ParetoPosStable (version 1.1)

PPS.fit: Fitting the Pareto Positive Stable (PPS) distribution

Description

PPS.fit() returns the fit of a PPS distribution to real data, allowing the scale parameter to be held fixed if desired.

Usage

PPS.fit(x, estim.method = "MLE", sigma = NULL, start, Pareto = FALSE, ...)

Arguments

x

a vector of observations

estim.method

the method of parameter estimation. It may be "MLE", "iMLE", "OLS", or "LMOM".

sigma

the value of the scale parameter, if it is known; if the value is NULL, the parameter is estimated.

start

a list with the initial values of the parameters for some of the estimation methods.

Pareto

a logical argument to constrain the PPS fit to a Pareto fit when the value is TRUE.

...

other arguments.

Value

A PPSfit Object, a list with

• estimateparameter estimates.
• loglikthe log-likelihood value.
• nthe number of observations.
• obsthe observations.
• obsNamethe name of the variable with the observations.
• estim.methodthe method of parameter estimation.

When this last value is "LMOM" the function also returns details about the convergence of the numerical method involved (convergence value).

Details

The maximum likelihood method implemented by the direct optimization of the log-likelihood is given by estim.method = "MLE". The numerical algorithm to search the optimum is the ``Nelder-Mead'' method implemented in the optim function, considering as initial values those given in the start argument or, if it is missing, those provided by the OLS method.

A different approximation of the maximum likelihood estimates is given by estim method = "iMLE"; it is an iterative methodology where optimize() function provides the optimum scale parameter value, while the uniroot() function solve normal equations for that given scale parameter.

The regression estimates ("OLS") searchs an optimum scale value (in a OLS criterion) by the optimize() function. Then the rest of the parameters are estimated also by OLS, as appears in Sarabia and Prieto (2009).

In the L-moments method ("LMOM") estimates are obtained searching parameters that equal the first three sample and theoretical L-moments by means of the ``Nelder-Mead'' algorithm implemented in optim(); the initial values are given in the start argument or, if it is missing, provided by the "iMLE".

References

Sarabia, J.M and Prieto, F. (2009). The Pareto-positive stable distribution: A new descriptive model for city size data, Physica A: Statistical Mechanics and its Applications, 388(19), 4179-4191. Hosking, J. R. M. (1990). L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society, Series B, 52, 105-124.

See Also

coef.PPSfit, print.PPSfit, plot.PPSfit, GoF

Examples

data(turkey)
fit <- PPS.fit(turkey$Pop2000)
print(fit)
coef(fit)
se(fit, k = 100, parallel = FALSE)
logLik(fit)
par(mfrow=c(2,2))
plot(fit)
GoF(fit, k = 100, parallel = FALSE)