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 .     

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.     

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 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].     

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.     

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().     

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.     

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.

     

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.     

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).     

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.     

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.     

A Nonclassical Law of the Iterated Logarithm for Functions of Positively Associated Random Variables

Let be a sequence of stationary positively associated random variables and a sequence of positive constants be monotonically approaching infinity and be not asymptotically equivalent to loglog n. Under some suitable conditions, a nonclassical law of the iterated logarithm is investigated, i.e.

Runs in an ordered sequence of random variables

Let be independent and identically distributed random variables with continuous distribution function. Denote by the corresponding order statistics. In the present paper, the concept of -neighbourhood runs, which is an extension of the usual run concept to the continuous case, is developed for the sequence of ordered random variables      

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.     

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 .

     

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.     

Convergence inr-mean of normalized partial sums in a separable Banach space

Ver {V _n } be a sequence of random elements in a separable Banach space. Conditions are obtained under which .     

Loss functions and admissible estimators in survey sampling

Summary Consider simple random sampling (without replacement) of a fixed size and lett ₀ be the sample mean, i.e. the arithmetic mean of all variate values observed in the sample. The class {t ₀: real} of estimators for the population mean (i.e. The arithmetic mean of all variate values) then surely is of interest.We discuss different types of variates, in particular variates with positive values only. For these variates the usual square error loss gives rise to a strange class of admissible estimators. An other type of loss functions seems far more appropriate. For this (logarithmic) type, t ₀ is admissible iff =1. We prove that there exists no other type of loss functions with the property that the unbiased estimatort ₀ is the only admissible element of the class {t ₀: >0}.

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Zusammenfassung Bei fest vorgegebenem Stichprobenumfang werde uneingeschränkt zufällig ausgewählt (ohne Zurücklegen);t ₀ bezeichne das Stichprobenmittel, d.h. das arithmetische Mittel aller Stichprobenbeobachtungen des Untersuchungsmerkmals. Von besonderen Interesse ist dann zweifellos die Klasse {t ₀: reell} von Schätzfunktionen für das Gesamtmittel, d.h. für das arithmetische Mittel aller Ausprägungen des Untersuchungsmerkmals.Wir betrachten verschiedene Typen von Untersuchungsmerkmalen, insbesondere Merkmale, die nur positive Ausprägungen besitzen. Für diese Merkmale führt die Verwendung der üblichen quadratischen verlustfunktion zu einer sehr merkwürdigen Klasse zulässiger Schätzfunktionen. Ein anderer Typ von Verlustfunktionen erscheint weit eher angebracht. Für diesen (logarithmischen) Typ ist t ₀ genaudann zulässig, wenn =1 gilt. Wir beweisen, daß für keinen anderen Typ von Verlustfunktionent ₀ das einzige zulässige Element der Klasse {t ₀:>0} ist.