Exact confidence sets and goodness-of-fit methods for stable distributions

Usual inference methods for stable distributions are typically based on limit distributions. But asymptotic approximations can easily be unreliable in such cases, for standard regularity conditions may not apply or may hold only weakly. This paper proposes finite-sample tests and confidence sets for tail thickness and asymmetry parameters (α and β) of stable distributions. The confidence sets are built by inverting exact goodness-of-fit tests for hypotheses which assign specific values to these parameters. We propose extensions of the Kolmogorov-Smirnov, Shapiro-Wilk and Filliben criteria, as well as the quantile-based statistics proposed by McCulloch (1986) in order to better capture tail behavior. The suggested criteria compare empirical goodness-of-fit or quantile-based measures with their hypothesized values. Since the distributions involved are quite complex and non-standard, the relevant hypothetical measures are approximated by simulation, and p-values are obtained using Monte Carlo (MC) test techniques. The properties of the proposed procedures are investigated by simulation. In contrast with conventional wisdom, we find reliable results with sample sizes as small as 25. The proposed methodology is applied to daily electricity price data in the US over the period 2001-2006. The results show clearly that heavy kurtosis and asymmetry are prevalent in these series.