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.     

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.     

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.      

Le XIImecongrès international d'actuaires

Abstract

We study the following inverse thinning problem for renewal processes: for which completely monotone functions f is f/(p+qf), 0?p?1, q=1-p, completely monotone? A characterisation of such f's is given. We also study the case when f comes from a gamma distribution, and present some ideas for more general results.

The intention of this note is to add some information to a paper by Yannaros (1985), in which thinned renewal processes are considered. Let X_n , n?1, be i.i.d. non-negative random variables, distributed according to a probability measure µ, and let S_n = X ₁+...+X_n (with S ₀=0) be the corresponding renewal process. Replacing µ by the probability measure ν=∑_n?1 pq^n-1 µ^n* (µ^n* =µ* ... µ*, n times) we get a new renewal process, obtained from the original one by independently at each stage preserving the process with probability p. Here and below q= 1-p, and to avoid trivialities we assume that 0 Let µ^(s) = ∫[0,∞) exp (-sx)µ(dx) , s?0, denote the Laplace transform of µ. Then ν^=pµ/(1-µ^). We will study the inverse problem: given a completely monotone function ψ, when does ψ(p+qψ) define a completely monotone function. A complete characterisation, and some of its consequences, is given in §§ 1–3 below. In §§ 4–5 we study the gamma distribution. It is proved that the inverse problem has a negative solution when the parameter a > 1, i.e. 1/(p + q(1 + s) ^a ) is not completely monotone then. In Yannaros (1985) this was proved for a=2, 3, ... with entirely different methods. (That 1/(p+q(1+s)^a is completely monotone for 0?a?1 is easily seen; cf. Yannaros (1985). Finally, in § 6 we give some suggestions to more general results related to thinning. Perhaps the most interesting problem is to find sufficiently general conditions for an absolutely monotone function to have a Bernstein function as its inverse.     

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.      

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.     

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 .     

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 .     

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 ).     

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.     

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 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.     

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      

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).     

Osculatory mechanical quadrature

Abstract

The literature contains many formulas of mechanical quadrature¹, most of which are expressible in the form where the A's are constants, f(a _v) represents the functional value of f(x) at each of the n + 1 points x=a _v (v=0,1,2,..., n), and R is the remainder term. Two general and important types of the above formula are the Newton-Cotes ² formula in which the points a _v are equally spaced from c to d, and the Gaussian ³ quadrature formula in which the a's are chosen so as to obtain the greatest accuracy. The Euler-Maclaurin ⁴ formula of summation and quadrature uses the functional values f(a _v ), and the odd ordered derivatives of f(x) at the end points of the interval of integration. Steffensen ⁵ developed a formula for approximate integration employing not only the functional values but the first derivatives, f'(a _v ).     

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 .     

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.     

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.     

On divided differences in terms of derivatives

Abstract

The problem of expressing a difference of a given order of a function in terms of successive derivatives of the function and the related problem of obtaining a manageable form of the remainder- term of a special expansion of this kind have on several occasions been treated in the literature. One of the best known results of investigations on this subject is Markoff's formula,I which may be written in a slightly modified form: where Δ^m0^µ = [Δ^mx^µ ]_{x = 0} and Δ^m _h denotes the descending difference of order m for a table-interval of length h.     

Remarks on semi-invariants and incomplete moments

Let us consider a general discontinuous frequency distribution where the x^pi -S are the values of the variable x, and f(x^pi) is the probability that x will take the value x^pi . We will assume that that is to say: x must take one of the values x^pi(i = 0, 1, 2, 3, ... n).