Suchergebnisse

A review is given of shrinkage and penalization as tools to improve predictive accuracy of regression models. The James‐Stein estimator is taken as starting point. Procedures covered are Pre‐test Estimation, the Ridge Regression of Hoerl and Kennard, the Shrinkage Estimators of Copas and Van Houwelingen and Le Cessie, the LASSO of Tibshirani and the Garotte of Breiman. An attempt is made to place all these procedures in a unifying framework of semi‐Bayesian methodology. Applications are briefly mentioned, but not amply discussed.

SummaryA procedure is given for the construction of a monotone estimator that dominates a given estimator for a class of discrete distributions with monotone likelihood ratio. This procedure is applied to some empirical Bayes estimators. Monte Carlo results are given that demonstrate the usefulness of monotonizing.

SummaryThis paper gives a short introduction into the empirical Bayes approach. To illustrate this approach the classical problem of testing a simple hypothesis against a simple alternative is discussed. Some explicit results are given for testing Ho: N(‐1, 1) against H,: N(1, 1). This paper is a further elaboration of section 3.1 of the author's dissertation Wan Houwelingen (1973)).

Inequality constrained regression involves the notion of a truncated parameter space, which was studied extensively by Moors (1985). His general results are extended here and applied to linear models. Using the invariance principle, for every observation x a set Vx is defined with the property that estimators taking values in Vx (with positive probability) are inadmissible. One of the main conclusions is that the usual estimators in inequality constrained regression are inadmissible; a method to obtain better estimators is indicated.

A sequence of logistic models is fitted to data from a Dutch follow‐up study on preterm infants (POPS). To examine the adequacy of the model, a recently developed non parametric method to check goodness of fit is applied (le Cessie and Van Houwelingen (1991)). This method uses a test statistic based upon kernel regression methods.In this paper the problem of choosing a "best" bandwidth, corresponding to the greatest power of the test statistic, is avoided by computing the test statistic for a range of different bandwidths. Testing is then based upon the asymptotic distribution of the maximum of the test statistics.The testing method is used as a goodness of fit criterion, and the contribution of each individual observation to the test statistic is used as a diagnostic tool to localize deviations of the model, and to determine directions in which the model can be improved.

The length of repeated hypercalcemia free periods of patients with bone metastasis of breast cancer with at least one hypercalcemic event was modelled according to a generalized linear mixed model formulated in terms of transition probabilities and according to a latent variable model. In the former case the periods were assumed to be lognormally distributed with two variance components (patients and residue). In the latter case the conditional intensity given a patient was assumed to be the intensity of the Weibull distribution, while the random patient effect (frailty) was assumed to be drawn from a gamma distribution. In both cases the selection of only patients with at least one hypercalcemic event was taken into consideration. In both models the variance of the patient effect turned out to be negligible. For the second and later periods the Weibull appeared to fit better than the lognormal model. For the first period there was almost no information available.