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.      

Some inequalities satisfied by incomplete beta functions

Abstract

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

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 .     

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 asymptotic properties of an estimate of a functional of a probability density

Abstract

Bhattacharyya & Roussas (1969) proposed to estimate the functional Δ = ∫ _?∞/^∞ f ²(x)dx by , where is a kernel estimate of the probability density f(x). Schuster (1974) proposed, alternatively, to estimate Δ by , where F _n (x) is the sample distribution function, and showed that the two estimates attain the same rate of strong convergence to Δ. In this note, two large sample properties of are presented, first strong convergence of to Δ is established under less assumptions than those of Schuster (1974), and second the asymptotic normality of established.     

Zwei Ungleichungen sur Ergänzung des Tchebycheffschen Lemmas

Abstract

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

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 .     

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 .     

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 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 .     

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.     

Notes on collective risk theory

Abstract

A complete proof of existence of a probability measure m the space Ω of all sample functions was given by Cramér [4]. For a finitc period, a simplified proof was given in my paper [2]. The latter proof could be restricted to the space of sample functions having only a finite number of jumps, as the probability of an infinite number of jumps is zero in this case. In fact, dividing the space Ω into disjunct subspaces Ω_n containing exactly n jumps we have: The measure of Ω_n m the case of a finite period of length x is: Thus and consequently P (Ω_∞) = 0. Therefore the set Ω_∞ and all its subsets can be neglected.     

Initiation of a formula for approximate valuation of premiums for disability benefits

Abstract

The premium for a deferred disability pension payable continually during maximum n years may be expressed by means of the well known fundamental symbolic letters as in recent times, the exact value (1) is often approximated ¹ This approximation formula (2), having been used for quite a length of time in Sweden and Norway, is now officially adopted in Switzerland for the new group insurance tariffs introduced in 1948. by the expression      

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.     

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 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.     

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:      

Sur un problème d'interpolation actuariel

Abstract

1. Dans une note dans ce journal M. Poukka 1 Über die Berechnung del Leibrente bei Verändemng des Zinsfusses. Skand. Aktuarietidskr. 1923, p. 137–152. a employé pour construire des formules, dormant approximativement le ehangement d'une rente viagère en augmentant ou diminuant le taux d'intérêt, le procédé de M. Lindelöf 2 Remarques sur un principe général de la théorie des fonctions analytiques. Acta Societatis Fennicae, t. XXIV, no 7. de rendre meilleur la convergence d'une série par la substitution de la variable x par une fonction formant une représentation conforme d'une aire plus grande que le cercle de convergence |x| < R de la série (1) sur ce cercle.     

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.