Data-Driven Confidence Interval Estimation Incorporating Prior Information with an Adjustment for Skewed Data

ABSTRACT Bayesian credible interval (CI) estimation is a statistical procedure that has been well addressed in both the theoretical and applied literature. Parametric assumptions regarding baseline data distributions are critical for the implementation of this method. We provide a nonparametric technique for incorporating prior information into the equal-tailed (ET) and highest posterior density (HPD) CI estimators in the Bayesian manner. We propose to use a data-driven likelihood function, replacing the parametric likelihood function to create a distribution-free posterior. Higher order asymptotic propositions are derived to show the efficiency and consistency of the proposed method. We demonstrate that the proposed approach may correct confidence regions with respect to skewness of the data distribution. An extensive Monte Carlo (MC) study confirms the proposed method significantly outperforms the classical CI estimation in a frequentist context. A real data example related to a study of myocardial infarction illustrates the excellent applicability of the proposed technique. Supplementary material, including the R code used to implement the developed method, is available online.

[1]  E F Schisterman,et al.  TBARS and Cardiovascular Disease in a Population-Based Sample , 2001, Journal of cardiovascular risk.

[2]  Jerome P. Reiter,et al.  A Note on Bayesian Inference After Multiple Imputation , 2010 .

[3]  L. Tierney,et al.  Accurate Approximations for Posterior Moments and Marginal Densities , 1986 .

[4]  Peter Hall,et al.  Inverting an Edgeworth Expansion , 1983 .

[5]  J. Lawless,et al.  Empirical Likelihood and General Estimating Equations , 1994 .

[6]  Peter Hall,et al.  Smoothed empirical likelihood confidence intervals for quantiles , 1993 .

[7]  Lyle D. Broemeling,et al.  Bayesian Biostatistics and Diagnostic Medicine , 2007 .

[8]  Ming-Hui Chen,et al.  Monte Carlo Estimation of Bayesian Credible and HPD Intervals , 1999 .

[9]  A. Owen.  Empirical likelihood ratio confidence intervals for a single functional , 1988 .

[10]  Daniel F. Heitjan.  Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis by DANIELS, M. J. and HOGAN, J. W , 2009 .

[11]  An evaluation of the power and conditionality properties of empirical likelihood , 1998 .

[12]  N M Laird,et al.  Missing data in longitudinal studies. , 1988, Statistics in medicine.

[13]  N. Lazar.  Bayesian empirical likelihood , 2003 .

[14]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[15]  Xiao-Hua Zhou,et al.  Confidence intervals for the log-normal mean . , 1997, Statistics in medicine.

[16]  Le Kang,et al.  Modifications of the Empirical Likelihood Interval Estimation with Improved Coverage Probabilities , 2009, Commun. Stat. Simul. Comput..

[17]  W. Stahel,et al.  Log-normal Distributions across the Sciences: Keys and Clues , 2001 .

[18]  A. Hutson,et al.  Posterior expectation based on empirical likelihoods , 2014 .

[19]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[20]  Xiaotong Shen,et al.  Empirical Likelihood , 2002 .

[21]  Ling Chen,et al.  Testing the Mean of Skewed Distributions , 1995 .

[22]  A. W. Vaart,et al.  Frequentist coverage of adaptive nonparametric Bayesian credible sets , 2013, 1310.4489.