Infinitely divisible statistical experiments
type: Item
instance of: scholarly article
title: Infinitely divisible statistical experiments
zbMATH Open document ID: 0566.62001
author: Hartmut Milbrodt
author: Helmut Strasser
author: Arnold Janssen
published in: Lecture Notes in Statistics
publication date: 1985-01-01T00:00:00Z
review text: Classical asymptotic statistical inference is based on the fact that for large sample sizes all regular statistical experiments can be approximated by a Gaussian shift experiment. However, the use of Gaussian shift approximations has to be questioned in most non-regular cases (such as a uniform shift experiment) and in regular cases when the parameter is close to the boundary of the parameter set (such as a Bernoulli experiment with small success probabilities). In order to find the correct approximations in these (and other) cases, the limiting behaviour of products of independent experiments is investigated. This Lecture Notes volume consists of four chapters. The first chapter, written by Milbrodt and Strasser, renders a careful elaboration of \textit{L. LeCam}'s theory of infinitely divisible experiments [Notes on asymptotic methods in statistical decision theory. Montreal (1974)]. This theory is quite similar to the theory of limit distributions for sums of independent random variables. So, e.g., compound Poisson experiments are used as accompanying experiments for triangular arrays of infinitesimal experiments. The theory yields a characterization of all possible limits of products of independent infinitesimal experiments and to the identification of certain types of limits. Furthermore, a ''Lévy-Khintchine representation'' is derived for the Hellinger transform of limit experiments. In the second part, written by Janssen, infinitely divisible experiments are characterized via the corresponding property of the distribution of the log-likelihood process. Each infinitely divisible experiment generates a semigroup admitting a Lévy-Khintchine representation. This approach leads to an interesting interpretation for the Gaussian and the Poisson part in the representation. In part three, written by Milbrodt, it is shown that two concepts of Poisson experiments, one introduced in part one, the second due to LeCam, are the same. In part four, written by Strasser, exper…
Mathematics Subject Classification ID: 62-02
Mathematics Subject Classification ID: 62A01
Mathematics Subject Classification ID: 62F99
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62B15
zbMATH DE Number: 3903727
zbMATH Keywords: asymptotic statistical inference
zbMATH Keywords: Gaussian shift approximations
zbMATH Keywords: non- regular cases
zbMATH Keywords: infinitely divisible experiments
zbMATH Keywords: compound Poisson experiments
zbMATH Keywords: triangular arrays of infinitesimal experiments
zbMATH Keywords: limits of products of independent infinitesimal experiments
zbMATH Keywords: Lévy-Khintchine representation
zbMATH Keywords: Hellinger transform of limit experiments
zbMATH Keywords: log-likelihood process
zbMATH Keywords: concepts of Poisson experiments
zbMATH Keywords: independent increments
zbMATH Keywords: binary subexperiments
zbMATH Keywords: discontinuous densities
zbMATH Keywords: weak limit
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3903727