# observed.varcov: Observed Fisher Information In mle.tools: Expected/Observed Fisher Information and Bias-Corrected Maximum Likelihood Estimate(s)

## Description

`observed.varcov` calculates the inverse of the observed Fisher Information. Analytical second-order partial log-density derivatives are used in the calculations.

## Usage

 `1` ```observed.varcov(logdensity, X, parms, mle) ```

## Arguments

 `logdensity` An expression with the log of the probability density function. `X` A numeric vector with the observations. `parms` A character vector with the parameter name(s) specified in the logdensity expression. `mle` A numeric vector with the parameter estimate(s).

## Details

The second-order partial log-density derivatives are calculated via `D` function.

## Value

`observed.varcov` returns a list with two components (i) mle: the inputted maximum likelihood estimate(s) and (ii) varcov: the observed variance-covariance evaluated at the inputted mle argument.

If the observed information is singular an error message is returned.

## Author(s)

Josmar Mazucheli jmazucheli@gmail.com

`deriv`, `D`, `expected.varcov`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59``` ```{library(mle.tools); library(fitdistrplus); set.seed(1)}; ##Normal distribution lpdf <- quote(-log(sigma) - 0.5 / sigma ^ 2 * (x - mu) ^ 2) x <- rnorm(n = 100, mean = 0.0, sd = 1.0) observed.varcov(logdensity = lpdf, X = x, parms = c("mu", "sigma"), mle = c(mean(x), sd(x))) ################################################################################ ## Weibull distribution lpdf <- quote(log(shape) - shape * log(scale) + shape * log(x) - (x / scale) ^ shape) x <- rweibull(n = 100, shape = 1.5, scale = 2.0) fit <- fitdist(data = x, distr = 'weibull') fit\$vcov observed.varcov(logdensity = lpdf, X = x, parms = c("shape", "scale"), mle = fit\$estimate) ################################################################################ ## Exponetial distribution lpdf <- quote(log(rate) - rate * x) x <- rexp(n = 100, rate = 0.5) fit <- fitdist(data = x, distr = 'exp') fit\$vcov observed.varcov(logdensity = lpdf, X = x, parms = c("rate"), mle = fit\$estimate) ################################################################################ ## Gamma distribution lpdf <- quote(-shape * log(scale) - lgamma(shape) + shape * log(x) - x / scale) x <- rgamma(n = 100, shape = 1.5, scale = 2.0) fit <- fitdist(data = x, distr = 'gamma', start = list(shape = 1.5, scale = 2.0)) fit\$vcov observed.varcov(logdensity = lpdf, X = x, parms = c("shape", "scale"), mle = fit\$estimate) ################################################################################ ## Beta distribution lpdf <- quote(lgamma(shape1 + shape2) - lgamma(shape1) - lgamma(shape2) + shape1 * log(x) + shape2 * log(1 - x)) x <- rbeta(n = 100, shape1 = 2.0, shape2 = 2.0) fit <- fitdist(data = x, distr = 'beta', start = list(shape1 = 2.0, shape2 = 2.0)) fit\$vcov observed.varcov(logdensity = lpdf, X = x, parms = c("shape1", "shape2"), mle = fit\$estimate) ```