Renext (version 3.1-4)

flomax: ML estimation of the Lomax distribution

Description

Fast Maximum Likelihood estimation of the Lomax distribution.

Usage

flomax(x,
       info.observed = TRUE,
       plot = FALSE,
       scaleData = TRUE,
       cov = TRUE)

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.

dloglik

Gradient of the log-likelihood at the optimum. Its two elements should normally be close to zero.

cov

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

info

The information matrix.

Arguments

x

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

info.observed

Should the observed information matrix be used or the expected one be used?

plot

Logical. If TRUE, a plot will be produced showing the derivative of the concentrated log-likelihood, function of the shape parameter.

scaleData

Logical. If TRUE observations in x (which are positive) are divided by their mean value. The results are in theory not affected by this transformation, but scaling the data could improve the estimation in some cases. The log-likelihood plots are shown using the scaled values so the returned estimate of the scale parameter is not the the abscissa of the maximum shown on the plot.

cov

Logical. If FALSE, a minimal estimation is performed with no covariance matrix or derivative returned. This can be useful when a large number of ML estimations are required, e.g. to sample from a likelihood ratio.

Author

Yves Deville

Details

The likelihood is concentrated with respect to the shape parameter. This function is increasing for small values of the scale parameter \(\beta\). For large \(\beta\), the derivative of the concentrated log-likelihood tends to zero, and its sign is that of \((1 - \textrm{CV}^2)\) where \(\textrm{CV}\) is the coefficient of variation, computed using \(n\) as denominator in the formula for the standard deviation.

The ML estimate does not exist when the sample has a coefficient of variation CV less than 1 and it may fail to be found when CV is greater than yet close to 1.

References

J. del Castillo and J. Daoudi (2009) "Estimation of the Generalized Pareto Distribution", Statist. Probab. Lett. 79(5), pp. 684-688.

D.E. Giles, H. Feng & R.T. Godwin (2013) "On the Bias of the Maximum Likelihood Estimator for the Two-Parameter Lomax Distribution" Comm. Statist. Theory Methods. Vol. 42, n. 11, pp. 1934-1950.

N. Johnson, S. Kotz and N. Balakrishnan Continuous Univariate Distributions vol. 1, Wiley 1994.

See Also

Lomax for the Lomax distribution.

Examples

## generate sample
set.seed(1234)
n <- 200
alpha <- 2 + rexp(1)
beta <- 1 + rexp(1)
x <- rlomax(n, scale = beta, shape = alpha)
res <- flomax(x, plot = TRUE)

## compare with a GPD with shape 'xi' and scale 'sigma'
xi <- 1 / alpha; sigma <- beta * xi  
res.evd <- evd::fpot(x, threshold = 0, model = "gpd")
xi.evd <- res.evd$estimate["shape"]
sigma.evd <- res.evd$estimate["scale"]
beta.evd <- sigma.evd / xi.evd 
alpha.evd <- 1 / xi.evd
cbind(Renext = res$estimate, evd = c(alpha = alpha.evd, beta = beta.evd))