Linear Empirical Bayes Estimation Research Articles

Robust Empirical Bayes Confidence Intervals

We construct robust empirical Bayes confidence intervals (EBCIs) in a normal means problem. The intervals are centered at the usual linear empirical Bayes estimator, but use a critical value accounting for shrinkage. Parametric EBCIs that assume a normal distribution for the means (Morris (1983b)) may substantially undercover when this assumption is violated. In contrast, our EBCIs control coverage regardless of the means distribution, while remaining close in length to the parametric EBCIs when the means are indeed Gaussian. If the means are treated as fixed, our EBCIs have an average coverage guarantee: the coverage probability is at least 1 − α on average across the n EBCIs for each of the means. Our empirical application considers the effects of U.S. neighborhoods on intergenerational mobility.

Revisiting Empirical Bayes Methods and Applications to Special Types of Data

Empirical Bayes methods have been around for a long time and have a wide range of applications. These methods provide a way in which historical data can be aggregated to provide estimates of the posterior mean. This thesis revisits some of the empirical Bayesian methods and develops new applications. We first look at a linear empirical Bayes estimator and apply it on ranking and symbolic data. Next, we consider Tweedie's formula and show how it can be applied to analyze a microarray dataset. The application of the formula is simplified with the Pearson system of distributions. Saddlepoint approximations enable us to generalize several results in this direction. The results show that the proposed methods perform well in applications to real data sets.

Simultaneous estimation based on empirical likelihood and general maximum likelihood estimation

One typical problem in simultaneous estimation of mean values is estimating means of normal distributions, however when normality or any other distribution is not specified, more robust estimation procedures are demanded. A new estimation procedure is proposed based on empirical likelihood which does not request any specific distributional assumption. The new idea is based on incorporating empirical likelihood with general maximum likelihood estimation. One well-known nonparametric estimator, the linear empirical Bayes estimator, can be interpreted as an estimator based on empirical likelihood under some framework and it is shown that the proposed procedure can improve the linear empirical Bayes estimator. Numerical studies are presented to compare the proposed estimator with some existing estimators. The proposed estimator is applied to the problem of estimating mean values corresponding to high valued observations. Simulations and real data example of gene expression are provided.

A CREDIBILITY APPROACH FOR COMBINING LIKELIHOODS OF GENERALIZED LINEAR MODELS

AbstractGeneralized linear models are a popular tool for the modelling of insurance claims data. Problems arise with the model fitting if little statistical information is available. In case that related statistics are available, statistical inference can be improved with the help of the borrowing-strength principle. We present a credibility approach that combines the maximum likelihood estimators of individual canonical generalized linear models in a meta-analytic way to an improved credibility estimator. We follow the concept of linear empirical Bayes estimation, which reduces the necessary parametric assumptions to a minimum. The concept is illustrated by a simulation study and an application example from mortality modelling.

Linear Bayes estimator for the two-parameter exponential family under type II censoring

For the two-parameter exponential family, a linear Bayes method is proposed to simultaneously estimate the parameter vector consisting of location and scale parameters. The superiority of the proposed linear Bayes estimator (LBE) over the classical UMVUE is established in terms of the mean square error matrix (MSEM) criterion. The proposed LBE is simple and easy to use compared with the usual Bayes estimator, which is obtained by the MCMC method. Numerical results are presented to verify that the LBE works well. In the empirical Bayes framework, the paper invokes a linear empirical Bayes estimator (LEBE) by using a linear combination of historical samples. It is shown under some mild regularity conditions that the LEBE is superior to the classical UMVUE and the maximum likelihood estimator in terms of MSEM. It is further shown with numerical results that the performance of LEBE gets better with the increase in the number of historical samples.

Adaptive nonparametric empirical Bayes estimation via wavelet series: The minimax study

In the present paper, we derive lower bounds for the risk of the nonparametric empirical Bayes estimators. In order to attain the optimal convergence rate, we propose generalization of the linear empirical Bayes estimation method which takes advantage of the flexibility of the wavelet techniques. We present an empirical Bayes estimator as a wavelet series expansion and estimate coefficients by minimizing the prior risk of the estimator. As a result, estimation of wavelet coefficients requires solution of a well-posed low-dimensional sparse system of linear equations. The dimension of the system depends on the size of wavelet support and smoothness of the Bayes estimator. An adaptive choice of the resolution level is carried out using Lepski et al. (1997) method. The method is computationally efficient and provides asymptotically optimal adaptive EB estimators. The theory is supplemented by numerous examples.

A parametric Bayesian combination of local and regional information in flood frequency analysis

Because of their impact on hydraulic structure design as well as on floodplain management, flood quantiles must be estimated with the highest precision given available information. If the site of interest has been monitored for a sufficiently long period (more than 30–40 years), at‐site frequency analysis can be used to estimate flood quantiles with a fair precision. Otherwise, regional estimation may be used to mitigate the lack of data, but local information is then ignored. A commonly used approach to combine at‐site and regional information is the linear empirical Bayes estimation: Under the assumption that both local and regional flood quantile estimators have a normal distribution, the empirical Bayesian estimator of the true quantile is the weighted average of both estimations. The weighting factor for each estimator is conversely proportional to its variance. We propose in this paper an alternative Bayesian method for combining local and regional information which provides the full probability density of quantiles and parameters. The application of the method is made with the generalized extreme values (GEV) distribution, but it can be extended to other types of extreme value distributions. In this method the prior distributions are obtained using a regional log linear regression model, and then local observations are used within a Markov chain Monte Carlo algorithm to infer the posterior distributions of parameters and quantiles. Unlike the empirical Bayesian approach the proposed method works even with a single local observation. It also relaxes the hypothesis of normality of the local quantiles probability distribution. The performance of the proposed methodology is compared to that of local, regional, and empirical Bayes estimators on three generated regional data sets with different statistical characteristics. The results show that (1) when the regional log linear model is unbiased, the proposed method gives better estimations of the GEV quantiles and parameters than the local, regional, and empirical Bayes estimators; (2) even when the regional log linear model displays a severe relative bias when estimating the quantiles, the proposed method still gives the best estimation of the GEV shape parameter and outperforms the other approaches on higher quantiles provided the relative bias is the same for all quantiles; and (3) the gain in performance with the new approach is considerable for sites with very short records.

A note on linear empirical Bayes estimation of finite population means

In this paper we consider linear empirical Bayes estimation of m finite population means with n j observations from the jth population and obtain generalizations of some of the theorems of Ghosh and Lahiri (J. Amer. Statist. Assoc. 82 (1987) 1153) to the case where n j 's are not necessarily bounded and may change with m. Our results strengthen, among other things, the estimators of Ghosh and Lahiri (1987) by showing asymptotical optimality of their estimators under more general conditions. They have also useful applications in credibility theory, which considers a similar problem with m populations that are not necessarily finite.

On asymptotic optimality in empirical Bayes credibility

In this paper we consider linear empirical Bayes estimation of the pure premiums in a Bühlmann–Straub model, with m risk contracts and n i observations from the ith contract, and provide sufficient conditions, analogous to the conditions of Norberg [Scandinavian Actuarial Journal (1980) 172–194], for a stricter notion of asymptotic optimality. The sufficient conditions are used in the paper to show asymptotic optimality of explicit non-parametric examples. Our results, among other things, strengthen the estimators proposed by Bühlmann and Straub (1970) and their generalization by Sundt (1983). They also strengthen the finite population mean estimators of Ghosh and Lahiri (1987) by showing the asymptotic optimality of their estimators under a weaker condition than the boundedness of the moments of fourth order.

Extended linear empirical bayes estimation

In this paper, the linear empirical Bayes estimation method, which is based on approximation of the Bayes estimator by a linear function, is generalized to an extended linear empirical Bayes estimation technique which represents the Bayes estimator by a series of algebraic polynomials. The extended linear empirical Bayes estimators are elaborated in the case of a location or a scale parameter. The theory is illustrated by examples of its application to the normal distribution with a location parameter and the gamma distribution with a scale parameter. The linear and the extended linear empirical Bayes estimators are constructed in these two cases and, then, studied numerically via Monte Carlo simulations. The simulations show that the extended linear empirical Bayes estimators have better convergence rates than the traditional linear empirical Bayes estimators.

Linear. empirical bayes estimation in the case of the wishart distribution

We consider independent pairs (X1,∑1), (X2,∑2),…,(Xn∑n), where each Si is distributed according to some unknown density function g(∑) and, given ∑i = ∑, X has a conditional density function g(x|∑) of the Wishart type. In each pair, the first component is observable but the second is not. After the (n + l)-th observation Xn+i is obtained, the objective is to estimate ∑ n+i corresponding to Xn+i. This estimator is called an empirical Bayes (EB) estimator of ∑. We construct a linear EB estimator of ∑ and examine its precision.

Using regional regression within index flood procedures and an empirical Bayesian estimator

Studies have illustrated the performance of at-site and regional flood quantile estimators. For realistic generalized extreme value (GEV) distributions and short records, a simple index-flood quantile estimator performs better than two-parameter (2P) GEV quantile estimators with probability weighted moment (PWM) estimation using a regional shape parameter and at-site mean and L-coefficient of variation (L-CV), and full three-parameter at-site GEV/PWM quantile estimators. However, as regional heterogeneity or record lengths increase, the 2P-estimator quickly dominates. This paper generalizes the index flood procedure by employing regression with physiographic information to refine a normalized T-year flood estimator. A linear empirical Bayes estimator uses the normalized quantile regression estimator to define a prior distribution which is employed with the normalized 2P-quantile estimator. Monte Carlo simulations indicate that this empirical Bayes estimator does essentially as well as or better than the simpler normalized quantile regression estimator at sites with short records, and performs as well as or better than the 2P-estimator at sites with longer records or smaller L-CV.

The best uniform convergence rate of linear empirical bayes estimators

In this paper the thought on the uniform convergence of an empirical Bayes estimator or linear empirical Bayes (l.e.B.) estimator is advanced. Under two different models the l.e.B. estimators of the parameter are constructed respectively. It is proved that the convergence rates and uniform convergence rates of these l.e.B. estimators are all one with respect to the corresponding prior families. It is shown that the uniform convergence rate one is the best, under mild assumptions imposed on the conditional density of the sample.

Empirical Bayes approach to multiparameter estimation: with special reference to multinomial distribution

Empirical Bayes approach to estimation of many parameters is considered. Special features of the techniques discussed are: (i) the handling of unequal sample sizes at various stages of an Empirical Bayes sampling scheme and (ii) a general iterative procedure for estimating the parameters of a parametric prior distribution based on the likelihood approach. Linear empirical Bayes estimation is also considered. Applica- tion of the general techniques is demonstrated with special reference to a multinomial data distribution. p where, for every i, 0 < Xi ~ m, 0 < 0i < 1 and i=1 ~ Xi : m, i=~=l 0i : 1. The above probability model is fundamental in the analysis of categorical data and contingency table analyses. In the above the quantity m often represents the sample size of a sample of objects classified into p classes. Suppose that in a current experiment, m observations are made on a p-category Bernoulli r.v. Y with probability 0i of belonging to category i. Let Xg be the number of Y's in i-th

A linear regression with unobserved dependent variables

Let (⊖,X) be a random vector such that E(X|⊖) = ⊖ and Var(x|⊖) a + b⊖ + c⊖2 for some known constants a, b and c. Assume X1,…,Xn are independent observations which have the same distribution as X. Let t(X) be the linear regression of ⊖ on X. The linear empirical Bayes estimator is used to approximate the linear regression function. It is shown that under appropriate conditions, the linear empirical Bayes estimator approximates the linear regression well in the sense of mean squared error.

On the bounded regret of empirical bayes estimators

Let (θ1,x1),…,(θn,xn) be independent and identically distributed random vectors with E(xθ) = θ and Var(x|θ) = a + bθ + cθ2. Let ti be the linear Bayes estimator of θi and θ~i be the linear empirical Bayes estimator of θi as proposed in Robbins (1983). When Ex and Var x are unknown to the statistician. The regret of using θ~i instead of ti because of ignorance of the mean and the variance is ri = E(θi − θi)2 −E(ti −θi)2. Under appropriate conditions cumulative regret Rn = r1+…rn is shown to have a finite limit even when n tends to infinity. The limit can be explicitly computed in terms of a,b,c and the first four moments of x.