ML estimation of the Gamma distribution

Fast Maximum Likelihood estimation of the Gamma distribution.

Usage

fgamma(x, check.loglik = FALSE)

Arguments

x

Sample vector to be fitted. Should contain only positive non-NA values.

check.loglik

If TRUE, the log-likelihood is recomputed using dgamma function with log = TRUE. The result is returned as a list element.

Details

The likelihood is concentrated with respect to the scale parameter. The concentrated log-likelihood is a strictly concave function of the shape parameter which can easily maximised numerically.

Value

A list with the following elements

estimate

Parameter ML estimates.

sd

Vector of (asymptotic) standard deviations for the estimates.

loglik

The maximised log-likelihood.

check.loglik

The checked log-likelihood.

cov

The (asymptotic) covariance matrix computed from theoretical or observed information matrix.

info

The information matrix.

Author

Yves Deville

Note

The distribution is fitted by using the scale parameter rather than rate (inverse of scale).

Examples

set.seed(9876)
alpha <- 0.06
beta <- rexp(1)
n <- 30
x <- rgamma(n, shape = alpha, scale = beta)
fit <- fgamma(x, check.loglik = TRUE)

## compare with MASS results
if (require(MASS)) {
   fit.MASS <- fitdistr(x, densfun = "gamma")
   rate <- 1 / fit$estimate["scale"]
   est <- c(fit$estimate, rate = rate)
   der <- rate * rate ## derivative of rate w.r.t scale
   sdest <- c(fit$sd, rate = der * fit$sd["scale"])
   tab <- rbind(sprintf(" %10.8f ", est),
                sprintf("(%10.8f)", sdest))
   colnames(tab) <- c("shape", "scale", "rate")
   rownames(tab) <- c("est", "sd")
   noquote(tab)
}
#> Loading required package: MASS
#>     shape        scale        rate         
#> est  0.05534680   0.05686669   17.58498593 
#> sd  (0.01037116) (0.04540000) (14.03912135)