PPS.fit: Fitting the Pareto Positive Stable (PPS) distribution
In ParetoPosStable: Computing, Fitting and Validating the PPS Distribution

Description Usage Arguments Details Value References See Also Examples

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

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

`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.

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".

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).

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.

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

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)

Data:      turkey$Pop2000 


Parameter estimates:
lambda   sigma      nu  
0.8905   20150   1.138  


Log-likelihood:
[1]  -3363

Sample size:
[1]  280


Estimation method:      MLE 
   lambda      sigma         nu  
0.8904793   20149.88   1.138317  

Standard errors:
      lambda    sigma         nu
1 0.04884881 211.2101 0.05228772

'log Lik.' -3362.861 (df=3)

Goodness of fit tests:
         Statistic.value   p.value
KS test       0.03260172 0.6336634
AD test       0.25692630 0.7524752
PPS test      7.47043879 0.8613861