Claims reserving: A correlated Bayesian model

Estimation of adequate reserves for outstanding claims is one of the main activities of actuaries in property/casualty insurance and a major topic in actuarial science. The need to estimate future claims has led to the development of many loss reserving techniques. There are two important problems that must be dealt with in the process of estimating reserves for outstanding claims: one is to determine an appropriate model for the claims process, and the other is to assess the degree of correlation among claim payments in different calendar and origin years. We approach both problems here. On the one hand we use a gamma distribution to model the claims process and, in addition, we allow the claims to be correlated. We follow a Bayesian approach for making inference with vague prior distributions. The methodology is illustrated with a real data set and compared with other standard methods.     

A linear Bayesian stochastic approximation to update project duration estimates

By relaxing the unrealistic assumption of probabilistic independence on activity durations in a project, this paper develops a hierarchical linear Bayesian estimation model. Statistical dependence is established between activity duration and the amount of resource, as well as between the amount of resource and the risk factor. Upon observation or assessment of the amount of resource required for an activity in near completion, the posterior expectation and variance of the risk factor can be directly obtained in the Bayesian scheme. Then, the expected amount of resources required for and the expected duration of upcoming activities can be predicted. We simulate an application project in which the proposed model tracks the varying critical path activities on a real time basis, and updates the expected project duration throughout the entire project. In the analysis, the proposed model improves the prediction accuracy by 38.36% compared to the basic PERT approach.     

On the sequence of Bayesian estimates of normal random walk process

Let (μ_t)^∞_t=0 be a k-variate (k?1) normal random walk process with successive increments being independently distributed as normal N(δ, R), and μ₀ being distributed as normal N(0, V₀). Let X_t have normal distribution N(μ_t, Σ) when μ_t is given, t = 1, 2,….Then the conditional distribution of μ_t given X₁, X₂,…, X_t is shown to be normal N(U_t, V_t) where U_t's and V_t's satisfy some recursive relations. It is found that there exists a positive definite matrix V and a constant θ, 0 < θ < 1, such that, for all t?1,

$\text{|R}\text{1}\text{2}\text{(V}{}^{\mathrm{?1}}{}_{\mathrm{t}}\text{?V}{}^{\mathrm{?1}}\text{R}\text{1}\text{2}\text{|<θ}{}^{\mathrm{t}}\text{|R}\text{1}\text{2}\text{(V}{}^{\mathrm{?1}}{}_0\text{?V}{}^{\mathrm{?1}}\text{)R}\text{1}\text{2}\text{|}$

where the norm |·| means that |A| is the largest eigenvalue of a positive definite matrix A. Thus, V_t approaches to V as t approaches to infinity. Under the quadratic loss, the Bayesian estimate of μ_t is U_t and the process {U_t}^∞_t=₀, U₀=0, is proved to have independent successive increments with normal N(θ, V_t?V_t+1+R) distribution. In particular, when V₀ =V then V_t = V for all t and {U_t}^∞_t=0 is the same as {μ_t}^∞_t=0 except that U₀ = 0 and μ₀ is random.     

Asymptotic theory of nonparametric regression estimates with censored data

For regression analysis, some useful information may have been lost when the responses are right censored. To estimate nonparametric functions, several estimates based on censored data have been proposed and their consistency and convergence rates have been studied in literature, but the optimal rates of global convergence have not been obtained yet. Because of the possible information loss, one may think that it is impossible for an estimate based on censored data to achieve the optimal rates of global convergence for nonparametric regression, which were established by Stone based on complete data. This paper constructs a regression spline estimate of a general nonparametric regression function based on right_censored response data, and proves, under some regularity conditions, that this estimate achieves the optimal rates of global convergence for nonparametric regression. Since the parameters for the nonparametric regression estimate have to be chosen based on a data driven criterion, we also obtain the asymptotic optimality of AIC, AICC, GCV, Cp and FPE criteria in the process of selecting the parameters.     

Correction in Bayesian nonparametric estimation in a series system or a competing-risk model

We revisit the Bayesian nonparametric estimator of the component reliability of a series system under a competing risk scenario. Unlike the estimator of Salinas-Torres et al. (2002), our estimator of the component survival function is the corresponding Bayes estimator under quadratic loss.     

On a class of asymptotically optimal nonparametric tests for grouped data II

Summary In a previous paper [5], the author has proposed a class of asymptotically optimal (in the sense of Wald [11]) nonparametric tests for testing the hypothesis of no regression in a multiple linear regression model. In the present paper, we are interested in testing that the intercept in the multiple (linear) regression model is zero along with the absence of regression. A class of permutationally distribution-free tests has been proposed and their asymptotic optimality has been established. These results generalize analogus findings of Puri and Sen [9] for ungrouped data. As an important application, an alternative test to the problem of paired comparison has been considered. Research supported by National Institutes of Health, Institute of General Medical Sciences GM 12868-05.     

Modified nonparametric kernel estimates of a regression function and their consistencies with rates

Summary Two sets of modified kernel estimates of a regression function are proposed: one when a bound on the regression function is known and the other when nothing of this sort is at hand. Explicit bounds on the mean square errors of the estimators are obtained. Pointwise as well as uniform consistency in mean square and consistency in probability of the estimators are proved. Speed of convergence in each case is investigated. Major work of this research was completed during the first author's two visits (November–December, 1983 and August–September 1984) to the second author at the Universite du Quebec a Montreal. Part of the work of the second author was supported by the Air Force Office of Scientific Research under contract F49620-85-C-0008 while he was at the University of Pittsburgh during Spring in 1985.     

Bayesian Nonparametric Analysis for a Generalized Dirichlet Process Prior

This paper considers a generalization of the Dirichlet process which is obtained by suitably normalizing superposed independent gamma processes having increasing integer-valued scale parameter. A comprehensive treatment of this random probability measure is provided. We prove results concerning its finite-dimensional distributions, moments, predictive distributions and the distribution of its mean. Most expressions are given in terms of multiple hypergeometric functions, thus highlighting the interplay between Bayesian Nonparametrics and special functions. Finally, a suitable simulation algorithm is applied in order to compute quantities of statistical interest.     

Bayesian robustness under a skew-normal class of prior distribution

We develop a global sensitivity analysis to measure the robustness of the Bayesian estimators with respect to a class of prior distributions. This class arises when we consider multiplicative contamination of a base prior distribution. A similar structure was presented by van der Linde [12]. Some particular specifications for this multiplicative contamination class coincide with well known families of skewed distributions. In this paper, we explore the skew-normal multiplicative contamination class for the prior distribution of the location parameter of a normal model. Results of a Bayesian conjugation and expressions for some measures of distance between posterior means and posterior variance are obtained. We also elaborate on the behavior of the posterior means and of the posterior variances through a simulation study.     

Bayesian testing of nonparametric hypotheses and its application to global optimization

Random distribution functions are the basic tool for solving nonparametric decision-theoretic problems. In 1974, Doksum introduced the family of distributions neutral to the right, that is, distributions such thatF(t ₁),[F(t ₂)–F(t ₁)]/[1 –F(t ₁)],...,[F(t _k)–F(t _{k – 1})]/[1 –F(t _{k – 1})] are independent whenevert ₁ < ... <t _kIn practice, application of distributions neutral to the right has been prevented by the lack of a manageable analytical expression for probabilities of the typeP(F(t)<q) for fixedt andq. A subclass of such distributions can be provided which allows for a close expression of the characteristic function of log[1–F(t)], given the sample. Then, thea posteriori distribution ofF(t) is obtained by numerical evaluation of a Fourier integral. As an application, the global optimization problem is formulated as a problem of inference about the quantiles of the distributionF(y) of the random variableY=f(X), wheref is the objective function andX is a random point in the search domain.The author thanks J. Koronacki and R. Zielinski of the Polish Academy of Sciences for their valuable criticism during the final draft of the paper.     

小批量生产的贝叶斯质量控制模型

本应用贝叶斯统计推断方法，研究了基于正态共轭先验分布和正态——逆伽玛共轭先验分布的小批量生产下的质量控制模型问题，根据不同控制对象的预报分布密度函数，分别构造了方差已知时的贝叶斯均值控制图和方差未知时的贝叶斯均值——标准差控制图，并与经典质量控制模型进行了比较。     

Gamma shared frailty model based on reversed hazard rate for bivariate survival data

The unknown or unobservable risk factors in the survival analysis cause heterogeneity between the individuals. Frailty models are used in the survival analysis to account for the unobserved heterogeneity in the individual risks to disease and death. In this paper, we suggest the shared gamma frailty model with the reversed hazard rate. We introduce the Bayesian estimation procedure using MCMC technique to estimate the parameters involved in the model and compare the frailty model with the baseline model. We apply the proposed models to Australian twin data set and suggest a better model.     

Bayesian nonparametrie inference and monte carlo optimization

As global or combinatorial optimization problems are not effectively tractable by means of deterministic techniques, Monte Carlo methods are used in practice for obtaining ”good“ approximations to the optimum. In order to test the accuracy achieved after a sample of finite size, the Bayesian nonparametric approach is proposed as a suitable context, and the theoretical as well as computational implications of prior distributions in the class of neutral to the right distributions are examined. The feasibility of the approach relatively to particular Monte Carlo procedures is finally illustrated both for the global optimization problem and the {0 - 1} programming problem.     

Tapered Covariance: Bayesian Estimation and Asymptotics

The method of maximum tapered likelihood has been proposed as a way to quickly estimate covariance parameters for stationary Gaussian random fields. We show that under a useful asymptotic regime, maximum tapered likelihood estimators are consistent and asymptotically normal for covariance models in common use. We then formalize the notion of tapered quasi-Bayesian estimators and show that they too are consistent and asymptotically normal. We also present asymptotic confidence intervals for both types of estimators and show via simulation that they accurately reflect sampling variability, even at modest sample sizes. Proofs, an example, and detailed derivations are provided in the supplementary materials, available online.     

On weighted estimates for a class of integral operators

Khabarovsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 4, pp. 177–183, July–August, 1993.     

On coefficient estimates for a class of holomorphic mappings

Let X be a complex Banach space with norm ‖ · ‖, B be the unit ball in X, D ⁿ be the unit polydisc in ℂ ⁿ . In this paper, we introduce a class of holomorphic mappings on B or D ⁿ . Let f(x) be a normalized locally biholomorphic mapping on B such that (Df(x))⁻¹ f(x) ∈ and f(x) − x has a zero of order k + 1 at x = 0. We obtain coefficient estimates for f(x). These results unify and generalize many known results. This work was supported by National Natural Science Foundation of China (Grant No. 10571164), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20050358052), the Jiangxi Provincial Natural Science Foundation of China (Grant No. 2007GZS0177) and Specialized Research Fund for the Doctoral Program of Jiangxi Normal University.     

Measuring the baseline sales and the promotion effect for incense products: a Bayesian state-space modeling approach

One of the most important research fields in marketing science is the analysis of time series data. This article develops a new method for modeling multivariate time series. The proposed method enables us to measure simultaneously the effectiveness of marketing activities, the baseline sales, and the effects of controllable/uncontrollable business factors. The critical issue in the model construction process is the method for evaluating the usefulness of the predictive models. This problem is investigated from a statistical point of view, and use of the Bayesian predictive information criterion is considered. The proposed method is applied to sales data regarding incense products. The method successfully extracted useful information that may enable managers to plan their marketing strategies more effectively.     

On a class of asymptotically optimal nonparametric tests for grouped data I

Summary Generalizing the results of Sen [8], a class of nonparametric tests for the hypothesis of no regression in the multiple linear regression model is obtained here. The asymptotic power properties of the proposed class of tests are studied, and the asymptotic optimality of the tests is established under the conditions of Wald [10]. Applications of the results are also considered. Research supported by National Institutes of Health, Institute of General Medical Sciences, Grant No. 12868-05.     

On the inconsistency of Bayesian non-parametric estimators in competing risks/multiple decrement models

In the competing risks/multiple decrement model, the joint distribution is often not identifiable given only the observed time of failure and the cause of failure. The traditional approach is consequently to assume a parametric model. In this paper we shall not do this, but rather assume a Bayesian stance, take a Dirichlet process as a prior distribution, and then calculate the posterior distribution given the data. In this paper we show that in dimensions ? 2, the posterior mean yields an inconsistent estimator of the joint probability law, contrary to the common assumption that the prior law ‘washes out’ with large samples. For single decrement mortality tables however, the non-parametric Bayesian method allows a flexible method for adjusting a standard mortality table to reflect mortality experience, or covariate information.