The characteristic function of the discrete Cauchy distribution

A new family of integer-valued Cauchy-type distributions is introduced, the {\it Cauchy-Cacoullos family}. The characteristic function is evaluated, showing some interesting distributional properties, similar to the ordinary (continuous) Cauchy scale family. The results are extendable to discrete Student-type distributions with odd degrees of freedom. Keywords: Fourier series; discrete Student distribution; Cauchy-Cacoullos family.Comment: Dedicated to Professor Theo Cacoullos (13 pages, 1 Figure

Self-Inverse and Exchangeable Random Variables

A random variable Z will be called self-inverse if it has the same distribution as its reciprocal 1/Z. It is shown that if Z is defined as a ratio, X/Y, of two rv's X and Y (with Pr[X=0]=Pr[Y=0]=0), then Z is self-inverse if and only if X and Y are (or can be chosen to be) exchangeable. In general, however, there may not exist iid X and Y in the ratio representation of Z.Comment: Statistics and Probability Letters (to appear, 6 pages

Linear Estimation of Location and Scale Parameters Using Partial Maxima

Consider an i.i.d. sample X^*_1,X^*_2,...,X^*_n from a location-scale family, and assume that the only available observations consist of the partial maxima (or minima)sequence, X^*_{1:1},X^*_{2:2},...,X^*_{n:n}, where X^*_{j:j}=max{X^*_1,...,X^*_j}. This kind of truncation appears in several circumstances, including best performances in athletics events. In the case of partial maxima, the form of the BLUEs (best linear unbiased estimators) is quite similar to the form of the well-known Lloyd's (1952, Least-squares estimation of location and scale parameters using order statistics, Biometrika, vol. 39, pp. 88-95) BLUEs, based on (the sufficient sample of) order statistics, but, in contrast to the classical case, their consistency is no longer obvious. The present paper is mainly concerned with the scale parameter, showing that the variance of the partial maxima BLUE is at most of order O(1/log n), for a wide class of distributions.Comment: This article is devoted to the memory of my six-years-old, little daughter, Dionyssia, who leaved us on August 25, 2010, at Cephalonia isl. (26 pages, to appear in Metrika

On the inversion of the Laplace transform (In Memory of Dimitris Gatzouras)

The Laplace transform is a useful and powerful analytic tool with applications to several areas of applied mathematics, including differential equations, probability and statistics. Similarly to the inversion of the Fourier transform, inversion formulae for the Laplace transform are of central importance; such formulae are old and well-known (Fourier-Mellin or Bromwich integral, Post-Widder inversion). The present work is motivated from an elementary statistical problem, namely, the unbiased estimation of a parametric function of the scale in the basic model of a random sample from exponential distribution. The form of the uniformly minimum variance unbiased estimator of a parametric function $h(\lambda)$, as well as its variance, are obtained as series in Laguerre polynomials and the corresponding Fourier coefficients, and a particular application of this result yields a novel inversion formula for the Laplace transform. MSC: Primary 44A10, 62F10. Key words and phrases: Exponential Distribution, Unbiased Estimation; Fourier-Laguerre Series; Inverse Laplace Transform; Laguerre Polynomials.Comment: 14 page