# Poisson MLE and Limiting Distribution

1. Jan 19, 2012

### mlarson9000

1. The problem statement, all variables and given/known data

Let yi denote the number of times individual i buys tobacco in a given month.
Suppose a random sample of N individuals is available, for which we observe values
0,1,2,... for yi.
Let xi be an observed characterisitc of these individuals (for example, gender). If we assume that for a given xi, yi has a Poisson distribution with mean
λi = exp ( β1 + β 2xi).
That is, the distribution function is:

Pr (yi = y |xi) =[exp(-λi)λi^y]/y! y = 0,1,2,...

(a) Obtain Bmle and derive its limiting distribution.
(b) Now suppose that yi does not take Poisson distribution. However, the orthogonality
condition holds:
E (yi - exp (β 1 + β 2xi) | xi) = 0:
Propose a consistent estimator for and derive its limiting distribution.

2. Relevant equations

3. The attempt at a solution

I'm not sure how to approach this. Do I just plug exp(β1+β2xi) into λi, and derive the MLE, or am I supposed to do something else? When I tried it that way, I get

dL/dβ1=ny-Ʃexp(β1+β2xi)=0

dL/dβ2=yƩxi-Ʃxi[exp(β1+β2xi)]=0

And I'm not sure how to solve for β1 and β2 with this.