Nils Ekholm

Abstract

The problem of χ² tests of a linear hypothesis H₀ for ‘matched samples’ in attribute data has been discussed earlier by the author (Bennett, 1967, 1968). This note presents corresponding results for the hypothesis that the multinomial probabilities p satisfy (c ?1) functional restrictions: F ₁(p) = 0, ... , F _C?1(p) = 0. An explicit relationship between the usual ‘goodness-of-fit’ χ² and the modified minimum χ² (=χ*²) of Jeffreys (1938) and Neyman (1949) is demonstrated for this situation. An example of the test for the 2 × 2 × 2 contingency table is given and compared with the solution of Bartlett (1935).     

Zur Variationsrechnung für die Verteilungsfunktionen

Abstract

Die Variationsrechnung hat zur Aufgabe, unter den Funktionen F(x), die einer gewissen Klasse C angehören, diejenige zu finden, welche ein Funktional J[F] zum Maximum oder Minimum macht. In der allgemeinen Variationsrechnung enthält die Definition der Klasse C gewöhnlich nur derartige Bedingungen, die erforderlich sind, urn die Existenz von J[F] zu sichern, z. B. Kontinuität oder Differentierbarkeit bis zu einer gewissen Ordnung, und ausserdem eventuell gewisse “Nebenbedingungen” von der Form J_v [F] = c_v (v = 0, 1, ..., n).     

On the transient behaviour of the GI/G/1 waiting-times

Abstract

Consider a single-server queuing system, where the arrival intervals T_i and the service-times U_i of consecutive customers form two independent sequences of independent and equally distributed random variables. Assume that customers arriving when the server is busy line up and that they are then served in order of arrival. Let W_n be the waiting-time of the nth customer and suppose that the server is idle at the start, i.e. W₁ = 0. Put W = lim _n ∞ W_n when the limit exists. Furthermore, let F_n (?) be the c.d.f. of W_n and put EW_n =ω _n .     

Large Sample Behavior of the CTE and VaR Estimators under Importance Sampling

Abstract

The α-level value at risk (Var) and the α-level conditional tail expectation (CTE) of a continuous random variable X are defined as its α-level quantile (denoted by q_α ) and its conditional expectation given the event {X > q_α }, respectively. Var is a popular risk measure in the banking sector, for both external and internal reporting purposes, while the CTE has recently become the risk measure of choice for insurance regulation in North America. Estimation of the CTE for company assets and liabilities is becoming an important actuarial exercise, and the size and complexity of these liabilities make inference procedures with good small sample performance very desirable. A common situation is one in which the CTE of the portfolio loss is estimated using simulated values, and in such situations use of variance reduction techniques such as importance sampling have proved to be fruitful. Construction of confidence intervals for the CTE relies on the availability of the asymptotic distribution of the normalized CTE estimator, and although such a result has been available to actuaries, it has so far been supported only by heuristics. The main goal of this paper is to provide an honest theorem establishing the convergence of the normalized CTE estimator under importance sampling to a normal distribution. In the process, we also provide a similar result for the Var estimator under importance sampling, which improves upon an earlier result. Also, through examples we motivate the practical need for such theoretical results and include simulation studies to lend insight into the sample sizes at which these asymptotic results become meaningful.     

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.      

Sequential estimation problems for the scale parameter of a Pareto distribution

Abstract

The Pareto distribution plays a central role in many areas of econometrics. So, we first consider sequential point estimation problems for the scale parameter of a Pareto distribution. Under a very general loss structure, we derive several asymptotic results regarding the associated “risk” and “regret” functions. Then, we consider the problem of constructing a fixed-ratio confidence interval for the scale parameter, and we propose various sampling techniques to achieve the intended goal. Most of our theoretical findings are asymptotic in nature for either problem, and thus we have presented extensive simulation studies to examine moderate sample performances of all the procedures. The findings in the point estimation problem are also supposed to fill many important gaps left in the paper of Wang (1973).     

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.     

Generalized log-normal chain-ladder

ABSTRACT

We propose an asymptotic theory for distribution forecasting from the log-normal chain-ladder model. The theory overcomes the difficulty of convoluting log-normal variables and takes estimation error into account. The results differ from that of the over-dispersed Poisson model and from the chain-ladder-based bootstrap. We embed the log-normal chain-ladder model in a class of infinitely divisible distributions called the generalized log-normal chain-ladder model. The asymptotic theory uses small σ asymptotics where the dimension of the reserving triangle is kept fixed while the standard deviation is assumed to decrease. The resulting asymptotic forecast distributions follow t distributions. The theory is supported by simulations and an empirical application.     

Holm's probability-paper tests in relation to the Kolmogorov—Smirnov one-sample test

Abstract

In [5] S. Holm proposed teststatistics for testing simple hypotheses by means of the probability paper for distribution functions (d.f.) of the form F ₀(x) = Φ[(x - μ₀)/σ₀], where μ₀ is location parameter, σ₀ scale parameter, and Φ is an absolutely continuous distribution function with Φ(0) = 1/2. If μ₀ and (σ₀ are known, the hypothesis H ₀ is:

• H ₀: H(x) = F ₀(x) = Φ[(x?μ₀)/σ₀],

while the three possible alternatives are

• H ₁: H(x) > F ₀(x)

• H ₂: H(x) < F ₀(x)

• H ₃: H(x) ≠ F ₀(x).

     

Semiparametric Estimation for Non-Ruin Probabilities

We consider the classical risk model with unknown claim size distribution F and unknown Poisson arrival rate u . Given a sample of claims from F and a sample of interarrival times for these claims, we construct an estimator for the function Z ( u ), which gives the probability of non-ruin in that model for initial surplus u . We obtain strong consistency and asymptotic normality for that estimator for a large class of claim distributions F . Confidence bounds for Z ( u ) based on the bootstrap are also given and illustrated by some numerical examples.     

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.     

A credibility theory for automobile bonus systems

Summary

This paper deals with the problem of designing experience rating systems of the bonus type, commonly used in automobile insurance. On the basis of a simple model the mean squared deviation between a policy's expected claim amount and its premium in the nth insurance period as n→∞, is taken as a measure of the efficiency of a bonus system. It is shown that to any set of bonus rules (which determines the bonus class transitions of the policies), there is an optimal premium scale, which coincides with the one proposed by Pesonen in 1963. Thus the problem of choosing an efficient bonus system reduces to choosing efficient bonus rules. Examples are given of comparison between different bonus rules. In one example the present Norwegian bonus system is compared to alternative systems. Comments are made on earlier papers on bonus systems. The credibility theoretic foundation is laid in a separate section.     

A note on different models of stochastic processes dealt with in the collective theory of risk

Abstract

1. For the definition of general processes with special regard to those concerned in Collective Risk Theory reference is made to Cramér (Collective Risk Theory, Skandia Jubilee Volume, Stockholm, 1955). Let the independent parameter of such a process be denoted by τ, with the origin at the point of departure of the process and on a scale independent of the number of expected changes of the random function. Denote with p(τ, n)dt the asymptotic expression for the conditional probability of one change in the random function while the parameter passes from τ to τ + dτ: relative to the hypothesis that n changes have occurred, while the parameter passes from 0 to τ. Assume further—unless the contrary is stated—that the probability of more than one change, while the parameter passes from τ to τ + dτ, is of smaller order than dτ.     

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).     

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.     

Exact moments of order statistics from a power-function distribution

Summary

In this note a problem on exact moments of order statistics from a power-function distribution is considered. The characteristic function of the kth order statistic is obtained and moments about the origin of the kth order statistic are expressed in terms of gamma functions. An exact expression for the covariance of any two order statistics Y_i < Y_j is obtained in terms of beta and gamma functions. Various recurrence relations between the expected values of order statistics are also obtained.     

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)     

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.     

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.     

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.