Infinitely divisible statistical experiments
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, experiments with independent increments are studied. Conditions are given under which a sequence of experiments has a limit experiment with independent increments. In this case the asymptotic behaviour of the full experiments is completely determined by its binary subexperiments. The concept of experiments with independent increments is particularly useful for the case of statistical problems with discontinuous densities. For the general case of independent not necessarily identically distributed observations, conditions are given which imply that every weak limit of the corresponding product experiments is a compound Poisson experiment (which has independent increments). The conditions are ''almost'' satisfied in the case considered by \textit{I. A. Ibragimov} and \textit{R. Z. Khas'minskij}, Statistical estimation. Asymptotic theory. (1981; Zbl 0467.62026). The reading is facilitated by a list of symbols. Errors from parts 1 to 3 are corrected on the last page.
triangular arrays of infinitesimal experiments
limits of products of independent infinitesimal experiments
Hellinger transform of limit experiments
scientific article; zbMATH DE number 3903727