Empirical Bayes Procedure Research Articles

Accuracy of genotypic value predictions for marker-based selection in biparental plant populations

The availability of cheap and abundant molecular markers has led to plant-breeding methods that rely on the prediction of genotypic value from marker data, but published information is lacking on the accuracy of genotypic value predictions with empirical data in plants. Our objectives were to (1) determine the accuracy of genotypic value predictions from multiple linear regression (MLR) and genomewide selection via best linear unbiased prediction (BLUP) in biparental plant populations; (2) assess the accuracy of predictions for different numbers of markers (N(M)) and progenies (N(P)) used in estimation; and (3) determine if an empirical Bayes approach for modeling of the variances of individual markers and of epistatic effects leads to more accurate predictions in empirical data. We divided each of four maize (Zea mays L.) datasets, one Arabidopsis dataset, and two barley (Hordeum vulgare L.) datasets into an estimation set, where marker effects were calculated, and a test set, where genotypic values were predicted based on markers. Predictions were more accurate with BLUP than with MLR. Predictions became more accurate as N(P) and N(M) increased, until sufficient genome coverage was reached. Modeling marker variances with the empirical Bayes method sometimes led to slightly better predictions, but the accuracy with different variants of the empirical Bayes method was often inconsistent. In nearly all cases, the accuracy with BLUP was not significantly different from the highest accuracy across all methods. Accounting for epistasis in the empirical Bayes procedure led to poorer predictions. We concluded that among the methods considered, the quick and simple BLUP approach is the method of choice for predicting genotypic value in biparental plant populations.

Bayesian detection of non-sinusoidal periodic patterns in circadian expression data

Cyclical biological processes such as cell division and circadian regulation produce coordinated periodic expression of thousands of genes. Identification of such genes and their expression patterns is a crucial step in discovering underlying regulatory mechanisms. Existing computational methods are biased toward discovering genes that follow sine-wave patterns. We present an analysis of variance (ANOVA) periodicity detector and its Bayesian extension that can be used to discover periodic transcripts of arbitrary shapes from replicated gene expression profiles. The models are applicable when the profiles are collected at comparable time points for at least two cycles. We provide an empirical Bayes procedure for estimating parameters of the prior distributions and derive closed-form expressions for the posterior probability of periodicity, enabling efficient computation. The model is applied to two datasets profiling circadian regulation in murine liver and skeletal muscle, revealing a substantial number of previously undetected non-sinusoidal periodic transcripts in each. We also apply quantitative real-time PCR to several highly ranked non-sinusoidal transcripts in liver tissue found by the model, providing independent evidence of circadian regulation of these genes. Matlab software for estimating prior distributions and performing inference is available for download from http://www.datalab.uci.edu/resources/periodicity/. Supplementary data are available at Bioinformatics online.

A Bayesian machine learning method for sensor selection and fusion with application to on-board fault diagnostics

In applications like feature-level sensor fusion, the problem of selecting an optimal number of sensors can lead to reduced maintenance costs and the creation of compact online databases for future use. This problem of sensor selection can be reduced to the problem of selecting an optimal set of groups of features during model selection. This is a more complex problem than the problem of feature selection, which has been recognized as a key aspect of statistical model identification. This work proposes a new algorithm based on the use of a Bayesian framework for the purpose of selecting groups of features during regression and classification. The hierarchical Bayesian formulation introduces grouping for the parameters of a generalized linear model and the model hyper-parameters are estimated using an empirical Bayes procedure. A novel aspect of the algorithm is its ability to simultaneously perform feature selection within groups to reduce over-fitting of the data. Further, the parameters obtained from this algorithm can be used to obtain a rank order among the selected sensors. The performance of the algorithm is first tested on a synthetic regression example. Finally, it is applied to the problem of fault detection in diesel engines (30,000 data records from 43 sensors, 8 classes) and used to compare the misclassification rates with a varying number of sensors.

신용등급전이행렬의 경험적 베이지안 추정과 비교

신용전이행렬을 추정함에 있어서 국내의 등급전이자료의 축적이 부족한 점을 극복하기 위하여 외국의 신용평가기관(무디스)의 전이행렬자료와 국내의 신용등급 부여자료를 이용하여 경험적 베이지안 추정방법에 의한 전이행렬을 도출하고, 이 전이행렬을 다른 전이행렬과 비교해보기 위하여 전이행렬의 동적인 요소를 평균전이확률의 개념으로 표시할 수 있는 특성척도를 개발하여 신용전이행렬의 시계열 특성과 통계적 특성을 비교한다. 시계열자료의 척도는 베이지안 추정행렬이 안정적임을 보여주는 반면 국내 행렬은 시간적으로 변화의 폭이 크고 무디스나 베이지안 행렬보다 상대적으로 인접전이의 비율이 높게 나타났다. 붓스트랩 검정을 통하여 세 가지 추정방법이 통계적으로 유의한 차이가 있음을 보이고 베이지안 행렬이 무디스 자료보다는 국내자료에 더 많은 영향을 받았음을 유추할 수 있다. 신용등급 전이에 따른 포트폴리오의 가치변화를 고려하는 몬테칼로 시뮬레이션을 통하여 신용 VaR를 구하여 비교하였다. 국내 전이행렬의 경우에 평균은 가장 크고 신용위험도 가장 큰 값을 보였다. 시뮬레이션에서도 베이지안 추정에 의한 결과가 국내자료에 의한 결과와 더 가깝다는 것을 알 수 있다. In order to overcome the lack of Korean credit rating migration data, we consider an empirical Bayes procedure to estimate credit rating migration matrices. We derive the posterior probabilities of Korean credit rating transitions by utilizing the Moody's rating migration data and the credit rating assignments from Korean rating agency as prior information and likelihood, respectively. Metrics based upon the average transition probability are developed to characterize the migration matrices and compare our Bayesian migration matrices with some given matrices. Time series data for the metrics show that our Bayesian matrices are stable, while the matrices based on Korean data have large variation in time. The bootstrap tests demonstrate that the results from the three estimation methods are significantly different and the Bayesian matrices are more affected by Korean data than the Moody's data. Finally, Monte Carlo simulations for computing the values of a portfolio and its credit VaRs are performed to compare these migration matrices.

Selecting the Best Process Based on Capability Index via Empirical Bayes Approach

Consider k (k ≥ 2) manufacturing processes whose mean θ i , variance and process capability index C pw (i), i = 1,…, k, are all unknown. For two given control values C pw (0) and , we are interested in selecting some process whose capability index is no less than C pw (0) and is the largest in the qualified subset in which each process variance is no larger than . Under a Bayes framework, we consider the normally distributed manufacturing processes taking normal-gamma as its conjugate prior. A Bayes approach is set up and an empirical Bayes procedure is proposed which has been shown to be asymptotically optimal. A simulation study is carried out for the performance of the proposed procedure and it is found practically useful.

Appraisal of the Interactive Highway Safety Design Model’s Crash Prediction and Design Consistency Modules: Case Studies from Pennsylvania

The interactive highway safety design model (IHSDM) is a suite of software analysis tools used to evaluate safety and operational effects of geometric design on two-lane rural highways. IHSDM checks existing or proposed two-lane highway designs against relevant design policy values and provides estimates of a design’s expected safety and operational performance. This paper describes efforts to apply IHSDM to two existing rural roadways (one an arterial and the other a collector) in Pennsylvania and to compare the crash prediction and speed profile results to observed speed and crash data for these same two roadway segments. When considering both case study roadways together, the use of site-specific crash history data, adjusted using the empirical Bayes procedure, and a county-level calibration factor, produced crash frequency predictions that most closely matched the observed crash frequencies. The design consistency module output on the arterial roadway closely matched the observed speeds. On the collec...

Multitask Compressive Sensing

Compressive sensing (CS) is a framework whereby one performs n non-adaptive measurements to constitute an n-dimensional vector v, with v used to recover an m-dimensional approximation ^ u to a desired m-dimensional signal u, with n ? m; this is performed under the assumption that u is sparse in the basis represented by the matrix “, the columns of which define discrete basis vectors. It has been demonstrated that with appropriate design of the compressive measurements used to define v, the decompressive mapping v ! ^ u may be performed with error kui^ uk 2 having asymptotic properties (large n and m > n) analogous to those of the best adaptive transform-coding algorithm applied in the basis “. The mapping v ! ^ constitutes an inverse problem, often solved using ‘1 regularization or related techniques. In most previous research, if multiple compressive measurements fvigi=1;M are performed, each of the associated f^ uigi=1;M are recovered one at a time, independently. In many applications the M “tasks” defined by the mappings vi ! ^ ui are not statistically independent, and it may be possible to improve the performance of the inversion if statistical inter-relationships are exploited. In this paper we address this problem within a multi-task learning setting, wherein the mapping vi ! ^ ui for each task corresponds to inferring the parameters (here, wavelet coefficients) associated with the desired signal ui, and a shared prior is placed across all of the M tasks. In this multi-task learning framework data from all M tasks contribute toward inferring a posterior on the hyperparameters, and once the shared prior is thereby inferred, the data from each of the M individual tasks is then employed to estimate the task-dependent wavelet coefficients. An empirical Bayes procedure and fast inference algorithm is developed. Example results are presented on several data sets.

Tests for gene‐environment interaction from case‐control data: a novel study of type I error, power and designs

To evaluate the risk of a disease associated with the joint effects of genetic susceptibility and environmental exposures, epidemiologic researchers often test for non-multiplicative gene-environment effects from case-control studies. In this article, we present a comparative study of four alternative tests for interactions: (i) the standard case-control method; (ii) the case-only method, which requires an assumption of gene-environment independence for the underlying population; (iii) a two-step method that decides between the case-only and case-control estimators depending on a statistical test for the gene-environment independence assumption and (iv) a novel empirical-Bayes (EB) method that combines the case-control and case-only estimators depending on the sample size and strength of the gene-environment association in the data. We evaluate the methods in terms of integrated Type I error and power, averaged with respect to varying scenarios for gene-environment association that are likely to appear in practice. These unique studies suggest that the novel EB procedure overall is a promising approach for detection of gene-environment interactions from case-control studies. In particular, the EB procedure, unlike the case-only or two-step methods, can closely maintain a desired Type I error under realistic scenarios of gene-environment dependence and yet can be substantially more powerful than the traditional case-control analysis when the gene-environment independence assumption is satisfied, exactly or approximately. Our studies also reveal potential utility of some non-traditional case-control designs that samples controls at a smaller rate than the cases. Apart from the simulation studies, we also illustrate the different methods by analyzing interactions of two commonly studied genes, N-acetyl transferase type 2 and glutathione s-transferase M1, with smoking and dietary exposures, in a large case-control study of colorectal cancer.

In-season prediction of batting averages: A field test of empirical Bayes and Bayes methodologies

Batting average is one of the principle performance measures for an individual baseball player. It is natural to statistically model this as a binomial-variable proportion, with a given (observed) number of qualifying attempts (called “at-bats”), an observed number of successes (“hits”) distributed according to the binomial distribution, and with a true (but unknown) value of pi that represents the player’s latent ability. This is a common data structure in many statistical applications; and so the methodological study here has implications for such a range of applications. We look at batting records for each Major League player over the course of a single season (2005). The primary focus is on using only the batting records from an earlier part of the season (e.g., the first 3 months) in order to estimate the batter’s latent ability, pi, and consequently, also to predict their batting-average performance for the remainder of the season. Since we are using a season that has already concluded, we can then validate our estimation performance by comparing the estimated values to the actual values for the remainder of the season. The prediction methods to be investigated are motivated from empirical Bayes and hierarchical Bayes interpretations. A newly proposed nonparametric empirical Bayes procedure performs particularly well in the basic analysis of the full data set, though less well with analyses involving more homogeneous subsets of the data. In those more homogeneous situations better performance is obtained from appropriate versions of more familiar methods. In all situations the poorest performing choice is the naïve predictor which directly uses the current average to predict the future average. One feature of all the statistical methodologies here is the preliminary use of a new form of variance stabilizing transformation in order to transform the binomial data problem into a somewhat more familiar structure involving (approximately) Normal random variables with known variances. This transformation technique is also used in the construction of a new empirical validation test of the binomial model assumption that is the conceptual basis for all our analyses.

Semi-Parametric Differential Expression Analysis via Partial Mixture Estimation

We develop an approach for microarray differential expression analysis, i.e. identifying genes whose expression levels differ between two or more groups. Current approaches to inference rely either on full parametric assumptions or on permutation-based techniques for sampling under the null distribution. In some situations, however, a full parametric model cannot be justified, or the sample size per group is too small for permutation methods to be valid. We propose a semi-parametric framework based on partial mixture estimation which only requires a parametric assumption for the null (equally expressed) distribution and can handle small sample sizes where permutation methods break down. We develop two novel improvements of Scott's minimum integrated square error criterion for partial mixture estimation [Scott, 2004a,b]. As a side benefit, we obtain interpretable and closed-form estimates for the proportion of EE genes. Pseudo-Bayesian and frequentist procedures for controlling the false discovery rate are given. Results from simulations and real datasets indicate that our approach can provide substantial advantages for small sample sizes over the SAM method of Tusher et al. [2001], the empirical Bayes procedure of Efron and Tibshirani [2002], the mixture of normals of Pan et al. [2003] and a t-test with p-value adjustment [Dudoit et al., 2003] to control the FDR [Benjamini and Hochberg, 1995].

Selection of the best Bernoulli population provided it is better than a control: An empirical Bayes approach

We investigate an empirical Bayes procedure δ n * for selecting the best Bernoulli population from among k competitors provided it is better than a control based on negative binomial sampling. The performance of δ n * is evaluated in terms of its associated Bayes risk. Under some conditions, it is shown that δ n * is asymptotically optimal, and the associated Bayes risk converges to the minimum Bayes risk at an exponential decay rate exp ( - cn ) for some positive number c, where n is the number of historical data at hand when the present selection problem is considered. A simulation is also carried out to study the performance of δ n * for small to moderate values of n.

Empirical Bayes Inference of Pairwise FST and Its Distribution in the Genome

Populations often have very complex hierarchical structure. Therefore, it is crucial in genetic monitoring and conservation biology to have a reliable estimate of the pattern of population subdivision. F(ST)'s for pairs of sampled localities or subpopulations are crucial statistics for the exploratory analysis of population structures, such as cluster analysis and multidimensional scaling. However, the estimation of F(ST) is not precise enough to reliably estimate the population structure and the extent of heterogeneity. This article proposes an empirical Bayes procedure to estimate locus-specific pairwise F(ST)'s. The posterior mean of the pairwise F(ST) can be interpreted as a shrinkage estimator, which reduces the variance of conventional estimators largely at the expense of a small bias. The global F(ST) of a population generally varies among loci in the genome. Our maximum-likelihood estimates of global F(ST)'s can be used as sufficient statistics to estimate the distribution of F(ST) in the genome. We demonstrate the efficacy and robustness of our model by simulation and by an analysis of the microsatellite allele frequencies of the Pacific herring. The heterogeneity of the global F(ST) in the genome is discussed on the basis of the estimated distribution of the global F(ST) for the herring and examples of human single nucleotide polymorphisms (SNPs).

Safety Index for Evaluation of Two-Lane Rural Highways

The methodological approach to safety evaluation of two-lane rural highway segments that is presented uses both analytical procedures referring to alignment design consistency models and safety inspection processes. A safety index (SI) that quantitatively measures the relative safety performance of a road segment is calculated from the procedure. The SI is formulated by combining three components of risk: the exposure of road users to road hazards, the probability of a vehicle's being involved in an accident, and the resulting consequences should an accident occur. This systematic and replicable procedure integrates two different, complementary approaches–one based on design consistency evaluations and the other on safety inspections–and makes it possible to address a wide variety of safety issues effectively. A further advantage of the procedure is its applicability on highways where crash data are either not available or unreliable. Validation of the procedure was carried out on a sample of roads by a comparison of the risk rank obtained by using the SI and accident history. The SI was assessed in 30 segments chosen from a sample of two-lane rural highways in Italy, and the actual accident situation was obtained with the empirical Bayes (EB) procedure. Spearman's rank correlation was used to determine the level of agreement between the rankings obtained with the two techniques. The results from the Spearman's rank-correlation analysis validate the SI, indicating that the ranking from the SI scores and the EB estimates agrees at the 99.9% level of significance with a correlation coefficient of 0.87.

Sampling Survey and Statistical Genetics in Fishery Resource Management and Conservation

This paper describes several methods for estimating essential parameters in fishery resource management and conservation, which we developed so far. First, an unbiased estimator for evaluating total landings and a ratio estimator for measuring contribution of hatchery releases were derived based on a two-stage random sampling and applied to a masu salmon stock enhancement program. Second, the efficacy of an integrated likelihood approach to estimate Wright's FST in a metapopulation was demonstrated using numerical simulations and real data analysis. Third, an empirical Bayes procedure for estimating genetic parameters is explained, which can simulate posterior distributions for any parametric functions of allelic frequencies. As an example, the effective population size of the Northern pike was estimated. Finally, a method for simultaneous estimation of mixing proportion and genetic drift based on series of genetic tagging data was presented with application to the data from a stock enhancement program of the mud crab.

Effect of missing values in detecting differentially expressed genes in a cDNA microarray experiment

The aim of this paper is to discuss the effect of missing values in detecting differentially expressed genes in a cDNA microarray experiment in the context of a one sample problem. We conducted a cDNA microarray experiment to detect differentially expressed genes for the metastasis of colorectal cancer based on twenty patients who underwent liver resection due to liver metastasis from colorectal cancer. Total RNAs from metastatic liver tumor and adjacent normal liver tissue from a single patient were labeled with cy5 and cy3, respectively, and competitively hybridized to a cDNA microarray with 7775 human genes. We used M=log2(R/G) for the signal evaluation, where R and G denoted the fluorescent intensities of Cy5 and Cy3 dyes, respectively. The statistical problem comprises a one sample test of testing E(M)=0 for each gene and involves multiple tests. The twenty cDNA microarray data would comprise a matrix of dimension 7775 by 20, if there were no missing values. However, missing values occur for various reasons. For each gene, the no missing proportion (NMP) was defined to be the proportion of non-missing values out of twenty. In detecting differentially expressed (DE) genes, we used the genes whose NMP is greater than or equal to 0.4 and then sequentially increased NMP by 0.1 for investigating its effect on the detection of DE genes. For each fixed NMP, we imputed the missing values with K-nearest neighbor method (K=10) and applied the nonparametric t-test of Dudoit et al. (2002), SAM by Tusher et al. (2001) and empirical Bayes procedure by Lonnstedt and Speed (2002) to find out the effect of missing values in the final outcome. These three procedures yielded substantially agreeable result in detecting DE genes. Of these three procedures we used SAM for exploring the acceptable NMP level. The result showed that the optimum no missing proportion (NMP) found in this data set turned out to be 80%. It is more desirable to find the optimum level of NMP for each data set by applying the method described in this note, when the plot of (NMP, Number of overlapping genes) shows a turning point. Corresponding author: Byung Soo Kim (Tel: +82-22123-4541, Fax:+82-2-313-5331, Email: bskim@yonsei.ac.kr) B.S. Kim’s study was supported by Yonsei University Research Fund of 2001. S.Y. Rha’s study was supported by a grant of the IMT-2000 project, Ministry of Health & Welfare, Republic of Korea (01-PJ11-PG9-01BT00A-0028). Introduction The DNA microarray has been established as a major tool in biological researches due to its ability of monitoring gene expression levels of thousands of genes simultaneously under different conditions (Jin et al., 2001: Gibson 2002; Hedenfalk, 2002; Olesiak 2002; Ramaswamy 2002; Huang 2003; Keshave and Ong, 2003). It is not trivial to analyze the data from microarray experiment, not because they just involve large amount of data, but because they comprise a non-standard statistical problem which is often referred to as a “large p, small n” problem (West, 2003). Typically, we have thousands of genes (=p) for a microarray experiment with tens of microarrays (=n). Several analysis tools including SAM (Tusher et al, 2001) and BRB-ArrayTools (Simon and Peng) have been introduced in the pubic domain to provide a guidance to laboratory scientists on the statistical analysis of microarray data. Microarray experiment data can be represented by a p x n matrix, where the (i, j)th element of the matrix indicates the i-th gene expression level for the j-th microarray, i=1, .., p, and j=1, ,n. It is quite often that we observe the missing values in the data of p x n matrix. Missing values occur for various reasons not only from the technical problems but also from the biological characteristics. Currently, as the chip quality and the hybridization techniques have been improved to a certain level, the missing values usually come from the biological reasons, such as no expression of the specific genes in the sample or the

Estimating the mean effect size in meta-analysis: bias, precision, and mean squared error of different weighting methods.

Although use of the standardized mean difference in meta-analysis is appealing for several reasons, there are some drawbacks. In this article, we focus on the following problem: that a precision-weighted mean of the observed effect sizes results in a biased estimate of the mean standardized mean difference. This bias is due to the fact that the weight given to an observed effect size depends on this observed effect size. In order to eliminate the bias, Hedges and Olkin (1985) proposed using the mean effect size estimate to calculate the weights. In the article, we propose a third alternative for calculating the weights: using empirical Bayes estimates of the effect sizes. In a simulation study, these three approaches are compared. The mean squared error (MSE) is used as the criterion by which to evaluate the resulting estimates of the mean effect size. For a meta-analytic dataset with a small number of studies, the MSE is usually smallest when the ordinary procedure is used, whereas for a moderate or large number of studies, the procedures yielding the best results are the empirical Bayes procedure and the procedure of Hedges and Olkin, respectively.

Uncertainty in most probable number calculations for microbiological assays.

Microbiological assays commonly use incubations of multiple tubes in a dilution series, and microorganism concentration is read as a most probable number (MPN) in standard tables for the observed pattern of positive tubes. Published MPN tables differ, sometimes substantially, because of use of approximate MPN calculation procedures, different rounding conventions in the results, and different methods of calculating confidence or credible intervals. We conclude that the first 2 issues can now be resolved by using recently developed exact MPN calculation methods and by reporting rounding conventions in standard tables. The third issue is not amenable to complete resolution, especially if credible interval (as opposed to confidence interval) limits are desired--as we think they most often are. In that case, Bayesian statistics are called for and the analyst must provide a distribution of concentration that was presumed to be true before the assay was performed. This is mathematically combined with the assay data, resulting in a posterior concentration distribution. These distributions may then be used to quantify the uncertainty in the MPN estimate, and the best approach is to use the highest posterior density regions of these distributions. If based on diffuse prior information (positing that, prior to an assay being performed, all positive concentrations are equally likely), then established procedures might be used to calculate the limits and publish them in standard tables. In the event that this prior assumption is held to be not satisfactory, we show results for an empirical Bayes procedure, with a Poisson prior distribution, giving credible interval widths much narrower than in the other cases examined.

Effects of clofibric acid on mRNA expression profiles in primary cultures of rat, mouse and human hepatocytes

The mRNA expression profile in control and clofibric acid (CLO)-treated mouse, rat, and human hepatocytes was analyzed using species-specific oligonucleotide DNA microarrays (Affymetrix). A statistical empirical Bayes procedure was applied in order to select the significantly differentially expressed genes. Treatment with the peroxisome proliferator CLO induced up-regulation of genes involved in peroxisome proliferation and in cell proliferation as well as down-regulation of genes involved in apoptosis in hepatocytes of rodent but not of human origin. CLO treatment induced up-regulation of microsomal cytochrome P450 4a genes in rodent hepatocytes and in two of six human hepatocyte cultures. In addition, genes encoding phenobarbital-inducible cytochrome P450s were also up-regulated by CLO in rodent and human hepatocyte cultures. Up-regulation of phenobarbital-inducible UDP-glucuronosyl-transferase genes by CLO was observed in both rat and human but not in mouse hepatocytes. CLO treatment induced up-regulation of l-fatty acid binding protein ( l-FABP) gene in hepatocytes of both rodent and human origin. However, while genes of the cytosolic, microsomal, and mitochondrial pathways involved in fatty acid transport and metabolism were up-regulated by CLO in both rodent and human hepatocyte cultures, genes of the peroxisomal pathway of lipid metabolism were up-regulated in rodents only. An up-regulation of hepatocyte nuclear factor 1α (HNF1α) by CLO was observed only in human hepatocyte cultures, suggesting that this trans-activating factor may play a key role in the regulation of fatty acid metabolism in human liver as well as in the nonresponsiveness of human liver to CLO-induced regulation of cell proliferation and apoptosis.

Non-parametric empirical Bayes procedure

Abstract In this paper we introduce an empirical Bayes procedure for estimating an unknown parameter, say θ. This procedure gives the empirical Bayes estimator for θ and its associated minimum posterior risk in closed forms without estimating the unknown prior density function of θ. In such procedure the posterior probability density function of θ is not required. A sufficient statistic for θ with conditional probability density function in the one parameter exponential family is required. Instead of estimating the unknown prior density function, the marginal density function of the sufficient statistic must be estimated. As special cases the empirical Bayes estimators and their respective minimum posterior risks of the failure rate for the exponential distribution, the unknown scale parameters of Weibull and gamma distributions are obtained in simple forms as special cases. Numerical results and a simulation study are introduced to (i) investigate how the number of available past experiments and the sample size of each influence the accuracy of the empirical Bayes estimator, (ii) make a comparison between the presented procedure and the Bayes procedure when the prior probability density function of the parameter θ is gamma.

An Empirical Bayes Group-Testing Approach to Estimating Small Proportions

Group testing has long been recognized as a safe and sensible alternative to one-at-a-time testing in applications wherein the prevalence rate p is small. In this article, we develop an empirical Bayes (EB) procedure to estimate p using a beta-type prior distribution and a squared-error loss function. We show that the EB estimator is preferred over the usual maximum likelihood estimator (MLE) for small group sizes and small p. In addition, we also discuss interval estimation and consider the use of other loss functions perhaps more appropriate in public health studies. The proposed methods are illustrated using group-testing data from a prospective hepatitis C virus study conducted in Xuzhou City, China.