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 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.     

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.     

Some inequalities satisfied by incomplete beta functions

Abstract

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

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 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 .     

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 .     

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.     

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      

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.     

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 .     

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)     

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.     

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.     

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).     

Neyman's test of goodness of fit when the null hypothesis is composite

Abstract

1. The need for a ?smooth? test of goodness of fit has long been felt by statisticians. In response to this J. Neyman (1937) put forward his tests. As he showed, they have many of the properties required of smooth tests. In spite of this they have not come into general use.     

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 .

     

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.     

Short note: Stein's two-stage procedure for the intra-class model

In his nice paper (Mykhopadhyay, 1982) as well as in his significant monograph (Mykhopadhyay & Solanky, 1994) N. Mykhopadhyay considers the following application of STEIN's two-stage procedure: Suppose that (X ₁,..., X_n ) ^T , n = 1, 2,..., is n-dimensional normal with mean vector µ = µ l and dispersion matrix Σ _n =σ ²(ρ_ij ) with ρ_ij = 1, ρ_ij = ρ ^*, i ≠ j = 1,..., n where (µ, Σ, ρ) ∈ ? × ?⁺ × (-1, 0); this is called the intra-class model. For given d > 0 and α ∈ (0, 1) one wants to construct a (sequential) confidence interval I for µ having width 2d and confidence coefficient at least (1 - α). It is claimed that where N is determined, according to Stein's two-stage procedure (Stein, 1945), as where m ? 2 is the first stage sample size and denotes the sample variance, fulfills this aim.