1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Poisson MLE and Limiting Distribution

  1. Jan 19, 2012 #1
    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
    λi = 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



    And I'm not sure how to solve for β1 and β2 with this.
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted