integrate_function

Applies `safe_integrate` to a given integrand over a vector of x values.

Provides various functions for parameter estimation of one-dimensional
stable distributions and their mixtures. It implements a diverse set of
estimation methods, including quantile-based approaches, regression methods
based on the empirical characteristic function (empirical, kernel, and
recursive), and maximum likelihood estimation. For mixture models, it provides
stochastic expectation–maximization (SEM) algorithms and Bayesian estimation
methods using sampling and importance sampling to overcome the long burn-in
period of Markov Chain Monte Carlo (MCMC) strategies. The package also includes
tools and statistical tests for analyzing whether a dataset follows a stable
distribution. Some of the implemented methods are described in
Hajjaji, O., Manou-Abi, S. M., and Slaoui, Y. (2024) <doi:10.1080/02664763.2024.2434627>.

Solym Manou-Abi

MixStable

Parameter Estimation for Stable Distributions and Their Mixtures

Adam Najib

Yousri Slaoui

integrate_function function

<dl><dt>f</dt>
<dd>Function. The integrand function to be evaluated.</dd>
<dt>x</dt>
<dd>Numeric vector. Observations at which the integral is computed.</dd>
<dt>alpha</dt>
<dd>Numeric scalar. Stability parameter of the stable distribution
(0 &lt; \(\alpha \le 2\)), controlling tail thickness.</dd>
<dt>beta</dt>
<dd>Numeric scalar. Skewness parameter of the stable distribution
(\(-1 \le \beta \le 1\)).</dd>
<dt>delta</dt>
<dd>Numeric scalar. Location parameter of the stable distribution.</dd>
<dt>omega</dt>
<dd>Numeric scalar. Scale parameter of the stable distribution
(omega &gt; 0).</dd>
<dt>upper</dt>
<dd>Numeric scalar. Upper limit of integration (default = 50).</dd>
<dt>eps</dt>
<dd>Numeric scalar. Tolerance for numerical integration (default = 1e-8).</dd></dl>

Arguments

Robust integration helper function — integrate_function

<dl>

<dt>f</dt>
<dd>Function. The integrand function to be evaluated.</dd>


<dt>x</dt>
<dd>Numeric vector. Observations at which the integral is computed.</dd>


<dt>alpha</dt>
<dd>Numeric scalar. Stability parameter of the stable distribution
(0 &lt; \(\alpha \le 2\)), controlling tail thickness.</dd>


<dt>beta</dt>
<dd>Numeric scalar. Skewness parameter of the stable distribution
(\(-1 \le \beta \le 1\)).</dd>


<dt>delta</dt>
<dd>Numeric scalar. Location parameter of the stable distribution.</dd>


<dt>omega</dt>
<dd>Numeric scalar. Scale parameter of the stable distribution
(omega &gt; 0).</dd>


<dt>upper</dt>
<dd>Numeric scalar. Upper limit of integration (default = 50).</dd>


<dt>eps</dt>
<dd>Numeric scalar. Tolerance for numerical integration (default = 1e-8).</dd>

</dl>

integrate_function: Robust integration helper function

Description

Usage

Value

Arguments