A Finite Sample Test for the Composite Hypothesis of Stable Law

The Family of stable distributions serves as an error model for a wide-and even widening -variety of applications. The importance of these distributions is due to the properties that sums of independent and identically distributed stable random variables are again stable and that these are the only possible limiting distributions of sums of independent and identically distributed random variables. There are many notable potential applications in economics, physics, insurance, risk management, and especially finance, to name just a few. Several types of parameter estimation procedures have been implemented and many empirical studies concerning financial time series have been conducted under the proposition that the stable distributions provide viable models of price behavior. The viability of stable models has remained in doubt because of the lack of quantitative test for fit - despite appealing stable-like characteristics exhibited by the data, e.g. estimates of shape parameter less than 2, pronounced skewness, etc. In this paper we propose and provide a (family of) finite sample test(s) for the composite hypothesis of stable distribution with all parameters unspecified. The test procedure is based on the property of stability under addition (sums of m random variables) as captured by the m ^th power of the empirical characteristic function, and a variant of the estimation procedure of Paulson and Delehanty. The test has attractive power properties against an array of alternatives including normal mixtures, the lognormal distributions, Student's t, and other competing model distributions. We provide percentage points for sample size n ≥ 50. Detailed studies of the behavior of the test under various conditions involving the skewness, scale and location parameters are provided. We briefly comment on several applications from financial markets