Transformationen,die die Normalverteilung charakterisieren

Zusammenfassung Es sei A: R ⁿ R ⁿ eine Abbildung mit für jedes sei einn-dimensionaler Zufallsvektor. Wir beschreiben die Klasse aller TransformationenA, für die unabhängige, nachN(0, 1) verteilte Komponenten hat, sofern nur die KomponentenX ₁,...,X _n des Zufallsvektors ebenfalls unabhängig und identish Gaußisch verteilt sind mit Erwartungswert Null und Varianz 1. Weiter sind Bedingungen angegeben, die sicherstellen, daß nachN(O, ²) verteilte KomponentenX ₁,...,X _n hat, sofern dieX ₁,...,X _n unabhängig und und identisch verteilt sind. Zwei vonBeer undLukacs behandelte Transformationen sind Spezialfälle der hier untersuchten Transformationen.

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Summary Let A: R ⁿ R ⁿ be a transformation with the property for every . We consider a random vector and characterize the class of all transformationsA such that has independentN (0, 1) distributed componentsY ₁,...,Y _n if has the same distribution. Furthermore in the paper there are given conditions which ensure that hasN(O, ² distributed components if and are identically distributed and the componentsX ₁,...,X _n are independent, identically distributed random variables. Two of the transformations tried byBeer andLukacs are special cases of our transformations.

     

Contributions to nonparametric generalized failure rate function estimation

For a specified distribution functionG with densityg, and unknown distribution functionF with densityf, the generalized failure rate function (x)=f(x)/gG ^–1 F(x) may be estimated by replacingf andF byf _n and , wheref _n is an empirical density function based on a sample of sizen from the distribution functionF, and . Under regularity conditions we show and, under additional restrictions whereC is a subset ofR and _n. Moreover, asymptotic normality is derived and the Berry-Esséen type bound is shown to be related to a theorem which concerns the sum of i.i.d. random variables. The order boundO(n^–1/2+c _n ^1/2 ) is established under mild conditions, wherec _n is a sequence of positive constants related tof _n and tending to 0 asn.Research was supported in part by the Army, Navy and Air Force under Office of Naval Research contract No. N00014-76-C-0608. AMS 1970 subject classifications. Primary 62G05. Secondary 60F15.     

A note on minimum variance

Summary Minimizing is discussed under the unbiasedness condition: and the condition (A):f _i (x) (i=1, ..., p) are linearly independent , and .     

Rate of convergence in the central limit theorem and in the strong law of large numbers for von mises statistics

This paper provides the rate of convergence in the central limit theorem and in the strong law of large numbers forvon Mises statistics , based on i.i.d. random variablesX ₁ ,..., X _N .The proofs rely on a decomposition ofvon Mises statistics into a linear combination ofU-statistics and then use (generalized) results on the convergence rates forU-statistics obtained byGrams/Serfling [1973] andCallaert/Janssen [1978].     

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.     

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().     

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.     

Reproducibility in the one-parameter exponential family

Summary LetX ₁, ...,X _n be i.i.d. random variables with common distribution an element of a linear one-parameter exponential family indexed by a natural parameter . It is proved that the distribution of is an element ofF, for all andn=1, 2, ... if and only ifF is a family of scale transformed Poisson distributions.     

A characterization of entropies of degree α

The functional equation has been studied by several authors under various assumptions onf and onk, l. Here we solve this equation iff is measurable andk 3,l 2 are fixed integers. Using the solution we characterize the entropies of degree for all real . Our results generalize the results ofBehara/Nath [1973],Kannappan [1974] andMittal [1976].     

Aspetti dinamici di leggi finanziarie scindibili

Vengono studiate le proprietà delle intensità istantanee di interesse di leggi finanziarie scindibili non necessariamente omogeneef(x, s, t). Esse risultano dipendenti dal montante e dal tempo finale secondo il modello . Ciò porta ad ottenere una naturale corrispondenza fra leggi finanziarie scindibili ed equazioni differenziali ordinarie. Si esaminano in dettaglio i casi particolari di leggi uniformi, leggi omogenee e leggi uniformi-omogenee, individuando la forma delle equazioni differenziali ad esse associate. Si estendono infine i risultati a leggi finanziarie del tipo , che dipendono anche dalla variabile istante decisionale .

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Summary We study the properties of the interest rates of the so-called scindibili financial laws (not necessarily homogeneous)f(x, s, t). They explicity depend on the value off andt only, according to the form . This suggests a natural correspondence between such financial laws and ordinary differential equations.The particular cases of uniform laws, homogeneous laws and uniform-homogeneous laws are examined and the structure of the associated differential equations are obtained.The previous results are extended to the financial laws of type which also depend on a decisional time .

     

Moving averages of ergodic processes

A necessary and sufficient condition for the almost everywhere convergence of the moving ergodic averages is given. The result is then generalized to ergodic flows, and finally constrasted with earlier results ofPfaffelhuber andJain.     

Proprietà degli operatori monotoni per problemi di ottimizzazione vettoriale

In questa nota si assegnano condizioni per la monotonia della funzione di ottimo e si dimostrano alcune relazioni fra problemi reciproci, sia nel caso in cui si considera un problema di ottimizzazione scalare vincolata, che in quello in cui si introduce una funzione obiettivo vettoriale, in presenza di uno o piu vincoli. Si formulano infine alcune applicazioni economiche.

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First of all we consider a scalar problem withk inequality constraints max {f(x):g _i (x)b _i ,i=1, ...,k, xD} and analyze the conditions under which the optimal solution is not strictly dominated with respect to the constraint operatorG(x)=[g ₁(x),...,g _k (x)] in the sense of the vector programming. Furthermore we link optimal solutions for reciprocal problems such as and min {G(x):f(x)=(b), xD}. Since the minimization problem is a vector one we pass to study monotonicity properties for the optimum-value correspondence individuated by a vector optimization problem with one or more constraints in order to establish on one hand that the constraint is active at the solution point, on the other hand that the optimal solution is not strictly dominated with respect to the constraint operator.These results allow us to give an interesting interpretation to some economic problems.

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Lavoro eseguito nell'ambito del Gruppo G.N.A.F.A. del C.N.R. per l'anno 1979.

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Presentato alle Giornate Attuariali Triestine e della A.M.A.S.E.S., Trieste, 14–16 Giugno 1979.     

Suffizienz bei verschiebungsinvarianten Schätzproblemen

Zusammenfassung Es wird gezeigt, daß beim Schätzen eines die Verteilung einer ZufallsgrößeX (mit Dichte) charakterisierenden Lageparameters verschiebungsinvariante FunktionenZ ₁=a ₁(X ₁,...,X _n ),...,Z _m =a _m (X ₁,...,X _n ) dern unabhängigen WiederholungenX ₁,...,X _n vonX genau dann suffizient sind, wenn für jede konvexe Schadensfunktion ein gleichmäßig bestes, nur vonZ ₁,...,Z _m abhängendes verschiebungsinvariantes Schätzverfahren existiert. Weiter wird bewiesen, daßX genau dann normalverteilt ist, wenn zu jeder konvexen Schadensfunktion ein existiert derart, daß ein gleichmäßig bestes verschiebungsinvariantes Schätzverfahren ist.

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Summary LetX ₁,...,X _n be independent random variables with density functionf(x–) and unknown location parameter R ¹; furthermore leta _i (x ₁,...,x _n ),i=1,..., m, be functions which are invariant with respect to translations. ThenZ _i =a _i (X ₁,...,X _n ),i=1,...,m, are sufficient iff for every convex loss functions (.) there exists a functionh(z ₁,...,z _m ) such thath(Z ₁,...,Z _m ) is a best invariant estimate for the location parameter . Furthermore we show thatX ₁,...,X _n is a sample from a normal distribution if for every convex loss functions (.) there exists a constant such that is a best invariant estimate for .

     

Asymptotic distributions of smoothed histograms

As an estimator for an unknown probability density functionf, concentrated on a known intervalI, one can use a histogram smoothed by a suitable family of lattice distributions. For such an estimator a uniform weak consistency result and a central limit theorem with an error bound are given. Further for the global deviation of fromf the asymptotic distribution is developed.Partially supported by the Natural Sciences and Engineering Research Council of Canada, grant A 2983, A4806, and A3988.     

Una nuova caratterizzatione dell'ammissibilità nella teoria delle decisioni

In questo lavoro viene fornita una nuova caratterizzazione dell'ammissibilità attraverso un adeguato uso della nozione di ammissibilità parziale. Questa caratterizzazione consente di affrontare le questioni riguardanti la completezza della classe delle decisioni ammissibiliti sotto condizioni «maneggevoli». Fornisce inoltre un approccio unificante al problema della completezza che consente di derivare, come casi particolari, alcuni risultati già noti nella letteratura sull'argomento.

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In this paper a new characterization of admissibility is given for general decision problems. It is based on an adequate use of the notion of partial admissibility.A general decision problem is usually synthetized by a triplet (, , ) where is the states (or parameters) space, the set of available decisions and is a family of real valued functions defined on and expressing numerically the consequences of choosing when the state is . The set is regarded as a subset of the space of all real valued functions on endowed with the topology of pointwise convergence.As for as admissibility is concerned all the pertinent information about decisions are contained in the corresponding functionsW .This allows to introduce a notion of partial admissibility through the neigh-bourhoods of this topology. Admissibile decisions are then shown to be limits of monotone non increasing sequences of partially admissible decisions.Moreover this topological characterization allows to prove the completeness of classes of admissible decisions under acceptable systems of conditions which contain as special cases, known results in literature.

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Lavoro svolto nell'ambito del Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni del C.N.R.     

A motivation for $$1/ \sqrt{n}$$-shrinking neighborhoods

In this paper we give a motivation for the shrinking rate let p ₀ and q _n be the outlier probability under the ideal model, and some member of a neighborhood about this ideal model of radius r _n , respectively. Assuming n i.i.d. observations, the critical rate of r _n may be defined such that the minimax test for outlier probability q _n =p ₀ versus q _n >p ₀ has asymptotic error probabilities bounded away from 0 and 1/2. Summarizing the neighborhoods to their upper probability, this leads to r _n of the exact rate . The result makes precise and simplifies ideas in Bickel (1981), Rieder (1994), and Huber (1997). Considering general probabilities of exact Hellinger distance r _n to P, this shrinking rate translates into , but leads to the same optimality theory as in the corresponding setup.     

Berry–Esseen bounds for density estimates under NA assumption

Let {X _j } be a strictly stationary sequence of negatively associated random variables with the marginal probability density function f(x). The recursive kernel estimators of f(x) are defined by

Asymptotically optimal nonparametric one- and two-way analysis of variance tests for the log linear model under censoring

We considerr ×c populations with failure ratesλ _ij(t) satisfying the condition