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.

     

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.     

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.     

Una classe di funzioni monotone generalizzate

Si considera la classe delle funzioni realiF(x,y) definite inS×S, conSR ^N , che soddisfano per ognix,yS la condizione di monotoniaF(x,y)+F(y,x)0. Indebolendo la precedente disuguaglianza si introducono classi di funzioni monotone generalizzate e, supponendo soddisfatta una opportuna condizione di omogeneità, si caratterizzano tali funzioni in base alla struttura del segno delle funzioni _{x, v} (t, s) = F(x + tv, x + sv), x S, v R ^N \{0}. Infine dopo aver definite le funzioni F-differenziabili, si introducono classi di funzioni conversse generalizzate, rispetto ad F, e si studiano i collegamenti tra queste classi e la monotonia generalizzata diF.

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Summary We consider the class of real valued functionF(x,y) defined inS×S, withSR ^N , satisfying x,yS the monotone conditionF(x,y)+F(y,x)0. Weakening the previous inequality we introduce the class of quasi-monotone, pseudo-monotone and strictly pseudo-monotone functions. Under a suitable assumption of homogeneity we characterize the generalized monotone functions studying the sign structure of the functions _{x, v} (t, s) = F(x + tv, x + sv), x S, v R ^N \{0}.Finally by means of the notion ofF-differentiability we introduce new classes of generalized convex functions (with respect toF) and we study the relationship between these classes and the generalized monotonicity ofF.

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Questa ricerca è stata parzialmente finanziata dal Ministero per l'Università e la Ricerca Scientifica.     

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.     

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 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].     

A curtailed test for the shape parameter of the Weibull distribution

Summary A procedure is proposed in this paper for testing the shape parameter, of the Weibull distribution. The test statistic which is based on the extremal quotient, possesses a monotone property which makes it possible for rejection earlier than the last planned observation of the null hypothesis,H ₀: =₀ when the alternative hypothesis isH _a: <₀ and early acceptance ofH ₀ whenH _a: >₀. The test being scale-free, does not require the scale parameter to be known.     

Realization of closed,convex, and symmetric subsets of the unit square as regions of risk for testing simple hypotheses

It is well-known that the region of risk for testing simple hypotheses is some closed, convex, and (1/2, 1/2)-symmetric subset of the unit square, which contains the points (0, 0) and (1, 1). It is shown that for any such subsetR of the unit square and any atomless probability measureP on some -algebra there exists some probability measureQ on the same -algebra such thatR is the corresponding region of risk for testingP againstQ. This generalizes a result of [4] and is as a first step derived here for the special case, whereP is equal to the uniform distribution on the unit interval. The corresponding distributionQ is given explicitly in this case and the general case is treated by some well-known measure-isomorphism. This method of proof shows thatQ might be chosen to be of typeQ=Q ₁+(1–)Q ₂ for some satisfying 01, whereQ ₁ is a probability measure, which is absolutely continuous with respect toP andQ ₂ is a one-point mass.     

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.     

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).     

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.

     

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.     

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.

     

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.     

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 .     

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     

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.     

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.     

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}, ....