This is the main interface for the maxLik package, and the function that performs Maximum Likelihood estimation. It is a wrapper for different optimizers returning an object of class "maxLik". Corresponding methods handle the likelihood-specific properties of the estimates, including standard errors.

```
maxLik(logLik, grad = NULL, hess = NULL, start, method,
constraints=NULL, ...)
```

logLik

log-likelihood function. Must have the parameter vector as the first argument. Must return either a single log-likelihood value, or a numeric vector where each component is log-likelihood of the corresponding individual observation.

grad

gradient of log-likelihood. Must have the parameter
vector as the first argument. Must return either a single gradient
vector with length equal to the number of parameters, or a matrix
where each row is the gradient vector of the corresponding individual
observation. If `NULL`

, numeric gradient will be used.

hess

hessian of log-likelihood. Must have the parameter
vector as the first argument. Must return a square matrix. If
`NULL`

, numeric Hessian will be used.

start

numeric vector, initial value of parameters. If it has names, these will also be used for naming the results.

method

maximisation method, currently either "NR" (for Newton-Raphson), "BFGS" (for Broyden-Fletcher-Goldfarb-Shanno), "BFGSR" (for the BFGS algorithm implemented in R), "BHHH" (for Berndt-Hall-Hall-Hausman), "SANN" (for Simulated ANNealing), "CG" (for Conjugate Gradients), or "NM" (for Nelder-Mead). Lower-case letters (such as "nr" for Newton-Raphson) are allowed. If missing, a suitable method is selected automatically.

constraints

either `NULL`

for unconstrained maximization
or a list, specifying the constraints. See `maxBFGS`

.

object of class 'maxLik' which inherits from class 'maxim'.
The structure is identical to that of the class “maxim”
(see `maxNR`

) but the methods differ.

The constrained maximum likelihood estimation should be considered experimental. In particular, the variance-covariance matrix is not corrected for constrained parameter space.

`maxLik`

supports constrained optimization in the sense that
constraints are passed further to the underlying optimization
routines, and suitable default method is selected. However, no
attempt is made to correct the resulting variance-covariance matrix.
Hence the inference may be wrong. A corresponding warning is issued
by the summary method.

`maxNR`

, `nlm`

and `optim`

for different non-linear optimisation routines, see
`maxBFGS`

for the constrained maximization examples.

# NOT RUN { ## Estimate the parameter of exponential distribution t <- rexp(100, 2) loglik <- function(theta) log(theta) - theta*t gradlik <- function(theta) 1/theta - t hesslik <- function(theta) -100/theta^2 ## Estimate with numeric gradient and hessian a <- maxLik(loglik, start=1, control=list(printLevel=2)) summary( a ) ## Estimate with analytic gradient and hessian a <- maxLik(loglik, gradlik, hesslik, start=1) summary( a ) ## ## Next, we give an example with vector argument: Estimate the mean and ## variance of a random normal sample by maximum likelihood ## loglik <- function(param) { mu <- param[1] sigma <- param[2] ll <- -0.5*N*log(2*pi) - N*log(sigma) - sum(0.5*(x - mu)^2/sigma^2) ll } x <- rnorm(100, 1, 2) # use mean=1, stdd=2 N <- length(x) res <- maxLik(loglik, start=c(0,1)) # use 'wrong' start values summary( res ) ## ## The previous example showing parameter names and fixed values ## resFix <- maxLik(loglik, start=c(mu=0, sigma=1), fixed="sigma") summary(resFix) # 'sigma' is exactly 1.000 now. # }