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.

     

Eine simultane Charakterisierung der geometrischen Verteilung und der logistischen Verteilung

Summary LetE denote the generating function of a non-degenerate probability distribution on the positive integers, letF be a non-degenerate distribution function, and let be a real valued function on the interval (0, 1]. In the present paper the solutions (E, F, ) of the functional equationE(F(x)/E()=F(x+()), –<x<+, (0, 1] are given. It is shown that ifF is symmetric about zero,E andF belong to a solution of the functional equation if and only ifF is a logistic distribution andE is the generating function of a geometric distribution.     

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.     

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     

A note on optimum spacing of observations from a continuous time simple Markov process

Summary Assume thatX() is a continuous time simple Markov process with a parameter . The problem is to choose observation points ₀ < ₁ <...<_T which provide with the maximum possible information on . Suppose that the observation points are equally spaced, that is, fort=1, ...,T, T, ;_t–_t–1 is constant. Then the optimum value fors is obtained.     

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.     

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     

Minimal efficiency of designs under the class of orthogonally invariant information criteria

Consider the linear regression model with uncorrelated errors and an experimental design . In the article, we address the problem of calculating the minimal efficiency of with respect to the class of orthogonally invariant information criteria, containing all Kiefers criteria of _p-optimality, among others. We show that the -minimal efficiency of is equal to the minimal efficiency of with respect to a finite class of criteria which generalize the criterion of E-optimality. We also formulate conditions under which a design is maximin efficient, i.e. the most efficiency-stable for criteria from . To illustrate the results, we calculated the -minimal efficiency of _p (in particular D, A and E) optimal designs for polynomial regression on [–1,1] up to degree 4. Moreover, for the quadratic model we explicitly constructed the -maximin efficient design.Acknowledgement. The author would like to thank prof. Pázman as well as an anonymous referee for useful and inspiring comments on earlier versions of this article.Supported by: VEGA grant of the Slovak Agency No. 1/0264/03     

Erneuerungsprozesse mit Wartungsstrategien

Summary The following renewal process is considrred: given intervals (kt ₁,(k+1)t ₁],k=0, 1, 2, ..., 0<t ₁<, there will be with probabilityp, 0p1, a renewal in each interval at a time selected by random. The costs for each of this renewals are a units, while the costs of the other renewals areb units each. The renewal function and the cost function are derivided and their asymptotic behavior is discussed.     

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().     

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}, ....     

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.     

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.     

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.     

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.     

Charakterisierung der zweiseitigen Exponentialverteilung

LetF () be a family of distribution functions with a translation parameter such thatF (0) has a densityf. It is well known that each sample median is a maximum likelihood estimate of , iff belongs to the classE of all bilateral exponential densities which are symmetric about 0. Here it is shown that, conversely,fE holds, either if there is an evenm such that for every sample of sizem each median is an MLE of , or if there is an infinite setM such that for every sample of any sizemM at least one median is an MLE of .     

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.     

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.     

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.     

Das Gesetz des iterierten Logarithmus für kumulative Prozesse

Summary Smith [1955] defined cumulative processes. He proved a strong law of large numbers and a central limit theorem for such processes. This paper deals with the Hartman-Wintner law of the iterated logarithm for cumulative processes using that one for sums [Hartmann/Wintner] and that one for renewal counting processes [Neffke, 1980; cf. alsoWalk].

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Für identisch verteilte Zufallsvariablen wird der Index in Ausdrücken wieE (),P fortgelassen, alsoP , steht fürP _i,i2.

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Im folgenden beschränkt man sich aufv _t()< für allet, also P. Dieses ist mit Wahrscheinlichkeit Eins erfüllt.