Tadeusz Inglot

Intermediate approach to comparison of some goodness-of-fit tests

In this paper we present the intermediate approach to investigating asymptotic power and measuring the efficiency of nonparametric goodness-of-fit tests for testing uniformity. Contrary to the classical Pitman approach, the intermediate approach allows the explicit quantitative comparison of powers and calculation of efficiencies. For standard tests, like the Cramér-von Mises test, an intermediate approach gives conclusions consistent with qualitative results obtained using the Pitman approach. For other more complicated cases the Pitman approach does not give the right picture of power behaviour. An example is the data driven Neyman test we present in this paper. In this case the intermediate approach gives results consistent with finite sample results. Moreover, using this setting, we prove that the data driven Neyman test is asymptotically the most powerful and efficient under any smooth departures from uniformity. This result shows that, contrary to classical tests being efficient and the most powerful under one particular type of departure from uniformity, the new test is an adaptive one.

Simple upper and lower bounds for the multivariate Laplace approximation

We propose a new proof of the Laplace approximation for multiple integrals and consequently find new bounds for the approximation error. The main advantages of our approach are its simplicity, an explicit form of the bounds and small coefficient in the main error term, which depend on the smallest eigenvalue of the Hessian matrix, only. On the other hand, the method seems indicating much larger values of the parameter then really necessary.

On large deviation theorem for data-driven Neyman's statistic

The aim of the paper is to show that for data-driven Neymans statistic large deviation theorem does not hold. We derive an explicit estimate from below for probabilities of large and moderate deviations. The main tool is a version of a lower exponential inequality recently obtained by Mogulskii.

Gaussian random series on metric vector spaces

On decrit les proprietes de base des mesures gaussiennes sur des espaces vectoriels metriques complets separables en termes de noyaux reproduisants