Efficient Estimation for the Stable Laws

This paper is concerned with Fourier procedures in inference which admit arbitrarily high asymptotic efficiency. The problem of estimation for the stable laws is treated by two different approaches. The first involves FFT inversion of the characteristic function. A detailed discussion is given of truncation and discretization effects with reference to the special structure of the stable densities. Some further results are give also concerning a second approach based on the empirical characteristic function (ecf). Finally we sketch an application of this method to testing for independence, and also present a stationary version of the ecf.