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.     

Some inequalities satisfied by incomplete beta functions

Abstract

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

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.      

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 .     

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).     

The Problem of Optimum Stratification

Abstract

Although most applications of stratified sampling represent sampling from a finite population, π(N), consisting of k mutually exclusive sub-populations or strata, n, (N,), it is for purposes of theoretical investigations convenient to deal with a hypothetical population n, represented by a distribution function f(y), a < y < b. This hypothetical population likewise consists of k mutually exclusive strata, π_i , i = 1,.2 ... k. The mean of this population is µ_i being the mean of ni. By means of a random sample of n observations, n_i of which are selected from π_i , µ, is estimated by: being the estimate of µ_i .     

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 .     

Optimal proportional reinsurance policies for diffusion models

Abstract

When applying a proportional reinsurance policy π the reserve of the insurance company is governed by a SDE =(a_π (t)u dt + a_π (t)σ dW_t where {W_t } is a standard Brownian motion, µ, π, > 0 are constants and 0 ? a_π (t) ? 1 is the control process, where a_π (t) denotes the fraction, that is reinsured at time t. The aim of this paper is to find a policy that maximizes the return function V_π (x) = where c > 0, τ_π is the time of ruin and x refers to the initial reserve.     

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)     

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.     

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      

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.     

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.     

Contribution to the Study of Grouped Observations

§ 1. Introduction.

a. Grouping. From a purely mathematical point of view practical observations are often more or less grouped, but in applied mathematics only fairly coarse groupings need be taken into account. The statures of the individuals in a population are commonly given as an example of grouped observations. A measurement of between x - unit and x + unit is referred to as x units, and in order to compensate for the inaccuracy of the estimates of the means and standard deviations calculated from such observations the use of certain corrections has been advised. The benefit hereof is, however, doubtful as usually grouping is comparatively fine and then it is not really disturbing. In the following an account will be given of some facts connected with far coarser grouping. As otherwise the subject would become too extensive, the exposition will here be confined to normally distributed observations. The problem of the “best estimates” of the mean and the standard deviation for the coarsely grouped normal observations will be solved in accordance with the principle of maximum likelihood.     

Die Grewinnbeteiligung bei hohen Versieherungssummen

Es ist üblich, den Gewinnplan für gemischte Versicherungen mit Gewinnbeteilung nach der Formel aufzustellen. Dabei ist die Versicherungssumme S = 1 angenommen. P bedeutet den Tarifprämiensatz. α, β, γ sind die Erwerbskosten-, Inkassokosten- und Verwaltungskostensätze, wie sie für die üblichen Rechnungsgrundlagen II. Ordnung Verwendung finden. d ist der Diskontsatz des Rechnungszinsfusses II. Ordnung. U hat die symboliscbe Bedeutung des Gewinnplanes. (Ia) ist der Barwert der “Einheit” des Gewinnplanes ¹ (Ia) hat hier und im Folgenden nieht die sonst übliche Bedeutung und ist der gewöhnliche temporäre vorsehüssige oder stetige Leibrentenbarwert. ² Eintrittsalter x und Versicherungsdauer n spielen hier nur die Kolle vou Parametern. Sie sind für jede einzelne Versicherung unveränderlich und werden daher, und um die Formeln nicht mit zuviel Indizes zu belasten, konsequent weggelassen. Aber auch die Versicherungssumme S, die im Folgenden als selbständiges Argument von gewissen Funktionen auftritt, wird, wo kein Missverständnis möglich ist, weggelassen. Sowohl (Ia) als auch a sind nach den Rechnungsgrundlagen II. Ordnung gerechnet.     

On the composition of elementary errors

Analysis of statistical distributions.

1. Let m and σ denote the mean and the standard deviation of a statistical variable X, and let W(x) be the probability function of that variable as defined in the first paper ¹ This journal, 1928, p. 13. We shall refer to that paper by the letter I. — The sense in which the words probability function and frequency function are used here must be carefully observed, If the probability that a certain variable lies between x and x+dx is f(x) dx, then f(x) is the frequency function of the variable. The probability function is, in cases where a finite frequency function ex-ists, equal to the integral of the latter, taken over the interval from -∞ to x — The notations of the present paper will, as a rule, correspond to those of I, the most important exception being the symbol n , which will here always denote the number of observations in a statistical series and not, as in I, the number of elementary components. , Art. 1. If we put (cf. I, formula (3)) F(x) is the probability function of the variable , with the mean value 0 and the standard deviation 1. Denoting by µ₂, µ₃, ... the moments of W(x) , taken about the mean (cf. I, Art. 7, where m is supposed to be zero), we put, following Charlier,      

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.     

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 .