Data-driven smooth tests when the hypothesis Is composite

In recent years several authors have recommended smooth tests for testing goodness of fit. However, the number of components in the smooth test statistic should be chosen well; otherwise, considerable loss of power may occur. Schwarz's selection rule provides one such good choice. Earlier results on simple null hypotheses are extended here to composite hypotheses, which tend to be of more practical interest. For general composite hypotheses, consistency of the data-driven smooth tests holds at essentially any alternative. Monte Carlo experiments on testing exponentiality and normality show that the data-driven version of Neyman's test compares well to other, even specialized, tests over a wide range of alternatives

Vanishing shortcoming and asymptotic relative efficiency

The shortcoming of a test is the difference between the maximal attainable power and the power of the test under consideration. Vanishing shortcoming, when the number of observations tends to infinity, is therefore an optimality property of a test. Other familiar optimality criteria are based on the asymptotic relative efficiency of the test. The relations between these optimality criteria are investigated. It turns out that vanishing shortcoming is seemingly slightly stronger than first-order efficiency, but in regular cases there is equivalence. The results are in particular applied on tests for goodness-of-fit

Data driven smooth tests for composite hypotheses comparison of powers

Rao's score statistic is a standard tool for constructing statistical tests.If departures from the null model are described by some k-dimensional exponential family the resulting score test is called also smooth test or Neyman's smooth test with k components. An important practical question in applying a smooth test in the goodness-of-fit problem is how large k should be taken. Since a wrong choice may give a considerable loss of power,it is important to make a careful selection.Renewed research in this area shows that the simple question has no simple deterministic answer. Therefore,edwina introduced,for testing a simple goodness-of-fit hypothesis,a data driven version of Neyman's smooth test. First,Schwarz's rule is applied to find a suitable dimension and then the smooth test statistic in the “right" dimension finishes the job. Simulation results and some theoretical considerations show that this data driven version of smooth tests performs well for a wide range of alternatives,and is competitive with other recently introduced (data driven) procedures.This data-dependent choice of the number of components is extended in this paper to testing the goodness-of-fit problem with composite null hypothesis,being of more practical interest.A k-dimensional exponential family for modelling departures from the null hypothesis is given and the related Rao's score test is described. A suitable version of Schwarz's rule is proposed and some simplifications of it are discussed.To check validity of the proposed construction,the method is applied to testing exponentiality and normality.In the extensive simulation study,presented in this paper,it turns out that the data driven version of smooth tests compares well for a wide range of alternatives with other,more specialized,commonly used tests

On the Bahadur efficiency of some tests of independence

.In the paper we give a short review of results connected with deriving of Bahadur efficiency. Moreover, calculations of the Bahadur efficiency of some tests of independence are discussed

On the asymptotic efficiency of estimators

W pracy przedstawiamy i dyskutujemy pojęcie asymptotycznej efektywności estymatorów w ujęciu Hajeka i Le Cama. Podajemy też ogólną konstrukcję pewnej klasy asymptotycznie optymalnych estymatorów dla parametrów z przestrzeni euklidesowej. Pokrótce szkicujemy uogólnienia dyskutowanych idei na przypadek semiparametryczny i pokazujemy, że techniczne wyniki uzyskane w teorii asymptotycznie efektywnej estymacji mogą być z powodzeniem wykorzystane w asymptotycznej teorii testowania. Wybór materiału jest wysoce subiektywny i tylko w niewielkim stopniu oddaje złożoność rozpatrywanych współcześnie zagadnień oraz ogrom wyników, jakie uzyskano w tej tematyce. Tekst jest skróconą wersją wykładu przygotowanego na zaproszenie Organizatorów Konferencji ze Statystyki Matematycznej – Wisła 2005. Głównym celem prezentacji jest pokazanie, że klasyczne podejście do definiowania asymptotycznej efektywności nie sprawdziło się i przedyskutowanie tego jak, dla pewnej klasy zagadnień, w naturalny i elgancki sposób został ten problem rozwiązany.We present and discuss the notion of asymptotic efficiency of estimators as introduced by Hajek and Le Cam. We give also some general construction of a class of asymptotically efficient estimators of Euclidean parameters. Moreover, we briefly indicate some generalizations of the discussed ideas to the case of semiparametric models. We show also that technical results obtained in the asymptotic theory of efficient estimation can be successfully used in asymptotic theory of testing. The selection of the material is highly subjective and to a little extent reflects complexity of several problems and range of results available in present-day literature. The paper is a shortened version of invited series of lectures presented at the Conference on Mathematical Statistics WISŁA 2005. Its main purpose is to show that classic approach to define efficiency was not satisfactory and to discuss how, for some class of problems, this question was solved in a natural and elegant way