Statistical Methods for Stable Distribution Using The Empirical Characteristic Function

Stable distributions are a class of distributions allowing heavy tail and skewness. Most of stable distributions lack closed-form expression for their densities, so that estimating parameters is a challenging problem. When distributions have no closed-form density expression, their characteristic function becomes a useful alternative to define the unique distribution. We use the empirical characteristic function, i.e. the sample analog of the characteristic function, for estimation and goodness-of-fit tests for data. The existing fixed interval empirical characteristic function method works well when the alpha parameter is large but performs poorly for small values of alpha. This study offers a modification based on an adaptive grid to improve the estimation result for small alpha parameter without having a negative effect on the estimation of the other parameters. Goodness-of-fit tests based on the empirical characteristic function are also given and compared to classical tests based on the empirical cumulative distribution function.