Estimating conditional tail expectation with actuarial applications in view

We develop statistical inferential tools for estimating and comparing conditional tail expectation (CTE) functions, which are of considerable interest in actuarial science. In particular, we construct estimators for the CTE functions, develop the necessary asymptotic theory for the estimators, and then use the theory for constructing confidence intervals and bands for the functions. Both parametric and non-parametric approaches are explored. Simulation studies illustrate the performance of estimators in various situations. Results are obtained under minimal assumptions, and the general Vervaat process plays a crucial role in achieving these goals.     

On a class of improved estimators of variance and estimation under order restriction

We consider the estimation of error variance and construct a class of estimators improving upon the usual estimators uniformly under entropy loss or under squared error loss. Through a Monte Carlo simulation study, the magnitude of the risk reduction of our improved estimator as compared with the usual one is examined in a context of a nested linear hypothesis testing of a linear regression model, where substantial risk reduction can be attained. We also construct a class of confidence intervals having larger coverage probabilities and not larger interval lengths than those of the usual ones. This allows us to construct a class of estimators universally dominating the usual ones. Further, we consider the estimation of order-restricted normal variances. We give a class of isotonic regression estimators improving upon the usual ones under various types of order restrictions. We also give a class of improved confidence intervals over the usual ones, and a class of estimators universally dominating the usual ones.     

Bias-Corrected Moment-Based Estimators for Parametric Models Under Endogenous Stratified Sampling

This paper provides an integrated approach for estimating parametric models from endogenous stratified samples. We discuss several alternative ways of removing the bias of the moment indicators usually employed under random sampling for estimating the parameters of the structural model and the proportion of the strata in the population. Those alternatives give rise to a number of moment-based estimators that are appropriate for both cases where the marginal strata probabilities are known and unknown. The derivation of our estimators is very simple and intuitive and incorporates as particular cases most of the likelihood-based estimators previously suggested by other authors.     

Semiparametric proportional means model for marker data contingent on recurrent event

In many biomedical studies with recurrent events, some markers can only be measured when events happen. For example, medical cost attributed to hospitalization can only incur when patients are hospitalized. Such marker data are contingent on recurrent events. In this paper, we present a proportional means model for modelling the markers using the observed covariates contingent on the recurrent event. We also model the recurrent event via a marginal rate model. Estimating equations are constructed to derive the point estimators for the parameters in the proposed models. The estimators are shown to be asymptotically normal. Simulation studies are conducted to examine the finite-sample properties of the proposed estimators and the proposed method is applied to a data set from the Vitamin A Community Trial.     

Nonparametric smooth estimation of the expected inactivity time function

The expected inactivity time (EIT) function (also known as the mean past lifetime function) is a well known reliability function which has application in many disciplines such as survival analysis, actuarial studies and forensic science, to name but a few. In this paper, we use a fixed design local polynomial fitting technique to obtain estimators for the EIT function when the lifetime random variable has an unknown distribution. It will be shown that the proposed estimators are asymptotically unbiased, consistent and also, when standardized, has an asymptotic normal distribution. An optimal bandwidth, which minimizes the AMISE (asymptotic mean integrated squared error) of the estimator, is derived. Numerical examples based on simulated samples from various lifetime distributions common in reliability studies will be presented to evaluate the performances of these estimators. Finally, three real life applications will also be presented to further illustrate the wide applicability of these estimators.     

On the $$L_p$$ norms of kernel regression estimators for incomplete data with applications to classification

We consider kernel methods to construct nonparametric estimators of a regression function based on incomplete data. To tackle the presence of incomplete covariates, we employ Horvitz–Thompson-type inverse weighting techniques, where the weights are the selection probabilities. The unknown selection probabilities are themselves estimated using (1) kernel regression, when the functional form of these probabilities are completely unknown, and (2) the least-squares method, when the selection probabilities belong to a known class of candidate functions. To assess the overall performance of the proposed estimators, we establish exponential upper bounds on the \(L_p\) norms, \(1\le p<\infty \), of our estimators; these bounds immediately yield various strong convergence results. We also apply our results to deal with the important problem of statistical classification with partially observed covariates.     

精算思想的演进与精算理论发展综述

精算是保险发展的基础,是保险经营的技术支持。精算在国外有四百年的发展历史,引入中国只有二十年。要使精算技术在中国得到发展创新并为社会需要服务,必须了解精算思想产生的历史背景,厘清精算理论发展的脉络,真正把握精算思想的实质。基于此,介绍了精算各发展时期的主要代表人物及其学术思想,阐述精算技术对各时期保险发展的影响,同时对精算学与复利理论、数学、统计学、计算技术、金融经济学交叉融合的历史过程进行了分析述评。     

Empirical Bayes confidence intervals shrinking both means and variances

Summary.  We construct empirical Bayes intervals for a large number p of means. The existing intervals in the literature assume that variances     are either equal or unequal but known. When the variances are unequal and unknown, the suggestion is typically to replace them by unbiased estimators     . However, when p is large, there would be advantage in 'borrowing strength' from each other. We derive double-shrinkage intervals for means on the basis of our empirical Bayes estimators that shrink both the means and the variances. Analytical and simulation studies and application to a real data set show that, compared with the t -intervals, our intervals have higher coverage probabilities while yielding shorter lengths on average. The double-shrinkage intervals are on average shorter than the intervals from shrinking the means alone and are always no longer than the intervals from shrinking the variances alone. Also, the intervals are explicitly defined and can be computed immediately.     

Nonparametric estimation of the conditional mean residual life function with censored data

The conditional mean residual life (MRL) function is the expected remaining lifetime of a system given survival past a particular time point and the values of a set of predictor variables. This function is a valuable tool in reliability and actuarial studies when the right tail of the distribution is of interest, and can be more informative than the survivor function. In this paper, we identify theoretical limitations of some semi-parametric conditional MRL models, and propose two nonparametric methods of estimating the conditional MRL function. Asymptotic properties such as consistency and normality of our proposed estimators are established. We investigate via simulation study the empirical properties of the proposed estimators, including bootstrap pointwise confidence intervals. Using Monte Carlo simulations we compare the proposed nonparametric estimators to two popular semi-parametric methods of analysis, for varying types of data. The proposed estimators are demonstrated on the Veteran’s Administration lung cancer trial.     

The generalized Poisson-binomial distribution and the computation of its distribution function

The Poisson-binomial distribution is useful in many applied problems in engineering, actuarial science and data mining. The Poisson-binomial distribution models the distribution of the sum of independent but non-identically distributed random indicators whose success probabilities vary. In this paper, we extend the Poisson-binomial distribution to a generalized Poisson-binomial (GPB) distribution. The GPB distribution corresponds to the case where the random indicators are replaced by two-point random variables, which can take two arbitrary values instead of 0 and 1 as in the case of random indicators. The GPB distribution has found applications in many areas such as voting theory, actuarial science, warranty prediction and probability theory. As the GPB distribution has not been studied in detail so far, we introduce this distribution first and then derive its theoretical properties. We develop an efficient algorithm for the computation of its distribution function, using the fast Fourier transform. We test the accuracy of the developed algorithm by comparing it with enumeration-based exact method and the results from the binomial distribution. We also study the computational time of the algorithm under various parameter settings. Finally, we discuss the factors affecting the computational efficiency of the proposed algorithm and illustrate the use of the software package.     

Utilizing auxiliary variables in sequential ratio and regression estimation schemes

Ratio and regression estimators for a mean are considered in conjunction with certain sequential sampling schemes. An auxiliary variable is assumed present and both fixed-cost and fixed- width confidence interval stopping rules are investigated. The asymptotic distributions of the estimators are derived as well as optimal probabilities pertinent to the schemes. Comparisons are made with results of certain double sampling procedures. Estimation of the ratio of two means is also considered and the results of a Monte Carlo simulation are included.     

On the small sample properties of norm-restricted maximum likelihood estimators for logistic regression models

This paper develops alternatives to maximum likelihood estimators (MLE) for logistic regression models and compares the mean squared error (MSE) of the estimators. The MLE for the vector of underlying success probabilities has low MSE only when the true probabilities are extreme (i.e., near 0 or 1). Extreme probabilities correspond to logistic regression parameter vectors which are large in norm. A competing “restricted” MLE and an empirical version of it are suggested as estimators with better performance than the MLE for central probabilities. An approximate EM-algorithm for estimating the restriction is described. As in the case of normal theory ridge estimators, the proposed estimators are shown to be formally derivable by Bayes and empirical Bayes arguments. The small sample operating characteristics of the proposed estimators are compared to the MLE via a simulation study; both the estimation of individual probabilities and of logistic parameters are considered.     

A two-phase sampling scheme and πps designs

A two-phase approach for sampling with unequal inclusions probabilities and fixed sample size is presented. The expansion estimator using target inclusion probabilities is suggested for estimation of population parameters. As an alternative, the estimator for two-phase sampling can be used for estimation. Inclusion probabilities are shown to be asymptotically equivalent to the targeted inclusion probabilities. By means of simulation associated estimators are shown to work well with respect to bias and precision.     

Bayes methods in the ecological fallacy cntext:estimation of individual correlation from aggregate data

The ecological fallacy is related to Simpson's paradox (1951) where relationships among group means may be counterintuitive and substantially different from relationships within groups, where the groups are usually geographic entities such as census tracts. We consider the problem of estimating the correlation between two jointly normal random variables where only ecological data (group means) are available. Two empirical Bayes estimators and one fully Bayesian estimator are derived and compared with the usual ecological estimator, which is simply the Pearson correlation coefficient of the group sample means. We simulate the bias and mean squared error performance of these estimators, and also give an example employing a dataset where the individual level data are available for model checking. The results indicate superiority of the empirical Bayes estimators in a variety of practical situations where, though we lack individual level data, other relevant prior information is available.     

Improved estimators for the common mean and ordered means of two normal distributions with ordered variances

We first consider the problem of estimating the common mean of two normal distributions with unknown ordered variances. We give a broad class of estimators which includes the estimators proposed by Nair (1982) and Elfessi et al. (1992) and show that the estimators stochastically dominate the estimators which do not take into account the order restriction on variances, including the one given by Graybill and Deal (1959). Then we propose a broad class of individual estimators of two ordered means when unknown variances are ordered. We show that in estimating the mean with larger variance, estimators which do not take into account the order restriction on variances are stochastically dominated by the proposed class of estimators which take into account both order restrictions. However, in estimating the mean with smaller variance, similar improvement is not possible even in terms of mean squared error. We also show a domination result in the simultaneous estimation problem of two ordered means. Further, improving upon the unbiased estimators of the two means is discussed.     

Nonparametric estimation of transition probabilities in a non-Markov illness–death model

In this paper we consider nonparametric estimation of transition probabilities for multi-state models. Specifically, we focus on the illness-death or disability model. The main novelty of the proposed estimators is that they do not rely on the Markov assumption, typically assumed to hold in a multi-state model. We investigate the asymptotic properties of the introduced estimators, such as their consistency and their convergence to a normal law. Simulations demonstrate that the new estimators may outperform Aalen–Johansen estimators (the classical nonparametric tool for estimating the transition probabilities) in non-Markov situation. An illustration through real data analysis is included.     

Inverse probability weighted estimators for single-index models with missing covariates

Abstract

In this article, we consider the inverse probability weighted estimators for a single-index model with missing covariates when the selection probabilities are known or unknown. It is shown that the estimator for the index parameter by using estimated selection probabilities has a smaller asymptotic variance than that with true selection probabilities, thus is more efficient. Therefore, the important Horvitz-Thompson property is verified for the index parameter in single index model. However, this difference disappears for the estimators of the link function. Some numerical examples and a real data application are also conducted to illustrate the performances of the estimators.     

中国地方政府债务的结构性风险

在或有债务不确定触发地方政府代偿的现实背景下,本文对地方政府显性债务和或有债务的结构性风险进行了模型刻画,给出了两类债务违约概率的显示解及其估计方法。基于地方政府报告的显性债务和审计署的有关公告等,本文对地方政府的显性债务和或有债务规模进行了结构性分解和估算,多视角估计了不同久期（或平均债务到期时间）下的两类债务的结构性风险状态,同时对比分析了其变动趋势。模型估计表明：地方政府债务风险的主要形态是或有债务的不确定触发,在部分时段内,地方政府的结构性代偿压力明显较大,但债券置换等旨在拉长债务久期的政策设计有助于缓释地方政府债务的信用风险边界,降低地方政府的结构性代偿压力。本文用具体数字揭示了地方政府债务的结构性风险状况及其变动趋势,以及地方政府债务久期之伸缩与其代偿压力之间的定量关系。     

General purpose multiply robust data integration procedures for handling nonprobability samples

In recent years, there has been an increased interest in combining probability and nonprobability samples. Nonprobability sample are cheaper and quicker to conduct but the resulting estimators are vulnerable to bias as the participation probabilities are unknown. To adjust for the potential bias, estimation procedures based on parametric or nonparametric models have been discussed in the literature. However, the validity of the resulting estimators relies heavily on the validity of the underlying models. Also, nonparametric approaches may suffer from the curse of dimensionality and poor efficiency. We propose a data integration approach by combining multiple outcome regression models and propensity score models. The proposed approach can be used for estimating general parameters including totals, means, distribution functions, and percentiles. The resulting estimators are multiply robust in the sense that they remain consistent if all but one model are misspecified. The asymptotic properties of point and variance estimators are established. The results from a simulation study show the benefits of the proposed method in terms of bias and efficiency. Finally, we apply the proposed method using data from the Korea National Health and Nutrition Examination Survey and data from the National Health Insurance Sharing Services.     

Estimation of two order restricted normal means with unknown and possibly unequal variances

Estimation of each of and linear functions of two order restricted normal means is considered when variances are unknown and possibly unequal. We replace unknown variances with sample variances and construct isotonic regression estimators, which we call in our paper the plug-in estimators, to estimate ordered normal means. Under squared error loss, a necessary and sufficient condition is given for the plug-in estimators to improve upon the unrestricted maximum likelihood estimators uniformly. As for the estimation of linear functions of ordered normal means, we also show that when variances are known, the restricted maximum likelihood estimator always improves upon the unrestricted maximum likelihood estimator uniformly, but when variances are unknown, the plug-in estimator does not always improve upon the unrestricted maximum likelihood estimator uniformly.