ESTIMATION FOR THE POSITIVE STABLE LAWS, I

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Abstract
Summary: Estimation of scale and index parameters of positive stable laws is considered. Maximum likelihood estimation is known to be efficient, but very difficult to compute, while methods based on the sample characteristic function are computationally easy, but have uncertain efficiency properties.In this paper an estimation method is presented which is reasonably easy to compute, and which has good efficiency properties, at least when the index α (0, 0.5). The method is based on an expression for the characteristic function of the logarithm of a positive stable random variable, and is derived by relating the stable estimation problem to that of location/scale estimation in extreme‐value‐distribution families, for which efficient methods are known.The proposed method has efficiency which →1 as α→,but on the other hand, efficiencies deteriorate after α >0.5, and in fact appear to →0 as α+ 1.