On the asymptotic normality of some unbiased estimates

Abstract

Let X _m(n) =(X _j , n, ..., X _{j m,n} ) be a subset of observations of a sample X_n = (X_1n X _2n ... , X _nn ). Here the X_jn 'S in Xn are not necessarily independent or identically distributed, and m(n) mayor may not tend to infinity as n tends to infinity. Suppose the joint density function h_n =h_n (x _m (n); θ) of the X _jn 's in Xm(n) is completely specified except the values of the parameters in the parameter vector θ = (θ1 θ2, ... , θ _k ), where θ belongs to a non-degenerate open subset H of the k-dimensional Euclidean space Rk and k?m(n).     

A characterization of geometric distributions by distributional properties of order statistics

Abstract

Let X ₁, X ₂ be independent identically distributed positive integer valued random variables. H the X _i 's have a geometric distribution, then the conditional distribution of R = max(X ₁, X ₂)-min(X ₁, X ₂), given R > 0, is the same as the distribution of X ₁. This property is shown to characterize the geometric distribution.     

Residual variables in regression and confluence analysis

Abstract

Let be the regression of X ₁ on X ₂, X ₃,… X_n (also called the first elementary regression in the set of variables X ₁, X ₂,…,X_n ).     

A characterization of the exponential distribution and of the geometric distribution

Abstract

Cook (1978) has proved that n positive random variables X ₁ ..., X _n are independent and follow the same exponential distribution iff the random vectors (X ₁ ..., X _s ) and (X _s+1, ..., X _n ) are independent for some s ∈ {1, ..., n-l} and E(Π} _j=1 ⁿ max {X _j -a _j , 0}) is a function of Σ _j=1 ⁿ a _j for a ₁, ..., a _n ∈ dR ⁺. In this paper a generalization of this characterization of the exponential distribution and an analogous characterization of the geometric distribution are given.     

A note on robust testing by indexing the parameter space

1. Introduction

A preliminary test is sometimes performed to obtain a simple model with few parameters. This should ostensibly result in a better performance in the subsequent treatment of the observations. For obvious reasons such a procedure is questionable. It is felt that one should keep to the original a priori model with the original parameter vector θ. In this note a third method is proposed. Adhere to original parameters θ, but concentrate attention on an index φ(θ) which sums up some essential features of the model. (This principle has been advocated by Goldstein (1981).)     

Consistency and asymptotic normality of maximum likelyhood estimators

Abstract

Let the random variable X denote the time taken in completion of a process. For a fixed a, if the observed value of X is less than a, the X is observable, but if X is greater than a, the process is tampered with and is accelerated or decelerated at time a by some unknown factor α, and Y=a+α(X-a) is observed. If the experimenter has only partial control over the experiment, it may be difficult to get several observations on Y corresponding to the same a value. Thus we have a set of independent but not identically distributed observations. The large sample behavior of m.l.e. of the unknown parameters based on tampered random variables Y ^b1 , ..., Y ^bn is studied. If X follows an exponential distribution with mean (1/--), ... the consistency and asymptotic normality of the m.l.e. of α and -- is established under mild conditions on a ^b1, a ^b2, ... The conditions needed for establishing the consistency of m.l.e. of lX are given when X follows a uniform distribution U(O, --) or when X has any known distributional form     

A Logical,Simple Method for Solving the Problem of Properly Indexing Social Security Benefits

Abstract

The problem of allocating responsibility for risk among members of a portfolio arises in a variety of financial and risk-management contexts. Examples are particularly prominent in the insurance sector, where actuaries have long sought methods for distributing capital (net worth) across a number of distinct exposure units or accounts according to their relative contributions to the total “risk” of an insurer’s portfolio. Although substantial work has been done on this problem, no satisfactory solution has yet been presented for the case of inhomogeneous loss distributions— that is, losses X ～ F _{X| λ} such that F _X|tλ (X) ≠ F _{tX| λ} (X) for some t > 0. The purpose of this article is to show that the value-assignment method of nonatomic cooperative games proposed in 1974 by Aumann and Shapley may be used to solve risk-allocation problems involving losses of this type. This technique is illustrated by providing analytical solutions for a useful class of multivariatenormal loss distributions.     

On the convergence rate of fixed-width sequential confidence intervals

Abstract

Let X _f1, X _f2, ... be a sequence of i.i.d. random variables with mean µ and variance σ²∈ (0, ∞). Define the stopping times N(d)=min {n:n ^?1 Σ ⁿ _i=1} (X _i&#x2212;X _n)²+n ^?1?nd ²/a ²}, d>0, where X n =n ^?1 Σ ⁿ _i=1} Xi and (2π)^?½ ∫ ^a _?a exp (?u ²/2) du=α ∈(0,1). Chow and Robbins (1965) showed that the sequence I_n,d =[X_n ?d, X _n + d], n=1,2, ... is an asymptotic level -α fixed-width confidence sequence for the mean, i.e. lim_d→0 P(µ∈I_N(d),d )=α. In this note we establish the convergence rate P(µ∈I_N(d),d )=α + O(d½?δ) under the condition E|X₁|^3+?+5/(28) < ∞ for some δ ∈ (0, ½) and ??0. The main tool in the proof is a result of Landers and Rogge (1976) on the convergence rate of randomly selected partial sums.     

Some thoughts on (re) insurance loadings under a ruin criterion

Abstract

Suppose a (re)insurer has free reserves of amount U at his disposal and a portfolio characterised by the distribution function F_x (z; µ σ²). ^X is a stochastic variable describing the accumulated loss during a certain time interval; µ, and σ²) = V are the expected value and the variance of X respectively.     

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      

Über normale stabile statistische Reihen

Abstract

Es sei X eine zufällige Variable, welche k verschiedene Werte ξ₁, ξ₂, ... ξ _k annehmen kann, wofür die Wahrscheinlichkeiten p ₁, p ₂, ... p _k bestehen, wo p ₁ + p ₂ + ... + p _k = 1; die Gesamtheilt der Werte ξ₁, ξ₂, ... ξ _k und der ihnen zugeordneten Wahrscheinlichkeiten p ₁, p ₂, ... p _k wird das Verteilungsgesetz von X genannt. Werden an der Variablen X Versuche in der Weise vorgenommen, dass das Verteilungsgesetz stets dasselbe bleibt und die einzelne Versuche alle von einander unabhängig sind, so wird die Reihe der unter solchen Bedingungen zu stande kommenden empirisch-zufälligen Werte von X als normal stabil bezeichnet.     

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 choice of statistics in credibility estimation

Abstract

The question on what statistics to base our credibility estimators, is discussed in a general model. We introduce concepts of sufficiency, completeness, θ-sufficiency, and θ-completeness that are useful in this connection, and use methods of Rao-Blackwell type. Some of the present results are closely related to results by Taylor (1977).     

Eksamen i Forsikringsvidenskab og Statistik ved Københavns Universitet

Abstract

Let X ₁, X ₂,... be a sequence of independent, identically distributed random variables with P(X?0)=0, and such that pκ = ?₀ ^∞ x ^κ dP(x)<∞, k= 1, 2, 3, 4. Assume that {N(t), t?0} is a Poission stochastic process, independent of the X ₁ with E(N(t))=t. For λ ? 0, let Z _T= max {Σ _t?1 ^N(t) X _t ?t(p ₁+λ)}. Expressions 0 ?t?T for E(Z _T ), E(Z _T ²), and P(Z _T =0) are derived. These results are used to construct an approximation for the finite-time ruin function Ψ(u, T) = P(Z _T >u) for u?0. An alternate method of approximating Ψ(u, T) was presented in [10] by Olof Thorin and exemplified in [11] by Nils Wikstad. One of the purposes of this paper is to compare the two methods for two distributions of claims where the number of claims is a Poisson variate. The paper will also discuss the advantages and disadvantages of the two methods. We will also present a comparison of our approximate figures with the exact figures for the claim distribution      

Über das Theorem von Tchebycheff

Abstract

Eine Grösse X hänge in der Weise vom Zufall ab, dass sie verschiedene Werte x ₁, x ₂, … XII annimmt, je nachdem das Ereignis E ₁, oder das Ereignis E ₂ oder … oder das Ereignis E _n eintritt, wofür die Wahrscheinlichkeiten p ₁, p ₂, … p _n bestehen sollen; p ₁ + p ₂ + … + p _n = 1. Mann nennt X eine von Zufall abhängige Grösse oder X eine variable Grösse mit dem Wertevorrat (x ₁, x ₂, … x _n), wobei jedem einzelnen dieser Werte eine bestimmte Wahrscheinlichkeit zukommt. Zwei Grössen X, Y mit den Individualwerten x ₁, x ₂, … x _n; y ₁, y ₂, … y _m heissen unabhängig von einander, wenn die Wahrscheinlichkeit p _i von x _i dieselbe bleibt, welches auch der Wert von Y sei, und wenn auch die Wahrscheinlichkeit q _p von y _p dieselbe bleibt, welchen Wert auch X annehmen möge.     

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.     

A stronger version of Gautschi's inequality satisfied by the gamma function

Abstract

Let X be a random variable defined on a probability space, and E denote the expectation operator.     

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.     

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.      

Simultaneous estimation of the location and scale parameters of the gamma distribution by linear functions of order statistics

Summary

Large sample estimation of the origin (α₁ and the scale parameter (α₂ of the gamma distribution when the shape parameter m is known is considered. Assuming both parameters are unknown, the optimum spacings (0<λ₁<λ₂<...λ _k <1) determining the maximum efficiences among other choices of the same number of observations are obtained. The coefficients to be used in computing the estimates, their variances and their asymptotic relative efficiencies (A.R.E.) relative to the Cramer Rao lower bounds are given.