribution is chi-squared with $df = 1$, when $\lambda_1$ and $\lambda_2$ are large.

The problem is that I have trouble arriving at this answer. I start with the poisson pmf, find the likelihood for each and take the ratio. But I dont see where the $\bar{y}$ comes from? Can someone show me how to get this? I'd really appreciate the help.

Edit: OK, what I got after taking the ratio of the likelihoods (at mle's) was $2n [\lambda_2 - \bar{y} + \bar{y}log(\bar{y} / \lambda_2)]$

• 1. Unless I made an error, I think the answer you give appears to be wrong; I think it's missing a term that can't be dropped if you're using the asymptotic approximation. 2. Note that $y_1+y_2=2\bar{y}$. – Glen_b -Reinstate Monica Nov 15 '16 at 8:49
• Please add the self-study tag, read its tag-wiki and modify your question to follow the guidelines on asking such questions. In particular, you'll need to clearly identify what you've done to solve the problem yourself, and indicate the specific help you need at the point you struck difficulty. – Glen_b -Reinstate Monica Nov 15 '16 at 8:49
• To add the tag you'd need to remove one. I think "generalized-linear-model" or "maximum-likelihood" could go, but if you want to keep both, just leave it out. – Glen_b -Reinstate Monica Nov 15 '16 at 9:46
• Thanks Glen- see my edits. Also, the answer I gave came from my book so I'm not sure...but I haven't matched it either. – kiring24 Nov 15 '16 at 11:33
• What happened to the factorial terms? – Glen_b -Reinstate Monica Nov 15 '16 at 11:39

Leaving a little work at the end for you, I got the following. This is a bit loose, but it's intended as an outline -- I'm sure you can tidy it up. Anyway here goes:

$\Lambda = \frac{\mathcal{L}_0(y_1,y_2)}{\mathcal{L}_1(y_1,y_2)}$

$\mathcal{L}_1=\exp(-\lambda_1)\lambda_1^{y_1}/y_1!\cdot \exp(-\lambda_2)\lambda_2^{y_2}/y_2!$

$\mathcal{L}_0=exp(-2\lambda)\lambda^{y_1+y_2}/(y_1+y_2)!$

$\hat{\lambda}_1=y_1\,,\hat{\lambda}_2=y_2\,,\hat{\lambda}=(y_1+y_2)/2=\bar{y}$

$\log(\Lambda)=\ell_0(y_1,y_2)-\ell_1(y_1,y_2)$

$= -2\lambda +2\bar{y}\log(\lambda)-\log((y_1+y_2)!)\\ \qquad - [-\lambda_1+y_1\log(\lambda_1)-log(y_1!)-\lambda_2+y_2\log(\lambda_2)-log(y_2!)]$

at MLE

$\log(\Lambda)=-2\bar{y} +2\bar{y}\log(\bar{y})-\log((y_1+y_2)!) \\ \qquad - [-y_1+y_1\log(y_1)-log(y_1!)-y_2+y_2\log(y_2)-log(y_2!)]$

$=-2\bar{y} +2\bar{y}\log(\bar{y})-\log((y_1+y_2)!)\\ \qquad - [-y_1+y_1\log(y_1)-log(y_1!)-y_2+y_2\log(y_2)-log(y_2!)]$

etc

If you simplify further, that gets to the answer in the question plus an additional term I don't believe you can leave out.