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Archive for the ‘fiducial inference’ Category

Empirical Bayes single-comparison procedure

1 July 2016 Leave a comment

D. R. Bickel, “Small-scale inference: Empirical Bayes and confidence methods for as few as a single comparison,” International Statistical Review 82, 457-476 (2014). Published version2011 preprint | Simple explanation (link added 21 June 2017)

Parametric empirical Bayes methods of estimating the local false discovery rate by maximum likelihood apply not only to the large-scale settings for which they were developed, but, with a simple modification, also to small numbers of comparisons. In fact, data for a single comparison are sufficient under broad conditions, as seen from applications to measurements of the abundance levels of 20 proteins and from simulation studies with confidence-based inference as the competitor.

Coherent inference after checking a prior

7 January 2016 Leave a comment

Fiducial nonparametrics

24 August 2015 Leave a comment

Sonderegger, Derek L.; Hannig, Jan
Fiducial theory for free-knot splines. Contemporary developments in statistical theory, 155–189,
Springer Proc. Math. Stat., 68, Springer, Cham, 2014.
62F12 (62F10 62F99 65D07)

This chapter provides both asymptotic and finite-sample properties of a fiducial solution to the problem of free-knot splines with four or more degrees, assuming a known number of knot points. The authors lay a foundation for the solution by proving the asymptotic normality of certain multivariate fiducial estimators. After demonstrating that the fiducial solution meets the sufficient conditions for asymptotic normality, they quantify small-sample performance on the basis of simulations. The authors conclude that fiducial inference provides a promising alternative to Bayesian inference for the free-knot spline problem addressed.
The research reported reflects the recent surge in developments of Fisher’s fiducial argument [S. Nadarajah, S. Bityukov and N. Krasnikov, Stat. Methodol. 22 (2015), 23–46; MR3261595]. The work of this chapter is carried out within the framework of generalized fiducial inference [J. Hannig, Statist. Sinica 19 (2009), no. 2, 491–544; MR2514173 (2010h:62071)], which is built on the functional-model formulation of fiducial statistics [A. P. Dawid, M. Stone and M. Stone, Ann. Statist. 10 (1982), no. 4, 1054–1074; MR0673643 (83m:62008)] rather than on the broadly equivalent confidence-based tradition beginning with [G. N. Wilkinson, J. Roy. Statist. Soc. Ser. B 39 (1977), no. 2, 119–171; MR0652326 (58 #31491)] and generalized by [E. E. M. van Berkum, H. N. Linssen and D. Overdijk, J. Statist. Plann. Inference 49 (1996), no. 3, 305–317; MR1381161 (97k:62007)].

{For the entire collection see MR3149911.}

Reviewed by David R. Bickel

This review first appeared at “Fiducial theory for free-knot splines” (Mathematical Reviews) and is used with permission from the American Mathematical Society.

Maximum entropy over a set of posteriors

10 August 2015 Leave a comment

D. R. Bickel, “Blending Bayesian and frequentist methods according to the precision of prior information with applications to hypothesis testing,” Statistical Methods & Applications 24, 523-546 (2015). Published article2012 preprint | 2011 preprint | Slides | Simple explanation

SMA

This framework of statistical inference facilitates the development of new methodology to bridge the gap between the frequentist and Bayesian theories. As an example, a simple and practical method for combining p-values with a set of possible posterior probabilities is provided.

In this general approach, Bayesian inference is used when the prior distribution is known, frequentist inference is used when nothing is known about the prior, and both types of inference are blended according to game theory when the prior is known to be a member of some set. (The robust Bayes framework represents knowledge about a prior in terms of a set of possible priors.) If the benchmark posterior that corresponds to frequentist inference lies within the set of Bayesian posteriors derived from the set of priors, then the benchmark posterior is used for inference. Otherwise, the posterior within that set that minimizes the cross entropy to the benchmark posterior is used for inference.

Fiducial model averages from model checks

10 May 2015 Leave a comment

Erratum: “Simple estimators of false discovery rates given as few as one or two p-values without strong parametric assumptions”

31 March 2015 Leave a comment

Small-scale empirical Bayes & fiducial estimators

22 March 2015 Leave a comment

M. Padilla and D. R. Bickel, “Empirical Bayes and fiducial effect-size estimation for small numbers of tests,” Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/32151 (2015). 2015 preprint