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.     

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.

     

On Bayesian optimum allocation in some two phase sampling problems

Summary For estimating certain parametric functions, we consider the problem of allocatingN _i, the size of the sample from theith population,i=1,2,...,k, at the second phase of sampling of a two phase sampling procedure, given that we taken _i observations from the population at the first phase. We consider that the observations from theith population follow the exponential distribution with mean _i,i=1,2,...,k, and the functions to be estimated are (i) (d_i/_i) and (ii) (d_ii). When the total cost of sampling at the second phase is c _iN_i and is fixed, allocations using the Bayes approach are obtained so that the estimation is as precise as is possible.     

Maximal asymptotic power and efficiency of two-sample tests based on generalized U-Statistics

In this article a systematic study is given of the asymptotic behavior of two-sample tests based on U-Statistics with arbitrary antisymmetric kernels . Besides the investigation under the hypothesis and under fixed alternatives we determine the local power as a function of as well as its maximizing value _opt. Moreover formulas for the asymptotic relative efficiency ARE(₂,₁) of the ₂-test with respect to the ₁-test are derived. It turns out that _opt also yields the most efficient test in the sense that ARE(_opt,)1 for all (admissible) kernels .     

Continuous representations of interval orders based on induced preorders

Conditions are given for the existence of a pair of continuous real valued functions (u, v) representing an interval order on a topological space (X, ), in the sense that [xy if and only ifu(x)].I am deeply indepted to Professor Alain Chateauneuf for many valuable suggestions.     

Hypocrisy pay

Emotional labor—putting on an act asrequired explicitly orimplicitly in work situations-is very often psychologically, emotionally, and mentally damaging to most employees. As for soldiers in wartime who receive combat pay, it is suggested that such employees be specially compensated for their emotional labor with hypocrisy pay. Another solution is to select only those who are immune to such stress for jobs involving emotional labor—an ideal that probably cannot be achieved in all cases. Another assessment is that such special compensation is unwarranted.     

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)).

     

Die asymptotische Verteilung des Prognosefehlers bei Prognosen mit Hilfe eines dynamischen ökonometrischen Modells höherer als erster Ordnung

Zusammenfassung In dieser Arbeit wird die asymptotische Verteilung des Prognosefehlers, wie er sich im Rahmen einer dynamischen Simulation eines allgemeinen autoregressiven ökonometrischen Modells der Ordnungp (einschließlich verzögerter exogener Variabler der Ordnungq) ergibt, abgeleitet. Daran anschließend werden einige Fragestellungen, die damit in unmittelbarem Zusammenhang stehen, diskutiert: Die Frage der relativen Effizienz der Prognoseschätzung, basierend auf der unrestricted- bzw. der derived reduced form, die Verwendung der asymptotischen Verteilung des Prognosefehlers für einen predictive test des Modells. Außerdem werden asymptotische simultane Prognoseintervalle abgeleitet.

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Summary The asymptotic distribution of the forecast error in the dynamic simulation of a higher than first order linear dynamic econometric model is derived and related topics are discussed.

     

Effect of outliers on the estimation of parameters

Summary The estimation of parameter in the type of distributionf(x)=b x ^{–1 /b} exp (–x ^b (/b),x>0, is considered, when several outliers of the type , ,r=1,2, ...,k, are present in the data. The estimates of as well as of 's are put in the closed form. Special cases, Weibull, Gamma and Exponential are considered for the case of single outlier. Actual estimates are calculated from the generated samples of size 2 and 3 for the Weibull and Exponential.     

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.     

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.     

Scharfe untere und obere Schranken für die Wahrscheinlichkeit der Realisation vonk untern Ereignissen

Summary LetA ₁,...,A _n be events in a probability space (,A,W). We denote byL _k the event, that at leastk events among then eventsA ₁,...A _n occur, and byK _k the event, that exactlyk events occur. If only the inequalities _i W(A _i ) _i ,i=1,...,n, are known, we calculate sharp lower and upper bounds forW(L _k ) andW(K _k ). These bounds only depend onn, k and _i , _i ,i=1,...,n. They are relevant, when treating combined tests or confidence procedures.     

Characterizations of distributions via linearity of regression of generalized order statistics

Let X (r, n, m, k), 1 r n, denote generalized order statistics based on an absolutely continuous distribution function F. We characterize all distribution functions F for which the following linearity of regression holds E(X(r+l,n,m,k) | X(r,n,m,k))=aX(r,n,m,k)+b.We show that only exponential, Pareto and power distributions satisfy this equation. Using this result one can obtain characterizations of exponential, Pareto and power distributions in terms of sequential order statistics, Pfeifers records and progressive type II censored order statistics. Received July 2001/Revised August 2002     

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.     

Die Charakterisierung der Normalverteilung nach Gauß

A proof is offered for the best possible version of the following Gauss type characterization of normality: LetF () be a family of distribution functions with translation parameter such thatF (0) has a densityf with certain regularity properties, and letM{1, 2, ...}. If the mean of every sample of any sizemM is a maximum likelihood estimate of , thenF (0) is normal with zero expectation. While in the best prior version of this theorem,f satisfies a continuity assumption andM={2, 3}, here no regularity condition is needed, andM can be any of the sets {3}, {4}, ....     

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.

     

Der diskontierte Einarmige Bandit

Summary A sequential stochastic process is observed in discrete time intervals; the process is generated by Bernoulli trials with unknown meanp. Given an a-priori distribution forp, regarded as a random variable, and discounting future payoffs with a factor , O<<1, optimal and suboptimal stopping rules (depending on ) are constructed. this leads to the connection of the process under consideration with another process of the same structure, but with known meanP _o for the Bernoulli trials, thus finally resulting in an One-Armed-Bandit problem.     

Sulla convergenza di un algoritmo di direzioni ammissibili per problemi di ottimo a vincoli lineari con funzione oggetto convessa non differenziabile

Se, essendof la funzione obiettivo del problema, {x ^k } e {f(x ^k )} sono le successioni delle approssimazioni rispettivamente di una soluzione ottimax ^* e dell' ottimof(x ^*) generate da un noto algoritmo di direzioni ammissibili a parametri antizigzag _k , mostriamo che per avere (a) lim _k f(x ^*)=f(x ^*) basta assumere lim _{k k} =0. Inoltre, ove si assuma in più la stretta convessità dif, si ha anche (b) lim _k x ^k =x ^*. Da quest'ultima condizione deriviamo infine specifiche ipotesi, in ordine alla (b), per il caso particolare del problema di trasporto stocastico.

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Summary The aim of the present paper is to analyze, without differentiability of the objective functionf, the convergence of a known «feasible directions» algorithm for constrained optimization problems having the constraints linear [8], 6.5.2.In these circumstances (i.e. iff is not differentiable) one must, almost in general, verify some preliminary conditions to obtain convergence [4]. Nevertheless, this work is not always easy to accomplish particularly in absence of differentiability.Here, we establish that under the convexity assumption forf, the only condition lim _{k k} =0, where the _k are the antizigzag parameters, suffices to obtain the convergence of the algorithm, i.e. lim _k f(x ^k )=opt., thex ^k being the approximate solutions to problem. The proof is obtained by application of the Th. 24.5, [6]. Successively, we consider the question if one has also the convergence of {x ^k } to optimal solution. By using now the Cor. 27.2.2, [6], we establish, for this purpose, that under an additional general qualification forf — precisely the strict convexity — the convergence of {x ^k } is also stated. Finally, we examine the above property for the stochastic transportation problem [1] for which we indicate special conditions in order to verify the latter convergence property.

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pervenuto il 28-4-82     

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.     

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