# Newton Raphson Method

Problem Statement:

There were 46 crude oil spills of at least 1000 barrels from tankers in U.S. waters during 1974-1999.

Data: Ni = the number of spills in the ith year

bi1 = the estimated amount of oil shipped through US waters as part of US import/export operations in the ith year, adjusted for spillage in international or foreign waters

bi2 = the amount of oil shipped through U.S. waters during domestic shipments in the ith year.

Oil shipment amounts are measured in billions of barrels (Bbbl). The volume of oil shipped is a measure of exposure to spill risk. Suppose we use the Poisson process assumption given by:

Ni|bi1, bi2 ∼Poisson(λi) where $$λi = α1*bi1 +α2*bi2$$

The parameters of this model are α1 and α2, which represent the rate of spill occurrence per Bbbl oil shipped during import/export and domestic shipments, respectively.

(a) Derive the Newton-Raphson update for finding the MLEs of α1 and α2.

(b) Derive the Fisher scoring update for finding the MLEs of α1 and α2.

My attempt so far:

For (a), I need to use the Newton-Raphson method so:

$$x_t+1 = x_t - g'(x_t)/g''(x_t)$$

I have the pdf for Poisson(λ) $$= (λ^k)*\exp(-λ)/k!$$ and I am assuming that is my $$g(x_t)$$ in this case.

If so, am I supposed to differentiate wrt λ now?

I am fairly comfortable finding the MLE for more straightforward problems when I understand what the parameter is I'm looking to solve for, but I am confused with this problem. I am not sure how to proceed with the problem, which is my reason for posting this. Can anyone offer any help or direction as to how I'm supposed to approach this?

$$L(\alpha)=\prod_i \frac{\lambda_i^{y_i}e^{-\lambda_i}}{y_i!}$$
$$l(\alpha)=\sum_iy_ilog(\lambda_i)-\lambda_i-log(y_i!)=\sum_i y_ilog(a_1b_{i1}+a_2b_{i2})-a_1b_{i1}-a_2b_{i2}-log(y_i!)$$