An Empirical Bayes Method for Chi-Squared Data. Issue 537 (2nd January 2022)

• Record Type: Journal Article
• Authors: Du, Lilun; Hu, Inchi
• Is Part Of: Journal of the American Statistical Association. Volume 117:Issue 537(2022)
• Page Start: 334
• Page End: 347
• Publication Date: 2022-01-02
• Subjects: False discovery rate -- High-dimensional data analysis -- Large-scale inference -- Post-selection inference -- Selection bias -- Tweedie's formula; Statistics -- Periodicals; Statistics -- Periodicals; Statistiques -- Périodiques; États-Unis -- Statistiques -- Périodiques; 519.5
• DOI: 10.1080/01621459.2020.1777137
• Languages: English
• ISSNs: 0162-1459

Abstract

Abstract: In a thought-provoking paper, Efron investigated the merit and limitation of an empirical Bayes method to correct selection bias based on Tweedie's formula first reported in the study by Robbins. The exceptional virtue of Tweedie's formula for the normal distribution lies in its representation of selection bias as a simple function of the derivative of log marginal likelihood. Since the marginal likelihood and its derivative can be estimated from the data directly without invoking prior information, bias correction can be carried out conveniently. We propose a Bayesian hierarchical model for chi-squared data such that the resulting Tweedie's formula has the same virtue as that of the normal distribution. Because the family of noncentral chi-squared distributions, the common alternative distributions for chi-squared tests, does not constitute an exponential family, our results cannot be obtained by extending existing results. Furthermore, the corresponding Tweedie's formula manifests new phenomena quite different from those of the normal distribution and suggests new ways of analyzing chi-squared data.