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.     

The restricted regularity of admissible minimax criteria for many well known parametric statistical problems

Summary The stability of test selection criteria is a question that was raised byLeCam in 1964 [seeLeCam, 1964], but which seems to have received very little subsequent attention. Roughly, the problem posed byLeCam was to decide whether a given criterion is regular at a given statistical problem, i.e. given that the criterion prescribes the test ₀ for the statistical problem, (₀,D ₀,R ₀), whether it prescribes a best which is nearly the same as ₀ for any statistical problem, (, D,R), near to (₀, D₀,R ₀).A more precise formulation ofLeCam's question is one of the objectives of this paper. This involves defining a suitable topology on the domain of problems where the criterion in question is to be applied. Elementary minimax results of a general nature are proved, and counter examples are given to show that the assumptions used are not completely superfluous. The generality of these results seems to lend support to the authors opinion, that the method of formulation is suitable for treating minimax questions. We then consider the standard elementary problems of parametric statistics, (monotone likelihood ratio, general exponential family) and prove that the admissible minimax criterion is regular with respect to certain types of perturbations in these problems. The treatment is, in a sense, restricted and departs fromLeCam in holding the risk function essentially constant while varying the parameter space. That the problem variations considered may be of some practical interest and that the general formulations used are not completely unrealistic is shown at the conclusion of the paper where we apply a rest, icted regularity theorem to obtain regularity results for the binomial case.     

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.     

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 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 .     

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.

     

Sulla convergenza di un algoritmo di direzioni ammissibili per problemi di ottimo a vincoli lineari con funzione oggetto convessa non differenziabile

Se, essendof la funzione obiettivo del problema, {x ^k } e {f(x ^k )} sono le successioni delle approssimazioni rispettivamente di una soluzione ottimax ^* e dell' ottimof(x ^*) generate da un noto algoritmo di direzioni ammissibili a parametri antizigzag _k , mostriamo che per avere (a) lim _k f(x ^*)=f(x ^*) basta assumere lim _{k k} =0. Inoltre, ove si assuma in più la stretta convessità dif, si ha anche (b) lim _k x ^k =x ^*. Da quest'ultima condizione deriviamo infine specifiche ipotesi, in ordine alla (b), per il caso particolare del problema di trasporto stocastico.

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Summary The aim of the present paper is to analyze, without differentiability of the objective functionf, the convergence of a known «feasible directions» algorithm for constrained optimization problems having the constraints linear [8], 6.5.2.In these circumstances (i.e. iff is not differentiable) one must, almost in general, verify some preliminary conditions to obtain convergence [4]. Nevertheless, this work is not always easy to accomplish particularly in absence of differentiability.Here, we establish that under the convexity assumption forf, the only condition lim _{k k} =0, where the _k are the antizigzag parameters, suffices to obtain the convergence of the algorithm, i.e. lim _k f(x ^k )=opt., thex ^k being the approximate solutions to problem. The proof is obtained by application of the Th. 24.5, [6]. Successively, we consider the question if one has also the convergence of {x ^k } to optimal solution. By using now the Cor. 27.2.2, [6], we establish, for this purpose, that under an additional general qualification forf — precisely the strict convexity — the convergence of {x ^k } is also stated. Finally, we examine the above property for the stochastic transportation problem [1] for which we indicate special conditions in order to verify the latter convergence property.

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pervenuto il 28-4-82     

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.     

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     

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.     

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.     

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.     

An identity concerning a Wishart matrix

Summary LetS be ap×p Wishart matrix with parametersn and . For a rational number =r/s withr ands integers ands positive, letS denote a positive definite matrix such that (S ) ^s =S ^r . Using a decision theoretic argument, we prove thatE[(trS)² trS ]=(np+2+2)E[trS trS ] when =I andnp+2+2 is positive.     

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.

     

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 .     

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.     

The large-sample power of the 195-1195-1195-1test for multidimensional contingency tables

Summary The asymptotic power of the frequency ² test depends on a noncentrality parameter, ,Mitra [1958] offered a general expression for , which is rather difficult to apply. This work provides simplified formulae for in various models associated with multidimensional contingency tables.     

On Bayesian optimum allocation in some two phase sampling problems

Summary For estimating certain parametric functions, we consider the problem of allocatingN _i, the size of the sample from theith population,i=1,2,...,k, at the second phase of sampling of a two phase sampling procedure, given that we taken _i observations from the population at the first phase. We consider that the observations from theith population follow the exponential distribution with mean _i,i=1,2,...,k, and the functions to be estimated are (i) (d_i/_i) and (ii) (d_ii). When the total cost of sampling at the second phase is c _iN_i and is fixed, allocations using the Bayes approach are obtained so that the estimation is as precise as is possible.     

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.

     

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().