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)).

     

Inference ony>y o by using a correlated variable: A three decision approach

Summary In an extension of the two decision approach [Bauer, Scheiber andWohlzogen, 1975] a Bayes solution is aimed at for the three decisiony>y _o,yy _o or no classification on the basic of the measurement of a positively correlated random variableX, which can be measured more easily and/or with smaller expense. Assuming a bivariate normal distribution forX andY optimal decision regions for the measuredx are derived in the case of constant or exponentially increasing losses.

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Zusammenfassung In Erweiterung des Zwei-Entscheidungsproblems [Bauer, Scheiber undWohlzogen, 1975] wird eine Bayes-Lösung für die drei Entscheidungeny>y ₀,yy ₀ oder keine Zuordnung aufgrund der Messung einer mitY positiv korrelierten, einfacher und/oder billiger zugänglichen ZufallsvariablenX angestrebt. Optimale Entscheidungsbereiche für die Messungenx werden bei Voraussetzung einer bivariaten Normalverteilung fürX undY unter der Annahme konstanter oder exponentiell wachsender Verluste bestimmt.

     

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.     

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.     

Über eine Variante der empirischen Verteilungsfunktion

Summary If ₁, ₂,..., _n and ₁, ₂,..., _–1 are two ordered samples from a population with continuous distribution functionF(x), then the points ( _r ,r/n),r=1, 2,..., n–1 provide a better approximation ofF(x) than the points ( _r ,r/n),r=1, 2,..., n, in the following sense:A maximal upper deviation and a maximal lower deviation of more theny have — contrary to the points ( _r ,r/n) — equal probability for anyy0, if we deal with the points ( _r ,r/n). This probability is at least for ally in the interval , 1 less than the probability for a maximal upper deviation of more thany in the case of the points ( _r ,r/n). This is shown by a comparison of the Smirnow-Birnbaum-Tingey — formula with an analogous formula for the maximal one-sided deviations of the points( _r ,r/n).     

Wrapped distributions and measurement errors

Summary For a random variableX and >0 letU n (X)–X, wheren (x)=nZ iffx(n–/2,n+/2]. Random variables of this type are important in the theory of measurement errors. We derive formulas for the distribution ofU and apply them to the case XN(,²). General conditions for the unimodality ofU are given. The correlation of the measurement errorsX–E (X) andU (X) is seen to beO (^j) withj depending on the smoothness and asymptotic behavior of the density ofX. This gives a precise sense to the assertion that scale errors upwards and downwards are averagely well-balanced. In the normal case the density ofU is shown to be constant up to , as 0.     

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.

     

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.     

Ulteriori contributi a un problema di Leo Moser

Si dimostrano condizioni necessarie e sufficienti relative a punti di Kuhn-Tucker per il «problema dei dadi truccati» e viene proposto un algoritmo per la ricerca di tali punti, tramite una successione di programmi lineari.

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The author's version of the «loaded dice problem» asks for x₁ to be maximum subject tox0 andx ^T H _i x1, whereH _i is the Hankel matrix of the (2n–1)-dimensional unity vectore ⁱ (i=1,..., 2n–1).Proofs are given here about necessary and sufficient conditions for Kuhn-Tucker points, together with an algorithm for finding them by means of a sequence of linear programs.

     

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 .     

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.     

On monotone optimal decision rules and the stay-on-a-winner rule for the two-armed bandit

Summary Consider the following optimization problem: Find a decision rule such thatw(x, (x))=max _a w(x, a) for allx under the constraint (x)D (x). We give conditions for the existence of monotone optimal decision rules . The term monotone is used in a general sense. The well-known stay-on-a-winner rules for the two-armed bandit can be characterized as monotone decision rules by including the stage number intox and using a special ordering onx. This enables us to give simple conditions for the existence of optimal rules that are stay-on-a-winner rules. We extend results ofBerry andKalin/Theodorescu to the case of dependent arms.     

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).     

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.     

Locally minimax tests for a multinormal data problem

LetX be ap-normal random vector with unknown mean and unknown covariance matrix and letX be partitioned asX=(X ₍₁₎ ,X ₍₂₎ , ...,X _(r) ) whereX _(j) is a subvector of dimensionp _j such that _j=1 ^r p _j =p. We show that the tests, obtained by Dahel (1988), are locally minimax. These tests have been derived to confront H_o: =0 versusH ₁: 0 on the basis of sample of sizeN, X ₁, ..., X_N, drawn fromX andr additional samples of sizeN _j, U _i ^(j) , i=1, ..., N_j, drawn fromX ₍₁₎, ...X _(r) respectively. We assume that the (r+1) samples are independent and thatN _j>p _j forj=0, 1, ..., r (N _oN andp _op). Whenr=2 andp=2, a Monte Carlo study is performed to compare these tests with the likelihood ratio test (LRT) given by Srivastava (1985). We also show that no locally most powerful invariant test exists for this 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     

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.

     

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     

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.     

Über die Existenz überall trennscharfer Tests im nicht-dominierten Fall

Einleitung Es sei (,A) ein Meßraum undP eine Familie von Wahrscheinlichkeitsmaßen auf . IstP durch ein -endliches Maß dominiert, so ist nachPfanzagl [1960] für die Existenz eines überall trennscharfen Tests zu jedem Niveau für die HypotheseP=P _o (P, P _o P) gegen die AlternativePP ₀ notwendig und hinreichend, daßP/{P ₀} isotonen Likelihood-Quotienten bzgl.P ₀ besitzt. Für den Fall, daßP total geordnet und dominiert ist, gilt nach [Pfanzagl, 1963] eine entsprechende Aussage: Genau dann existiert zu jedem Niveau ein überall trennscharfer Test, wennP isotonen Likelihood-Quotienten besitzt.Die vorliegende Arbeit zeigt, daß auf die Annahme der Dominiertheit verzichtet werden kann, und liefert darüber hinaus einen einheitlichen Beweis für die beiden oben zitierten Sätze vonPfanzagl.

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Summary Two theorems ofPfanzagl [1960, 1963] about necessary and sufficient conditions for the existence of uniformly most powerful tests are generalized to the undominated case. Moreover a unified proof for the two theorems ofPfanzagl is given.