Zwei Ungleichungen sur Ergänzung des Tchebycheffschen Lemmas

Abstract

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

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.     

On the solution of a functional equation concerning mortality formulae

Abstract

1. If (x) and (y) are lives whose remaining lifetimes are stochasticallyindependent, and if the mortality of each of the lives is given by a Makeham expression, then as a well known fact (see e.g. P. F. Hooker & L. H. Longley-Cook, Life and Other Contingencies, Cambridge 1957, vol. II, pp. 137&138) the evaluation of joint-life endowments and joint-life annuities on the lives (x) and (y) may be performed by substituting a single life (u) for (x) and (y) and altering the force of interest, provided that and with the same value of the parameter c( > 1).     

Aktuareksamen ved Københavns Universitet

Abstract

Extract

d1. Vis, at for fast x er en ikke voksende funktion af y og for fast y en ikke aftagende funktion af x, når dødsintensiteterne μg og μ ^I for de to live er ikke aftagende. Det forudsættes, at dødsaldrene for (x) og (y) er stokastisk uafhængige.     

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 .     

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.     

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.     

Eksamen i Statistik ved Københavns Universitet

Abstract

Extract

d1. Bestern karakteristikkerne for den partielle differentialigning Gør rede for, at der ved begyndelsebetingelsen z=2√x for 0<x<+∞,y=0 fastlægges netop en løsning til (*) i et passende område ω i xy-planen, og bestem denne løsning (herunder et brugbart område ω).     

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.      

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.     

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 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.     

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.     

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.     

The prohlem of optimum stratification in a special type of design

Abstract

The concept of optimum stratification was introduced in a paper published in this journal 1950 (1), where as well a technique was devised to determine the condition for optimum stratification. This technique is based upon the representation of the population by a density function ? (y). The variance V ( y ) of the estimate y for a specific type of allocation of the sample to the k strata is then dealt with as a function of the points Y_i of stratification. By solving the equations: the conditions that these points Y_i must fulfil in order to make V ( y ) a minimum are arrived at. In a second paper (2) further results were given.     

Calculation of Death-Rates with Regard to Migrations

Abstract

As is known, the death-rates of a population are usually worked out by the aid of the formula where D_x denotes the number of deaths at the age x - x + 1 year and where M_x signifies the average population in the same age group or the total observed risktime (expressed in years).     

Zur Kritik der Ersatzmethoden der direkten Bestimmung von Erbzahlen bei nicht repräsentativem Material

Abstract

Für die Untersuchung auf bestimmte Erbzahlen bei unvollständigem einseitig ausgelesenem Material gibt es zwei Methoden : 1. den Vergleich der in dem un vollständigen Material gefundenen Häufigkeit bestimmter Merkmalträger mit der Erwartung auf Grund der Annahme einer bestimmten Erbregel , wobei k die Grösse der Sippschaften, p die erwartete Erbzahl und q = 1 ? p ist. Diese Formel ist von mir schon 1912 aufgestellt. Bernstein nennt diese Methode nicht sehr glücklich die apriorische. Erstmals ist sie 1916 praktisch von Apert angewandt. Man wird sie besser als direkte Vergleichsmethode bezeichnen.

2. Die Feststellung der Häufigkeit s des Merkmals bei den Gesehwistern seiner Träger, T. Ist deren Zahl t, k die Sippschaftsgrösse, t_x die Zahl der Träger mit x Trägergeschwisten, so ergibt sich theoretisch .

     

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:      

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.     

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).