A general APL program for maximum likelihood estimation or other function maximization using numerical differentiation
✍ Scribed by S. Olesen Larsen
- Publisher
- Elsevier Science
- Year
- 1982
- Weight
- 262 KB
- Volume
- 15
- Category
- Article
- ISSN
- 0010-468X
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✦ Synopsis
A description and a fisting are made of an APL program for general maximum likelihood estimation based on ad hoc programming of the log-likelihood function and on lqewton-Raphson's iterative procedure using numerical differentiation to obtain first-and second-order derivative.~. The program can also be used to maximize other functions having a well-defined and well-shaped maximum.
APL-program Function maximization Maximum likelihood Numerical differentiation 1. PROBLEM DESCRIPTION AND MATHE-MATICAL BACKGROUND
We consider a log-likelihood function L = L(X, A), where X stands for the observations and A for the unknown parameters at, a2,...,a,. We want to find the a-values which maximize L. We therefore differentiate L with regard to al, .... a, to obtain the vector F = (f~,...,fn) of first-order derivatives. Usually it is impossible directly to solve the equations f~ = 0, i = 1, ...,n and we therefore have to use an iterative procedure. According to the Newton-Raphson method one finds the matrix S = (s~j) of second-order derivatives of L with regard to the parameters. The procedure is now from one value A, of the vector A to construct a new one Ak+ I by calculating Ak+ j = A k -FS -i. Upon convergence we have the solution for which F = 0, and we can use -S-m as an estimate of the variance-covariance matrix of the estimates.
Instead of a log-likelihood function we could consider any other function which we might wish to maximize. An assumption not always valid even for likelihood functions is however that there exists a unique maximum within reach of the iterative procedure.