Some inequalities satisfied by incomplete beta functions

Abstract

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

Zwei Ungleichungen sur Ergänzung des Tchebycheffschen Lemmas

Abstract

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

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).     

More about invalidity functions

Abstract

1. In the discussion that followed the reading to the Danish Actuarial Society of the paper quoted below1 it was suggested by Mr N. E. Andersen that the hypothesis T. F. (49), or , employed in the second half of the paper, might with advantage be replaced by x_o being the initial age. In this way it is obtained that and it then follows, by T. F. (6), that      

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 inequality for gamma functions

Abstract

For all real values of α and λ satisfying the following inequality holds. When compared with a similar inequality due to Gurland [3] this is seen to be stronger for a certain range of α.     

On the connection between tests of significance for correlation coefficients and for differences between means

1. Some questions about the connection between statistical tests of significance for simple and multiple correlation coefficients and for differences between sample means (and between sample means and population means) of variables of one or several dimensions are treated in this paper. The distributions of the random variables that are considered in such tests are given, under certain conditions, by frequency functions of the following types ¹ the recently published treatise “Mathematical Methods of Statistics” by Professor Harald Cramér (Uppsala 1945). : where - ∞ < t < ∞, n≧1; where where 0 < t < ∞, k≧1, n≧k; and where .     

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.     

On the test of the hypothesis that the probability of an event is contained within given limits

Asbtract

The hypernormal (or Lexian) frequency function can be defined by the integral where θ(p) is the frequency (or density) function of p defined in the interval. We have, of course, that and that .     

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.      

Aktuareksamen og Eksamen i Statistik ved Københavns Universitet

Abstract

Extract

d1. Vis, at man for n ? 2 har når x ikke antager nogen af værdierne 0, ?1, ..., ?n+1, og når x ikke antager nogen af værdierne 0, 1, ..., n+1.     

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.     

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      

The mean credibility formula is a bayes rule

Abstract

Recent advances in statistical decision theory and stochastic processes provide the machinery for showing that the celebrated mean credibility formula is a Bayes rule within a nonparametric context. The credibility factor is obtained as a simple function of the parameter that characterizes the prior distribution. A natural estimator of leads to a credibility formula having a form similar to the James-Stein estimator.     

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      

A theorem on translations by Hille,and its interpretation from the point of view of the theory of probability

Abstract

The Taylor expansion of a function around a point a may, as is well-known, be formally written      

On some discrete frequency functions

Abstract

The hypernormal (or Lexian) frequency functions are defined by the integral .     

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).     

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.     

Fourier inversion—distribution functions—long tails

Abstract

Given a characteristic functionφ(t) we want to calculate the corresponding distribution function. For the sake of simplicity we will assume that the mean value of the distribution is zero, i.e. that φ'(0) =0. For these calculations we will use the following formula where The dash on the summation sign indicates that the term corresponding to k = 0 is missing.