A moment inequality with an application to the central limit theorem

Abstract

Consider a sequence of independent random variables (r.v.) X ₁ X ₂, …, X_n , … , with the same distribution function (d.f.) F(x). Let E (X_n ) = 0, _E , E (?(X)) denoting the mean value of the r.v. ? (X). Further, let the r.v. where have the d.f. F _n (x). It was proved by Berry [1] and the present author (Esseen [2], [4]) that Φ(x) being the normal d.f.      

On the remainder in the central limit theorem

Abstract

Let X^bv (v = 1,2, ..., n) be independent random variables with the distribution functions F_bvx) and suppose . We define a random variable by where and denote the distribution function of X by F (x.     

Zwei Ungleichungen sur Ergänzung des Tchebycheffschen Lemmas

Abstract

Sei ?(x) eine für ? ∞ < x < + ∞ definierte reelle nichtnegative Funktion und      

Das Restglied der Taylorschen Reihe des Rentenbarwertes und einige Formeln des Zinsfussproblems für grosse Zinsspannungen

Abstract

Im Zinsfussproblem spielen eine wichtige Rolle drei Hilfsfunktionen der Summen der diskontierten Zahlen, die wir vorweg kurz erwahnen wollen. Unter der nten Summe der diskontierten Zahlen D_x verstehen wir SpezieU haben wir:      

On an analogue of Cramér-Rao's inequality

Abstract

Rao [1] and simultaneously Cramér [2, 3] have shown that if f (x, θ) is the probability density function of a distribution involving an unknown parameter θ and distributed over the range α ? x ? b, where a and b are independent of θ, and if x ₁ x ₂ ... x _n is a random sample of n independent observations from this distribution, the variance of any estimate unbiased for Ψ (θ), satisfies the inequality where E denotes mathematical expectation and is Fisher's information index about θ. In (1), equality holds if, and only if, θ* is sufficient for θ. This inequality is further generalized to the multi-parametric case.     

Remarks on semi-invariants and incomplete moments

Let us consider a general discontinuous frequency distribution where the x^pi -S are the values of the variable x, and f(x^pi) is the probability that x will take the value x^pi . We will assume that that is to say: x must take one of the values x^pi(i = 0, 1, 2, 3, ... n).     

The Problem of Optimum Stratification

Abstract

Although most applications of stratified sampling represent sampling from a finite population, π(N), consisting of k mutually exclusive sub-populations or strata, n, (N,), it is for purposes of theoretical investigations convenient to deal with a hypothetical population n, represented by a distribution function f(y), a < y < b. This hypothetical population likewise consists of k mutually exclusive strata, π_i , i = 1,.2 ... k. The mean of this population is µ_i being the mean of ni. By means of a random sample of n observations, n_i of which are selected from π_i , µ, is estimated by: being the estimate of µ_i .     

Sur un point de la théorie des erreurs accidentelles

Abstract

Dans son traité de la théorie des erreurs ¹ Theorie der Beobachtungsfehler, Leipzig 1891. M. E. Czuber s'occupe page 202–204 du calcul de la valeur probable de la plus petite erreur dans une série d'observations. En admettant la loi de Gauss il en trouve l'expression , où n est le nombre des observations et . Afin d'évaluer l'intégrale qui représente σ, M. Czuber la remplace par , eu remarquant que la valeur de est très petite, dès que k est en quelque façon considérable. Pour les petites valeurs de x la fonction θ(x) ne diffère que légèrement de hx, dit-il ensuite, de sorte que nous aurons comme valeur approchée de σ . En supposant le nombre n très grand, il est clair, dit M. Czuber, qu'on peut choisir k de manière que (1-hk)ⁿ⁺¹ soit négligeable; il parvient ainsi à la valeur definitive .     

Some inequalities satisfied by incomplete beta functions

Abstract

Let t (x, n) being defined by Max and .     

Über den Einfluss einer Änderung der Sterblichkeit auf die Prämienreserve

Abstract

In einer Note über die Theorie des Deekungskapitales habe ieh für das reduzierte Kapital der gemisehten Versicherung auf die Beträge At den Ausdruck gebraueht (1) wobei als Deckungsintensität bezeichnet wurde.     

Contribution to the study of grouped observation. II. Loss of information caused by grouping of normally distributed observations

Abstract

1. Introduction

(a) Maximum Likelihood.—In a previous paper (THIS JOURNAL, vol. XXXII, 1949, pp. 135–159) the author gave tables of the functions and where ?(x) denotes the normal law of distribution, φ(x) its integral and ?′(x) its first derivative. With the aid of these tables it is practicable to solve the maximum likelihood equations for coarsely grouped normal observations. The procedure was illustrated by examples.     

Die wahrscheinliche Le bensdauer und die Sterblichkeitsmessung

Im Gegensatz zur durchschnittlichen Lebensdauer hat die wahrscheinliche Lebensdauer in der Versicherungstechnik wohl kaum je eine wirkliche Anwendung gefunden. Der Grund hierfür dürfte hauptsächlich in der recht schwierigen mathematischen Definition dieser Masszahl zu such en sein. Während die durchschnittliche Lebensdauer bekantlich explicite definiert ist, so ist die wahrscheinliche Lebensdauer — wir werden sie mit t^tx bezeichnen — implicite durch die Gleichung definiert.     

Über die Berechnung der Leibrente bei Veränderung des Zinsfusses.

Abstract

Herr K. A. POUKKA hat in einer unter dem gleichem Titel erschienenen Arbeit¹ fü die zum Zinsfuss i+h zu berechnende Leibrente eine Näherungsformel. abgeleitet, welche sehr zufriedenstellende Resultate gibt. Die Formel wird aus der Reihenentwicklung gewonnen, für welche Herr POUKKA eine Ableitung mitteilt.     

Programme of the Tenth International Congress of Actuaries

Abstract

The problem of “optimum stratification” was discussed by the firstmentioned author in an earlier paper (1). The discussion in that paper was limited to sampling from an infinite population, represented by a density function f{y). The optimum points y_i of stratification, for estimating the mean µ using were determined by solving the equations: which gives the stratification points Y_i that minimize the sampling variance V _y (provided the usual condition for the minimum is fulfilled)     

On the optimum spacing of sample quantiles from a normal distribution. Part I

Abstract

Assume that a large number of observations are made on a normal random variable with the density function where σ σ 0, When the sample is very large the ordinary estimates of µ and a involve considerable computational work. In order to simplify the estimation of µ and/or σ it is sometimes convenient to select a small number of sample quantiles and to use estimates which are linear functions of these sample quantiles, Such a procedure is particularly convenient when the observations occur naturally in order of magnitude, which happens in life testing, for instance, Let      

On the composition of elementary errors

Analysis of statistical distributions.

1. Let m and σ denote the mean and the standard deviation of a statistical variable X, and let W(x) be the probability function of that variable as defined in the first paper ¹ This journal, 1928, p. 13. We shall refer to that paper by the letter I. — The sense in which the words probability function and frequency function are used here must be carefully observed, If the probability that a certain variable lies between x and x+dx is f(x) dx, then f(x) is the frequency function of the variable. The probability function is, in cases where a finite frequency function ex-ists, equal to the integral of the latter, taken over the interval from -∞ to x — The notations of the present paper will, as a rule, correspond to those of I, the most important exception being the symbol n , which will here always denote the number of observations in a statistical series and not, as in I, the number of elementary components. , Art. 1. If we put (cf. I, formula (3)) F(x) is the probability function of the variable , with the mean value 0 and the standard deviation 1. Denoting by µ₂, µ₃, ... the moments of W(x) , taken about the mean (cf. I, Art. 7, where m is supposed to be zero), we put, following Charlier,      

On the solution of a maximum-likelihood equation of the negative binomial distribution

Summary

In a paper in Biometrika, Anscombe (1950) considered the question of solving the equation with respect to x. Here “Log” denotes the natural logarithm, while N _s , where N _k >0 and N _s =0 for s>k, denotes the number of items ?s in a sample of independent observations from a population with the negative binomial distribution and m denotes the sampling mean: it can in the case k ? 2 be shown that the equation (*) has at least one root. In vain search for “Gegenbeispiele”, Anscombe was led to the conjecture (l.c., 367) that (*) has no solution, if m ² > 2S, and a unique solution, if k ? 2 and m ² < 2S. In the latter case, x equals the maximum-likelihood estimate of the parameter ?.

In the present paper it will, after some preliminaries, be shown that the equation (*) has no solution, if k=l, or if k?2 and m ² ? 2S, whereas (*) has a unique solution, if k ? 2 and m ² < 2S.     

Analytical studies in stop-loss reinsurance. II

Abstract

I

In an earlier paper [5] we discussed the problem of finding an unbiased estimator of where p (x, 0) is a given frequency density and 0 is a (set of) parameter(s). In general, will not be an unbiased estimator of (1), when Ô is an unbiased estimate of O. In [5] it was shown that is an unbiased estimator of (1), if we define y_i , as the larger of 0 and X _j - c. It was emphasized that the resulting estimate may very well be zero, even when it is unreasonable to assume that the premium for a stop.loss reinsurance. defined by a frequency p (x, 0) of claims x and a critical limit c, should be zero when the critical limit has not been exceeded during the n years considered for the determination of the premium.     

The use of efficient estimates in multiple tests for goodness of fit

Abstract

Let be Pearson's statistics for testing goodness of fit in various marginal distributions associated with a categorized array of N objects. This study is concerned with disturbances in the limiting joint distribution of when maximum likelihood estimates from the original ungrouped data are used instead of the usual estimates from the cell frequencies after grouping. Under regularity conditions the limiting distributions of , and are shown to satisfy for each positive {c^b1 x ... x c^bT }, where A(c) is the Cartesian product set A(c) = (0, c^b1 ] x ... x (0, c^bT ]. The limiting distributions are characterized in terms of partitioned Wishart matrices having unit rank and parameters as appropriate. These results are extensions of work by Chernoff and Lehmann (1954) and Jensen (1974).     

Consistent estimators of the parameters of a linear structural relation

Abstract

1. Summary of results. Let E and Eo be chance variables at least one of which is not normally distributed (throughout the present paper a chance variable which is constant with probability one will be considered to be normally distributed with variance zero), and whose distribution is otherwise unknown, except that it is known that with probability one, where 0 and p are unknown constants, . Let (u; v) be jointly normally distributed chance variables with unknown covariance matrix, distributed independently of (ε, ε₀). Without loss of generality we assume that the expected values E u and E v, of u and v respectively, are both zero. Define