# [R] maximum likelihood, 1st and 2nd derivative

francogrex francogrex at mail.com
Thu Jan 11 17:21:25 CET 2007

```Hi guys again, it seems I haven't been doing the maximum likelihood
estimation correctly. I quote below, can someone explain to me please what
does it mean that the 2nd and 3rd derivatives of the function equals zero
and how to compute that in R.

"We have our initial estimated, subjective parameters for the gamma mixture
and we have our likelihood that is the mixture of negative binomials
representing the distribution of actual observed values. We 'pool' these
distributions and determine which expression for the parameters would be
most likely to produce the sample of observed negative binomial counts
(determine the MLE). This maximisation involves a search in five-dimensional
parameter space {θ: α1,α2, β1, β2, P} for the vector that maximises the
likelihood as evidenced by the first and second derivatives of the function
being zero. The likelihood is L(θ) = Πij {P f (Nij; α1, β1, Eij) + (1-P) f
(Nij; α2, β2, Eij)} This involves millions of calculations. The
computational procedures required for these calculations are based on the
Newton-Raphson method. This is an old calculus-based technique that was
devised to find the roots of an equation (e.g. the values of the independent
variable (e.g. x) for which the value of the function (e.g. f(x)) equals
zero."

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