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Actuarial statistics I: likelihood characterizations
230
Abstracts and Reviews
074004 (M02) Some rece developments in statistical theory. Cox D.R.,4 ~uffield_._..College, England, Scandinavian ~_~_ Actuarial Journal, Harald Cramer Symposium, nr. I, 1995, pp. 29-34. ARer some brief comments on Harald Cramer’s statistical work, one view is given of developments in statistical theory since the publication of Cramer’s book. The purpose of some recent work on higher-order asymptotic theory is sketched. Keywords: Theory of inference, conditional inference, asymptotic theory.
MI& PROBABILITY THEORY AND MATHEMATICAL STATISTICS IN INSURANCE, GENERAL AND MISCELLANEOUS 074005 (MlO) Applications of risk theory and multivariate analysis in insurance practice. Niggemeyer II., Radtke M., Reich A., Applied Stochastic Models and Data Analysis, Vol. II, nr. 3, 1995, pp. 231-244. Segmentation strategies and differentiated preselection in underwriting call for new portfolio management techniques, the use of which is becoming increasingly widespread amongst insurance companies faced with growing competitive pressure. In recent years, mathematical procedures have been developed for this purpose and the applicability of existing procedures for use in insurance practice has been recognized. The present paper elucidates various methods for determining the aggregate claims distribution, which describes the performance and volatility of a portfolio. There then follows a presentation of multivariate methods, particularly generalized linear _models and methods applying variance and discriminant analysis, which facilitate the analysis of more narrowly defined segments and subportfolios. Finally, the paper contains a description of applications used by numerous insurance companies, primarily for motor and property portfolios. Keywords: multivariate statistics, generalized linear models, aggregate claims. 074006 (MlO) Utilisation de modeles sur les montants cumulds pour determiner la loi de probabilite des provisions pour sinistres a payer en assurance IARD.
Parisi A., Martignolles S., Actuariat IARD AXA, France, Astin Colloquium Leuven, 1995. The object of this article is to show that riski models on random sums can be used to find the distribution of the total amount of reserves in non-life insurance. Let V be the random variable total amount of reserves. The idea consists in choosing a distribution function for V. We can then build an interval of probable values for V or calculate the probability that the amount of reserves is not sufficient. After describing our approach and several models used to determine a distribution function for random sums we give two examples. The first one is applied to damages in motor insurance and the second to bodily injury. Keywords: Reserve, Random Sums. ~. 074007 (MlO) Actuarial statistics I: likelihood characterizations. Hiirlimann W., Winterthur, Switzerland, Astin Colloquium Leuven, 1995. The present work establishes some likelihood properties of Bernoulli mixed discrete-continuous models with special emphasis on actuarial models. Potential statistical applications include one-parametric deformations from discrete to continuous families of distributions, perturbed models for Robust Statistics, models with unobservable events of use in the (re)insurance of large claims, insurance claims models with outliers, distribution-free reinsurance models via stop-loss extremal distributions (Section 1). Based on mean equivalence relation for parametric random functions, a mean equivalence class of loglikelihoods is considered. The corresponding systems of sample mean equations lead to a class of maximum likelihood mean equivalent estimators, or shortly likelihood estimators (Section 2). For Bernoulli mixed discrete-continuous models alikelihood estimator, which h is asymptotically equal to the maximum likelihood estimator, is constructed. Sometimes it is equal to the maximum likelihood estimator. This is the case for a insurance claims model with outliers. specific Furthermore this estimator appears to be useful as estimation method for the unknown parameters of a stop-loss ordered extremal distribution associated to the class of positive risks with fixed mean and variance (Section 3). The mainly theoretical Section 4 studies the relationship between pseudo-estimators and orthogonal parameters. After some motivation for the use of pseudo-estimators has been given, the well-known information inequality
Abstracts and Reviews
for pseudo-estimators is recalled. It extends the classical Cramer-Rao lower bound for unbiased estimators. Several examples,,which ..~~_ illustrate the usefulness of the generalized inequality, are presented. As a main result, a substantial generalization of a characterization obtained previously by the author is derived. It characterizes unbiased pseudo-estimators related to the class of maximum likelihood mean equivalent estimators (introduced in Section 2) under an orthogonal condition of the parameter of interest. A thorough discussion and several examples follow. In particular a complete likelihood characterization of one-parametric maximum likelihood estimators is formulated (Corollary 4.2). The latter result is applied in Section 5 to provide “rational” justifications for the use of some categories of statistical models with location or scale parameters encountered in the fields of Economy and Social Sciences. First two simpler variants of classical results Then practically quite useful are formulated. characterizations ofthe compound Poisson, binomial and negative binomial gamma models in collective risk theory are derived.... In Section 6 sufficient conditions for the sample mean to be a somewhat robust likelihood estimator in a Bernoulli mixed discrete-continuous model are given. These conditions characterize in Section 7 a mean scaled gamma outlier model encountered in the recent actuarial literature. Keywords: Bernoulli mixedmodel, outliers, distributionfree reinsurance, log-likelihood mean equivalence, maximum likelihood, pseudo-estimators, information inequality, orthogonal parameters, location parameter, scale parameter, sample mean, robustness. 074008 (MlO,E51) A recursive scheme for perpetuities with random positive interest rates. Part I. Analytical results. De Schepper A. (*), Goovaerts M.J. (**), Kaas R., RUCA, University of Antwerp (*), Katholieke Universiteit Leuven, Belgium (**), University of Amsterdam, The Netherlands, Astin Colloquium Leuven, 1995. Recently, the authors showed how interest randomness in actuarial functions can be described by means of Wiener processes using path integrals. This paper wants to present an extension of this kind of models, by investigating the situation of interest rates that cannot become negative. The case of an annuity certain and in particular that of a perpetuity will be dealt with in detail.
Keywords: Interest randomness, Annuity Perpetuity, Wiener processes, Distribution.
231 certain,
074009 (MlO) Actuarial statistics II: exponential characterizations. Hiirlimann W., Winterthur, Switzerland, Astin Colloquium Leuven, 199.5. The present paper is based on the well-known characterization of regular exponential families by minimum variance unbiased estimators. Four simple equivalent statements, two of which do not seem to be formulated so far in the literature, recall this fact (Proposition 2.1). Inspired by a recent work of Targhetta (1990), conditions, under which the 2x2 Bhattacharya matrix of a regular exponential family is diagonal, are derived (Theorem 3.1). Interesting theoretical insights related to the special class of regular exponential families with diagonal 2x2 Bhattacharya matrix and quadratic variance of the observation are discussed. It is shown how this class can be generated using commutative transformations starting from a single elementary distribution (expanding on a previous idea by Morris (1982)). Of special actuarial interest, one observes that this class contains the shifted gamma process (Example 3.1) and a continuous version of the two-parametric Panjer family of discrete distributions (Remarks 4.1). Keywords: exponential family, information inequality, MVUE, Bhattacharya matrix, quadratic variance function, elementary distributions, shifted gamma process, continuous Panjer family.
Mll: STOCHASTIC MODELS FOR CLAIM FREQUENCY, CLAIM SIZE AND AGGREGATE CLAIMS 074010 (Mll) A time-continuous Markov chain interest model with applications to insurance. Norberg R., Laboratory of Actuarial Mathematics, Denmark Applied Stochastic Models and Data Analysis, Vol. II, nr. 3, 1995, pp. 245-256. The force of interest is modelled by a homogeneous time-continuous Markov chain with finite state space. Ordinary differential equations are obtained for expected values of various functionals of this process, in particular for moments of present values of payment streams that may be deterministic or, possibly, also stochastic and driven by a time-continuous Markov
Abstracts and Reviews
074004 (M02) Some rece developments in statistical theory. Cox D.R.,4 ~uffield_._..College, England, Scandinavian ~_~_ Actuarial Journal, Harald Cramer Symposium, nr. I, 1995, pp. 29-34. ARer some brief comments on Harald Cramer’s statistical work, one view is given of developments in statistical theory since the publication of Cramer’s book. The purpose of some recent work on higher-order asymptotic theory is sketched. Keywords: Theory of inference, conditional inference, asymptotic theory.
MI& PROBABILITY THEORY AND MATHEMATICAL STATISTICS IN INSURANCE, GENERAL AND MISCELLANEOUS 074005 (MlO) Applications of risk theory and multivariate analysis in insurance practice. Niggemeyer II., Radtke M., Reich A., Applied Stochastic Models and Data Analysis, Vol. II, nr. 3, 1995, pp. 231-244. Segmentation strategies and differentiated preselection in underwriting call for new portfolio management techniques, the use of which is becoming increasingly widespread amongst insurance companies faced with growing competitive pressure. In recent years, mathematical procedures have been developed for this purpose and the applicability of existing procedures for use in insurance practice has been recognized. The present paper elucidates various methods for determining the aggregate claims distribution, which describes the performance and volatility of a portfolio. There then follows a presentation of multivariate methods, particularly generalized linear _models and methods applying variance and discriminant analysis, which facilitate the analysis of more narrowly defined segments and subportfolios. Finally, the paper contains a description of applications used by numerous insurance companies, primarily for motor and property portfolios. Keywords: multivariate statistics, generalized linear models, aggregate claims. 074006 (MlO) Utilisation de modeles sur les montants cumulds pour determiner la loi de probabilite des provisions pour sinistres a payer en assurance IARD.
Parisi A., Martignolles S., Actuariat IARD AXA, France, Astin Colloquium Leuven, 1995. The object of this article is to show that riski models on random sums can be used to find the distribution of the total amount of reserves in non-life insurance. Let V be the random variable total amount of reserves. The idea consists in choosing a distribution function for V. We can then build an interval of probable values for V or calculate the probability that the amount of reserves is not sufficient. After describing our approach and several models used to determine a distribution function for random sums we give two examples. The first one is applied to damages in motor insurance and the second to bodily injury. Keywords: Reserve, Random Sums. ~. 074007 (MlO) Actuarial statistics I: likelihood characterizations. Hiirlimann W., Winterthur, Switzerland, Astin Colloquium Leuven, 1995. The present work establishes some likelihood properties of Bernoulli mixed discrete-continuous models with special emphasis on actuarial models. Potential statistical applications include one-parametric deformations from discrete to continuous families of distributions, perturbed models for Robust Statistics, models with unobservable events of use in the (re)insurance of large claims, insurance claims models with outliers, distribution-free reinsurance models via stop-loss extremal distributions (Section 1). Based on mean equivalence relation for parametric random functions, a mean equivalence class of loglikelihoods is considered. The corresponding systems of sample mean equations lead to a class of maximum likelihood mean equivalent estimators, or shortly likelihood estimators (Section 2). For Bernoulli mixed discrete-continuous models alikelihood estimator, which h is asymptotically equal to the maximum likelihood estimator, is constructed. Sometimes it is equal to the maximum likelihood estimator. This is the case for a insurance claims model with outliers. specific Furthermore this estimator appears to be useful as estimation method for the unknown parameters of a stop-loss ordered extremal distribution associated to the class of positive risks with fixed mean and variance (Section 3). The mainly theoretical Section 4 studies the relationship between pseudo-estimators and orthogonal parameters. After some motivation for the use of pseudo-estimators has been given, the well-known information inequality
Abstracts and Reviews
for pseudo-estimators is recalled. It extends the classical Cramer-Rao lower bound for unbiased estimators. Several examples,,which ..~~_ illustrate the usefulness of the generalized inequality, are presented. As a main result, a substantial generalization of a characterization obtained previously by the author is derived. It characterizes unbiased pseudo-estimators related to the class of maximum likelihood mean equivalent estimators (introduced in Section 2) under an orthogonal condition of the parameter of interest. A thorough discussion and several examples follow. In particular a complete likelihood characterization of one-parametric maximum likelihood estimators is formulated (Corollary 4.2). The latter result is applied in Section 5 to provide “rational” justifications for the use of some categories of statistical models with location or scale parameters encountered in the fields of Economy and Social Sciences. First two simpler variants of classical results Then practically quite useful are formulated. characterizations ofthe compound Poisson, binomial and negative binomial gamma models in collective risk theory are derived.... In Section 6 sufficient conditions for the sample mean to be a somewhat robust likelihood estimator in a Bernoulli mixed discrete-continuous model are given. These conditions characterize in Section 7 a mean scaled gamma outlier model encountered in the recent actuarial literature. Keywords: Bernoulli mixedmodel, outliers, distributionfree reinsurance, log-likelihood mean equivalence, maximum likelihood, pseudo-estimators, information inequality, orthogonal parameters, location parameter, scale parameter, sample mean, robustness. 074008 (MlO,E51) A recursive scheme for perpetuities with random positive interest rates. Part I. Analytical results. De Schepper A. (*), Goovaerts M.J. (**), Kaas R., RUCA, University of Antwerp (*), Katholieke Universiteit Leuven, Belgium (**), University of Amsterdam, The Netherlands, Astin Colloquium Leuven, 1995. Recently, the authors showed how interest randomness in actuarial functions can be described by means of Wiener processes using path integrals. This paper wants to present an extension of this kind of models, by investigating the situation of interest rates that cannot become negative. The case of an annuity certain and in particular that of a perpetuity will be dealt with in detail.
Keywords: Interest randomness, Annuity Perpetuity, Wiener processes, Distribution.
231 certain,
074009 (MlO) Actuarial statistics II: exponential characterizations. Hiirlimann W., Winterthur, Switzerland, Astin Colloquium Leuven, 199.5. The present paper is based on the well-known characterization of regular exponential families by minimum variance unbiased estimators. Four simple equivalent statements, two of which do not seem to be formulated so far in the literature, recall this fact (Proposition 2.1). Inspired by a recent work of Targhetta (1990), conditions, under which the 2x2 Bhattacharya matrix of a regular exponential family is diagonal, are derived (Theorem 3.1). Interesting theoretical insights related to the special class of regular exponential families with diagonal 2x2 Bhattacharya matrix and quadratic variance of the observation are discussed. It is shown how this class can be generated using commutative transformations starting from a single elementary distribution (expanding on a previous idea by Morris (1982)). Of special actuarial interest, one observes that this class contains the shifted gamma process (Example 3.1) and a continuous version of the two-parametric Panjer family of discrete distributions (Remarks 4.1). Keywords: exponential family, information inequality, MVUE, Bhattacharya matrix, quadratic variance function, elementary distributions, shifted gamma process, continuous Panjer family.
Mll: STOCHASTIC MODELS FOR CLAIM FREQUENCY, CLAIM SIZE AND AGGREGATE CLAIMS 074010 (Mll) A time-continuous Markov chain interest model with applications to insurance. Norberg R., Laboratory of Actuarial Mathematics, Denmark Applied Stochastic Models and Data Analysis, Vol. II, nr. 3, 1995, pp. 245-256. The force of interest is modelled by a homogeneous time-continuous Markov chain with finite state space. Ordinary differential equations are obtained for expected values of various functionals of this process, in particular for moments of present values of payment streams that may be deterministic or, possibly, also stochastic and driven by a time-continuous Markov