Estimation of a regression coefficient after two preliminary tests of significance

Summary The estimation of the regression coefficient of a population, defined byE (y)= +x, incorporating two preliminary tests of significance has been discussed. The experimenter has two random samples of different sizes from two such populations, as defined above, with regression coefficients ₁ and ₂ respectively, where ₂ may possibly be equal to ₁. Besides this, it is also conjectured that the common conditional variance ² of the two populations has a specified value ₀ ² . The two preliminary tests are used to resolve these two uncertainties.The author has rejoined Lucknow University, Lucknow India on Oct. 4, 1976 after availing leave for two years.     

Über die Anzahl von Zufallspunkten mit typ-gleichem nächsten Nachbarn und einen multivariaten Zwei-Stichproben-Test

Summary For independents-variate samplesX ₁, ...,X _m i.i.d.f. (.),Y ₁, ...,Y _n i.i.d. g. (.), where the densitiesf (.),g (.) are assumed to be continuous on their respective sets of positivity, consider the numberT _m,n of pointsZ of the pooled sample (which are either of typeX or of typeY) such that the nearest neighbor ofZ is of the same type asZ. We show that, as , independently of (.). An omnibus test for the two sample problem f(.)g(.) orf(.)g(.)? may be obtained by rejecting the hypothesisf(.)g(.) for large values ofT _m,n.     

A numerical representation of semiorders on a countable set

Let be a semiorder on a countable setX and letx⁰ y if and only if either there existsx withxxy or there existsx withxxy. Then ⁰ is a preference relation with transitive indifference, which can be represented by a utility functionf of the usual sort. It is well known that is represented by a pair of real-valued functionsu, v, in the sense thatxy if and only ifu(x)>v(y). We prove that there exists a pair of functionsu, v, representing , such thatu+v is the utility function which represents in the usual sense. Moreover it is easily seen that, for such a pair of functionsu, v, we havex⁰ y if and only if eitheru(x)>u(y) or (u(x)=u(y) andv(x)>v(y)).

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Sommario Consideriamo unsemiordine su un insiemeX numerabile e poniamox⁰ y se e solo se esistex tale chexxy, oppure esistex tale chexxy. In questo caso ⁰ è unordine debole, che può essere rappresentato da una funzione di utilitàf nel senso usuale. D'altra parte è rappresentato da una coppia di funzioniu, v, nel senso chexy se e solo seu(x)>v(y). In questo lavoro si prova che ammette una rappresentazioneu, v tale chex⁰ y se e solo seu(x)+v(x)>u(y)+v(y). Si dimostra altresì che, con riguardo ad una siffatta rappresentazioneu, v di , riescex⁰ y se e solo seu(x)>u(y) oppure (u(x)=u(y) ed anchev(x)>v(y)).

     

Alcune proprieta′ di monotonia generalizzata

Questo lavoro pone la monotonia generalizzata come un opportuno strumento per uno sviluppo controllato di funzioni scalari e vettoriali. E' cosi possibile caratterizzare la relazione tra su spazi vettoriali.

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Summary Some generalized monotonicity properties are analyzed both for scalar and vector functions. In particular the relation between and the class of functions, which are able to preserve it, are studied.

     

On functionals of order statistics

Summary Let g_n be real functions,U _ni, 1in, the ordered sample ofn independentU(0,1) distributed random variables, andc _ni(), 1in, 01 be (known) real numbers,n=1, 2, ... The random quantity , 01, is studied. Based on a method proposed byShorack [1972] the main result is the weak convergence of to Gaussian processes, where , 01. The convergence is with respect to theSkorokhod [1956]-topologiesM ₂,M ₁ onD (I) and the -topology onC(I), depending on the conditions imposed on thec _ni().     

A dual approach for matrix-derivatives

Summary A dual approach for matrix by matrix derivatives is proposed, based on the work ofBalestra and the idea ofDwyer/MacPhail. The two derivative concepts are calledB-type derivative for the form B/A = (b _kl /A), because it was studied fully inBalestra, andA-type derivative for the form B/ /A = (B/a _ij ). Both derivative concepts are linked by permutation matrices, which also reveil the duality aspect more clearly, and can be transformed to each other very easily. The derivatives are applied to least squares estimates and posterior means in the general regression model and the seemingly unrelated regression (SUR)-system, introduced byZellner. The derivatives with respect to the covariance matrix is related to the local sensitivity concept ofLeamer [1978], while the derivatives with respect to the data-matrix, also called local resistance, is linked with the robustness concept ofTukey. The newly definedB-derivative enables an easier interpretation of the results.     

Goodness-of-fit test with nuisance regression and scale

In the linear model Y _i = x _i + e _i, i=1,,n, with unknown (, ), {\open R}^p, >0, and with i.i.d. errors e ₁,,e _n having a continuous distribution F, we test for the goodness-of-fit hypothesis H ₀:F(e)F ₀(e/), for a specified symmetric distribution F ₀, not necessarily normal. Even the finite sample null distribution of the proposed test criterion is independent of unknown (,), and the asymptotic null distribution is normal, as well as the distribution under local (contiguous) alternatives. The proposed tests are consistent against a general class of (nonparametric) alternatives, including the case of F having heavier (or lighter) tails than F ₀. A simulation study illustrates a good performance of the tests. Received July 2001     

Das maximale signifikanzniveau des Tests: “LehneH o ab,wennk untern gegebenen tests zur ablehnung führen”

Summary On one sample space there aren tests with critical regionsK ₁ and levels of significance _i ,i=1, ...,n (resp.n eventsK _i in a probability space with probabilities not greater than _i ,i=1, ...,n). In this paper we calculate the smallest upper bound of the level of significance of the test reject the hypothesis, if at leastk among the,n tests do so (resp. of the probability of the event at leastk among then events are realized). By the way, we will show, that this smallest upper bound does not change, if we replace at leastk by exactlyk.     

Two-dimensional Bernstein polynomial density estimators

A Bernstein polynomial estimator for f_nN(x, y) for an unknown probability density functionf(x, y) concentrated on the triangle ={(x, y): 0x, y<1,x+y<1} or on the square =(x, y):0 x, y 1 is developed. As a measure of quality the exact order of magnitude for the pointwise mean squared error is established. It is seen that the quality of these Bernstein polynomial estimators is comparable with the quality of the so-called kernel estimators. Further for such estimators uniform weak consistency results and central limit theorems are developed.     

Charakterisierung der zweiseitigen Exponentialverteilung

LetF () be a family of distribution functions with a translation parameter such thatF (0) has a densityf. It is well known that each sample median is a maximum likelihood estimate of , iff belongs to the classE of all bilateral exponential densities which are symmetric about 0. Here it is shown that, conversely,fE holds, either if there is an evenm such that for every sample of sizem each median is an MLE of , or if there is an infinite setM such that for every sample of any sizemM at least one median is an MLE of .     

On the power of nonparametric changepoint-tests

We consider a sequenceX _1n,..., X_nn, n N, of independent random elements. Suppose there exists a [0, 1) such thatX _1n,...,X _(n),n have the distribution v₁ andX _[n]+1.n ,...,X _nn have the distribution v₂v₁. We construct consistent level- tests forH ₀:=0 versusH ₁:(0, 1), which are based on certainU-statistic type processes. A detailed investigation of the power function is also provided.     

Estimation problems for rectangular distributions (or the taxi problem revisited)

The problem of estimating the unknown upper bound on the basis of a sample of sizen from a uniform or rectangular distribution on [0, ] has considerable interest. This or the analogous discrete version is variously known as the Taxi-problem or the German bomb (or Tank) problem and has a long history. The emphasis here is on estimation of through the lengths of the observed gaps or spacings which seem natural for this problem.     

Minimal sufficient statistics for the group divisible partially balanced incomplete block design (GD-PBIBD) with interaction under an Eisenhart Model II

Summary In this paper, an Eisenmhart Model II with interaction for a GD-PBIB design withp replicates per cell is considered. Specifically the Model Y_ijl=µ+_i+_j+()_ij+e_ijl is assumed, wherei=1, 2, ...,b; j=1, 2, ...,t andl=0, 1, 2, ...p s _ij wheres _ij=1, if treatmentj appears in blocki, 0, otherwise.If _i, _j, ()_ij ande _ijl are normally and independently distributed, then a minimal sufficient (Vector-valued) statistic for the class of densities for this model is found, together with the distribution of each component in the minimal sufficient statistic. It is also shown that the minimal sufficient statistic for this class densities is not complete. Hence the solution of the problem of finding minimum variance unbiased estimators of the variance components is not straightforward.     

r-te Momente von Punktprozessen

Zusammenfassung {X (t): tR ⁺} sei ein Punktprozeß,H (x) eine konvexe nicht-negative Funktion. Mit Hilfe der bedingten Wahrscheinlichkeitenp _n (t) für genaun Ereignisse (Punkte) im Zeitpunkt punktt unter der Bedingung, daß im Zeitpunktt mindestens ein Ereignis eintritt, wird eine Beziehung formuliert, die für die Existenz des ErwartungswertesE (H (X (t ₀))) notwendig ist. Hat der Punktprozeß unabhängige Zuwächse, und erfüllt die FunktionH (x) einige weitere Bedingungen, so ist die angegebene Beziehung auch hinreichend für die Existenz dieses Erwartungswertes. Für Punktprozesse mit unabhängigen Zuwächsen ergibt sich als unmittelbare Anwendung dieser Aussagen eine notwendige und hinreichende Bedingung für die Existenz vonE X (t ₀) ^r für reellesr1.

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Summary Let {X (t): tR ⁺} be a point process andH (x) a convex non-negative function. Using the conditional probabilitiesp _n (t) thatn events (points) occur at timet given that at least one event occurs att a condition is formulated which is necessary for the existence ofE (H (X (t ₀))). This condition is sufficient, too, if the point process has independent increments and the functionH (x) fulfils some further conditions. Using these statements one gets a necessary and sufficient condition for the existence ofE X (t ₀) ^r for realr1.

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Herrn ProfessorWeissinger zum 65. Geburtstag am 12. Mai 1978 gewidmet     

Vergleich des verallgemeinerten nichtzentralent-Tests mit dem Binomialtest

Zusammenfassung Es sei {F _,(x); –<<, >0} mitF _,(x)=F((x–)/)–F(x) eine standardisierte Verteilungsfunktion — die Familie der zulässigen Verteilungsfunktionen. Der (früher eingeführte) verallgemeinerte nichtzentralet-Test für die Hypothese {PP ₀} mitP:=F _,(x ₀) gegen die Alternative {P>P ₀} zum Niveau wird mit dem entsprechenden nichtparametrischen Test (Test für die Hypothese {pP ₀} über den Parameterp einer Binomialverteilung gegen die Alternative {p>P ₀}) verglichen. Für dent-Test wird die relative asymptotische Effizienz bestimmt.Beide Tests lassen sich als Tests für das zur WahrscheinlichkeitP ₀ gehörende Quantil einer Verteilungsfunktion interpretieren. Der klassische zentrale Student-Test ergibt sich als Spezialfall (F(x)=(x),P ₀=0,5).

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Summary Let {F _,(x);–<<, >0} withF _,(x ₀):=F((x–)/–F(x) a standarized distribution function — the family of admissible distribution functions. The (earlier introduced) generalized noncentralt-test for the hypothesis {PP ₀} withP:=F _,(x ₀) against the alternative {P>P ₀} at level of significance is compared with the corresponding nonparametric test (Binomial test). The relative asymptotic efficiency of thet-test is determined. Both kinds of tests can be interpreted as quantiltests. In caseF(x)=(x),P ₀=0,5 one gets the classical central Student-test.

     

Confronto fra eventi condizionati di probabilità nulla nell'inferenza statistica Bayesiana

Dopo aver introdotto delle operazioni logiche fra eventicondizionati (che estendono opportunamente quelle usuali fra eventi), si definisce il concetto di iperprobabilità condizionata Si può così dare una interpretazione significativa della pseudodensità condizionata, introdotta da R. Scozzafava nell'inferenza statistica bayesiana.

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By a suitable extension of the usual algebra of events, logical operations forconditional events are introduced. This leads to a definition of conditional iperprobability, which is the natural tool for a sensible interpretation of the concept of conditional pseudodensity, introduced by R. Scozzafava in Bayesian statistical inference.

     

Die messende Prüfung bei Abweichung von der Normalverteilungsannahme

Summary For sampling inspection by variables in the one-sided case (item bad if variablex>a) under the usual assumption of normality with known variance ₂ the operating characteristic is given by , wherep denotes the fraction defective. If instead of a normal distribution ((·–a–)/) there is a distributionF((·–a–)/) whereF is sufficiently regular and normed like , one has the approximative operating characteristic . It is shown that for arbitrarily fixed parametersn andc the function takes the valueL _n,c ⁽⁾ (p) at the pointp _F (p)=1–F(–^–1(p)). Sufficient conditions for a simple behavior of the differencep _F (p)–p are given. In the cases of rectangular and symmetrically truncated normal distribution these conditions are shown to be fulfilled.     

Sulla dipendenza continua della probabilità di rovina dal caricamento dei premi

Si studia un modo di approssimare la probabilità di rovina relativa a un caricamento ₀ con le probabilità di rovina relative a una successione di caricamenti ( _k ) _k , che approssimano ₀ quandok tende all'infinito.

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Summary In this paper we study a way of approximating the probability of ruin related to a loading ₀, by the probabilities of ruin related of a sequence of loadings ( _k ) _k which «approximate» ₀ ask converges to infinity.

     

RAM: A Range Adjusted Measure of Inefficiency for Use with Additive Models,and Relations to Other Models and Measures in DEA

Generalized Efficiency Measures (GEMS) for use in DEA are developed and analyzed in a context of differing models where they might be employed. The additive model of DEA is accorded a central role and developed in association with a new measure of efficiency referred to as RAM (Range Adjusted Measure). The need for separately treating input oriented and output oriented approaches to efficient measurement is eliminated because additive models effect their evaluations by maximizing distance from the efficient frontier (in ₁, or weighted ₁, measure) and thereby simultaneously maximize outputs and minimize inputs. Contacts with other models and approaches are maintained with theorems and accompanying proofs to ensure the validity of the thus identified relations. New criteria are supplied, both managerial and mathematical, for evaluating proposed measures. The concept of approximating models is used to further extend these possibilities. The focus of the paper is on the physical aspects of performance involved in technical and mix inefficiencies. However, an Appendix shows how overall, allocative and technical inefficiencies may be incorporated in additive models.     

Eine simultane Charakterisierung der geometrischen Verteilung und der logistischen Verteilung

Summary LetE denote the generating function of a non-degenerate probability distribution on the positive integers, letF be a non-degenerate distribution function, and let be a real valued function on the interval (0, 1]. In the present paper the solutions (E, F, ) of the functional equationE(F(x)/E()=F(x+()), –<x<+, (0, 1] are given. It is shown that ifF is symmetric about zero,E andF belong to a solution of the functional equation if and only ifF is a logistic distribution andE is the generating function of a geometric distribution.