Eine Bemerkung zur Existenz erwartungstreuer Schätzer

Summary A theorem ofBickel andLehmann concerning the existence of unbiased estimates for parameters on the convex family of all absolutely continuous probability distributions is generalized to arbitrary convex families of absolutely continuous distributions. In contrast to the result ofBickel andLehmann, however, we give necessary and sufficient conditions for the existence of only approximately unbiased estimates. An example, which is borrowed from density estimation, shows that in general the term approximately cannot be ommitted.     

On the measurability and consistency of minimum contrast estimates

Summary The concept of minimum contrast (m.c.) estimates used in this paper covers maximum likelihood (m.l.) estimates as a special case. Section 1 contains sufficient conditions for the existence of measurable m.c. estimates and for their consistency. The application of these results to m.l. estimates (section 2) yields the existence of m.l. estimates for families ofp-measures (probability measures) which are compact metric or locally compact with countable base, admitting upper semicontinuous densities, whereas the classical results refer to continuous densities. This generalization is insofar of interest as upper semicontinuous versions of the densities exist whenever the densities areμ-upper semicontinuous (whereasμ-continuity does not, in general, entail the existence of continuous versions). Under appropriate regularity conditions, consistency of asymptotic maximum likelihood estimates is proven for compact (and also locally compact) separable metric families ofp-measures with upper semicontinuous densities and for arbitrary families having uniformly continuous densities with respect to the uniformity of vague convergence. The conditions sufficient for consistency are shown “indispensable” by counterexamples. Section 3 contains auxiliary results. Besides their relevance for sections 1 and 2, some of them may also be of interest in themselves, e.g. Theorem (3.4) on the selection of semicontinuous functions from semicontinuous equivalence classes.     

Note on minimum contrast estimates forMarkov processes

Summary In the present paper it is shown that the concept of minimum contrast estimates (m.c.e.) considered inPfanzagl [1969a] for independent and identically distributed (iid) observations can be modified to cover stationary discrete timeMarkov processes admitting a unique stationary distribution which dominates the transition probabilities (Condition (S)). Sufficient conditions on the measurability and strong consistency of m.c.e. stated inPfanzagl [1969a] for the idd case are reformulated to give sufficient conditions for the existence of measurable m.c.e. and their strong consistency for such processes (section 1). The proofs of the main theorems are only sketched, because they are nearly the same as those given inPfanzagl [1969a] for the iid case.The concept of m.c.e. covers maximum likelihood estimates (m.l.e.) as a special case; therefore an application of the results to m.l.e. yields sufficient conditions for the existence of measurable m.l.e. and their strong consistency if the parameter space is compact metrizable or locally compact with countable base (Section 2). These conditions are weaker than the usual regularity conditions (see for exampleBillingsley [1961b] and the references cited there) and under the assumption that the transition probabilities as well as the stationary distribution are absolutely continuous with respect to a -finite measure they can be expressed in terms of the corresponding equivalence classes of transition densities. This seems to be more transparent than the conditions given byRoussas [1965]. In Section 3 asymptotic normality of m.c.e. is proved under conditions which correspond to those used inRoussas [1968] for the case of m.l.e.     

Compound gamma,beta and F distributions

Summary In this paper a compound gamma distribution has been derived by compounding a gamma distribution with another gamma distribution. The resulting compound gamma distribution has been reduced to the Beta distributions of the first kind and the second kind and to theF distribution by suitable transformations. This includes theLomax distribution as a special case which enjoys a useful property. Moment estimators for two of its parameters are explicitly obtained, which tend to a bivariate normal distribution. The paper contains expressions for a bivariate probability density function, its conditional expectation, conditional variance and the product moment correlation coefficient. Finally, all the parameters of the compound gamma distribution are explicitly expressed in terms of the functions of the moments of the functions of random variables in two different ways. This note is based on a technical report prepared by the author while he was with the Procter and Gamble Company.     

On the non-null distribution ofWilcoxon's signed rank test

Summary The properties of theWilcoxon signed rank sum testW ⁺ [Wilcoxon, 1945] for the hypothesis of symmetryH _o are discussed for alternativesH toH _o. The probability generating function and cumulant generating function ofW ⁺ are derived and a limiting form of the distribution is determined.     

On unordering Horvitz and Thompson'sT 1 class of linear estimators

Summary When elements of a finite population are sampled with varying probability selection at each draw,Horvitz andThompson [1952] have formulated certain classes of linear estimators to bear on the problem of providing a smaple appraisal of the population total.Horvitz andThompson's T ₁ class is an ordered one, which was examined by the present author [1967 b]. For some sampling procedures a best estimator exists for theT ₁ class. Subsequently the present author [1967 c] appliedMurthy's technique [Murthy 1967] of unordering an ordered estimator and derived a more efficient estimator. The present paper is concerned with applyingMurthy's technique to theT ₁ class itself, and examining the unorderedT ₁ class. Curiously enough, it is noted that the condition of unbiasedness is sufficient to completely specify the unorderedT ₁ class for the sampling procedure considered here.Research sponsored by Marathwada University, Aurangabad, India; under Grant No. Research-12-68-69/3314-16.     

Differentiability properties of thes-Meanm s ats=1

It is known that thes-meanm _s of a probability measure converge fors 1 to a medianm ₁ called the natural median. For purposes of estimation it is important to know whetherm _s is not only continuous but also differentiable ats=1. We show that the functionsm _s is indeed differentiable ats=1 for the case of a unique as well as a non unique median. We also give an explicit formula for the derivative ats=1.     

Zur Parameter- und Prozentpunktschätzung von Lebensdauerverteilungen bei kleinem Stichprobenumfang

Summary Some of the many methods developed for estimating parameters or percentage points of the Weibull distribution are compared. It is shown that the known estimation of the reciprocal shape parameter with the aid of a straight line in the extremal probability paper is rather biased for small sample sizes. To avoid the bias, correction factors are given, and the efficiency of the resulting unbiased estimator is calculated for sample sizesn=2, 3, …, 9. Results ofJ. Lieblein concerning the double exponential distribution are slightly modified in order to get best linear unbiased estimators for parameters and for the logarithms of percentage points of the Weibull distribution. Other methods are shortly discussed and a median-unbiased estimator for the shape parameter is derived.      

Bestimmung kostenoptimaler Parameter für den kontinuierlichen Stichprobenplan vonDodge

Summary Dodge's sampling plan for a continuous production process is determined by two constantsi andk. Under some assumptions about the statistical properties of the production process the average loss due to the costs for inspection and for replacing defective units by good ones is described by a loss function. It is shown that there are always constantsi andk which minimize the loss function in a special sense. Tables of these optimum parameters are computed.     

Uniform minimum variance unbiased estimators for bivariate families

Summary We consider uniform minimum variance unbiased (UMVU) estimation of an unbiased estimable function of distribution parameters for bivariate truncation (non-regular) parameter families. In particular, we derive the UMVU estimator of the probability thatY is less thanX.     

Estimation in the pareto distribution

The unique minimum variance unbiased (UMVU) estimate of the probability distribution function of the Pareto distribution is derived. It is shown that the distribution function and ther ^th moment associated with the UMVU estimate are also UMVU estimators. The p.d.f. and its estimator are compared graphically. An estimate of the 100p ^th percentile is given. It is seen that a function of this estimator has a chi-square distribution.     

Zur Definition und Schätzung von Mittelwerten für zufällige Variable auf der Sphäre

Zusammenfassung Diese Arbeit ist die gekürzte Fassung einer Dissertation, die vom Autor 1968 unter gleichem Titel in Würzburg ver?ffentlicht wurde. Es ist nicht sinnovll, Mittelwerte für zuf?llige Variable auf der Sph?re in der üblichen Weise, wie z. B. für zuf?llige Variable auf der reellen Achse, zu definieren. W. Uhlmann [1964] bediente sich entscheidungstheoretischer Begriffe, um für zirkul?re zuf?llige Variable mittlere Winkel zu definieren, die ihre Richtung unabh?ngig von der Wahl der Null-Richtung beibehalten. Die analoge Invarianzeigenschaft wird für alle hier definierten Mittelwertbegriffe (mittlere Richtung, mittlerer Gro?kreis, Mittelachse, Mittelkreis) gesichert, indem einfache Forderungen an die zu verwendenden Verlustfunktionen gestellt werden. Da als Sch?tzungen für diese Mittelwerte stets die entsprechenden Mittelwerte der empirischen Verteilung auftreten, haben diese auch die gleiche Invarianzeigenschaft. Um die Diskrepanz zwischen einer solchen Sch?tzung und dem zu sch?tzenden Mittelwert zu messen, werden neue Verlustfunktionen eingeführt. Es wird gezeigt, da? alle eingeführten Sch?tzungen bezüglich mindestens einer Verlustfunktion unverf?lscht sind, d. h. der erwartete Verlust wird minimal, wenn wir aus allen in Frage kommenden Objekten gerade den betreffenden Mittelwert gesch?tzt werden lassen. Dieser minimale Verlust wird die Dispersion der Sch?tzung bezüglich dieser Verlustfunktion genannt. Es wird bewiesen, da? alle ermittelten Dispersionen mindestens wien ^−1/2 gegen Null gehen, wenn n gegen Unendlich strebt.

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Summary This paper is shortened from an equally entitled dissertation which has been published by the author in 1968 at Würzburg. For random variables on the sphere it would make no sense to define means in the usual way as it is done e. g. for random variables on the real line. Introducing concepts of decision theory,W. Uhlmann [1964] defined mean angles for circular random variables the direction of which does not depend on the choice of the zero direction. Setting up simple conditions for the loss functions to be used, we ensure that all the means defined in the paper (mean directions, mean great circles, mean axes, mean circles) have the analogous invariance property. The estimators of these means are always the corresponding means of the empirical distribution, defined with respect to the same loss function and therefore they have the invariance property too. To measure the discrepance between an estimator and the estimated mean, new loss functions are introduced. It is shown that all the established estimators are unbiased with respect to at least one loss function, i. e. the expected loss is a minimum, if we take just the mean from all the things in question to be estimated by the regarded estimator. This minimum loss is called the dispersion of the estimator with respect to this loss function. It is proved, that all the calculated dispersions go to zero at least asn ^−1/2, ifn tends to infinity.

     

On the inequality Σp i f(p i )≥ p i f(q i )

Summary We shall deal with the inequality (2) and we prove among others the following results. Every solution of the inequality (2) is monotone increasing. Every solution of the inequality (2) is differentiable ifn≥3. A functionf satisfies the inequality (3) if and only if its saltus part and absolutely continuous part satisfy also the inequality (3). We give the general solution of (3) in the field of saltus function. Finally by the help of [Muszély] we give the general solution of (3). This results are generalisations of one ofJ. Aczél andJ. Pfanzagl.

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Zusammenfassung Wir betrachten die Ungleichung (2) und beweisen die folgenden Ergebnisse. Die allgemeine L?sung der Ungleichung (2) ist monoton wachsend. Die allgemeine L?sung der Ungleichung (2) ist differenzierbar im Falle wennn≥3 ist. Wir geben also die allgemeine L?sung der Ungleichung (3) mit Hilfe von [Muszély]. Diese Ergebnisse sind Verallgemeinerungen des Resultats vonJ. Aczél undJ. Pfanzagl.

     

About probability-like measures for entire theories

5. Summary It is discursively argued that a much closer rapport between the methodologies implicitly taught in Economic Theory and in Statistics is required and that the necessary interdisciplinary bridge can be provided, and clarification attained, through the study of logical measure functions of theKemeny-Carnap type. It is also argued that subjectivistic axiom systems of probability, while valuable in their own right and as bases for behavior theories, cannot by themselves render the study of logical measure functions superfluous. Elementary aspects of these functions are then explained, with somewhat more detailed references toKemeny’s measuresm ands and a proposed degree-of-theoretization measure, whose applicability to economics is negatively appraised. Finally,Carnap’s degree-of-confirmation functions are briefly dealt with, but throughout the paper the emphasis is on those properties of logical measure functions which donot primarily bear on statistical inference and estimation). “...I think, on the whole, one theory fits nearly everything. That is, if you admit one coincidence—and I think one coincidence is allowable. More than one, of course, is unlikely ...” Prof. Dr. E. M. Fels, 2126 CL, University of Pittsburgh, Pittsburgh 13, Penna., USA. The Murder at the Vicarage. New York: Dell, 1961, p. 197 (Dodd, Mead & Co., 1930).     

Eine Momentengleichung bei Exponentialfamilien

Zusammenfassung Eine beiKendall undStuart [S. 122 bzw. S. 126] für Binomial- bzw. Poissonverteilungen und beiNoack [S. 128] für Potenzreihenverteilungen aufgestellte Momentengleichung wird auf einparametrige Exponentialfamilien verallgemeinert, und es wird gezeigt, da? eine weitere Verallgemeinerung nicht m?glich ist. Als Anwendung ergibt sich eine Charakterisierung der Normalverteilung.

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Summary A recurrence relation for the moments about the mean, which has been given byKendall andStuart [p. 122 and p. 126] for Binomial and Poisson distributions and byNoack [p. 128] for power series distributions, is generalized to exponential families, and it is shown, that a further generalization is impossible. As an application a characterization of the normal distribution is given.

     

A characterization of the exponential distribution by higher order gap

Summary SupposeX is a non-negative random variable with an absolutely continuous (with respect to Lebesgue measure) distribution functionF (x) and the corresponding probability density functionf(x). LetX ₁,X ₂,...,X _n be a random sample of sizen fromF andX _i,n is thei-th smallest order statistics. We define thej-th order gapg _i,j(n) asg _i,j(n)=X _i+j,n–X_i,n 1i<n, 1nn–i. In this paper a characterization of the exponential distribution is given by considering a distribution property ofg _i,j(n).     

A measure of mutual complete dependence

Two random variables X and Y on a common probability space are mutually completely dependent (m.c.d.) if each one is a function of the other with probability one. For continuous X and Y, a natural approach to constructing a measure of dependence is via the distance between the copula of X and Y and the independence copula. We show that this approach depends crucially on the choice of the distance function. For example, the L ^p -distances, suggested by Schweizer and Wolff, cannot generate a measure of (mutual complete) dependence, since every copula is the uniform limit of copulas linking m.c.d. variables. Instead, we propose to use a modified Sobolev norm, with respect to which mutual complete dependence cannot approximate any other kind of dependence. This Sobolev norm yields the first nonparametric measure of dependence which, among other things, captures precisely the two extremes of dependence, i.e., it equals 0 if and only if X and Y are independent, and 1 if and only if X and Y are m.c.d. Examples are given to illustrate the difference to the Schweizer–Wolff measure.     

Zur Theorie der Rangtests

Zusammenfassung Bekanntlich lassen sich eine Reihe wichtiger einseitiger Rangtests für verschiedene Problemklassen als lokal beste invariante Tests herleiten [vgl. etwaLehmann], so u. a. für den Vergleich zweier Stichproben, das Symmetrietestproblem und für die Prüfung auf Unabh?ngigkeit. In 2.1 bis 2.5 werden die Voraussetzungen angegeben, die für die Herleitung einer für den Vergleich vonk-Stichproben zuerst vonHoeffding angegebenen Formel ben?tigt werden und die u. a. bei den obigen drei Testproblemen erfüllt sind (vgl. hierzu 3.1 bis 3.3). Die in 4.5 angegebene, vom speziellen Problem unabh?ngigeHoeffding-Formel erm?glicht, wie in 4.2 kurz angedeutet werden soll, nach der Reduktion durch Invarianz eine einheitliche (optimale) Herleitung von Rangtests für die verschiedenen Problemklassen. Eine entsprechende Systematisierung, die überdies die bei der praktischen Durchführung von Rang-und Permutationstests bestehenden Analogien widerspiegelt, ist bei Permutationstests m?glich; jedoch sind hierzu gewisse Zusatzüberlegungen erforderlich [vgl.Witting 1969]. Diese Arbeit ist aus Diskussionen mitG. N?lle † bei der Abfassung vonWitting u.N?lle entstanden. Ihm sei hierfür auch an dieser Stelle nochmals gedankt.

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Summary Assumptions are formulated which are necessary for deriving a formula originally given byHoeffding for the comparison ofk samples. An analogous formula, which is independent of the special problem, makes it possible to derive in a consistent manner locally most powerful invariant tests for several classes of one-sided nonparametric test problems.

     

On a linear stochastic differential equation

Summary Stochastic differential (s. d.) equations had been considered in [Nasr, 1960] and [Nasr]. We consider here, the s. d. equationf(D)x(t)=m(t)+v(t)z(t) wherem(t),v(t) are real functions oft,f(D) is a polynomial inD withD=d/dt, andz(t) is a random function. In particular,z(t) is assumed here, to be of the stationary type, while other types namely whenz(t) is of theGaussian or of thePoisson type, are considered in [Nasr]. A particular integral of the stated equation, and an associated covariance function of this integral are given in the form of generalized (g-)functions; [Nasr, 1965]. The equationdx/dt=v(t)z(t) wherez(t) is stationary in the wide sense is considered as a special case.     

Clarifying the Boundaries of Sexual Harassment and Employer Liability: Judicial Application of Harris v. Forklift Systems, Inc.

Many had anticipated that Harris v. Forklift Systems, Inc., the first significant Supreme Court ruling on sexual harassment since 1986, would clarify the boundary between merely offensive conduct and unlawful conduct, and would offer greater guidance on when an employer is liable for the creation of an abusive (or hostile) work environment. However, the Harris court handed down only a short, narrow decision that left these areas largely unresolved, and, accordingly, it was widely criticized for circumventing contentious issues and for perpetuating the vague framework under which abusive work environment claims are now analyzed. This article examines a representative cross-section of federal and state decisions that have interpreted and applied Harris to evaluate whether courts have achieved any consensus on what constitutes an abusive work environment and on when an employer is liable for this type of sexual harassment.