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.     

An inequality satisfied by the Gamma function

Abstract

1. Summary

For all real values of α and λ satisfying α + λ > 0, α ≠ 0, α ≠ 1, λ > 0 the following inequality involving the Gamma function holds:

This follows from a general inequality for the variance of regular unbiased estimators given by Cramér [1].     

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 .     

Zwei Ungleichungen sur Ergänzung des Tchebycheffschen Lemmas

Abstract

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

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 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 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.     

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 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.     

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.     

A general formula for the normal mean errors of the coefficients in parabolic least squares graduation

Abstract

A set of observations contains N elements, and the two measures x and y are observed for each element. Searching the ? best value ? of y as dependent on x, we put      

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 .     

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 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.     

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.     

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      

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.