Smooth empirical Bayes estimation for continuous distributions

Estimation of several normal means, with emphasis on the normal regression empirical Bayes model, is considered. Within this model, the empirical Bayes estimator is of Stein type. A hierarchical Bayes analysis provides a family of minimax estimators. The performances of the empirical Bayes and hierarchical Bayes estimators are also compared. The shrinkage behavior of those estimators is shown in an application to the two-way classification.

Assessments of the potential health impacts of contaminants and other environmental risk factors are often based on comparisons of disease rates among collections of spatially aligned areas. These comparisons are valid only if the observed rates adequately reflect the true underlying area‐specific risk. In areas with small populations, observed incidence values can be highly unstable and true risk differences among areas can be masked by spurious fluctuations in the observed rates. We examine the use of Bayes and empirical Bayes methods for stabilizing incidence rates observed in geographically aligned areas. While these methods improve stability, both the Bayes and empirical Bayes approaches produce a histogram of the estimates that is too narrow when compared to the true distribution of risk. Constrained empirical Bayes estimators have been developed that provide improved estimation of the variance of the true rates. We use simulations to compare the performance of Bayes, empirical Bayes, and constrained empirical Bayes approaches for estimating incidence rates in a variety of multivariate Gaussian scenarios with differing levels of spatial dependence. The mean squared error of estimation associated with the simulated observed rates was, on average, five times greater than that of the Bayes and empirical Bayes estimates. The sample variance of the standard Bayes and empirical Bayes estimates was consistently smaller than the variance of the simulated rates. The constrained estimators produced collections of rate estimates that dramatically improved estimation of the true dispersion of risk. In addition, the mean square error of the constrained empirical Bayes estimates was only slightly greater than that of the unconstrained rate estimates. We illustrate the use of empirical and constrained empirical Bayes estimators in an analysis of lung cancer mortality rates in Ohio.

Comparison of nonparametric methods in nonlinear mixed effects models

Investigating the effects of the fixed and varying dispersion parameters of Poisson-gamma models on empirical Bayes estimates

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

The relative 'performances of improved ridge estimators and an empirical Bayes estimator are studied by means of Monte Carlo simulations. The empirical Bayes method is seen to perform consistently better in terms of smaller MSE and more accurate empirical coverage than any of the estimators considered here. A bootstrap method is proposed to obtain more reliable estimates of the MSE of ridge esimators. Some theorems on the bootstrap for the ridge estimators are also given and they are used to provide an analytical understanding of the proposed bootstrap procedure. Empirical coverages of the ridge estimators based on the proposed procedure are generally closer to the nominal coverage when compared to their earlier counterparts. In general, except for a few cases, these coverages are still less accurate than the empirical coverages of the empirical Bayes estimator.

under entropy loss function, the E-Bayes estimation and empirical bayes estimation to the parameter of inverse Weibull distribution used conjugate prior are discussed. And we prove the empirical bayes estimation is asymptotically optimal. At last, the MSE of the estimations are compared based on Monte Carlo simulation study. According to these comparisons, it is suggested that the accuracy of E-Bayes estimation is close to the empirical bayes estimation.

Data on job histories is commonly available from study subjects and worksites, therefore jobs are often used for assigning exposures in historical epidemiological studies. Exposure estimates are often derived by offering jobs as fixed effects in statistical models. An alternative approach would be to offer job as a random effect to obtain empirical Bayes estimates of exposure. This approach is more efficient since it weights exposure estimates according to the within-job and between-job variability and the number of measurements for each job. We assess three models for predicting historical dust exposures of sawmill workers. Models were developed using 407 inhalable dust measurements collected from 58 jobs in four sawmills. The first model incorporated all variables as fixed effects; the second added a random term to account for correlation within workers; and the third offered random terms for worker, job and mill (hierarchical model). Empirical Bayes estimates were used to calculate job-specific exposures from the hierarchical model. The fixed effects and random worker mixed models performed nearly identically because there was low within-worker correlation (r = 0.26). The Bayesian exposure predictions from the hierarchical model were slightly more correlated with the observed mill-job arithmetic means than those from the models where jobs were fixed effects (0.74 versus 0.70). While we observed no large differences in exposure estimates by treating job as a fixed or random effect, treating job as a random effect allowed for job-specific coefficients to be estimated for every job while borrowing strength in the presence of sparse data by assuming that the job means are normally distributed around the group mean. In addition, empirical Bayes job estimates can be used for a posteriori job grouping. The use of this method for retrospective exposure assessment should continue to be examined.

Estimation of finite population means is considered when samples are collected using a stratified sampling design. Finite populations for different strata are assumed to be realizations from different superpopulations. The true means of the observations lie on a regression surface with random intercepts for different strata. The true sampling variances are also different and random for different strata. The strata are connected through two common prior distributions, one for the intercepts and another for the sampling variances for all the strata. The model is appropriate in two important survey situations. First, it can be applied to repeated surveys where the physical characteristics of the sampling units change slowly over time. Second, the model is appropriate in small-area estimation problems where a very few samples are available for any particular area. Empirical Bayes estimators of the finite population means are shown to be asymptotically optimal in the sense of Robbins. The proposed empirical Bayes estimators are also compared to the classical regression estimators in terms of the relative savings loss due to Efron and Morris. A measure of variability of the proposed empirical Bayes estimator is considered based on bootstrap samples. This measure of variability incorporates all sources of variations due to the estimation of various model parameters. A numerical study is conducted to evaluate the performance of the proposed empirical Bayes estimator compared to rival estimators.

In this paper we construct sequences of estimators for a density function and its derivatives, which are not assumed to be uniformly bounded, using classes of kernel functions. Utilizing these estimators, a sequence of empirical Bayes estimators is proposed. It is found that this sequence is asymptotically optimal in the sense of Robbins (Ann. Math. Statist. 35 (1964) 1-20). The rates of convergence of the Bayes risks associated with the proposed empirical Bayes estimators are obtained. It is noted that the exact rate is $n^{-q}$ with $q \leqq \frac{1}{3}$. An example is given and an explicit kernel function is indicated.

Cancer is a complex disease driven by numerous (bio)molecular mechanisms and clinical risk factors. Prediction models aim to capture such cancer drivers to accurately predict patient-specific cancer outcomes. Omics technologies, which analyze the human genome, enable the integration of molecular data into prediction models. However, these data are often high-dimensional as they usually comprise over 10,000 variables (e.g., gene expression measurements) for only ~100 patients. This high-dimensionality poses substantial challenges in finding the relevant variables in a large pool of candidates. This thesis introduces three methods that address specific challenges in high-dimensional omics-based cancer prediction. Several prediction models can predict cancer outcomes from high-dimensional data. The canonical learner for these prediction applications is penalized regression which considers a linear relationship for the variables. Tree-based methods offer a promising alternative class of learners. This thesis focuses on advancing tree-based learning approaches to improve prediction accuracy in cancer outcomes. A key aspect in comparing learners is assessing their predictive performance, particularly in the small-sample settings that are common in omics research. Chapter 2 presents a novel framework to estimate this performance when no independent test dataset is available. In such a scenario, one usually employs resampling techniques such as cross-validation and bootstrapping. A problem for such techniques is determining how to allocate samples for model fitting and for estimating the performance. The proposed framework addresses this by fitting a monotonic curve that models predictive performance as a function of sample size. This approach enables estimation of a more informative lower confidence bound for the performance and facilitates a comprehensive comparison between learners. Additionally, the curve provides insight into how different models behave as the sample size varies, thus guiding model selection and data collection strategies in future studies. Chapter 3 presents a method to incorporate external information on the covariates into Bayesian additive regression trees (BART), a commonly used tree-based method. The Bayesian specification of BART allows for flexible incorporation of external information into the model fitting process. To do so, an empirical Bayes estimator is developed for the prior variable weights of BART. This estimator recognizes how informative the external information is for the application at hand. If deemed informative, the estimator helps prioritize predictive variables from the large candidate pool, thereby improving both predictive performance and variable selection. Chapter 4 presents an easy-to-use software package for the methodology developed in Chapter 3. Chapter 4 also shows how the empirical Bayes estimator may be used for hyperparameter tuning of BART. Tuning by empirical Bayes is an computationally efficient alternative to the widely-used cross-validation. Chapter 4 also illustrates how modern variable selection techniques for BART may be integrated with the prior variable weight estimator developed in Chapter 3. Chapter 5 presents a novel cancer prediction model designed to handle data with both low-dimensional clinical risk factors and high-dimensional omics features. The set of clinical risk factors usually contains a disease progression measure such as tumor stage. Omics profiles are expected to differ across stage categories, therefore potentially favoring a prediction model with stage-omics interactions. To incorporate interactions, we combine a regression tree built using the clinical risk factors with omics-based regressions fitted within the leaf nodes of the tree. To fit the model, we present a novel penalized likelihood estimator that adapts to 1) the difference in predictive signal between clinical and omics variables and 2) the interaction strength between stage and omics variables through a fusion penalty. This fusion penalty enables information exchange across leaf nodes. Importantly, the proposed model also facilitates straightforward evaluation of the added predictive effect of omics variables in different patient subgroups defined by the fitted tree.

For estimatingp(⩾ 2) independent Poisson means, the paper considers a compromise between maximum likelihood and empirical Bayes estimators. Such compromise estimators enjoy both good componentwise as well as ensemble properties.

Pure robust versus robust portfolio unbiased—Credibility and asymptotic optimality

A compound sampling model, where a unit-specific parameter is sampled from a prior distribution and then observed are generated by a sampling distribution depending on the parameter, underlies a wide variety of biopharmaceutical data. For example, in a multi-centre clinical trial the true treatment effect varies from centre to centre. Observed treatment effects deviate from these true effects through sampling variation. Knowledge of the prior distribution allows use of Bayesian analysis to compute the posterior distribution of clinic-specific treatment effects (frequently summarized by the posterior mean and variance). More commonly, with the prior not completely specified, observed data can be used to estimate the prior and use it to produce the posterior distribution: an empirical Bayes (or variance component) analysis. In the empirical Bayes model the estimated prior mean gives the typical treatment effect and the estimated prior standard deviation indicates the heterogeneity of treatment effects. In both the Bayes and empirical Bayes approaches, estimated clinic effects are shrunken towards a common value from estimates based on single clinics. This shrinkage produces more efficient estimates. In addition, the compound model helps structure approaches to ranking and selection, provides adjustments for multiplicity, allows estimation of the histogram of clinic-specific effects, and structures incorporation of external information. This paper outlines the empirical Bayes approach. Coverage will include development and comparison of approaches based on parametric priors (for example, a Gaussian prior with unknown mean and variance) and non-parametric priors, discussion of the importance of accounting for uncertainty in the estimated prior, comparison of the output and interpretation of fixed and random effects approaches to estimating population values, estimating histograms, and identification of key considerations in the use and interpretation of empirical Bayes methods.

본 연구에서는 질병지도작성(disease mapping)을 위해 인접지역의 정보를 효과적으로 활용할 수 있는 EB(empirical Bayes) 추정 법과 HGLM(hierarchial generalized linear model)을 기초로 한 추정법을 다룬다. 사례연구로 이 추정방법들을 이용하여 2000년 사망원인통계자료를 이용해 경상도 및 전라도의 112개 시<TEX>$.$</TEX>군<TEX>$.$</TEX>구 단위 행정자치구역별 45세 이상 폐암 사망률을 산출하고, 경상도 및 전라도 지역 폐암 사망률 지도를 작성한다. 아울러 제시된 방법들에 위해 얻어진 추정치들의 변동과 3년간 평균 사망률을 기준으로 구한 MSD(mean square deviation)를 이용하여 추정방법들의 특성을 비교 분석한다. For the purpose of disease mapping, we consider the four small area estimation techniques to estimate the mortality rate of small areas; direct, Empirical estimation with total moment estimator and local moment estimator, Estimation based on hierarchial generalized linear model. The estimators are compared by empirical study based on lung cancer mortality data from 2000 Annual Reports on the Cause of Death Statistics in Gyeongsang-Do and Jeonla-Do published by Korean National Statistical Office. Also he stability and efficiency of these estimators are investigated in terms of mean square deviation as well as variation of estimates.