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

 `1` ```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.

## 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"`.

## 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).

## 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.

`coef.PPSfit`, `print.PPSfit`, `plot.PPSfit`, `GoF`
 ```1 2 3 4 5 6 7 8 9``` ```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) ```