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

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.

MSE superiority of Bayes and empirical Bayes estimators in two generalized seemingly unrelated regressions

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

Pawel, D. J., Preston, D. L., Pierce, D. A. and Cologne, J. B. Improved Estimates of Cancer Site-Specific Risks for A-Bomb Survivors. Radiat. Res. 169, 87-98 (2008). Simple methods are investigated for improving summary site-specific radiogenic risk estimates. Estimates in this report are derived from cancer incidence data from the Life Span Study (LSS) cohort of A-bomb survivors that are followed up by the Radiation Effects Research Foundation (RERF). Estimates from the LSS of excess relative risk (ERR) for solid cancer sites have typically been derived separately for each site. Even though the data for this are extensive, the statistical imprecision in site-specific (organ-specific) risk estimates is substantial, and it is clear that a large portion of the site-specific variation in estimates is due to this imprecision. Empirical Bayes (EB) estimates offer a reasonable approach for moderating this variation. The simple version of EB estimates that we applied to the LSS data are weighted averages of a pooled overall estimate of ERR and separately derived site-specific estimates, with weights determined by the data. Results indicate that the EB estimates are most useful for sites such as esophageal or bladder cancer, for which the separately derived ERR estimates are less precise than for other sites.

Pure robust versus robust portfolio unbiased—Credibility and asymptotic optimality

Conflicting evidence exists regarding whether hormone therapy for prostate cancer is associated with neurotoxicity. Thus, we aim to characterize the association between different types of hormone therapy and neurocognitive impairment in a real-world pharmacovigilance database. We queried VigiBase, the World Health Organization's international pharmacovigilance database, for reports of neurocognitive impairment among men who took hormone therapy from 1968 to 2021. We performed disproportionality analysis comparing rates of neurocognitive impairment with different types of hormone therapy versus other VigiBase drugs. Traditional hormonal therapy was defined as androgen deprivation therapy (ADT: gonadotropin-releasing-hormone agonists or antagonists) or first-generation androgen receptor (AR) antagonists. Novel AR signaling inhibitors (ARSIs) were defined as ARSIs with or without ADT. Differences were assessed using reporting odds ratio (ROR) with 95% confidence intervals (CI) and Empirical Bayes Estimator (EBE) values ≥1.0 signifying statistical significance. Odds of neurocognitive impairment were significantly elevated with traditional hormone therapy (ROR 1.47, 95% CI 1.34-1.62, EBE = 1.35) and novel ARSIs (ROR 2.40, 95% CI 2.28-2.54, EBE = 2.26). Odds of neurocognitive impairment were significantly elevated with enzalutamide (ROR 2.89, 95% CI 2.73-3.05, EBE = 2.70) and numerically increased with apalutamide (ROR 3.31, 95% CI 1.57-7.00, EBE = 0.98), but were decreased with abiraterone (ROR 0.68, 95% CI 0.55-0.84, EBE = 0.57). This study demonstrates elevated odds of neurocognitive impairment with hormone therapy in a real-world data set. Neurotoxicity risk was higher with novel ARSIs than traditional agents, and higher with enzalutamide than abiraterone. Due to limitations inherent to disproportionality analysis (measuring associations, not risk) and incomplete data prohibiting the ability to control for factors such as age or use of secondary drugs (e.g., concurrent use of novel ARSIs with ADT), results are exploratory in nature. The amalgamation of these and other conflicting data may contribute to clinical decision-making for men with prostate cancer eligible for treatment with these therapies, especially those with significant neurologic comorbidities.

Although the conditions for application of the technique of empirical Bayes estimation of the parameter of a distribution may exist, the form of the prior distribution is generally unknown. This often creates major difficultics in the determination of empirical Bayes estimators. The simple device of approximating the prior distribution by a stop function is used to overcome this problem and to obtain smooth empirical Bayes estimators. This paper supplements earlier work by considering continuous distributions and multiple past and current observations. The effectiveness of the proposed smooth empirical Bayes estimators is examined, and the results indicate that they can be substantially better than optimum non-Bayes estimators, even when the amount of past data is small.

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.

In this paper, the Bayes estimator and the parametric empirical Bayes estimator (PEBE) of mean vector in multivariate normal distribution are obtained. The superiority of the PEBE over the minimum variance unbiased estimator (MVUE) and a revised James-Stein estimators (RJSE) are investigated respectively under mean square error (MSE) criterion. Extensive simulations are conducted to show that performance of the PEBE is optimal among these three estimators under the MSE criterion.

For binomial data analysis, many methods based on empirical Bayes interpretations have been developed, in which a variance‐stabilizing transformation and a normality assumption are usually required. To achieve the greatest model flexibility, we conduct nonparametric Bayesian inference for binomial data and employ a special nonparametric Bayesian prior—the Bernstein–Dirichlet process (BDP)—in the hierarchical Bayes model for the data. The BDP is a special Dirichlet process (DP) mixture based on beta distributions, and the posterior distribution resulting from it has a smooth density defined on [0, 1]. We examine two Markov chain Monte Carlo procedures for simulating from the resulting posterior distribution, and compare their convergence rates and computational efficiency. In contrast to existing results for posterior consistency based on direct observations, the posterior consistency of the BDP, given indirect binomial data, is established. We study shrinkage effects and the robustness of the BDP‐based posterior estimators in comparison with several other empirical and hierarchical Bayes estimators, and we illustrate through examples that the BDP‐based nonparametric Bayesian estimate is more robust to the sample variation and tends to have a smaller estimation error than those based on the DP prior. In certain settings, the new estimator can also beat Stein's estimator, Efron and Morris's limited‐translation estimator, and many other existing empirical Bayes estimators.The Canadian Journal of Statistics40: 328–344; 2012 © 2012 Statistical Society of Canada

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.