#### Newton-Raphson Algorithm

Let

be the gradient vector and the Hessian matrix, where is the log likelihood for the jth observation. With a starting value of , the pseudo-estimate of is obtained iteratively until convergence is obtained:

where and are evaluated at the ith iteration . If the log likelihood evaluated at is less than that evaluated at , then is recomputed by step-halving or ridging. The iterative scheme continues until convergence is obtained—that is, until is sufficiently close to . Then the maximum likelihood estimate of is .