On a rank-order test for the equality of probability of an event

Abstract

The following situation is considered. A fixed number (= n) or sequence of independent trials T ₁ T ₂,…, T _n is given, and in each of these an event E mayor may not occur, It is further observed that the event E occurs a total of k times amongst the n trials T _i , (i = l,…, n). It is then required to test the hypothesis H ₀ that the probability of the occurrence of E is constant from trial to trial, i.e. H ₀ is the hypothesis: p ₁ = p ₂ = ? = p _n = p, if p _n (i = 1, …, n) represents the probability that E occurs on the ith trial.     

Risk theory in a Markovian environment

Abstract

We consider risk processes _{t t?0} with the property that the rate β of the Poisson arrival process and the distribution of B of the claim sizes are not fixed in time but depend on the state of an underlying Markov jump process {Z_t } _t?0 such that β=β _i and B=B_i when Z_t=i . A variety of methods, including approximations, simulation and numerical methods, for assessing the values of the ruin probabilities are studied and in particular we look at the Cramér-Lundberg approximation and diffusion approximations with correction terms. The mathematical framework is Markov-modulated random walks in discrete and continuous time, and in particular Wiener-Hopf factorisation problems and conjugate distributions (Esscher transforms) are involved.     

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 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.     

On time-heterogeneous credibility estimation

Abstract

Let χ_i be the total claim amount of an insurance policy in calendar year i. We assume that the χ_i's are conditionally independent given an unknown random parameter ø, and that for all i. In the present paper it is under these assumptions shown how to calculate the credibility estimator of m(ø) by recursive updating. We also give estimators for the unknown parameters α_i, β_i, and ?_i based on portfolio data. Finally we mention some related models.     

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      

Estimation of waiting-time parameters in the GI/G/1 queuing system

Abstract

In a previous paper [2] the author has studied the distribution of the average value of n consecutive observations of the waiting-time in an M/G/1 queuing system, i.e. a system where the arrival epochs form a Poisson process with constant intensity. The observations were assumed to be made during the equilibrium state of the process.     

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.     

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 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.     

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.     

One Approach to Dual Randomness in Life Insurance

Abstract

1. The Unnatural Hypothesis of a Constant Rate of Interest

There are loan contracts which assume a constant interest during several years and thereafter payment of the amount borrowed, but nowadays clauses are as a rule admitted giving the debtor right of conversion or repayment after a certain period, generally ten years. Low interest loans can be considered as perpetuities from a practical point of view, as long as no possibility is meant to exist that the market rate will fall under their nominal rate. Such a loan—as e.g. Consols—with the nominal rate i ₀ ought to be valued at a discount if the market rate is higher, say i > i ₀, the value being equal to the fraction i ₀ : i. But constant rates are no rule in practice.     

On the analysis of a multi-threshold Markovian risk model

We consider a class of Markovian risk models perturbed by a multiple threshold dividend strategy in which the insurer collects premiums at rate c _i whenever the surplus level resides in the i-th surplus layer, i=1, 2, …,n+1 where n<∞. We derive the Laplace-Stieltjes transform (LST) of the distribution of the time to ruin as well as the discounted joint density of the surplus prior to ruin and the deficit at ruin. By interpreting that the insurer, whose gross premium rate is c, pays dividends continuously at rate d _i =c?c _i whenever the surplus level resides in the i-th surplus layer, we also derive the expected discounted value of total dividend payments made prior to ruin. Our results are obtained via a recursive approach which makes use of an existing connection, linking an insurer's surplus process to an embedded fluid flow process.     

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.     

Statistical approximation of plane convex sets

Summary

Given a convex set F in the plane with a sufficiently smooth boundary we try to approximate it by polygons in the following way. Using some specified sampling procedure we pick out n points on the boundary. Through each such point we draw the tangent. Consider the polygon F*_n spanned by all these tangents. If n is large we would expect F*_n to be close to F. Measuring the deviation by the area of F*_n — F we will derive an asymptotic expression for this area when n becomes large. This expression can be used to choose the optimum sampling procedure in the sense of smallest asymptotic deviation.

The problem arose from a problem of statistical approximation in propositional calculus, see section 1.     

Analytical studies in stop-loss reinsurance

Abstract

Introduction.

Consider a unit of risk, say the whole portfolio of an office, or a comprehensive contract of a branch of casualty insurance, which can give rise to a variety of total amounts of claims during a chosen period, say one year. The total claims of the years i =- 1, 2, ... will be denoted by x ₁. They follow some frequency distribution and we assume that during the years considered they are independent from year to year and subject to the same parent distribution. This means, implicitly, that the volume of business and the value of money have remained unaltered and this assumption will be made, since the adjustments otherwise needed are technically trivial and we are not dealing here with the commercial aspect (dif. ficult though it may be of solution) arising out of changes in monetary value. The frequency distribution mentioned can then be regarded as given by a sample from a population whose probability distribution is given by p (x), say, so that      

The regression problem involving non-random variates in the case of stratified sample from normal parent populations with varying regression coefficients

Abstract

1. Introduction.

A sample of N independently observed points (x_{o 1} | x₁₁ , x₂₁ ,... , x_pt ), i = 1, 2, ... , N≥ p is given, where xk, k = 1, 2, ... , p are known, possibly choosable, non-random variates. Suppose now that, for any fixed values of x₁ , x₂ ..., x_p the random variable _o is normally distributed with the mean and the variance λ_o α _x and λ_o are unknown parameters, not involving xk, the regression coefficients and the residual variance of the parent population respectively.     

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