Some inequalities satisfied by incomplete beta functions

Abstract

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

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      

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 .     

Aktuareksamen 1925 ver Universitetet i Oslo

Abstract

Let be the order statistics in an ordered random sample of size n from the normal population N(±, σ² with mean ± and standard deviation σ. The present paper (1) provides the optimum ranks coefficients and efficiencies of Lloyd's [6] best linear unbiased estimates (BLUE) of ±, σ (when one is known) and (±, σ) based on the k = 1,2, 3, 4 order statistics selected from (Ll) which give the highest efficiencies and (2) demonstrates that efficiencies are only slightly reduced if the BLUE of ± and σ (when one is known) based on the order statistics with ranks are replaced by the BLUE based on the order statistics with ranks , where are the optimum spacings for Ogawa's [8] asymptotic best linear estimates (ABLE) of ± and σ when one parameter is known, or even are replaced directly by the ABLE with their corresponding optimum spacings.     

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 .     

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).     

On the central limit theorem in the space Rk,k> 1.

Abstract

Introduction. In an earlier paper ¹ BergströM (1) I proved the inequality for the difference between the normal d. f. ² Distribution function Φ (χ) and the d. f. of the sum of n equally distributed random variables with the mean value O. Here σ denotes the dispersion, β₃ the absolute third moment of the variable X_i and C is an absolute constant. To establish the inequality I gave an identical expansion of the convolution , when the dispersion for F(χ) was 1, and a lemma for Weierstrass' singular integral. I also remarked that this method could be used for d. f.'s in the space R_k , k> 1. In fact there is very little to be changed when I now give the generalization for the space R_k .     

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.     

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 .     

Über eine Ungleichung der mathematischen Statistik

Abstract

1. Es sei ?(x) eine für ? ∞ < x < + ∞ reelle nichtnegative Funktion, die für x = a ihr Maximum hat und von a nach beiden Seiten monoton abnimmt. ¹ d. h. ?(x) ≦ ?(x′) für jedes x und jedes zwischen α und x gelegene x′. Ferner sei und konvergent.     

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      

On the ω test of dependence between statistical variables

Abstract

1. In an earlier Note¹ I have suggested to measure the dependence between statistical variables by the expression where p_ij is the probability that x assumes the value x_i and y the value y_j , while By is meant summation with respect to all i and j for which p_ij > p_i* p_*j .     

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.      

Eine approximative Technik der Invaliditätsversicherung

Abstract

Die wichtigsten Rechnungsgründe der Invaliditätsversicherung sind, neben dem Zinsfuss, die Invaliditätshäufigkeit der Aktiven und der Abgang der Invaliden (infolge Sterblichkeit und Reaktivierung). Dieser wird gewöhnlich vom Alter und von der verflossenen Dauer der Invalidität, jene (i_x ) dagregen nur vom Lebensalter abhängig angenommen. Neben diesen Hauptrechnungsgründen ist noch die Anzahl Aktiven, nötig.     

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)