On universal admissibility of scale parameter estimators

A characterization result of Kushary (1998) regarding universal admissibility of equivariant estimators in the one parameter gamma distribution is generalized to a scale family of distributions with monotone likelihood ratio. New examples are given, among them the F-distribution with a scale parameter. In particular, universal admissibility is characterized within the class of location-scale equivariant estimators of the ratio of the variances of two normal distributions with unknown means. In this context the maximum likelihood estimator is shown to be universally inadmissible by virtue of a general sufficient condition for universal inadmissibility of a scale equivariant estimator. Received: January 2000     

On bootstrapping the mode in the nonparametric regression model with random design

In the nonparametric regression model with random design and based on i.i.d. pairs of observations (X _i, Y _i), where the regression function m is given by m(x)=?(Y _i|X _i=x), estimation of the location θ (mode) of a unique maximum of m by the location of a maximum of the Nadaraya-Watson kernel estimator for the curve m is considered. In order to obtain asymptotic confidence intervals for θ, the suitably normalized distribution of is bootstrapped in two ways: we present a paired bootstrap (PB) where resampling is done from the empirical distribution of the pairs of observations and a smoothed paired bootstrap (SPB) where the bootstrap variables are generated from a smooth bivariate density based on the pairs of observations. While the PB requires only relatively small computational effort when carried out in practice, it is shown to work only in the case of vanishing asymptotic bias, i.e. of “undersmoothing” when compared to optimal smoothing for mode estimation. On the other hand, the SPB, although causing more intricate computations, is able to capture the correct amount of bias if the pilot estimator for m oversmoothes. Received: May 2000     

Designs for smoothing spline ANOVA models

Smoothing spline estimation of a function of several variables based on an analysis of variance decomposition (SS-ANOVA) is one modern nonparametric technique. This paper considers the design problem for specific types of SS-ANOVA models. As criteria for choosing the design points, the integrated mean squared error (IMSE) for the SS-ANOVA estimate and its asymptotic approximation are derived based on the correspondence between the SS-ANOVA model and the random effects model with a partially improper prior. Three examples for additive and interaction spline models are provided for illustration. A comparison of the asymptotic designs, the 2^d factorial designs, and the glp designs is given by numerical computation. Received May 2000     

Optimality and efficiency of incomplete split-block designs

In this paper, a balanced incomplete split-block design (BISBD) is shown to be universally optimum for the estimation of the interaction effects with respect to each of the intra-plot stratum estimation and the GLSE. Furthermore, the efficiency factor of an optimal design is investigated.     

Analysis of a change-point regression problem in quality control by partial sums processes and Kolmogorov type tests

In practice, it is an important problem (especially in quality control) to secure that a known regression function occurs during a certain period in time. In the present paper, we consider the change-point problem that under the null hypothesis this known regression function occurs. As alternative, we consider a certain non-parametric class of functions that is of particular interest in quality control. We analyze this test problem by using partial sums of the data. Asymptotically, we get Brownian motion and Brownian motion with trend (≠0) under the hypothesis and under the alternative, respectively. We prove that tests based on partial sums have a larger power when the partial sums are taken from the time reversed data. This can be quantitatively determined in an asymptotic way by some new results on Kolmogorov type tests for Brownian motion with trend. We illustrate our results by a certain model that is interesting in quality control and by an example with real data.Supported in part by the Deutsche Forschungsgemeinschaft Grant Bi655.Supported in part by the Deutsche Forschungsgemeinschaft Grant Bi655 and by the Swiss National Science Foundation Grant 20-55586.98.     

Characterizations of the Pareto and related distributions

In this note, a class of Pareto distributions is characterized based on the Shannon entropy of k-record statistics. As a consequence of that characterizations of the uniform and exponential distributions are given. Received: October 1999     

Finite sample tail behavior of the multivariate trimmed mean based on Tukey-Donoho halfspace depth

Finite sample tail behavior of the Tukey-Donoho halfspace depth based multivariate trimmed mean is investigated with respect to a tail performance measure. It turns out that the tails of the sampling distribution of the α-depth-trimmed mean approach zero at least ⌊αn⌋ times as fast as the tails of the underlying population distribution and could be n−⌊αn⌋+ 1 times as fast. In addition, there is an intimate relationship among the tail behavior, the halfspace depth, and the finite sample breakdown point of the estimator. It is shown that the lower tail performance bound of the depth based trimmed mean is essentially the same as its halfspace depth and the breakdown point. This finding offers a new insight into the notion of the halfspace depth and extends the important role of the tail behavior as a quantitative assessment of robustness in the regression (He, Jurečková, Koenker and Portnoy (1990)) and the univariate location settings (Jurečková (1981)) to the multivariate location setting. Received: 1 July 1999     

Change Point Analysis based on Empirical Characteristic Functions

Test procedures for detection of a change in the distribution of a sequence of independent observations based on empirical characteristic functions are developed and their limit properties are studied. Theoretical results are accompanied by a simulation study.The work of the first author was partially supported by grants GAČR 201/03/0945 and MSM 113200008     

On the choice of support of Re-descendingψ-Functions in linear models with asymmetric error distributions

InM-estimation of the regression parameter vector in the linear model, we discuss the choice of the support of certain re-descendingψ-functions for both cases when the distribution of the i.i.d. errors is partially known and when it is completely functionally unknown. Research supported in part by the Natural Sciences and Engineering Research Council of Canada.     

D-optimal weighing designs for six objects

For all integers m≥6, we determine the maximum value of det X ^T X, where X is an m×6 (0, 1)-matrix, and exhibit (D-optimal) matrices X for which the maximum occurs. For D-optimal matrices X, the uniqueness of the Gram matrix X ^T X is discussed. Received: May 2000     

Statistical inference on seemingly unrelated varying coefficient partially linear models

This paper is concerned with the statistical inference on seemingly unrelated varying coefficient partially linear models. By combining the local polynomial and profile least squares techniques, and estimating the contemporaneous correlation, we propose a class of weighted profile least squares estimators (WPLSEs) for the parametric components. It is shown that the WPLSEs achieve the semiparametric efficiency bound and are asymptotically normal. For the non‐parametric components, by applying the undersmoothing technique, and taking the contemporaneous correlation into account, we propose an efficient local polynomial estimation. The resulting estimators are shown to have mean‐squared errors smaller than those estimators that neglect the contemporaneous correlation. In addition, a class of variable selection procedures is developed for simultaneously selecting significant variables and estimating unknown parameters, based on the non‐concave penalized and weighted profile least squares techniques. With a proper choice of regularization parameters and penalty functions, the proposed variable selection procedures perform as efficiently as if one knew the true submodels. The proposed methods are evaluated using wide simulation studies and applied to a set of real data.     

Recent and classical tests for exponentiality: a partial review with comparisons

A wide selection of classical and recent tests for exponentiality are discussed and compared. The classical procedures include the statistics of Kolmogorov-Smirnov and Cramér-von Mises, a statistic based on spacings, and a method involving the score function. Among the most recent approaches emphasized are methods based on the empirical Laplace transform and the empirical characteristic function, a method based on entropy as well as tests of the Kolmogorov-Smirnov and Cramér-von Mises type that utilize a characterization of exponentiality via the mean residual life function. We also propose a new goodness-of-fit test utilizing a novel characterization of the exponential distribution through its characteristic function. The finite-sample performance of the tests is investigated in an extensive simulation study.Received: January 2002/Revised: January 2004     

The discrepancy distribution of stationary multiplier rules for rounding probabilities

The problem of rounding finitely many (continuous) probabilities to (discrete) proportions N _i/n is considered, for some fixed rounding accuracy n. It is well known that the rounded proportions need not sum to unity, and instead may leave a nonzero discrepancy D=(∑N _i) −n. We determine the distribution of D, assuming that the rounding function used is stationary and that the original probabilities follow a uniform distribution. Received: April 1999     

Three‐period,two‐treatment crossover design under long‐term carryover effect

The paper describes non‐parametric approach for analysis of a three‐period, two‐treatment, four‐sequence crossover design in which test procedure for interchangeability of the treatment effects is obtained. The proposed procedure is based on a non‐parametric model, which incorporates, along with the direct treatment effects and the usual carryover effects, the long‐term carryover effects. Relevant competitors are obtained. Related asymptotic results are given. By performing simulation study, we compared the procedures with respect to type I error rate and power. Furthermore, confidence intervals for treatment differences are studied. The procedures are illustrated with a data study.     

Robustness issues regarding content-corrected tolerance limits

This article reviews the content-corrected method for tolerance limits proposed by Fernholz and Gillespie (2001) and addresses some robustness issues that affect the length of the corresponding interval as well as the corrected content value. The content-corrected method for k-factor tolerance limits consists of obtaining a bootstrap corrected value p ^* that is robust in the sense of preserving the confidence coefficient for a variety of distributions. We propose several location/scale robust alternatives to obtain robust corrected-content k-factor tolerance limits that produce shorter intervals when outlying observations are present. We analyze the Hadamard differentiability to insure bootstrap consistency for large samples. We define the breakdown point for the particular statistic to be bootstrapped, and we obtain the influence function and the value of the breakdown point for the traditional and the robust statistics. Two examples showing the advantage of using these robust alternatives are also included.     

On multivariate higher order von Mises expansions

In this paper we review the von Mises calculations of higher order for statistical functionals of one variable. For the functionals of several variables, the higher order Gateaux differentials are defined leading to the corresponding multivariate higher order von Mises expansions. These expansions are used to analyze the bias of the corresponding statistical functionals. The second and third order derivatives are computed for M-estimates. Applications of these expansions to study the bias of M-estimates of location and to simultaneous M-estimates of location and scale are also given. Received: July 2000     

Distributions of exact tests in the exponential family

The aim of this paper is to give some results on the exact density of the I-divergence in the exponential family with gamma distributed observations. It is shown in particular that the I-divergence can be decomposed as a sum of two independent variables with known distributions. Since the considered I-divergence is related to the likelihood ratio statistics, we apply the method to compute the exact distribution of the likelihood ratio tests and discuss the optimality of such exact tests. One of these tests is the exact LR test of the model which is asymptotically optimal in the Bahadur sense. Numerical examples are provided to illustrate the methods discussed. Received: January 2002 Acknowledgements. I am grateful to Prof. Andrej Pázman for helpful discussions during the setup and the preparation of the paper and to the referees for constructive comments on earlier versions of the paper. Research is supported by the VEGA grant (Slovak Grant Agency) No 1/7295/20     

Two-level search design for main-effect plus two plan

One important class of screening designs is the search design first proposed by Srivastava (1975). A new class of two-level factorial search designs which are capable of estimating all main-effect plus two interactions is provided. We first give a necessary and sufficient condition for the main-effect plus two plan and then show that the proposed search design always satisfies such a condition. Received October 2000