Question of stability estimation in the problem of reconstructing the type of a distribution

A quantitative estimate is obtained for the stability in the problem of reconstructing the complete type of a distribution according to the distribution of a maximal invariant.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 87, pp. 7–17, 1979.     

Determination of the distribution function of the dielectric relaxation times of polymer materials

An exposition is presented of a phenomenological theory of dielectric relaxation. A brief review is given of the existing procedures for finding the distribution function of the dielectric relaxation times. A procedure is suggested for determining the distribution of the relaxation times of polymer materials from an experimentally measured current-decay function by means of the numerical solution of an integral equation on a computer.Institute of the Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 348–353, March–April, 1976.     

Reconstruction of the error distribution in the linear regression scheme

Under a certain condition, the characterization of the additive type of an error distribution by the distribution of a maximal invariant is proved in the linear regression scheme. With an added condition on the design matrix, the possibility of reconstructing the error distribution itself is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 87, pp. 36–40, 1979.The author is grateful to L. B. Klebanov for his interest in this article.     

A characterization of the rectangular distribution

It is assumed that the statistics S and T (given by formula (1.2) and (1.3)) have constant regression on Λ= ∑ⁿ_j=1X_j. Under cetain additonal assumptions this leads either to a symmetric two point distribution or the rectangular distribution over (?1, +1). The two point distribution can be excluded by assuming that the population distribution function is continuous. A characterization theorem for the rectangular distribution over (?1, +1), is then obtained.     

Stability of a theorem of uniqueness of a characteristic function that is analytic in the neighborhood of the origin

Quantitative estimates are obtained for the closeness of distribution functions in terms of the closeness of their characteristic functions considered in finite intervals only. A detailed analysis is presented for the characteristic function of the standard normal distribution on a straight line.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei — Trudy Seminara, pp. 88–94, 1980.     

Limit distribution of the number of consecutive records

A succinct series expression is derived for describing the limit distribution of the number of times r consecutive elements are all records (in a sequence of independent and identically distributed random variables with a common continuous distribution) for all r ≥ 2. Previously, only the limit distributions for r = 1, 2, and 3 were known. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Univariate and multivariate versions of the negative binomial-inverse Gaussian distributions with applications

In this paper we propose a new compound negative binomial distribution by mixing the p negative binomial parameter with an inverse Gaussian distribution and where we consider the reparameterization p=exp(−λ). This new formulation provides a tractable model with attractive properties which make it suitable for application not only in the insurance setting but also in other fields where overdispersion is observed. Basic properties of the new distribution are studied. A recurrence for the probabilities of the new distribution and an integral equation for the probability density function of the compound version, when the claim severities are absolutely continuous, are derived. A multivariate version of the new distribution is proposed. For this multivariate version, we provide marginal distributions, the means vector, the covariance matrix and a simple formula for computing multivariate probabilities. Estimation methods are discussed. Finally, examples of application for both univariate and bivariate cases are given.     

On the development of manpower number and recruitment in the civil service

We present a model for manpower planning which describes the dependence between stocks, flows and age distributions. An unbalanced age distribution results in the following conflict: either a recruitment is chosen which produces minimal deviations from a given stock. Then the unbalanced age distribution will be reproduced. Or a recruitment is selected which produces a balanced age distribution, but then oscillations of the stock will occur until a stationary age distribution will be reached. A mixture of these two recruitment policies is possible.

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Zusammenfassung Das vorliegende Personalplanungsmodell beschreibt den Zusammenhang zwischen Beständen, Personalflüssen und Altersstrukturen. Eine unausgewogene Altersverteilung des Personalbestandes führt zu einem Zielkonflikt: Entweder werden Einstellungen so vorgenommen, daß Abweichungen von einem stabilen Soll-Bestand minimal werden, dann aber bleibt die Altersverteilung unausgewogen. Oder Einstellungen werden so gewählt, daß eine stabile Altersstruktur erreicht wird, dann aber schwankt der Gesamtbestand, bis eine stabile Altersverteilung erreicht wird. Mischungen zwischen diesen Einstellungsstrategien sind möglich.

     

The generalized near-integer Gamma distribution: a basis for ‘near-exact’ approximations to the distribution of statistics which are the product of an odd number of independent Beta random variables

In this paper the concept of near-exact approximation to a distribution is introduced. Based on this concept it is shown how a random variable whose exponential has a Beta distribution may be closely approximated by a sum of independent Gamma random variables, giving rise to the generalized near-integer (GNI) Gamma distribution. A particular near-exact approximation to the distribution of the logarithm of the product of an odd number of independent Beta random variables is shown to be a GNI Gamma distribution. As an application, a near-exact approximation to the distribution of the generalized Wilks Λ statistic is obtained for cases where two or more sets of variables have an odd number of variables. This near-exact approximation gives the exact distribution when there is at most one set with an odd number of variables. In the other cases a near-exact approximation to the distribution of the logarithm of the Wilks Lambda statistic is found to be either a particular generalized integer Gamma distribution or a particular GNI Gamma distribution.     

Some distribution results of the Oppenheim continued fractions

We study some distribution properties of the Oppenheim continued fraction expansions. A Gauss–Kuzmin–Lévy type theorem is established. Based on this, a Fréchet law concerning the partial maxima of the growth rate of the digit sequence is derived, which extends previous work of Galambos and Philipp on the regular continued fraction expansion. Besides, a uniform distribution modulo 1 result is obtained.     

On the theory of biaxial orientation of amorphous polymers

The macromolecule orientation distribution function for biaxial orientation is calculated on the basis of a network model of a linear amorphous polymer. The dependence of the distribution function on the biaxial stretch ratio, orientation temperature, and certain other factors is investigated. A relation is established between the distribution function and the experimentally observed birefringence. The birefringence of biaxially oriented polymethyl methacrylate is measured in relation to the degree of deformation. The experimental data are compared with theory.Moscow Lenin State Pedagogical Institute, Problem Laboratory of Polymer Physics. Translated from Mekhanika Polimerov, No. 5, pp. 771–779, September–October, 1970.     

Asymptotic behavior of the number of distinct values in a sample from the geometric stick-breaking process

Discrete random probability measures are a key ingredient of Bayesian nonparametric inference. A sample generates ties with positive probability and a fundamental object of both theoretical and applied interest is the corresponding number of distinct values. The growth rate can be determined from the rate of decay of the small frequencies implying that, when the decreasingly ordered frequencies admit a tractable form, the asymptotics of the number of distinct values can be conveniently assessed. We focus on the geometric stick-breaking process and we investigate the effect of the distribution for the success probability on the asymptotic behavior of the number of distinct values. A whole range of logarithmic behaviors are obtained by appropriately tuning the prior. A two-term expansion is also derived and illustrated in a comparison with a larger family of discrete random probability measures having an additional parameter given by the scale of the negative binomial distribution.

     

Dominance of the positive-part version of the James-Stein estimator

James-Stein estimators are shown to be dominated by positive-part versions when estimating the mean of a p-variate normal distribution (p ⩾ 3), where the covariance matrix is known up to a constant. A Monte Carlo study is undertaken to compare their risks.     

Distribution of k-th record values in the discrete case

A result is found for discrete random variables, which lets one express the distribution of k-th record values in terms of the distribution of sums of independent summands.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 133–137, 1987.     

Wigner quasi-probability densities and characterization of the Gaussian distribution

A new characterization of the Gaussian distribution was given by R. Hudson in 1974. A sharpening and some extensions of this result are proposed, and connections with other characterizations of the Gaussian distribution are given. Linnik's “ridge principle” is substantially used. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 142–153.     

An analytic refinement of the Heyde theorem on linear forms of independent random variables

In terms of the characteristic function of the joint distribution of two linear forms of independent random variables, one refines Heyde's known theorem on the characterization of the Gaussian distribution by the property of symmetry of the conditional distribution of one linear form under a given second form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 54–59, 1988.     

On the unimodality of multivariate symmetric distribution functions of class L

A theorem is proved that characterizes multivariate distribution functions of class L. This theorem is used to show that every n-dimensional, symmetric distribution function of class L is unimodal in the sense of Kanter.     

Asymptotic properties of some goodness-of-fit tests based on the L 1-norm

Some goodness-of-fit tests based on the L ₁-norm are considered. The asymptotic distribution of each statistic under the null hypothesis is the distribution of the L ₁-norm of the standard Wiener process on [0,1]. The distribution function, the density function and a table of some percentage points of the distribution are given. A result for the asymptotic tail probability of the L ₁-norm of a Gaussian process is also obtained. The result is useful for giving the approximate Bahadur efficiency of the test statistics whose asymptotic distributions are represented as the L ₁-norms of Gaussian processes.