On the variance of the ratio estimator

Summary In an earlier paper [Rao 1966] an exact expression for the variance of the ratio estimator under theMidzuno-Sen sampling scheme is obtained and here we study some of the interesting properties of the coefficients involved in this expression which depend on the auxiliary information. Use of these coefficients is made of in finding out an exact expression for the Bias and Mean Square Error of the ratio estimator under Simple Random Sampling With-Out Replacement (SRSWOR) scheme.     

On classes of estimators of the variance function of a linear estimator

Summary The variance function of a linear estimator can be expressed into a quadratic form. The present paper presents classes of estimators of this quadratic form along the lines implicitly suggested byHorvitz andThompson [1952] while formulating the classes of linear estimators. Accordingly it is noted that there exist nine principal classes of estimators out of which one principal class is examined in detail. Furthermore to illustrate the theory an example is considered where the expression for a unique estimator variance of the best estimator in theT ₁ class is derived.     

On the estimation of location parameters in the multivariate one sample and two sample problems

Summary In this paper we consider the problem of estimating the vectors of location parameters in the multivariate one sample and two sample problems. These estimators are obtained through the use of the multivariate rank order statistics such as theWilcoxon or the normal scores statistic considered by the authors inPuri, Sen [1966] andSen, Puri [1967] for the corresponding testing problems. The distribution of these estimators is shown to be symmetric with respect to the parameters being estimated. These estimators are translation invariant, robust and asymptotically normal. Their asymptotic relative efficiencies with respect to the estimators based on the vector of means and medians are discussed by applying the criterion ofWilks generalized variance [Anderson, p. 166]. In particular, it is shown that the estimators based on the multivariate normal scores statistics are asymptotically as efficient as the ones based on the method of least squares when the parent distributions are normal. Research sponsored by National Science Foundation Grant No. GP-12462, and by Research Grant, GM-12868 from the N.I.H., Public Health Service.     

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.

     

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.     

On the linear combinations of stochastic variables

Summary In this article we give a general distribution of the linear combinations of independent non-negative stochastic variables. This result generalizes a class of distributions of the linear combinations of independent non-negative variates and further a number of other particular cases can be obtained from the general distribution discussed in this article. In deriving the general distribution we used the properties of the most generalized special function, namely, the H-function ofFox [1961] andBraaksma [1964].     

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.     

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.     

Further results on asymptotic normality I

Summary Starting fromLe Cam [1956], it was shown inMichel andPfanzagl [1970] that — under certain regularity conditions — a dominated family of probability measures withEuclidean parameter space behaves approximately like a family of normal distributions, if each probability measure is the independent product of a great number of identical components. It is the purpose of this paper to estimate the accuracy of such a normal approximation.     

A further note on the asymptotic efficiency of friedman’s x r 2 -test

Summary For independently distributed error components, the asymptotic relative efficiency (A.R.E.) ofFriedman’sx _r ² -tests with respect to the classical analysis of variance test has been studied byElteren andNoether andSen [1967]. The present note extends these results to the case of correlated errors arising in some random-effects or mixed-effects models. Work supported by the U.S. Army Research Office, Durham, Grant DA-ARO-D-31-124-G 746.     

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.

     

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.     

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.

     

Über einN-elementiges Modell der Reiz-Stichproben-Theorie mit stetigem Zeitparameter und zwei Antwortklassen

Zusammenfassung Es wird ein Modell der Reiz-Stichproben-Theorie [vgl.Atkinson u.Estes] mit stetigem Zeitparameter untersucht. Dabei handelt es sich um ein sogenanntes Pattern-Modell mitN Reizelementen und zwei Antwortklassen. Die den Systemablauf beherrschenden Differentialgleichungen werden hergeleitet und gel?st. Es werden Resultate über das Grenzverhalten der verknüpften stochastischen Prozesse erzielt, insbesonders über den effektiven Antwortproze?. Ferner werden Methoden zur Punktsch?tzung der unbekannten ParameterN (Anzahl der Reizelemente) undc (Lernrate) behandelt.

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Summary A model of stimulus sampling theory [cp.Atkinson andEstes] with continuous time parameter is investigated. We are concerned with a so-called pattern model withN stimulus elements and two response classes. The differential equations governing the system are derived and solved. Some results on the limit behaviour of associated stochastic processes are obtained, concerning especially the actual response process. Furtheron methods are treated for point estimation of the unknown parametersN (number of stimulus elements) andc (learning rate).

     

On the power function of the exact test for ther×c contingency table

Summary LetN=[n _ij ] (i=1, …,r;j=1, …,c) be the matrix of observed frequencies in anr×c contingency table fromr possibly different multinomial populations with respective probabilitiesp _i =(p _i1, …,p _ic ).Freeman andHalton have proposed an exact conditional test for the hypothesisH ₀ :p _i =(p ₁, …p _c ) of the exact test is derived. Numerical values forβ(p) were previously computed for the special case:r=3,c=2 [Bennett andNakamura, 1964].     

On the positivity of a variance-estimator

Summary Murthy’s variance — estimator for the estimator obtained by him by unorderingDes Raj’s estimator is shown to be non-negative.     

A note on the optimality ofk-stage tests

Summary This note deals with sequential tests requiring at mostk observations. First a lemma is proved concerning the form of Bayes solutions. Then it is demonstrated by giving a counter-example that the theorem ofWald andWolfowitz has no correspondence fork-stage tests.

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Zusammenfassung Es wird gezeigt, daß man beik-stufigen Tests zwar für die Gestalt vonBayes-Lösungen ein Analogon zu den unbeschränkten Sequenztests hat, daß aber der Satz vonWald undWolfowitz über die gleichmäßige Optimalität bzgl. des Stichprobenumfangs keine Entsprechung besitzt.

     

On the active approach of stochastic linear programming

Summary The problem of statistical distribution of the optimal objective function under the so-called active approach of stochastic linear programming is investigated here from two interrelated aspects. First, the active approach is viewed as a method of decomposition. Second, some results on the asymptotic form of distribution of extreme values are utilized to derive the asymptotic form of the distribution of the maximand under the active approach. Research done under the partial support of the U.S. National Science Foundation Grant No. 420-04-62 at the Department of Economics, Iowa State University. Some of the work related to this paper may be found in the following references:Sengupta, J. K., G. Tintner, andC. Millham: “On Some Theorems of Stochastic Linear Programming”. Management Science, Vol. 10, October 1963, pp. 143–159.Sengupta, J. K.: “The stability of truncated solutions of stochastic linear programming”. Econometrica, Vol. 34, January 1966. pp. 77–104.Sengupta, J. K.: “On the stability of solution under recursive programming”. Metrika 1966.Sengupta, J. K. andT. Kumar: “An application of sensitivity analysis to a linear programming problem”. Unternehmensforschung, Vol. 9, 1965.     

Über eine Anwendung der Potentialtheorie transienter Markoff-Ketten

Zusammenfassung Bei diskretenMarkoff-Ketten sind die Bedingungen: Gleichheit der superharmonischen Funktionen, Gleichheit der Trefferwahrscheinlichkeiten sowie Gleichheit der Sprungmatrizen äquivalent. Einen einfachen Beweis erhält man mit Hilfe von Aussagen der Potentialtheorie transienterMarkoff-Ketten. Die Aussagen des diskreten Falles kann man benutzen, um entsprechende Aussagen für gewisse minimaleMarkoff-Ketten herzuleiten.

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Summary For discreteMarcov chains the following conditions are equivalent: Identical superharmonic functions, identical hitting probabilities, identical jump matrices. A simple proof can be given by using the potential theory of transientMarcov chains. With the help of the statements in the discrete case it is possible to get some of the corresponding results for certain classes of minimalMarcov chains.

     

Zum Erschöpftheitsbegriff vonD. Blackwell

Zusammenfassung Der vonBlackwell mittelsMarkoffscher Kerne eingeführte Begriff der Erschöpftheit eines klassischen Experiments für ein zweites mit gleicher Indexmenge wird mit dem Begriff der erschöpfenden Statistik vonHalmos undSavage verglichen. In der Tat gibt es Bedingungen, unter denen beide Begriffe zusammenfallen. Der Äquivalenzsatz wird durch EigenschaftenMarkoffscher Kerne sowie derBlackwellschen Erschöpftheitsrelation vorbereitet. Gegenbeispiele ermöglichen es, die Notwendigkeit der Regularitätsbedingungen einzuschen.

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Summary Blackwell's studies concerning comparisons of experiments yield a concept of sufficiency which in general is not equivalent to the popular notion of a sufficient statistic made precise byHalmos andSavage. In fact regularity conditions can be given under which the two concepts coincide. Properties ofMarkov kernels and ofBlackwell's sufficiency relation are examined first. After the proof of the equivalence theorem counterexamples are presented to indicate the significance of the regularity conditions.

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Das Manuskript ist am 14. 10 1968 bei den Herausgebern eingegangen.