logistic models help!

Jan 2010
39
0
The independent random variables \(\displaystyle Y_{1},Y_{2},...,Y_{n},\) each \(\displaystyle Y_{i}\in\{0,1\}\), are postulated to depend on covariates \(\displaystyle x_{1},x_{2},...,x_{n}\) through the model,
\(\displaystyle log\frac{\pi_{i}}{1-\pi_{i}}=\beta x_{i}\)

a) Derive the equation which the maximum likelihood estimator of \(\displaystyle \beta\) satisfies.

b) For the special case where each \(\displaystyle x_{i} \in \{-1,0,1\}\) solve the equation in part (a). You may assume that the number of \(\displaystyle x_{i} = j\) is \(\displaystyle n_{j}\), \(\displaystyle j=-1,0,1\) and that \(\displaystyle y_{i}=1\) for \(\displaystyle r_{j}\) of the cases where \(\displaystyle x_{i}=j\)


I'm really stuck on the part (b), can anyone either point me in the right direction or come up with a solution? Thanks!