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     

Estimation of a regression coefficient after two preliminary tests of significance

Summary The estimation of the regression coefficient of a population, defined byE (y)= +x, incorporating two preliminary tests of significance has been discussed. The experimenter has two random samples of different sizes from two such populations, as defined above, with regression coefficients ₁ and ₂ respectively, where ₂ may possibly be equal to ₁. Besides this, it is also conjectured that the common conditional variance ² of the two populations has a specified value ₀ ² . The two preliminary tests are used to resolve these two uncertainties.The author has rejoined Lucknow University, Lucknow India on Oct. 4, 1976 after availing leave for two years.     

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.     

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.     

Optimality of block designs with variable block sizes and random block effects

We consider the simple block model with random block effects, the block effects having variance _b ² =²;, with ² the variance of the errors. It is assumed that the experimenter can vary the sizes of the blocks. The universal optimality of certain designs for all over all designs with the same number of blocks and the same number of observations is shown. It is of interest to note that if Balanced Incomplete Block Designs compete, then they perform equally well for =0 and for =, i.e. in the one way classification model and in the simple block model with fixed block effects, but they perform worse for every (0, ).The result is, however, theoretical in nature. It treats a situation which is not very likely to happen in practice. The interest lies in the fact that it provides a counterexample to a conjecture on optimality of designs in mixed models.     

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.     

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.     

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.

     

A conventional statistical problem with irregular minimax behavior

Summary The problem of testing the mean vector of the two dimensional circularly symmetrical normal distribution with unit variances, where the data consists of just one sample point inR ², is examined for stability of -maximin criteria. If the null hypothesisH ₀ is the one point set containing the origin and the alternative set equal to the whole ofR ²–H ₀, then the -maximin is not unique. If a zone of indifference _I containingH ₀ is introduced, then the problem of testingH ₀ againstR ² – _I can turn out to have a unique -maximin test. In the present paper we show a class of such _I for which this is the case. We show further that, given any -maximin test for testingH ₀ againstR ²–H ₀, there is a decreasing sequence of _I , with intersection equal toH ₀, for which the corresponding sequence of -maximin tests forH ₀ againstR ² – _I approaches a limit (in the usual weak star topology) which is not equivalent to .     

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.     

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.     

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)).

     

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.     

Berechnung des maximalen Signifikanzniveaus des Testes „LehneH 0 ab,wennk untern gegebenen Tests zur Ablehnung führen”

Summary If for a simple hypothesis there aren tests with critical regionsK _i and level of significance _i a new test may be constructed with the rule reject the hypothesis if at leastk among then tests reject. The level of significance of this compound test was calculated byRüger [1978] in this journal; a simple proof is supplied here.     

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.     

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.     

Hypocrisy pay

Emotional labor—putting on an act asrequired explicitly orimplicitly in work situations-is very often psychologically, emotionally, and mentally damaging to most employees. As for soldiers in wartime who receive combat pay, it is suggested that such employees be specially compensated for their emotional labor with hypocrisy pay. Another solution is to select only those who are immune to such stress for jobs involving emotional labor—an ideal that probably cannot be achieved in all cases. Another assessment is that such special compensation is unwarranted.     

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).     

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.