Tadeusz Bednarski

ABSTRACT An asymptotic robust testing problem for parametric models contaminated by neighbourhoods which are generated by special capacities is studied. The construction of the optimal 10 statistic is described and in particular it is proved that under natural conditions, IC is always a superposition of the derivative of log-likelihood ratio and some wounded reai iimction, ±n& result is xirst proven ior a contammateu. exponential i amity, which approximates locally the original parametric model and then transformed to the original robust model. The paper indicates the dependence between the types ot contamina¬tion and the form of optimal IC for a large class of capacities. It also gives bases for studying estimates derived from tests as it was done by RIEDEB for e-contamination and total variation neighbourhoods

An estimation method is presented which compromises robust efficiency with computational feasibility in the case of the generalized Poisson model. The formal setup is built on flexible nonparametric extensions of the underlying model. The estimation efficiency is expressed via minimax properties of tests resulting from expansions of estimators. The nonparametric neighborhoods related to the proposed score function are exemplified and a real data case is analysed. The resulting method balances several qualitative features of statistical inference: strong differentiability (asymptotic derivations are more accurate), efficiency and natural model extension (quality of formal basic assumptions).

ABSTRACT A smooth parametric model with shrinking contamination given in terms of special capacities is considered. It is proved that logarithms of likelihood ratios between the capacities have expansions given by superpositions of certain bounded nondecreasing functions and the derivaties of log-likelihood ratios for the model.

The problem of simultaneous robust estimation of regression and scale parameters in the linear regression model is studied in the context of experimental design. Optimal M-estimates are given for a modified optimization problem of minimizing the asymptotic variances under bounded influence functions. This is done by reducing the multidimensional regression problem to the problem of estimating one-dimensional location and scale. For the location-scale case two subfamilies of optimal score functions are described in detail along with comparisons of the asymptotic variances and gross-error-sensitivities of the corresponding M-estimators. It turns out that, even for small gross-error-sensitivities, one of the subfamilies provides variances which are close to those of the nonrobust maximum likelihood estimators.

ABSTRACT A computationally simple method of robust estimation in the generalized Poisson model is presented. Estimators are proved to be optimal in the sense of local minimax testing, conditionally on the explanatory variable. Results of a Monte Carlo experiment are supplemented where robust and efficient estimators are compared.

Solutions to minimax test problems between neighbourhoods generated by specially defined capacities are discussed. The capacities are superpositions of probability measures and concave functions, so the paper covers most of the earlier results of Huber and Rieder concerning minimax testing between ɛ-contamination and total variation neighbourhoods. It is shown that the Neyman-Pearson lemma for 2-alternating capacities, proved by Huber and Strassen, can be applied to test problems between noncompact neighbourhoods of probability measures. It turns out that the Radon-Nikodym derivative between the special capacities is usually a nondecreasing function of the truncated likelihood ratio of some probability measures.

Robustness properties of Sasieni's estimators are studied via the functional approach. Conditions for the weight functions are given which lead to strong Fréchet differentiability of estimators functional. The efficiency of Sasieni estimators is compared with Bednarski (Scand. J. Statist. 20 (1993) 213) robust propositions. A two-step estimator facilitating computations of Sasieni estimators is proposed.

... Application of Theorem 4.4 of BILLINGSLEY (1968) completes the proof for the case when u is replaced by u, and EX is arbitrary finite. ... defined by f(p)= =E(-~I~Y,,~J -pEX. P Lemnia 3. If the quuntiles of 1 are u~niqz~ely detei.r)iinerE then n-JD, - f in D [E, I-&]. Proof. ...

The paper presents an unbiased and invariant test for testing the hypothesis H0:F=dispG against H1:F ⩽dispG, where G is a known distribution. Asymptotic law of the test statistic under the null hypothesis is established and the consistency of the test under a class of alternatives is verified.

Summary  Model selection methods have shown to be useful in the process of econometric modelling. The paper studies robust Akaike–Schwarz type information criteria of model choice within the Cox model. The criteria are based on a smooth modification of the partial likelihood function. Apart from asymptotic results, a Monte Carlo study is presented, which shows the finite sample behaviour of the procedure under discrepancies from the Cox model. Analysis of a real unemployment data case is also included.

It is shown how the method of Fréchet differentiability can simplify the asymptotic derivations in an important range of robust inferential problems for stationary and related time series models. The uniform root-n consistency of the empirical distribution function for the Cramer von Mises norm under a weak mixing condition is indicated. Various regularity conditions naturally implemented and leading to the differentiability are discussed. A simulation study supplementing the theoretical discussion is included.

Consideration of multivariate statistical procedures leads to description of robust methods of estimation of location and scale for real random variables. This paper outlines asymptotic theory for estimating location and scale by viewing it as a trimmed likelihood estimator. An appeal is made to results from empirical processes linking the proofs to compact differentiability of estimating functionals. The estimator for location does not depend on the scale estimate and is robust against asymmetric contamination. Coincidentally the location estimator denned at the normal distribution is similar to the least trimmed squares estimator. Simulations corroborate the asymptotic theory and illustrate robustness against asymmetric contamination.