The asymptotic distribution of the partial attributable risk in cross-sectional studies

The partial attributable risk (PAR) has been introduced as a tool for partitioning the responsibility for causing an adverse event between various risk factors. It has arisen from epidemiology, but it is also a valid general risk allocation concept, which can, for example, be applied to data from customer satisfaction surveys. So far, a variance formula for the PAR has been missing so that the confidence intervals were not directly available. This paper provides the asymptotic normal distribution for the PAR determined from a cross-sectional study.     

Parameter selection in modified histogram estimates

A multivariate modified histogram density estimate depending on a reference density g and a partition P has been proved to have good consistency properties according to several information theoretic criteria. Given an i.i.d. sample, we show how to select automatically both g and P so that the expected L ₁ error of the corresponding selected estimate is within a given constant multiple of the best possible error plus an additive term which tends to zero under mild assumptions. Our method is inspired by the combinatorial tools developed by Devroye and Lugosi [Devroye, L. and Lugosi, G., 2001, Combinatorial Methods in Density Estimation (New York, NY: Springer–Verlag)] and it includes a wide range of reference density and partition models. Results of simulations are also presented.     

Reducing component estimation for varying coefficient models

The estimation problem for varying coefficient models has been studied by many authors. We consider the problem in the case that the unknown functions admit different degrees of smoothness. In this paper we propose a reducing component local polynomial method to estimate the unknown functions. It is shown that all of our estimators achieve the optimal convergence rates. The asymptotic distributions of our estimators are also derived. The established asymptotic results and the simulation results show that our estimators outperform the the existing two-step estimators when the coefficient functions admit different degrees of smoothness. We also develop methods to speed up the estimation of the model and the selection of the bandwidths.     

Non-negative estimation of variance components in heteroscedastic one-way random-effects ANOVA models

There is a considerable amount of literature dealing with inference about the parameters in a heteroscedastic one-way random-effects ANOVA model. In this paper, we primarily address the problem of improved quadratic estimation of the random-effect variance component. It turns out that such estimators with a smaller mean squared error compared with some standard unbiased quadratic estimators exist under quite general conditions. Improved estimators of the error variance components are also established.     

Mathematische aspekte zu einer methode zur numerischen bestimmung individueller thermodynamischer aktivitätskoeffizienten einzelner ionensorten in konzentrierten elektrolytlösungen

The mathematical problems of the – in an communication [3] described – principle for the calculation of individual thermodynamic activity coefficients of single ionic species in concentrated electrolyte solutions are specified. It is the Newtonian approximation method that makes possible the evaluation of the constants b ₁,…b ₄ in the concentration function (0.1) for the product of the activity coefficients.

The efficiency of the method is represented by the example of the activity coefficients of pure and of – with other electrolytes – mixed solutions of NaCIO₄. The individual activity coefficients of the single ionic species are evaluated for several electrolytes of the concentration range from m = 0 to m = 10 mole/kg and published at another place [3, 17, 18].     

Bounded influence estimation for regression and scale

Bednarski and Müller [Optimal bounded influence regression and scale M-estimators in the context of experimental design, Statistics 35 (2001), pp. 349–369] introduced a class of bounded influence M estimates for the simultaneous estimation of regression and scale in the linear model with normal errors by solving the corresponding normal location and scale problem at each design point. This limits the proposal to regressor distributions with finite support. Based on their approach, we propose a slightly extended class of M estimates that is not restricted to finite support and is numerically easier to handle. Moreover, we employ the even more general class of asymptotically linear (AL) estimators which, in addition, is not restricted to normal errors. The superiority of AL estimates is demonstrated by numerical comparisons of the maximum asymptotic mean-squared error over infinitesimal contamination neighbourhoods.     

A note on Stein-type shrinkage estimator in partial linear models

In this paper, an exact sufficient condition for the dominance of the Stein-type shrinkage estimator over the usual unbiased estimator in a partial linear model is exhibited. Comparison result is then done under the balanced loss function. It is assumed that the vector of disturbances is typically distributed according to the law belonging to the sub-class of elliptically contoured models. It is also shown that the dominance condition is robust. Furthermore, a nonparametric estimation after estimation of the linear part is added for detecting the efficiency of the obtained results.     

Bayes sequential estimation for a Poisson process under a LINEX loss function

In this paper, within the framework of a Bayesian model, we consider the problem of sequentially estimating the intensity parameter of a homogeneous Poisson process with a linear exponential (LINEX) loss function and a fixed cost per unit time. An asymptotically pointwise optimal (APO) rule is proposed. It is shown to be asymptotically optimal for the arbitrary priors and asymptotically non-deficient for the conjugate priors in a similar sense of Bickel and Yahav [Asymptotically pointwise optimal procedures in sequential analysis, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, University of California Press, Berkeley, CA, 1967, pp. 401–413; Asymptotically optimal Bayes and minimax procedures in sequential estimation, Ann. Math. Statist. 39 (1968), pp. 442–456] and Woodroofe [A.P.O. rules are asymptotically non-deficient for estimation with squared error loss, Z. Wahrsch. verw. Gebiete 58 (1981), pp. 331–341], respectively. The proposed APO rule is illustrated using a real data set.