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Parametric Empirical Bayes Inference: Theory and Applications
Abstract
This article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical Bayes viewpoint, particularly in the case of parametric prior distributions. Some successful applications of major importance are considered. Recent results concerning estimates of error and confidence intervals are described and illustrated with data.
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“The highly parallel nature of gene expression experiments lends itself to a particular class of statistical methods, called parametric empirical Bayes, that borrow information between genes in a dynamic way (14,15).”
From limma powers differential expression analyses for RNA-sequencing and microarray studies (2015) · cites this paper as [15]
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limma powers differential expression analyses for RNA-sequencing and microarray studies
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“These methods all help improve inference at both the gene and gene set level in small experiments. genes in a dynamic way (14,15).”
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“…estimator (DerSimonian and Laird 1986; Raudenbush 2009), the Sidik-Jonkman estimator (Sidik and Jonkman 2005a,b), the maximum-likelihood or restricted maximum-likelihood estimator (Viechtbauer 2005; Raudenbush 2009), or the empirical Bayes estimator (Morris 1983; Berkey et al. 1995).”
BackgroundBlei, D. M., Ng, A. Y., & Jordan, M. I. (2002) · The MIT Press eBooks · doi:10.7551/mitpress/1120.003.0082 · Cited by 12,683
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“On the theoretical side, hierarchical models allow a more “objective” approach to inference by estimating the parameters of prior distributions from data rather than requiring them to be specified using subjective information (see James and Stein, 1960, Efron and Morris, 1975, and Morris, 1983).”
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“Method of moments and restricted maximum likelihood are those most commonly used and have been implemented in STATA’s metan command along with an empirical Bayes method [32,33].”
MethodBayes Empirical Bayes Inference of Amino Acid Sites Under Positive Selection
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“Several procedures have been suggested in the statistics literature to correct for the bias in the NEB approach to achieve approximately correct frequentist coverage probabilities (e.g., Morris 1983; Laird and Louis 1987; Carlin and Gelfand 1990).”
MethodRao, J. N. K., & Molina, I. (2003) · Encyclopedia of Statistical Sciences · doi:10.1002/0471667196.ess0731 · Cited by 2,137
Improved tests for a random effects meta‐regression with a single covariate
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“[1] uses the empirical Bayes (EB) estimator [13] of the between-trial variance which is also iteratively given as the solution of the equation”
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Colditz, G. A. (1994) · JAMA · doi:10.1001/jama.1994.03510330076038 · Cited by 2,021
Bias and Efficiency of Meta-Analytic Variance Estimators in the Random-Effects Model
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“It appears that Equation 19 originated with Morris (1983), who suggested it as an “approximate” REML estimate (p. 53).”
BackgroundThat BLUP is a Good Thing: The Estimation of Random Effects
Robinson, G. K. (1991) · Statistical Science · doi:10.1214/ss/1177011926 · Cited by 1,694
Methods to estimate the between‐study variance and its uncertainty in meta‐analysis
Veroniki, A. A., Jackson, D., Viechtbauer, W., Bender, R., Bowden, J., Knapp, G., Kuss, O., Higgins, J. P., Langan, D., & Salanti, G. (2016) · Research Synthesis Methods · doi:10.1002/jrsm.1164 · Cited by 1,685
“However, several investigators claim that in practice the differences between frequentist and Bayesian approaches appear to be small (Morris, 1983; Thompson and Sharp, 1999).”
An Overview of the Logic and Rationale of Hierarchical Linear Models
Hofmann, D. (1997) · Journal of Management · doi:10.1177/014920639702300602 · Cited by 1,681
“Bryk and Raudenbush (1992) drawing on the OLS regression rule of thumb of 10 observations per predictor, mentioned that analogous rules for hierarchical linear models can be developed.”
BackgroundMethodInfluentialSchäfer, J., & Strimmer, K. (2005) · Statistical Applications in Genetics and Molecular Biology · doi:10.2202/1544-6115.1175 · Cited by 1,556
Prior distributions for variance parameters in hierarchical modelsPreprint
Gelman, A. (2004) · RePEc: Research Papers in Economics · Cited by 1,558
Application of hierarchical linear models to assessing change.
Bryk, A. S., & Raudenbush, S. W. (1987) · Psychological Bulletin · doi:10.1037/0033-2909.101.1.147 · Cited by 1,505
Modeling Multilevel Data Structures
Steenbergen, M. R., & Jones, B. (2002) · American Journal of Political Science · doi:10.2307/3088424 · Cited by 1,521
“The first, is to simply adjust the GLS and EB estimators by incorporating estimates of the sampling variance of the (co)variance components (Morris 1983; Kackar and Harvill e 1984), possibly via bootstrapping (Laird and Louis 1987).”
MethodGroup representations in probability and statistics
Diaconis, P. (1988) · doi:10.1214/lnms/1215467407 · Cited by 1,381
Machine Learning Methods That Economists Should Know About
Athey, S., & Imbens, G. W. (2019) · Annual Review of Economics · doi:10.1146/annurev-economics-080217-053433 · Cited by 1,163
“, Morris [1983]).”
Bayesian Statistics and Marketing
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A comparison of heterogeneity variance estimators in simulated random‐effects meta‐analyses
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Blei, D. M., & Jordan, M. I. (2003) · International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR) · doi:10.1145/860435.860460 · Cited by 1,078
“With K draws from the prior distribution, however, we are in a good position to take the empirical Bayes perspective [9] and compute a maximum likelihood estimate of η.”
MethodPhipson, B., Lee, S., Majewski, I. J., Alexander, W. S., & Smyth, G. K. (2016) · The Annals of Applied Statistics · doi:10.1214/16-aoas920 · Cited by 1,013
“Morris, C. N. (1983). Parametric empirical Bayes inference: theory and applications. Journal of the American Statistical Association 78, 47–55. Murie, C., Woody, O., Lee, A. Y. and Nadon, R. (2009). Comparison of small n statistical tests of differential expression applied to microarrays. BMC Bioinformatics 10, 45. Pi…”
BackgroundInfluentialIllustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling
Gelfand, A. E., Hills, S. E., Racine-Poon, A., & Smith, A. F. M. (1990) · Journal of the American Statistical Association · doi:10.1080/01621459.1990.10474968 · Cited by 970
Bayes-Stein Estimation for Portfolio Analysis
Jorion, P. (1986) · Journal of Financial and Quantitative Analysis · doi:10.2307/2331042 · Cited by 961
The Handbook of Research Synthesis
Ziegel, E. R., Cooper, H., & Hedges, L. V. (1995) · Technometrics · doi:10.2307/1269651 · Cited by 936
Harbord, R. M., & Higgins, J. P. T. (2008) · The Stata Journal: Promoting communications on statistics and Stata · doi:10.1177/1536867x0800800403 · Cited by 897
“It is found by iterating the following equation (Morris 1983; Berkey et al. 1995):”
InfluentialA random‐effects regression model for meta‐analysis
Berkey, C. S., Hoaglin, D. C., Mosteller, F. C., & Colditz, G. A. (1995) · Statistics in Medicine · doi:10.1002/sim.4780140406 · Cited by 896
Regression discontinuity inference with specification error
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A Hierarchical Model for Studying School Effects
Raudenbush, S. W., & Bryk, A. S. (1986) · Sociology of Education · doi:10.2307/2112482 · Cited by 812
Statistical Modelling Issues in School Effectiveness Studies
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Estimating Teacher Impacts on Student Achievement: An Experimental Evaluation
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The Estimation of the Mean Squared Error of Small-Area Estimators
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Effects of omega-3 fatty acids on serum markers of cardiovascular disease risk: a systematic review
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League Tables and Their Limitations: Statistical Issues in Comparisons of Institutional Performance
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International Portfolio Diversification with Estimation Risk
Jorion, P. (1985) · The Journal of Business · doi:10.1086/296296 · Cited by 672
Guidelines for reporting mortality and morbidity after cardiac valve interventions
Akins, C. W., Miller, D. C., Turina, M. I., Kouchoukos, N. T., Blackstone, E. H., Grunkemeier, G. L., Takkenberg, J. J. M., David, T. E., Butchart, E. G., Adams, D. H., et al. (2008) · Journal of Thoracic and Cardiovascular Surgery · doi:10.1016/j.jtcvs.2007.12.002 · Cited by 664
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Cressie, N., Calder, C. A., Clark, J. S., Hoef, J. M. V., & Wikle, C. K. (2009) · Ecological Applications · doi:10.1890/07-0744.1 · Cited by 541
“One could try to account for the variability in P̂, but formal inference is usually approximate (e.g., Morris 1983a, b); one commonly used approximation is based on multivariable differential calculus and Taylor-series expansions of estimators (e.g., Prasad and Rao 1990, Cressie 1992, Rao…”
MethodThe Long-Term Impacts of Teachers: Teacher Value-Added and Student Outcomes in Adulthood
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“Frequentist reasoning was intimately involved in Robbins' original formulation of empirical Bayes, and in significant implementations of the paradigm, such as Morris (1983) for hierarchical models .”
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“Note that we do not discuss the empirical Bayes estimator proposed by Morris [28], due to its practical similarity to the model error variance estimators [18].”
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Akins, C. W., Miller, D. C., Turina, M. I., Kouchoukos, N. T., Blackstone, E. H., Grunkemeier, G. L., Takkenberg, J. J. M., David, T. E., Butchart, E. G., Adams, D. H., et al. (2008) · The Annals of Thoracic Surgery · doi:10.1016/j.athoracsur.2007.12.082 · Cited by 482
Meta-analysis: The Effect of Dietary Counseling for Weight Loss
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“In addition, all data were analyzed in a random-effects model meta-regression—a meta-analytic technique of multivariable linear regression across studies—according to the method of Morris (21) as described by Berkey and colleagues (22).”
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What Do Test Scores Miss? The Importance of Teacher Effects on Non–Test Score Outcomes
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Bayesian Statistics and MarketingPreprint
Allenby, G. M., & Rossi, P. E. (2002) · SSRN Electronic Journal · doi:10.2139/ssrn.320740 · Cited by 428
A comparison of heterogeneity variance estimators in combining results of studies
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“An EB estimate of 2 is obtained as an iterative solution to the equation [19, 20]”
MethodInfluentialTweedie’s Formula and Selection Bias
Efron, B. (2011) · Journal of the American Statistical Association · doi:10.1198/jasa.2011.tm11181 · Cited by 414
“(4.11) [Morris (1983) presented a hierarchical Bayes analysis of James–Stein estimation accuracy; see Remark I of Section 7.]”
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Viechtbauer, W. (2006) · Statistics in Medicine · doi:10.1002/sim.2514 · Cited by 399
Bayesian Estimation of a Multilevel IRT Model Using Gibbs Sampling
Fox, J., & Glas, C. A. W. (2001) · Psychometrika · doi:10.1007/bf02294839 · Cited by 398
Type S error rates for classical and Bayesian single and multiple comparison procedures
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Value-added modeling: A review
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“6 A detailed discussion of shrinkage estimators can be found in Morris (1983).”
BackgroundMeasuring Extreme Response Style
Greenleaf, E. A. (1992) · Public Opinion Quarterly · doi:10.1086/269326 · Cited by 368
Kass, R. E., & Steffey, D. (1989) · Journal of the American Statistical Association · doi:10.1080/01621459.1989.10478825 · Cited by 474
A mixture model approach for the analysis of microarray gene expression data
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“(See Efron and Morris 1975 and Morris 1983 for discussions of pooling and shrinkage in hierarchical or “empirical Bayes” inference.)”
BackgroundBryk, A. S., & Raudenbush, S. W. (1988) · American Journal of Education · doi:10.1086/443913 · Cited by 346
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Mixed model prediction and small area estimation
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“This data set has been analyzed by several authors in the past, including Efron and Morris (1975), Efron (1975), Morris (1983b), Gelman et al. (1995), Datta and Lahiri (2000), Rao (2003), Lahiri and Li (2005), among others.”
BackgroundMethodInfluentialCharter school quality and parental decision making with school choice
Hanushek, E. A., Kain, J. F., Rivkin, S. G., & Branch, G. F. (2007) · Journal of Public Economics · doi:10.1016/j.jpubeco.2006.09.014 · Cited by 322
Education and Economic Growth: A Meta-Regression Analysis
Benos, N., & Zotou, S. (2014) · World Development · doi:10.1016/j.worlddev.2014.06.034 · Cited by 334
“As suggested by Thompson and Sharp (1999), the unknown variance of the random-effect model can be computed by an iterative residual (restricted) maximum likelihood process (REML), the Empirical Bayes (EB) method (see also Morris, 1983), or a momentestimator (MM).”
MethodVisualizing Assumptions and Results in Network Meta-analysis: The Network Graphs Package
Chaimani, A., & Salanti, G. (2015) · The Stata Journal Promoting communications on statistics and Stata · doi:10.1177/1536867x1501500402 · Cited by 323
“…2005) be used for the estimation of the loop-specific heterogeneity (when tau2(loop) has been specified). eb specifies that the empirical Bayes method (Morris 1983) be used for the estimation of the loop-specific heterogeneity (when tau2(loop) has been specified). random specifies that the random-effects…”
Testing for Convergence Clubs in Income Per Capita: A Predictive Density Approach*
Canova, F. (2004) · International Economic Review · doi:10.1111/j.1468-2354.2004.00117.x · Cited by 322
“…a unknown break point in time series (see e.g., Ploberger et al., 1989; Bai, 1997; Polasek and Ren, 1997) and to the EB tradition of constructing posterior estimates of the coefficients of a model by plugging in estimates of the parameters of the prior (see Morris, 1983; Berger, 1985; Efron, 1996).”
BackgroundMethodInfluentialEmpirical bayes versus fully bayesian analysis of geographical variation in disease risk
Bernardinelli, L., & Montomoli, C. (1992) · Statistics in Medicine · doi:10.1002/sim.4780110802 · Cited by 302
Educational Applications of Hierarchical Linear Models: A Review
Raudenbush, S. W. (1988) · Journal of Educational Statistics · doi:10.3102/10769986013002085 · Cited by 301
Heagerty, P. J., & Zeger, S. L. (2000) · Statistical Science · doi:10.1214/ss/1009212671 · Cited by 298
Santesso, N., Akl, E. A., Bianchi, M., Mente, A., Mustafa, R., Heels-Ansdell, D., & Schünemann, H. J. (2012) · European Journal of Clinical Nutrition · doi:10.1038/ejcn.2012.37 · Cited by 292
Guidelines for reporting mortality and morbidity after cardiac valve interventions
Akins, C. W., Miller, D. C., Turina, M. I., Kouchoukos, N. T., Blackstone, E. H., Grunkemeier, G. L., Takkenberg, J. J. M., David, T. E., Butchart, E. G., Adams, D. H., et al. (2008) · European Journal of Cardio-Thoracic Surgery · doi:10.1016/j.ejcts.2007.12.055 · Cited by 289
“These methods include mixed models, random and fixed effects models, generalized estimating equation approach, and hierarchic models (such as currently used for The Society of Thoracic Surgeons National Database risk assessment) [31—33].”
MethodJacob, B. A., & Lefgren, L. (2008) · Journal of Labor Economics · doi:10.1086/522974 · Cited by 494
Grossman, P., Loeb, S., Cohen, J., & Wyckoff, J. (2013) · American Journal of Education · doi:10.1086/669901 · Cited by 296
Empirical bayes meta-analysis.
Raudenbush, S. W., & Bryk, A. S. (1985) · Journal of Educational Statistics · doi:10.3102/10769986010002075 · Cited by 299
Estimating the Effect of Leaders on Public Sector Productivity: The Case of School Principals
Branch, G. F., Hanushek, E. A., & Rivkin, S. G. (2012) · National Bureau of Economic Research · doi:10.3386/w17803 · Cited by 287
A smoothness priors time-varying AR coefficient modeling of nonstationary covariance time series
Kitagawa, G., & Gersch, W. (1985) · IEEE Transactions on Automatic Control · doi:10.1109/tac.1985.1103788 · Cited by 263
Health Care Exceptionalism? Performance and Allocation in the US Health Care Sector
Chandra, A., Finkelstein, A. N., Sacarny, A., & Syverson, C. (2016) · American Economic Review · doi:10.1257/aer.20151080 · Cited by 258
“Second, we restrict our sample to hospitals with a minimum number of patients per condition over the three-year measurement period; the cutoff threshold varies across our quality measures as described in Appendix B.”
Simple Heterogeneity Variance Estimation for Meta-Analysis
Sidik, K., & Jonkman, J. N. (2005) · Journal of the Royal Statistical Society Series C (Applied Statistics) · doi:10.1111/j.1467-9876.2005.00489.x · Cited by 254
“…of Hedges (1983) and Hedges and Olkin (1985), the maximum likelihood estimator of Hardy and Thompson (1996), the restricted maximum likelihood (REML) estimator of Raudenbush and Bryk (1985) and the empirical Bayes estimator of Morris (1983), which was used for meta-analysis by Berkey et al. (1995).”
MethodNatural Exponential Families with Quadratic Variance Functions: Statistical Theory
Morris, C. N. (1983) · The Annals of Statistics · doi:10.1214/aos/1176346158 · Cited by 253
On model prespecification in confirmatory randomized studies
Edwards, D. (1999) · Statistics in Medicine · doi:10.1002/(sici)1097-0258(19990415)18:7 · Cited by 32
Robust Bayes and Empirical Bayes Analysis with $_\epsilon$-Contaminated Priors
Berger, J. O., & Berliner, L. M. (1986) · The Annals of Statistics · doi:10.1214/aos/1176349933 · Cited by 246
Theory & Methods: Spatially‐adaptive Penalties for Spline Fitting
Ruppert, D., & Carroll, R. (2000) · Australian & New Zealand Journal of Statistics · doi:10.1111/1467-842x.00119 · Cited by 286
“…but then pretend that the hyperparameters were known so they do not account for extra variability in the posterior distribution caused by the estimation of hyperparameters in the prior; for discussion see, e.g., Morris (1983), Laird & Louis (1987), Kass & Steffey (1989) or Carlin & Louis (1996).”
A comparison of procedures to test for moderators in mixed-effects meta-regression models
Viechtbauer, W., López-López, J. A., Sánchez-Meca, J., & Marín-Martínez, F. (2015) · Psychological Methods · doi:10.1037/met0000023 · Cited by 241
“This estimator can be derived based on EB methods (Morris, 1983) and will therefore be denoted by u0007̂EB 2 .”
MethodInfluentialRaudenbush, S. W. (1993) · Journal of Educational Statistics · doi:10.2307/1165158 · Cited by 239
Empirical Bayes Confidence Intervals Based on Bootstrap Samples
Laird, N. M., & Louis, T. A. (1987) · Journal of the American Statistical Association · doi:10.1080/01621459.1987.10478490 · Cited by 233
Hobbs, B. P., Sargent, D. J., & Carlin, B. P. (2012) · Bayesian Analysis · doi:10.1214/12-ba722 · Cited by 232
“Empirical Bayesian In this subsection we consider parametric empirical Bayesian (EB) estimation (see e.g. Morris 1983; Kass and Steffey 1989; Carlin and Louis 2009) in the context of our proposed commensurate prior models.”
MethodWho Leaves? Teacher Attrition and Student Achievement
Boyd, D. J., Grossman, P., Lankford, H., Loeb, S., & Wyckoff, J. (2008) · National Bureau of Economic Research · doi:10.3386/w14022 · Cited by 230
“6 See Jacob and Lefgren (2005) for a clear summary of the approach we use as well as Kane and Staiger (2002) and Morris (1983) for earlier references.”
MethodDo Parents Value School Effectiveness?
Abdulkadiroğlu, A., Pathak, P. A., Schellenberg, J., & Walters, C. (2020) · American Economic Review · doi:10.1257/aer.20172040 · Cited by 232
“Relative to the unbiased but noisy estimate θ̂j , this EB shrinkage estimator uses the prior distribution to reduce sampling variance at the cost of increased bias, yielding a minimum mean squared error (MSE) prediction of θj (Robbins, 1956; Morris, 1983).”
MethodPrediction in Multilevel Generalized Linear Models
Skrondal, A., & Rabe‐Hesketh, S. (2009) · Journal of the Royal Statistical Society Series A (Statistics in Society) · doi:10.1111/j.1467-985x.2009.00587.x · Cited by 220
“Similarly, empirical Bayesians evaluate inferences with respect to joint sampling of ζ j and y j (e.g. Morris (1983)).”
BackgroundMethodA multivariate mixed linear model for meta-analysis.
Kalaian, H. A., & Raudenbush, S. W. (1996) · Psychological Methods · doi:10.1037/1082-989x.1.3.227 · Cited by 229
Metadata
- Authors
- Morris, C. N.
- Affiliations
- The University of Texas at Austin (US)
- Corresponding author
- Morris, Carl N.
- Venue
- Journal of the American Statistical Association
- Publisher
- Informa UK Limited
- Published
- 1983-03
- Volume
- 78
- Issue
- 381
- Pages
- 47-55
- ISSN
- 0162-1459 (ISSN-L)
- 1537-274X
- OpenAlex
- W1983607152
- Cited by
- 1,448
- References
- 36
- Citations by year
- 2026: 32 · 2025: 30 · 2024: 52 · 2023: 42 · 2022: 40 · 2021: 51 · 2020: 62 · 2019: 44 · 2018: 49 · 2017: 43 · 2016: 54 · 2015: 63 · 2014: 66 · 2013: 49 · 2012: 48
- FWCI
- 24.44
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of the American Statistical Association (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
- Keywords
- parametric empirical Bayes, confidence intervals, shrinkage estimation
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