R: Maximum Likelihood Estimation

mle {stats4} | R Documentation

## Maximum Likelihood Estimation

### Description

Estimate parameters by the method of maximum likelihood.

### Usage

    mle(minuslogl, start = formals(minuslogl), method = "BFGS",
        fixed = list(), nobs, ...)

### Arguments

`minuslogl` | Function to calculate negative log-likelihood.
`start` | Named list. Initial values for optimizer.
`method` | Optimization method to use. See `optim`.
`fixed` | Named list. Parameter values to keep fixed during optimization.
`nobs` | optional integer: the number of observations, to be used for e.g. computing `BIC`.
`...` | Further arguments to pass to `optim`.

### Details

The `optim` optimizer is used to find the minimum of the negative log-likelihood. An approximate covariance matrix for the parameters is obtained by inverting the Hessian matrix at the optimum.

### Value

An object of class `mle-class`.

### Note

Be careful to note that the argument is -log L (not -2 log L). It is for the user to ensure that the likelihood is correct, and that asymptotic likelihood inference is valid.

### See Also

`mle-class`

### Examples

    ## Avoid printing to unwarranted accuracy
    od <- options(digits = 5)
    x <- 0:10
    y <- c(26, 17, 13, 12, 20, 5, 9, 8, 5, 4, 8)

    ## Easy one-dimensional MLE:
    nLL <- function(lambda) -sum(stats::dpois(y, lambda, log=TRUE))
    fit0 <- mle(nLL, start = list(lambda = 5), nobs = NROW(y))
    # For 1D, this is preferable:
    fit1 <- mle(nLL, start = list(lambda = 5), nobs = NROW(y),
                method = "Brent", lower = 1, upper = 20)
    stopifnot(nobs(fit0) == length(y))

    ## This needs a constrained parameter space: most methods will accept NA
    ll <- function(ymax = 15, xhalf = 6) {
        if(ymax > 0 && xhalf > 0)
          -sum(stats::dpois(y, lambda = ymax/(1+x/xhalf), log = TRUE))
        else NA
    }
    (fit <- mle(ll, nobs = length(y)))
    mle(ll, fixed = list(xhalf = 6))
    ## alternative using bounds on optimization
    ll2 <- function(ymax = 15, xhalf = 6)
        -sum(stats::dpois(y, lambda=ymax/(1+x/xhalf), log = TRUE))
    mle(ll2, method = "L-BFGS-B", lower = rep(0, 2))

    AIC(fit)
    BIC(fit)

    summary(fit)
    logLik(fit)
    vcov(fit)
    plot(profile(fit), absVal = FALSE)
    confint(fit)

    ## Use bounded optimization
    ## The lower bounds are really > 0,
    ## but we use >=0 to stress-test profiling
    (fit2 <- mle(ll, method = "L-BFGS-B", lower = c(0, 0)))
    plot(profile(fit2), absVal=FALSE)

    ## a better parametrization:
    ll3 <- function(lymax = log(15), lxhalf = log(6))
        -sum(stats::dpois(y, lambda=exp(lymax)/(1+x/exp(lxhalf)), log=TRUE))
    (fit3 <- mle(ll3))
    plot(profile(fit3), absVal = FALSE)
    exp(confint(fit3))

    options(od)

[Package stats4 version 2.15.1 Index]