# Bias-Corrected Maximum Likelihood Estimate(s) `coxsnell.bc` calculates the bias-corrected maximum likelihood estimate(s) using the bias formula introduced by Cox and Snell (1968). ```r coxsnell.bc(density, logdensity, n, parms, mle, lower = "-Inf", upper = "Inf", ...) ``` ## Arguments - `density`: An expression with the probability density function. - `logdensity`: An expression with the logarithm of the probability density function. - `n`: A numeric scalar with the sample size. - `parms`: A character vector with the parameter name(s) specified in the density and logdensity expressions. - `mle`: A numeric vector with the parameter estimate(s). - `lower`: The lower integration limit (lower = ``-Inf'' is the default). - `upper`: The upper integration limit (upper = ``Inf'' is the default). - `...`: Additional arguments passed to `integrate` function. ## Returns `coxsnell.bc` returns a list with five components (i) mle : the inputted maximum likelihood estimate(s), (ii) varcov : the expected variance-covariance evaluated at the inputted mle argument, (iii) mle.bc : the bias-corrected maximum likelihood estimate(s), (iv) varcov.bc : the expected variance-covariance evaluated at the bias-corrected maximum likelihood estimate(s) and (v) bias : the bias estimate(s). If the numerical integration fails and/or the expected information is singular an error message is returned. ## Details The first, second and third-order partial log-density derivatives are analytically calculated via `D` function. The expected values of the partial log-density derivatives are calculated via `integrate` function. ## Examples ```r {library(mle.tools); library(fitdistrplus); set.seed(1)}; ## Normal distribution pdf <- quote(1 / (sqrt(2 * pi) * sigma) * exp(-0.5 / sigma ^ 2 * (x - mu) ^ 2)) lpdf <- quote(- log(sigma) - 0.5 / sigma ^ 2 * (x - mu) ^ 2) x <- rnorm(n = 100, mean = 0.0, sd = 1.0) {mu.hat <- mean(x); sigma.hat = sqrt((length(x) - 1) * var(x) / length(x))} coxsnell.bc(density = pdf, logdensity = lpdf, n = length(x), parms = c("mu", "sigma"), mle = c(mu.hat, sigma.hat), lower = '-Inf', upper = 'Inf') ################################################################################ ## Weibull distribution pdf <- quote(shape / scale ^ shape * x ^ (shape - 1) * exp(-(x / scale) ^ shape)) 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$vcov coxsnell.bc(density = pdf, logdensity = lpdf, n = length(x), parms = c("shape", "scale"), mle = fit$estimate, lower = 0) ################################################################################ ## Exponentiated Weibull distribution pdf <- quote(alpha * shape / scale ^ shape * x ^ (shape - 1) * exp(-(x / scale) ^ shape) * (1 - exp(-(x / scale) ^ shape)) ^ (alpha - 1)) lpdf <- quote(log(alpha) + log(shape) - shape * log(scale) + shape * log(x) - (x / scale) ^ shape + (alpha - 1) * log((1 - exp(-(x / scale) ^ shape)))) coxsnell.bc(density = pdf, logdensity = lpdf, n = 100, parms = c("shape", "scale", "alpha"), mle = c(1.5, 2.0, 1.0), lower = 0) ################################################################################ ## Exponetial distribution pdf <- quote(rate * exp(-rate * x)) lpdf <- quote(log(rate) - rate * x) x <- rexp(n = 100, rate = 0.5) fit <- fitdist(data = x, distr = 'exp') fit$vcov coxsnell.bc(density = pdf, logdensity = lpdf, n = length(x), parms = c("rate"), mle = fit$estimate, lower = 0) ################################################################################ ## Gamma distribution pdf <- quote(1 /(scale ^ shape * gamma(shape)) * x ^ (shape - 1) * exp(-x / scale)) 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$vcov coxsnell.bc(density = pdf, logdensity = lpdf, n = length(x), parms = c("shape", "scale"), mle = fit$estimate, lower = 0) ################################################################################ ## Beta distribution pdf <- quote(gamma(shape1 + shape2) / (gamma(shape1) * gamma(shape2)) * x ^ (shape1 - 1) * (1 - x) ^ (shape2 - 1)) 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$vcov coxsnell.bc(density = pdf, logdensity = lpdf, n = length(x), parms = c("shape1", "shape2"), mle = fit$estimate, lower = 0, upper = 1) ``` ## Author(s) Josmar Mazucheli jmazucheli@gmail.com ## See Also `deriv`, `D`, `expected.varcov`, `integrate`, `observed.varcov`.

coxsnell.bc function