# Methods for parameter estimation of skew stable distributions. A set of functions are provided that perform the parameter estimation of skew stable distributions with different methods. ```r stable_fit_init(rnd, parametrization = 0L) stable_fit_koutrouvelis(rnd, pars_init = as.numeric(c()), parametrization = 0L) ``` ## Arguments - `rnd`: Random sample - `parametrization`: Parametrization used for the skew stable distribution, as defined by JP Nolan (1997). By default, parametrization = 0. - `pars_init`: Vector with an initial estimation of the parameters. `pars_init = c(alpha, beta, sigma, mu)`, where * alpha: shape / stability parameter, with 0 \< alpha \<= 2. * beta: skewness parameter, with -1 \<= beta \<= 1. * sigma: scale parameter, with 0 \< sigma. * mu: location parameter, with mu real. ## Returns A numeric vector. ## Details * `stable_fit_init()` uses McCulloch's method of quantiles [3]. This is usually a good initialization for the rest of the methods. * `stable_fit_koutrouvelis()` implements Koutrouvellis' method based on the characteristic function [4]. * `stable_fit_mle()` implements a Maximum likelihood estimation. * `stable_fit_mle2()` implements a modified maximum likelihood estimation as described in [1]. ## Examples ```r # Set alpha, beta, sigma and mu stable parameters in a vector pars <- c(1.5, 0.9, 1, 0) # Generate random values set.seed(1) rnd <- stable_rnd(100, pars) head(rnd) # Estimate the parameters of the skew stable distribution given # the generated sample: # Using the McCulloch's estimator: pars_init <- stable_fit_init(rnd) # Using the Koutrouvelis' estimator, with McCulloch estimation # as a starting point: pars_est_K <- stable_fit_koutrouvelis(rnd, pars_init) # Using maximum likelihood estimator: pars_est_ML <- stable_fit_mle(rnd, pars_est_K) # Using modified maximum likelihood estimator (see [1]): pars_est_ML2 <- stable_fit_mle2d(rnd, pars_est_K) ``` ## References * [1] Royuela-del-Val J, Simmross-Wattenberg F, Alberola López C (2017). libstable: Fast, Parallel and High-Precision Computation of alpha-stable Distributions in R, C/C++ and MATLAB. Journal of Statistical Software, 78(1), 1-25. doi:10.18637/jss.v078.i01 * [2] Chambers JM, Mallows CL, Stuck BW (1976). A Method for Simulating Stable Random Variables. Journal of the American Statistical Association, 71(354), 340-344. doi:10.1080/01621459.1976.10480344. * [3] McCulloch JH (1986). Simple Consistent Estimators of Stable Distribution Parameters. Communications in Statistics - Simulation and Computation, 15(4), 1109-1136. doi:10.1080/03610918608812563. * [4] Koutrouvelis IA (1981). An Iterative Procedure for the Estimation of the Parameters of Stable Laws. Communications in Statistics - Simulation and Computation, 10(1), 17-28. doi:10.1080/03610918108812189. * [5] Nolan JP (1997). Numerical Calculation of Stable Densities and Distribution Functions. Stochastic Models, 13(4) 759-774. doi:10.1080/15326349708807450. ## Author(s) Javier Royuela del Val, Federico Simmross Wattenberg and Carlos Alberola López Maintainer: Javier Royuela del Val [jroyval@lpi.tel.uva.es](mailto:jroyval@lpi.tel.uva.es)

stable_fit function