Links
Cox, D.R. (1958). Some problems with statistical inference. The Annals of Mathematical Statistics 29: 357-372.
Cox, D.R. (2006). Principles of Statistical Inference. Cambridge University Press.
Schweder, T., and N.L. Hjort (2002). Confidence and likelihood. Scandinavian Journal of Statistics 29: 309–332.
Singh, K., M. Xie and W.E. Strawderman (2005). Combining information from independent sources through confidence distributions. The Annals of Statistics 33: 159–183.
Probability boxes (p-boxes) and probability bounds analysis
Ferson, S., V. Kreinovich, L. Ginzburg, K. Sentz and D.S. Myers. 2003. Constructing probability boxes and Dempster−Shafer structures. Sandia National Laboratories, SAND2002-4015, Albuquerque, New Mexico. http://www.ramas.com/unabridged.zip. Abstract: This report summarizes a variety of the most useful and commonly applied methods for obtaining Dempster-Shafer structures, and their mathematical kin probability boxes, from empirical information or theoretical knowledge. The report includes a review of the aggregation methods for handling agreement and conflict when multiple such objects are obtained from different sources.
<<Manski>>
Ferson, S., V. Kreinovich, J. Hajagos, W.L. Oberkampf and L. Ginzburg 2007. Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty. SAND2007-0939, Sandia National Laboratories, Albuquerque, New Mexico. http://www.ramas.com/intstats.pdf. Abstract: This report addresses the characterization of measurements that include epistemic uncertainties in the form of intervals. It reviews the application of basic descriptive statistics to data sets which contain intervals rather than exclusively point estimates. It describes algorithms to compute various means, the median and other percentiles, variance, interquartile range, moments, confidence limits, and other important statistics and summarizes the computability of these statistics as a function of sample size and characteristics of the intervals in the data (degree of overlap, size and regularity of widths, etc.). It also reviews the prospects for analyzing such data sets with the methods of inferential statistics such as outlier detection and regressions. The report explores the tradeoff between measurement precision and sample size in statistical results that are sensitive to both. It also argues that an approach based on interval statistics could be a reasonable alternative to current standard methods for evaluating, expressing and propagating measurement uncertainties.
Nguyen, H.T., V. Kreinovich, B. Wu and G. Xiang (2012). Computing Statistics under Interval and Fuzzy Uncertainty. Springer Verlag.
Imprecise Dirichlet model and imprecise beta model
Walley, P. (1996). Inferences from multinomial data: learning about a bag of marbles. Journal of the Royal Statistical Society, Series B 58: 3–57.
Walley, P., L. Gurrin and P. Barton (1996). Analysis of clinical data using imprecise prior probabilities. The Statistician 45: 457–485.
Datta, G.S., and T.J. Sweeting (2005). Probability matching priors. Research Report No.252, Department of Statistical Science, University College London, United Kingdom. Abstract: A probability matching prior is a prior distribution under which the posterior probabilities of certain regions coincide with their coverage probabilities, either exactly or approximately. Use of such a prior will ensure exact or approximate frequentist validity of Bayesian credible regions. Probability matching priors have been of interest for many years but there has been a resurgence of interest over the last twenty years. In this article we survey the main developments in probability matching priors, which have been derived for various types of parametric and predictive region.
Staicu, A.-M., and N. Reid (2008). On probability matching priors. The Canadian Journal of Statistics 36 <<pages>>. Abstract: First order probability matching priors are priors for which Bayesian and frequentist inference, in the form of posterior quantiles, or con.dence intervals, agree to a second order of approximation. The present paper shows that the class of matching priors developed by Peers (1965) and Tibshirani (1989) are readily (and uniquely) implemented in a third order approximation to the posterior marginal density. The authors show how strong orthogonality of parameters simpli.es the arguments, and illustrate their results on several examples