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 distributions of functions of samples from a normally distributed infinite population

Abstract

7. The joint distribution of the moments a ₁₁, a ₂₂,…, a_nn and a ₁₂, …, a _1n may be deduced explicitly in the case, in which the variates χ₁, …, χ _n in (1) are mutually uncorrelated. In this case we have for the population values of the moments: α_μv = 0 for μ ═ v and, consequently, A_μv = 0 for μ ═ v, so that according to (6) λ_μv = 0 for μ ═ v; the distribution (5) of the moments α_μv is then      

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 .     

On the central limit theorem

Abstract

If X and Y are mutually independent random variables whith the d. f. ¹ Distribution function(s) F ₁(χ) and F ₂(χ), it is known ² CRAMÉR (1), p. 35. that the sum X + Y has the d. f. F ₂(χ), defined as the convolution where the integrals are Lebesgue-Stiltjes integrals. One uses the abbreviation More generally the sum X ₁ + X ₂ + … + X _n of n mutually independent random variables with the d. f. ¹ Distribution function(s) F ₁(χ), F ₂(χ) , … , F _n has the d. f.      

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.     

Optimal stratification according to the method of least squares

Abstract

A one-dimensional random variable X is given. We have L points, µ₁, µ₂, …, µ _L , and define the random variable Z = min_{µ h} | X — µ _h |, that is the distance to the nearest of the L points µ₁, …, µ _L . We want to find that set of points µ _h for which the function has a minimum. As we shall see in section 2, this problem is equivalent to finding L strata with the set of points of stratification x ₁, x ₂, …, x _L?1 that makes a minimum. w_h is the probability mass and σ² _h the variance of the hth stratum. By differentiation of φ with respect to x_h one can show [3] that a necessary condition for minimum is where µh is the mean of the hth stratum. In section 2 we obtain this condition in another way, which at the same time gives a method of finding the points µh and x_h .     

A note on the extended elementary renewal theorem

Abstract

Let X ₁ ^(µ), X ₂ ^(µ), ... be an infinite sequence of independent and identically distributed random variables defined on the whole real axis and with EX₁ ^(µ) = µ > 0. Put M_n ^(µ) = max (S₀ ^(µ), S₁ ^(µ), ..., S_n ^(µ) , where S_n ^(µ) = X₁ ^(µ) + ... + X_n ^(µ) for n = 1 , 2, ... and S₀ ^(µ) = 0, and define      

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.     

Über die Darstellung willkürlicher Funktionen durch Charliersche Differenzreihen

Abstract

Let us survey an economic subject A₀ who at the point t₀ is planning to offer for sale a number q of lots during a space of time = selling period of lottery ticket.     

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.     

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.     

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 .     

A Note on the Numerical Computation of the Integral of a Pearson Type III Density

Let the distribution function of X, where a?X ? ∞, defined by the Pearson type III density function be designated by _aH(x;p,β).     

Sur une loi de probabilité considérée par J. F. Steffensen

Abstract

Dans ce même périodique, vous avez considéré¹, à la page 7, la loi de probabilité de deux variables aléatoires X, Y,² où la probabilité élémentaire ?(x, y) dx dy pour que X et Y soient respectivement compris entre x et x + dx, y et y + dy, est de la forme où K, a ₁, a ₂, b ₁, b ₂ sont des constantes. Nous nous proposons, dans ce qui suit, d'apporter quelques compléments à votre exposé.     

Sur le calcul numérique des moments ordinaires et des moments composés d'une distribution statistique

Abstract

Considérons une distribution statistique (empirique ou théorique) x _v (v = 1, 2 ... n) désignant les valeurs que peut assumer une variable fortuite une-dimensionelle x, et P_v (v= 1,2 ... n) désignant les fréquences observées (absolues ou relatives) ou bien les probabilités des valeurs x_v .     

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 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 .     

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.     

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.