• Confidence distribution;
  • constrained maximum likelihood estimation;
  • empirical Bayes;
  • foundations of statistics;
  • Lindley's paradox;
  • local false discovery rate;
  • multiple comparison procedure;
  • multiple testing;
  • observed confidence level;
  • restricted parameter space


Empirical Bayes methods of estimating the local false discovery rate (LFDR) by maximum likelihood estimation (MLE), originally developed for large numbers of comparisons, are applied to a single comparison. Specifically, when assuming a lower bound on the mixing proportion of true null hypotheses, the LFDR MLE can yield reliable hypothesis tests and confidence intervals given as few as one comparison. Simulations indicate that constrained LFDR MLEs perform markedly better than conventional methods, both in testing and in confidence intervals, for high values of the mixing proportion, but not for low values. (A decision-theoretic interpretation of the confidence distribution made those comparisons possible.) In conclusion, the constrained LFDR estimators and the resulting effect-size interval estimates are not only effective multiple comparison procedures but also they might replace p-values and confidence intervals more generally. The new methodology is illustrated with the analysis of proteomics data.