Showing papers in "Scandinavian Actuarial Journal in 1995"
TL;DR: After the first world war, Cramer began studying the distribution of prime numbers, guided by Riesz and Mittag-Leffler as mentioned in this paper, and later in the mid-thirties, have had a profound influence on the way mathematicians think about the distribution.
Abstract: After the first world war, Cramer began studying the distribution of prime numbers, guided by Riesz and Mittag-Leffler. His works then, and later in the mid-thirties, have had a profound influence on the way mathematicians think about the distribution of prime numbers. In this article, we shall focus on how Cramer's ideas have directed and motivated research ever since.
146 citations
TL;DR: A survey of results in the theory of large deviations, including Cramer's Theorem, the Donsker-Varadhan theory, and other modern developments can be found in this paper.
Abstract: We survey a number of results in the theory of large deviations, including Cramer's Theorem, the Donsker-Varadhan theory, and other modern developments We then apply the large deviation theorems to three models in statistical mechanics, the Curie-Weiss model, the Curie-WeissPotts model, and the Ising model These models are analyzed by the three respective levels of the Donsker-Varadhan theory: the sample means (level 1), the empirical measures (level 2), and the empirical processes and fields (level 3) In the last section a general approach to the large deviation analysis of models in statistical mechanics is formulated
101 citations
TL;DR: In this paper, a general risk model for portfolios with delayed claims is introduced, which is a natural extension of the classical Poisson model and investigated ruin problems for different premium principles and provided approximations for the ruin probability.
Abstract: We introduce a general risk model for portfolios with delayed claims which is a natural extension of the classical Poisson model. We investigate ruin problems for different premium principles and provide approximations for the ruin probability. We conclude with some specific models, for example, for IBNR portfolios and portfolios where the pay-off process depends on the claim size.
47 citations
TL;DR: In this paper, saddlepoint techniques are applied to obtain approximations for the probability of ruin both in finite and in infinite time for the classical Cramer-Lundberg model.
Abstract: Saddlepoint techniques are applied to obtain approximations for the probability of ruin both in finite and in infinite time for the classical Cramer-Lundberg model. The resulting approximations are compared to exact values.
21 citations
TL;DR: In this article, a gamma density f(x; θ) = ((λ/θ λ /Γ(λ))x λ-1 exp( -λ/λ)x) with λ known.
Abstract: 1. NORMAL APPROXIMATIONS It is routine among applied statisticians to use normal approximations. One of the most classical results being for the maximum likelihood estimate θ^ based on n i.i.d. observations from a density with expected information i(θ). This result is not due to Cramer, but in Mathematical Methods of Statistics a precise mathematical treatment is given, and the assumptions used there are now standard. Behind (1) is the central limit theorem applied to the score function. We can see this easily in the case of a gamma density f(x; θ) = ((λ/θ λ /Γ(λ))x λ-1 exp( -λ/θ)x) with λ known. Here θ^ = X¯ = 1/n Σ Xi and (1) becomes For λ = 0.1 Table 1 shows the quality of the approximation (2). Table clearly shows that in some cases an improvement to (1) is called for.
6 citations
TL;DR: Harald Cramer had a multifaceted personality: in addition to being a mathematician and teacher, he was a university administrator, he played an important role in insurance, and was a scholar with humanistic inclinations as discussed by the authors.
Abstract: Harald Cramer had a multifaceted personality: in addition to being a mathematician and teacher he was a university administrator, he played an important role in insurance, he was a scholar with humanistic inclinations. Others are better qualified to describe some of these aspects and I shall only describe his personality as a scientist and teacher.
4 citations
TL;DR: A review of Cramer's contributions concerning factorizations of probability distributions is given in this article, and more recent results are also discussed in the paper "Cramer's Contributions concerning Factorization of Probabilistic Models".
Abstract: A review of Cramer's contributions concerning factorizations of probability distributions is given. More recent results are also discussed.
2 citations
TL;DR: The authors reviewed Harald Cramer's work on extremes and crossings of stationary processes during the 1960's, and very briefly outlined some of the directions of later development of the probability model.
Abstract: This paper reviews Harald Cramer's work on extremes and crossings of stationary processes during the 1960's, and very briefly outlines some of the directions of later development of the probability...
2 citations
TL;DR: A talk given at the Centenary Meeting for Harald Cramer, there taking full advantage of the opportunity for informality and personal recollections, with slides and audio tape excerpts, is described in this article.
Abstract: This paper is based on a talk given at the Centenary Meeting for Harald Cramer, there taking full advantage of the opportunity for informality and personal recollections, with slides and audio tape excerpts. Initially I felt that these reflections may be less appropriate in written form amongst formal scientific articles. My change of heart arises from the conviction that there are valuable lessons to be learned from the life, as well as the printed work of this giant of our field. It is my hope that some of these will become apparent to an interested reader in the course of the somewhat anectodal (and necessarily first person singular) recollections of one person's view of a statistical pioneer in his later, but still enormously productive years. It is a pleasure to record my appreciation to Professor Anders Martin-Lof and his committee for their splendid organization of the very fitting Centenary Meeting, for the invitation to participate in it personally, and to subsequently give these reminis...
2 citations
TL;DR: In this paper, a review of recent developments in risk theory is presented from the historical perspective of Harald Cramer's fundamental contributions to risk theory, including the emergence of insurance products based on ideas coming from finance, and the modelling of extremal events (catastrophes).
Abstract: Over the recent years, we have witnessed a growing need for probabilistic methodology in order to describe new models in insurance. Examples are the emergence of insurance products based on ideas coming from finance, and the modelling of extremal events (catastrophes). Some of these recent developments will be reviewed from the historical perspective of Harald Cramer's fundamental contributions to risk theory.
1 citations
TL;DR: In this article, Cramers work on stochastic processes (SP) has two phases: the first phase, beginning at the start of World War II is devoted to extending the 1934 results of Khinchin on univariate stationary SPs to multivariate stationarySPs, and the second phase, which began around 1950 and lasted until the early 1980s, was devoted to the analysis of nonstationary processes, specifically to determining the extent to which the representations available for stationary processes survive for non-stationary ones.
Abstract: oIntroduction Cramers work on stochastic processes (SP) has two phases. The first phase, beginning at the start of World War II is devoted to extending the 1934 results of Khinchin on univariate stationary SPs to multivariate stationary SPs, and to studying the connections between Khinchin's work and the earlier cognate work on Generalized Harmonic Analysis ( GHA) by Wiener ( 1930). The second phase, which began around 1950 and lasted until the early 1980s, is devoted to the analysis of non-stationary processes, specifically to determining the extent to which the representations available for stationary processes survive for non-stationary ones.
TL;DR: Mykhopadhyay and Solanky as mentioned in this paper considered the following application of STEIN's two-stage procedure: Suppose that (X 1,..., Xn ) T, n = 1, 2,..., is n-dimensional normal with mean vector µ = µ l and dispersion matrix Σ n =σ 2(ρij ) with ρij = 1.
Abstract: 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,..., Xn ) T , n = 1, 2,..., is n-dimensional normal with mean vector µ = µ l and dispersion matrix Σ n =σ 2(ρij ) with ρij = 1, ρij = ρ *, i ≠ j = 1,..., n where (µ, Σ, ρ) ∈ ℝ × ℝ+ × (-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.