Search

We obtain designs which are optimally robust against possibly misspecified regression models, assuming that the parameters are to be estimated by one of several types of M-estimation. Such designs minimize the maximum mean squared error of the predicted values, with the maximum taken over a class of departures from the fitted response function. One purpose of the study is to determine if, and how, the designs change in response to the robust methods of estimation as compared to classical least squares estimation. To this end, numerous examples are presented and discussed.

We consider the properties of the trimmed mean, as regards minimax-variance L-estimation of a location parameter in a Kolmogorov neighbourhood K() of the normal distribution: We first review some results on the search for an L-minimax estimator in this neighbourhood, i.e. a linear combination of order statistics whose maximum variance in Kt() is a minimum in the class of L-estimators. The natural candidate – the L-estimate which is efficient for that member of Kt,() with minimum Fisher information – is known not to be a saddlepoint solution to the minimax problem. We show here that it is not a solution at all. We do this by showing that a smaller maximum variance is attained by an appropriately trimmed mean. We argue that this trimmed mean, as well as being computationally simple – much simpler than the efficient L-estimate referred to above, and simpler than the minimax M- and R-estimators – is at least “nearly” minimax. Nous considerons les proprietes de la moyenne tronquee, en ce qui concerne l'estimation-L- de variance minimax d'un parametre de position dans un voisinage Kolmogorov Kt,() de la distribution normale: Nous passons premierement en revue quelques resultats sur la recherche d'un estimateur L-minimax dans ce voisinage, c'est a dire une combinaison lineaire de statistiques d'ordre dont la variance maximale dans Kt() est un minimum dans la classe des estimateurs-L. Le candidat naturel – l'estimation-L qui est efficace pour ce membre de Kt() avec information Fisher minimale – est reconnu comme n'etant pas une solution point-selle au probleme minimax. Nous montrons ici que ce n'est pas du tout une solution. Nous faisons cela en montrant qu'une variance maximum plus petite est atteinte par une moyenne tronquee de facon appropriee. Nous soutenons que cette moyenne tronquee, en plus d'ětre facile a calculer – beaucoup plus simple que l'estimation-L efficace a laquelle nous nous referrons plus haut – et plus simple que les estimateurs minimax M et R – est au moins presque minimax.

We consider minimax-bias M-estimation of a location parameter in a Kolmogorov neighbourhood K() of a normal distribution. The maximum asymptotic bias of M-estimators for the Kolmogorov normal neighbourhood is derived, and its relation with the gross-error sensitivity of the estimator at the nominal model (the Gaussian case) is found. In addition, efficient bias-robust M-estimators Ti are constructed. Numerical results are also obtained to show the percentage of increase in maximum asymptotic bias and the efficiency we can achieve for some well-known -functions. Nous considerons l'estimation-M a biais minimax d'un parametre de position dans un voisinage Kolmogorov K() d'une distribution normale. Nous derivons le biais asymptotique maximum d'estimateurs-M pour le voisinage Kolmogorov normal et trouvons sa relation avec la sensibilite a l'erreur grossiere de l'estimateur au modele nominal (le cas Gaussien). De plus, nous construisons des estimateurs-M T*, robustes au biais. Nous obtenons egalement des resultats numeriques arm de montrer le pourcentage d'accroissement du biais asymptotique maximum et l'efficience que nous pouvons atteindre pour certaines fonctions bien connues.

Summary This paper asks whether or not the efficient L-estimator of scale corresponding to the least informative distribution in e-contamination and Kol-mogorov neighbourhoods of certain distributions possesses the saddlepoint property. This is of interest since the saddlepoint property implies the mini-max property, namely, that the supremum of the relative asymptotic variance of an L-estimator is minimized by the efficient estimator corresponding to that member of the distributional class with minimum Fisher information for scale. Our findings are negative in all cases investigated.

We consider the problem of finding those distributions minimizing Fisher information for scale over e-contamination neighbourhoods Ge(G) = \s{F:F = (1 − e)G + eH, e fixed, H arbitrary\s} of distribution functions G satisfying certain mild conditions. Huber (1981) worked out the solution when G is normal. The general form of the solution is being discussed in this paper.