Distribution of Dixon’s statistics in the case of an exponential population

Summary The distributions of Dixon’s statistics for the case of the sample from an exponential population are found. For some of these statistics the upper 100 α percentage points (α=0,1; 0,05 and 0,01) are tabulated.

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Zusammenfassung Die Verteilungen der Variablen von Dixon werden für den Fall der Stichproben aus einer exponentialen Grundgesamtheit gefunden. Für einige dieser Variablen werden die oberen 100 α Prozentpunkte (α=0,1; 0,05 und 0,01) tabelliert.

     

Generalized entropies in coding theory

Equivalence of the generalized entropyH ^β (P, Φ _t ) defined in this paper andKapur’s entropy of orderα and typeβ, ie.H _α ^β (P), is established. The results given recently byCampbell follow as special cases. International Conference on System Sciences, Honolulu, January 1968.     

Social networks and labour productivity in Europe: an empirical investigation

This paper uses firm-level data recorded in the Amadeus database to investigate the distribution of labour productivity in different European countries. We find that the upper tail of the empirical productivity distributions follows a decaying power-law, whose exponent α is obtained by a semi-parametric estimation technique recently developed by Clementi et al. [Physica A 370(1):49–53, 2006]. The emergence of “fat tails” in productivity distribution has already been detected in Di Matteo et al. [Eur Phys J B 47(3):459–466, 2005] and explained by means of a model of social network. Here we show that this model is tested on a broader sample of countries having different patterns of social network structure. These different social attitudes, measured using a social capital indicator, reflect in the power-law exponent estimates, verifying in this way the existence of linkages among firms’ productivity performance and social network.     

The <Emphasis Type="Italic">K</Emphasis><Subscript>ϕ</Subscript>-Divergence Statistic for Categorical Data Problems

In this work a new family of statistics based on K _ϕ -divergence (Burbea and Rao (1982) On the convexity of divergence measures based on entropy function. IEEE Trans Inf Theory 28, 489–495) are obtained by either replacing both distributions involved in the argument by their samples estimates or replacing one distribution and considering the other as given. Asymptotic distributions of these statistics are obtained and test for goodness-of-fit and for homogeneity with a known distribution, can be constructed on the basis of these results     

Further results on asymptotic normality I

Summary Starting fromLe Cam [1956], it was shown inMichel andPfanzagl [1970] that — under certain regularity conditions — a dominated family of probability measures withEuclidean parameter space behaves approximately like a family of normal distributions, if each probability measure is the independent product of a great number of identical components. It is the purpose of this paper to estimate the accuracy of such a normal approximation.     

Queuing with balking and reneging in M|G|1 systems

Summary AM|G|1 queuing process in which units balk with a constant probability (1−β) and renege according to a negative exponential distribution has been considered. The busy period process is first investigated making use of the supplementary variable technique and discrete transforms. The expression for the joint distribution of the number of customers serviced during a busy period and the length of the busy period has been derived. FollowingGaver (1959) the general process is investigated and making use of renewal theory the ergodic properties of the general process have been studied. It has been shown that as long as reneging is permitted (α>0), the steady states always exist, but when no reneging is permitted (α=0), the steady states exist only whenλ β η<1.     

„Wahrscheinlichkeitslernen“ in der statistischen Lerntheorie

Zusammenfassung In diesem Beitrag zur statistischen Lerntheorie (Stimulus Sampling Theorie) wird einN-elementiges „Pattern“-Modell mitr verfügbaren Antworten und nichtkontingenter Verst?rkungsvorschrift diskutiert. Um von der lerntheoretischen Literatur nicht zu sehr abh?ngig zu sein, wird zun?chst ein kanpper überblick über die Grundbegriffe der Stimulus Sampling Theorie gegeben. Der Vorgang des „Wahrscheinlichkeitslernens“ wird mittels des Reiz-Antwort-Schemas der statistischen Lerntheorie erkl?rt. Durch die Einführung der sogenannten bedingenden Zust?nde gelangt die Theorie derMarkovketten zur Anwendung. Schlie?lich wird das Problem der übereinstimmung von empirischen Daten aus Lernexperimenten mit den aus der Theorie abgeleiteten anhand von Sequentialstatistiken studiert.

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Summary In this contribution to the statistical learning theory (stimulus sampling theory) aN-element pattern model withr availables responses and noncontingent reinforcement schedule is discussed. First a short survey of the principles of the stimulus sampling theory is given. The process of „probability learning“ is explained by the stimulus-response model of the statistical theory of learning. By the introduction of the so-called conditioning states the theory ofMarkov chains may be applicated. Finaly, the goodness-of-fit between empirical data of learning experiments and predicted statistics is studied on the basis of sequential predictions.

     

On a noncentral chi-square distribution in the analysis of weapon systems effectiveness

Summary A natural conjugate prior distribution for the parameters involved in the noncentral chi-square leads to many known distributions. The applications of the distributions thus obtained are briefly pointed out in evaluating the ‘kill’ probability in the analysis of weapon systems effectiveness. The ‘kill’ probabilities or the expected coverage are obtained associated with a gamma prior distribution and compared with those obtained byMcnolty. This paper is read in a symposium on Mathematical Sciences held under the auspices of Delhi University, Delhi im January 1966.     

Minima of vector quadratic forms with applications to statistics

Summary Bush andOlkin (1959) give some results concerning the minimum values of quadratic forms and show some applications of their results to statistics. We generalize some results ofBush andOlkin’s to vector quadratic forms and consider some applications of the results to statistics.     

On the estimation of location parameters in the multivariate one sample and two sample problems

Summary In this paper we consider the problem of estimating the vectors of location parameters in the multivariate one sample and two sample problems. These estimators are obtained through the use of the multivariate rank order statistics such as theWilcoxon or the normal scores statistic considered by the authors inPuri, Sen [1966] andSen, Puri [1967] for the corresponding testing problems. The distribution of these estimators is shown to be symmetric with respect to the parameters being estimated. These estimators are translation invariant, robust and asymptotically normal. Their asymptotic relative efficiencies with respect to the estimators based on the vector of means and medians are discussed by applying the criterion ofWilks generalized variance [Anderson, p. 166]. In particular, it is shown that the estimators based on the multivariate normal scores statistics are asymptotically as efficient as the ones based on the method of least squares when the parent distributions are normal. Research sponsored by National Science Foundation Grant No. GP-12462, and by Research Grant, GM-12868 from the N.I.H., Public Health Service.     

On the active approach of stochastic linear programming

Summary The problem of statistical distribution of the optimal objective function under the so-called active approach of stochastic linear programming is investigated here from two interrelated aspects. First, the active approach is viewed as a method of decomposition. Second, some results on the asymptotic form of distribution of extreme values are utilized to derive the asymptotic form of the distribution of the maximand under the active approach. Research done under the partial support of the U.S. National Science Foundation Grant No. 420-04-62 at the Department of Economics, Iowa State University. Some of the work related to this paper may be found in the following references:Sengupta, J. K., G. Tintner, andC. Millham: “On Some Theorems of Stochastic Linear Programming”. Management Science, Vol. 10, October 1963, pp. 143–159.Sengupta, J. K.: “The stability of truncated solutions of stochastic linear programming”. Econometrica, Vol. 34, January 1966. pp. 77–104.Sengupta, J. K.: “On the stability of solution under recursive programming”. Metrika 1966.Sengupta, J. K. andT. Kumar: “An application of sensitivity analysis to a linear programming problem”. Unternehmensforschung, Vol. 9, 1965.     

Multivariate gamma distributions with one-factorial accompanying correlation matrices and applications to the distribution of the multivariate range

Summary A new representation for the characteristic function of the joint distribution of the Mahalanobis distances betweenk independentN(μ, Σ)-distributed points is given. Especially fork=3 the corresponding distribution function is obtained as a special case of multivariate gamma distributions whose accompanying normal distribution has a positive semidefinite correlation matrix with correlationsϱ _ij=−a _i a _j. These gamma distribution functions are given here by one-dimensional parameter integrals. With some further trivariate gamma distributions third order Bonferroni inequalities are derived for the upper tails of the distribution function of the multivariate range ofk independentN(μ, I)-distributed points. From these inequalities very accurate (conservative) approximations to upperα-level bounds can also be computed for studentized multivariate ranges.     

Einige Bemerkungen zur statistischen Messung der Konzentration

Zusammenfassung Nach einer kritischen Analyse der Begriffe „absolute“ und „relative“ Konzentration werden einige Kriterien gegeben, die Ma?zahlen der absoluten oder der relativen Konzentration zu erfüllen haben. Hierauf wird untersucht, inwieweit die gebr?uchlichsten Ma?zahlen der Konzentration (Gini, Herfindahl, Münzner) diese Kriterien erfüllen. Dann wird gezeigt, da? sich eine vonAdam angegebene, sehr allgemeine Klasse von Ma?zahlen als Ma?zahlen der absoluten Konzentration qualifizieren; einige wichtige Spezialf?lle werden genauer untersucht. Im vorletzten Teil der Arbeit werden Ma?zahlen für die Ver?nderung der Konzentration diskutiert und schlie?lich an einem praktischen Beispiel erprobt.

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Summary After analysing the concepts of “absolute concentration” versus “relative concentration”, some criteria for coefficients of concentration are given to qualify them either as measures for absolute or for relative concentration. On the basis of these criteria, some well-known coefficients (Gini, Herfindahl, Münzner) are examined. Furthermore, it is shown that the measures of “predictivity” as given byAdam can be regarded as coefficients of absolute concentration; some important special cases (one of which the Herfindahl index) are investigated. Finally, measures for the change in concentration are developed and applied to a practical example.

     

On the positivity of a variance-estimator

Summary Murthy’s variance — estimator for the estimator obtained by him by unorderingDes Raj’s estimator is shown to be non-negative.     

Distribution of product statistics from aPareto population

Summary Distributions are derived of the product of sample values, the sample geometric mean, the product of two minimum values from sample of unequal size and product ofk minimum values from sample of equal size from aPareto population. The distributions can be conveniently transformed tox ². Paper presented at the Eastern Regional meeting of the Institute of Mathematical Statistics, Upton, Long Island, New York, April 27–29, 1966. Work done when the author was on the faculty of Western Reserve University, Cleveland, Ohio, U.S.A.     

About probability-like measures for entire theories

5. Summary It is discursively argued that a much closer rapport between the methodologies implicitly taught in Economic Theory and in Statistics is required and that the necessary interdisciplinary bridge can be provided, and clarification attained, through the study of logical measure functions of theKemeny-Carnap type. It is also argued that subjectivistic axiom systems of probability, while valuable in their own right and as bases for behavior theories, cannot by themselves render the study of logical measure functions superfluous. Elementary aspects of these functions are then explained, with somewhat more detailed references toKemeny’s measuresm ands and a proposed degree-of-theoretization measure, whose applicability to economics is negatively appraised. Finally,Carnap’s degree-of-confirmation functions are briefly dealt with, but throughout the paper the emphasis is on those properties of logical measure functions which donot primarily bear on statistical inference and estimation). “...I think, on the whole, one theory fits nearly everything. That is, if you admit one coincidence—and I think one coincidence is allowable. More than one, of course, is unlikely ...” Prof. Dr. E. M. Fels, 2126 CL, University of Pittsburgh, Pittsburgh 13, Penna., USA. The Murder at the Vicarage. New York: Dell, 1961, p. 197 (Dodd, Mead & Co., 1930).     

One optional observation inflates α by $${100/\sqrt{n}}$$ per cent

For one-sample level α tests ψ _m based on independent observations X ₁, . . . , X _m , we prove an asymptotic formula for the actual level of the test rejecting if at least one of the tests ψ _n , . . . , ψ _n+k would reject. For k = 1 and usual tests at usual levels α, the result is approximately summarized by the title of this paper. Our method of proof, relying on some second order asymptotic statistics as developed by Pfanzagl and Wefelmeyer, might also be useful for proper sequential analysis. A simple and elementary alternative proof is given for k = 1 in the special case of the Gauss test.     

On the power function of the exact test for ther×c contingency table

Summary LetN=[n _ij ] (i=1, …,r;j=1, …,c) be the matrix of observed frequencies in anr×c contingency table fromr possibly different multinomial populations with respective probabilitiesp _i =(p _i1, …,p _ic ).Freeman andHalton have proposed an exact conditional test for the hypothesisH ₀ :p _i =(p ₁, …p _c ) of the exact test is derived. Numerical values forβ(p) were previously computed for the special case:r=3,c=2 [Bennett andNakamura, 1964].     

Reliability with preventive maintenance

A system is generally required to undergo preventive maintenance in addition to the irregular repairs in order to prevent frequent failures. This paper aims to study the analytic behaviour of a system with preventive maintenance. System is assumed to undergo preventive maintenance after everyM number of failures. After each preventive maintenance the system regenerates and a new cycle starts again. It may be noted that the supplementary variable technique developed byKielson andKooharian andCox has been employed to obtain the solutions when all the distributions follow different general laws.