Contributions to Stochastics

Frontmatter

On the Distribution of the Suprema of Weighted Empirical Processes

Summary

A complete characterization of the distributions of the suprema of weighted empirical processes is given. The limit distributions are expressed in terms of weighted Brownian bridge, Wiener, Ornstein-Uhlenbeck and Poisson processes.

Miklós Csörgö, Josef Steinebach, Lajos Horváth

Estimating Percentile Residual Life Under Random Censorship

Summary

Asymptotic confidence intervals and simultaneous confidence bands are constructed for percentile residual lifetimes under random censorship.

Sàndor Csörgö

An Introduction to Linear Dynamic Errors-in-Variables Models

Summary

In this paper we give a short survey on identification of linear systems where both inputs and outputs are possibly contaminated by noise. Such systems are called errors-in-variables- or latent variables-systems.

Manfred Deistler

On Improved Versions of General Erdös-Rényi Laws and Large Deviation Asymptotics

Summary

Somme rather general versions of the (so-called) Erdös-Rényi [14] law of large numbers have been derived by S. Csörgö [7] and Steinebach [21] for sequences of random variables satisfying a fist order large deviation theorem together with certain independence and stationarity properties. In view of recent second order large deviation asymptotics due to Dersch [13], corresponding refinements of the forementioned general Erdös-Rényi laws are discussed in this paper. The latter results can be viewed as convergence rate improvements of their earlier counterparts, typically providing the best rates, but not necessarily the best constants. Some specific examples are presented to demonstrate the applicability of our general approach.

E. Dersch, J. Steinebach

Sequential Analysis: Exact Values for the Bernoulli Distribution

Summary

For the Bernoulli case of Wald’s sequential analysis it is shown that the following quantities can be calculated with any desired precision: First, the Operating Characteristic (OC-function) P(po accepted | p true), especially the errors of the first and second kind α and β. Furthermore, the exact distribution of the number of trials, especially the average sample number CASH-function) and higher moments of the number of trials.

U. Dieter

A Remark on Realizations of Stochastic Processes

Summary

Many with respect to their path properties unpleasant realizations of stochastic processes with continuous time parameterare universal, i.e. do not depend on the distributions of the processes. In particular, a process can be realized in each Baire class if the time parameter set contains a nonvoid interval.

Thomas Drisch, Wolfgang Sendler

Nonparametric Methods: The History, the Reality, and the Future (with Special Reference to Statistical Selection Problems)

Summary

After giving definition(s) of “nonparametric” methods, we note that some problems are “naturally” nonparametric, and that for parametric problems there are strong reasons for desiring nonparametric solutions. “Nonparametric” and “robust” methods are contrasted, and relationships with “rank-based,” “bootstrap,” and “simulation and Monte Carlo” nonparametric procedures are noted. Work on nonparametric selection is reviewed: the needs for it, the early hopes for easy solution (and the Rizvi-Woodworth counterexample to Lehmann’s work), Sobel’s procedure (and the invalidity of more intuitive procedures), dual problems, tables, computer science applications, and robust two-stage solutions. Possible directions of new research are noted.

Edward J. Dudewicz

On the Power of Entropy-Based Tests Against Bump-Type Alternatives

Summary

Recently bump-type alternatives to uniformity have generated interest among practitioners, especially in physics and reliability, and hence statistical tests for uniformity versus bump-type alternatives are currently of strong interest. In this paper we report the power of a new entropy-based test of uniformity under a bump alternative. The results are contrasted to previous statements of Cressie based on a statistic involving logarithms of high-order spacings.

Edward J. Dudewicz, Edward C. van der Meulen

Robust Statistical Methods Applied in the Analysis of Geochemical Variables

Summary

A preliminary case study of the analysis of geochemical variables is reported. Some ideas of resistant analysis and robust statistics are incorporated in certain statistical methods employed in geoscience: Descriptive statistics, principal components analysis, grouping of data, two-dimensional presentation, multiple regression analysis, canonical correlation, outlier detection. The main objective in this study is the outlier-resistent analysis of the data and the report of the outliers which are interesting for exploration.

R. Dutter

Invariant Sufficiency, Equivariance and Characterizations of the Gamma Distribution

Summary

In the first section it is shown for any sub-σ-algebra G of the σ-algebra of all scale invariant Borel subsets of IRⁿ that an equi-variant statistic Sis G-partially sufficient iff the generated σ-alge- bra S⁻¹(B) and G are independent and that S being invariantly sufficient and equivariant, the Pitman estimator is given by S E₁(S)/E₁(S²). For independent X₁,...,X_n the existence of an invariantly sufficient statistic ∑ c_j X _j ^k is characterized by X ₁ ^k ,...X _n ^k having gamma distributions. In the second section there are established some characterizations of the gamma distribution by properties (admissibility, optimality) of the minimum variance unbiased linear estimator where X₁,...,X_n are required to be independent. Finally, the indepen-dence of X₁,...,X_n is replaced by a certain linear framework and a method is presented for carrying over the characterizations previously stated for the independent case to this linear set-up.

Walther Eberl Jr.

Efficient Sequential Estimation for an Exponential Class of Processes

Summary

In this paper a general exponential class of random processes is introduced on the basis of a special exponential form of the likelihood function. Many widely used models for Markov processes are of this type. Martingale properties are proved for the corresponding score process. Sequential estimation procedures based on a finite stopping time τ are considered. A Cramer-Rao inequality is given in the sequential case and the efficiency of sequential estimators is discussed. As an application special results are given for Poisson branching processes.

Jürgen Franz, Wolfgang Winkler

On Convergence in Law of Random Elements in Certain Function Spaces

Summary

The aim of the present paper is to popularize the applicability of a model for convergence in law of random elements in certain (non-separable) function spaces being at first especially appropriate for simplifying the presentation of known functional limit theorems for univariate empirical processes (like the uniform one) and which at the same time allows for a straightforward generalization in handling also empirical processes based on multivariate observations up to empirical processes based on random data in arbitrary sample spaces and being indexed by certain classes of sets or functions, respectively.

Peter Gaenssler, Wilhelm Schneemeier

An Application of the Mixture Theory to the Decomposition Problem of Characteristic Functions

Summary

In more papers author dealt with a new foundation of the mixture theory of probability distribution functions. In this paper a part of the obtained results is used to give necessary and sufficient condition in order to decide whether a given characteristic function is a factor of another one, or not.

Béla Gyires

Construction of Minimax-Tests for Bounded Families of Distribution- Functions

Summary

For the composite testing problem H₀: F ε f₀ against H₁: F ε f₁ where f_i i=0,1 are families of distribution-functions on (R,B) defind by: \(\begin{array}{*{20}{c}} {{f_{i}} = \left\{ {F:{F_{{ - j}}} \leqslant F \leqslant {{\bar{F}}_{j}}} \right\}} & {i = 0,1,} \\ \end{array}\) least favourable pairs of distributions and the famlly of minimax-tests are constructed.

Robert Hafner

An Asymptotic χ 2-Test for Variance Components

Summary

In this paper we investigate an asymptotic χ ²-test of a general linear hypothesis of variance components. First a formulation of the model is given and some basic results on exact χ ²- and F-tests are presented. Then, repeated variance component models are introduced and an asymptotic χ ²-test for a general linear hypothesis of variance components is derived. As an example the two way nested classification model with random effects is considered and some tests are stated explicitly.

Joachim Hartung, Bernard Voet

Probabilities on Groups: Submonogeneous Embedding and Semi-Stability

Summary

There are different generalizations of the concept of stable and semistable probability distributions on the real line. In this paper we show that two natural generalizations coincide for large classes of Lie groups. Whereas however for general locally compact groups these concepts are different.

Wilfried Hazod

On the Use of Predictive Distributions in Portfolio Theory

Summary

In this paper we consider a multiperiod portfolio problem from a Bayesian point of view. Subjective estimates and/or estimation risk enter the predictive distributions which are the foundations for the determination of optimal portfolios. Predictive distributions are adjusted from period to period. It is shown that this adjustment process exhibits some natural requirements for the behaviour of investors. Contrary to other multiperiod models this approach can be seen to be endogeneously defined since transition probabilities are implied by Bayes formula.

Rudolf Henn, Peter Kischka

A Short Proof for the Contraction-Property of Markov Chains with Attractive States

Summary

A short proof for the contraction-property of the map TP = P.P for homogeneous Markov chains with attractive states is presented ¹). It enables the application of Banach’s fixed point theorem to show the existence of the stable distribution. This approach may serve as an alternative for the classic approach (cf. [1], chapter v.2) presented in postcalculus texts on the basics of Markov chains.

Ferdinand Österreicher

A Characteristic Property of Atomic Measures with Finitely Many Atoms Resp. Atomless Measures

Summary

It is proved that a finite measure on a σ-algebra is atomic with a finite number of atoms if and only if there does not exist a {0,1}-valued pure charge on the same σ-algebra whose system of zero sets is larger than the family of zero sets of the finite measure. Furthermore, an example is given which shows that there exists a {0,1}-valued pure charge whose system of zero sets is not larger than the family of zero sets of any finite measure. Finally it is proved that a finite measure on a σ-algebra is atomless if and only if the support of the regular Borel measure of the corresponding Stone representation does not contain a σ-additive measure.

Detlev Plachky

Estimation in the Presence of Nuisance Parameters

Abstract

We consider MVUE and median unbiased estimation in the presence of nuisance parameters. For several different situations sufficient assumptions are formulated guaranteeing that conditionality and marginalization principles lead to optimal estimators. We also discuss some related questions such as the maximality of ancillary statistics, completeness in markovian models, and a generalization of the independence of complete sufficient statistics and ancillary statistics due to Basu.

Ludger Rüschendorf

Stability of Filtered Experiments

Summary

A family of probability measures on a filtered probability space is called a filtered experiment. It is shown that sequences of filtered experiments, which are obtained by rescaling a fixed filtered experiment, can only have weak limits satisfying an invariance property called stability. This property allows a simplified approach to the problem of determining the sample size needed for separating parameter points by critical functions. In case of independent, identically distributed observations, the result covers previous assertions obtained by the author, [3]. In general, the result covers the case of dependent observations. It can be explained, how so-called mixed-normal situations arise in the limit. As a by-product we show, how an increasing family of experiments can be represented by a filtered experiment.

Helmut Strasser

Nonparametric Selection Procedures in Complete Factorial Experiments

Summary

The behavior of many real-world systems depends on two or more factors which can be set at various levels. In such systems factorial experiments are usually conducted so that one can select or rank factor-level combinations, and study the performance of the system at those selected factor-level combinations. For the goal of selecting the best factor-level combination, all the existing theory in ranking and selection assumes normality of the observations. In this paper, we consider selection procedures for the above goal, in two-factor factorial experiments without relying on the assumption of normality. These procedures are then campared under no-interaction and interaction cases, and adaptive procedures are formulated.

Baldeo K. Taneja

Bayesian Statistics in the Regional and Information Sciences

Summary

The development of Statistics shows growing importance of Bayesian Inference. Especially in applications where all available information has to be used the Bayesian paradigm is superior by the possibility of using expert information in the measurable form of a-priori distributions. Different inference techniques for “Regional- and Information Statistics” are pointed out where Bayesian methods are of advantage compared with classical statistical inference.

Reinhard Viertl

Inequalities for Convex Functions

Summary

In the present paper a completed form of Jensen’s inequality along with some of its consequences is given. As an application a stronger form of an inequality by G. J. MINTY [1] is obtained, which claims that the “gradient” of a convex function defined in a (certain kind of) linear space is also “monotone”, like the derivative in the case of a real, convex function.

István Vincze

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