Parameter-estimation methods for symmetric stable distributions: Application to small samples of spatial fluctuations of rainfall

The estimation of parameters of stable distributions is hampered by the lack of closed-form expressions for the density and distribution functions. Moreover, the accuracy of estimation methods is affected noticeably by small samples that fail to provide sufficiently descriptive information about the empirical distribution. This study reviewed the methods most commonly used and examined the possibility of improving the estimation of parameters of symmetric stable distributions using mathematical properties derived from the absolute value of symmetric stable variables. On one hand, considering the absolute value might be beneficial to the adjustment of small samples because insufficient data in one tail of the empirical distribution could be complemented with data from the other. Thus, the performance of techniques that rely on distribution quantiles might be improved. On the other hand, the evaluation of the distribution function at fractional absolute moments was found to be a good estimator of the characteristic exponent of stable distributions. This technique is faster than regression-type methods and returns remarkably similar results. The performances of these parameter-estimation methods were evaluated by analyzing the results of an application to small samples from spatial fluctuations of rainfall in which extreme variations are frequently found.