R help - Jan 2007 - maximum likelihood, 1st and 2nd derivative

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 {?: ?1,?2, ?1, ?2, 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(?) = ?ij {P f (Nij; ?1, ?1, Eij) + (1-P) f
(Nij; ?2, ?2, 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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francogrex <francogrex <at> mail.com> writes:
> [SNIP]
> This maximisation involves a search in five-dimensional
> parameter space {?: ?1,?2, ?1, ?2, 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(?) = ?ij {P f (Nij; ?1, ?1, Eij) + (1-P) f
> (Nij; ?2, ?2, 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."
  I'm sure someone will correct me if I'm wrong, but this seems wrong
to me.  We only want the first derivatives to be zero.  It wouldn't be
impossible for second and higher derivatives to be zero, but it would
be somewhat pathological.
  While optimization will be faster and more stable if you can compute
the first derivatives (the _gradient_) analytically (and R has the
D() and deriv() functions for doing so), R will compute derivatives
numerically by finite differences if you don't.  See ?optim.

  Blatant plug:  www.zoo.ufl.edu/bolker/emdbook/chap7A.pdf  pp. 3-4
might be helpful too.

  Ben Bolker