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.      

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      

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).

     

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.      

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.     

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.     

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.     

On definition of compound poisson processes

Abstract

Some authors define the (elementary) compound Poisson process in wide sense {χ _t , 0 ? t < ∞} with help of probability distributions where τ is a so-called operational time, a continuous non-decreasing function of t vanishing for t = 0, and V(q, t) is a non-negative distribution function for every t.     

Nonparametric estimation for a stochastic volatility model

Consider discrete-time observations (X _{? δ} )_1≤?≤n+1 of the process X satisfying $dX_{t}=\sqrt{V_{t}}dB_{t}Consider discrete-time observations (X _{ℓ δ} )_{1≤ℓ≤n+1} of the process X satisfying dX_t=?{V_t}dB_tdX_{t}=\sqrt{V_{t}}dB_{t} , with V a one-dimensional positive diffusion process independent of the Brownian motion B. For both the drift and the diffusion coefficient of the unobserved diffusion V, we propose nonparametric least square estimators, and provide bounds for their risk. Estimators are chosen among a collection of functions belonging to a finite-dimensional space whose dimension is selected by a data driven procedure. Implementation on simulated data illustrates how the method works.     

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.     

Sur la notion de prime d'épargne

Abstract

Il est bien connu que la réserve mathématique W (t) d'une assurance très générale sur la vie satisfait à une équation différentielle de Thiele, à savoir où l'on a

• δ (t) = le taux instantané ou l'intensité d'intérêt,

• μ (t) = l'intensité de mourir,

• P (t = la prime par unité du temps,

• S (t) = le capital assuré en cas de décès.

     

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.     

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.     

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 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 .     

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 ).     

Über die Konvergenz des Iterationsverfahrens bei der Berechnung des effektiven Zinsfusses der Anleihen

Abstract

1. Auf den Seiten dieser Zeitschrift hat man mehrmals die Frage nach der Anwendbarkeit der Iteration zur Errechnung der Effektivverzinsung der Anleihen diskutiert. ¹ In erster Linie sind hier die folgenden Aufsätze zu erwahnen : Man ist dabei von der Formel ausgegangen, wo i ₁ nomineller Zinsfuss, k Ausgabekurs, F_t Rückzahlung nach t Jahren und ist. Es hat sich ergeben, dass der direkte Iterationsprozess in der Regel, d. h. in den in der Praxis wirklich vorkommenden Fällen, zu konvergieren scheint, jedoch lassen sich auch Beispiele konstruieren, wo dies nicht der Fall ist ² Ein solches hat Herr von Mises a. a. O. vorgeführt. , und es dürfte, wie Herr Holme bemerkt hat, sehr schwer sein, dasjenige Gebiet abzugrenzen, innerhalb dessen die genannte Konvergenz sich bewührt. Man kan zwar, wenigstens teilweise, der Meinung des Herrn Steffensen ¹ In erster Linie sind hier die folgenden Aufsätze zu erwahnen : beitreten, dass die Frage nach der mathematischen Konvergenz des Verfahrens für die Praxis von untergeordneter Bedeutung ist, weil es dem Praktiker in erster Linie nur darauf ankommt, wie schnell eine qenüqende Annäherung zu erreichen ist. Die Frage besitzt jedoch unleugbares theoretisches Interesse, wozu noch kommt, dass es auch dem Praktiker nicht gleichgültig sein kann, ob und in welchen Fällen ein Verfahren überhaupt verwendbar ist. Es dürfte daher nicht überflüssig sein, eine möglichst allgemeingültige Lösung der Frage zu suchen. Dies ist der Zweck der folgenden Zeilen.     

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.