libstable4u: Fast and accurate evaluation, random number generation and parameter estimation of skew stable distributions.

Description

libstable4u provides functions to work with skew stable distributions in a fast and accurate way [1]. It performs:

Details

• Fast and accurate evaluation of the probability density function (PDF) and cumulative density function (CDF).

• Fast and accurate evaluation of the quantile function (inverse CDF).

• Random numbers generation [2].

• Skew stable parameter estimation with:

  • McCulloch's method of quantiles [3].

  • Koutrouvellis' method based on the characteristic function [4].

  • Maximum likelihood estimation.

  • Modified maximum likelihood estimation as described in [1]. *The evaluation of the PDF and CDF is based on the formulas provided by John P Nolan in [5].

Author(s)

Javier Royuela del Val, Federico Simmross Wattenberg and Carlos Alberola López;

Maintainer: Javier Royuela del Val jroyval@lpi.tel.uva.es

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

Examples

# Set alpha, beta, sigma and mu stable parameters in a vector
pars <- c(1.5, 0.9, 1, 0)

# Generate an abscissas axis and probabilities vector
x <- seq(-5, 10, 0.05)
p <- seq(0.01, 0.99, 0.01)

# Calculate pdf, cdf and quantiles
pdf <- stable_pdf(x, pars)
cdf <- stable_cdf(x, pars)
xq  <- stable_q(p, pars)

# 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)

Methods for parameter estimation of skew stable distributions.

Description

A set of functions are provided that perform the parameter estimation of skew stable distributions with different methods.

Usage

stable_fit_init(rnd, parametrization = 0L)

stable_fit_koutrouvelis(rnd, pars_init = as.numeric(c()), parametrization = 0L)

Arguments

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].

Value

A numeric vector.

Author(s)

Javier Royuela del Val, Federico Simmross Wattenberg and Carlos Alberola López

Maintainer: Javier Royuela del Val jroyval@lpi.tel.uva.es

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.

Examples

# 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)

PDF and CDF of a skew stable distribution.

Description

Evaluate the PDF or the CDF of the skew stable distribution with parameters pars = c(alpha, beta, sigma, mu) at the points given in x.

parametrization argument specifies the parametrization used for the distribution as described by JP Nolan (1997). The default value is parametrization = 0.

tol sets the relative error tolerance (precision) to tol. The default value is tol = 1e-12.

Usage

stable_pdf(x, pars, parametrization = 0L, tol = 1e-12)

Arguments

Value

A numeric vector.

Author(s)

Javier Royuela del Val, Federico Simmross Wattenberg and Carlos Alberola López

Maintainer: Javier Royuela del Val jroyval@lpi.tel.uva.es

References

Nolan JP (1997). Numerical Calculation of Stable Densities and Distribution Functions. Stochastic Models, 13(4) 759-774.

Examples

pars <- c(1.5, 0.9, 1, 0)
x <- seq(-5, 10, 0.001)

pdf <- stable_pdf(x, pars)
cdf <- stable_cdf(x, pars)

plot(x, pdf, type = "l")

Quantile function of skew stable distributions

Description

Evaluate the quantile function (CDF^-1) of the skew stable distribution with parameters pars = c(alpha, beta, sigma, mu) at the points given in p.

parametrization argument specifies the parametrization used for the distribution as described by JP Nolan (1997). The default value is parametrization = 0.

tol sets the relative error tolerance (precission) to tol. The default value is tol = 1e-12.

Usage

stable_q(p, pars, parametrization = 0L, tol = 1e-12)

Arguments

Value

A numeric vector.

Author(s)

Javier Royuela del Val, Federico Simmross Wattenberg and Carlos Alberola López

Maintainer: Javier Royuela del Val jroyval@lpi.tel.uva.es

Skew stable distribution random sample generation.

Description

stable_rnd(N, pars) generates N random samples of a skew stable distribuiton with parameters pars = c(alpha, beta, sigma, mu) using the Chambers, Mallows, and Stuck (1976) method.

Usage

stable_rnd(N, pars, parametrization = 0L)

Arguments

Value

A numeric vector.

Author(s)

Javier Royuela del Val, Federico Simmross Wattenberg and Carlos Alberola López

Maintainer: Javier Royuela del Val jroyval@lpi.tel.uva.es

References

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.

Examples

N <- 1000
pars <- c(1.25, 0.95, 1.0, 0.0)
rnd <- stable_rnd(N, pars)

hist(rnd)