Bayesian Analysis

Recent advances on Bayesian inference for $P(X < Y)$

Laura Ventura and Walter Racugno

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We address the statistical problem of evaluating $R = P(X \lt Y)$, where $X$ and $Y$ are two independent random variables. Bayesian parametric inference is based on the marginal posterior density of $R$ and has been widely discussed under various distributional assumptions on $X$ and $Y$. This classical approach requires both elicitation of a prior on the complete parameter and numerical integration in order to derive the marginal distribution of $R$. In this paper, we discuss and apply recent advances in Bayesian inference based on higher-order asymptotics and on pseudo-likelihoods, and related matching priors, which allow one to perform accurate inference on the parameter of interest $R$ only, even for small sample sizes. The proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals. From a theoretical point of view, we show that the used prior is a strong matching prior. From an applied point of view, the accuracy of the proposed methodology is illustrated both by numerical studies and by real-life data concerning clinical studies.

Article information

Bayesian Anal., Volume 6, Number 3 (2011), 411-428.

First available in Project Euclid: 13 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F15: Bayesian inference
Secondary: 62F10: Point estimation

Asymptotic expansions Frequentist coverage probability Matching prior Modified likelihood root Nuisance parameter ROC curve Stochastic precedence Stress-strength model Tail area probability


Ventura, Laura; Racugno, Walter. Recent advances on Bayesian inference for $P(X &lt; Y)$. Bayesian Anal. 6 (2011), no. 3, 411--428. doi:10.1214/11-BA616.

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  • Azuma, J., L. Maegdefessel, T. Kitagawa, R. Lee Dalman and M.V. McConnell 2010. Assessment of elastase-induced murine abdominal aortic aneurysms: comparison of ultrasound imaging with in situ video microscopy. Journal of Biomedicine and Biotechnology, in press.
  • Barndorff-Nielsen, O.E. 1983. On a formula for the distribution of the maximum likelihood estimator. Biometrika 70: 343–365.
  • Barndorff-Nielsen, O.E. and S.R. Chamberlin 1994. Stable and invariant adjusted directed likelihoods. Biometrika 81: 485–499.
  • Barndorff-Nielsen, O.E. and D.R. Cox 1994. Inference and Asymptotics. London: Chapman and Hall.
  • Bertolino, F. and W. Racugno 1994. Robust Bayesian analysis of variance and the $\chi^2$-test by using marginal likelihoods. The Statistician 43: 191–201.
  • Brazzale, A.R., A.C. Davison and N. Reid 2007. Applied Asymptotics. Cambridge: Cambridge University Press.
  • Chang, H. and R. Mukerjee 2006. Probability matching property of adjusted likelihoods. Statistics and Probability Letters 76: 838–842.
  • Chang, H., B.H. Kim and R. Mukerjee 2009. Bayesian and frequentist confidence intervals via adjusted likelihoods under prior specification on the interest parameter. Statistics 43: 203–211.
  • Chen, Z. and D.B. Dunson 2004. Bayesian estimation of survival functions under stochastic precedence. Lifetime Data Analysis 10: 159–173.
  • Cortese, G. and L. Ventura 2009. Accurate likelihood on the area under the ROC curve for small samples. Working Papers, 2009.17, Department of Statistics, University of Padova. Submitted.
  • Datta, G.S. and R. Mukerjee 2004. Probability Matching Priors: Higher Order Asymptotics. Berlin: Springer.
  • DiCiccio, T.J. and M.A. Martin 1991. Approximations of marginal tail probabilities for a class of smooth functions with applications to Bayesian and conditional inference. Biometrika 78: 891–902.
  • DiCiccio, T.J. and S.E. Stern 1994. Frequentist and Bayesian Bartlett corrections to test statistics based on adjusted profile likelihoods. Journal of the Royal Statistical Society B 56: 397–408.
  • Dunson, D.B. and S.D. Peddada 2008. Bayesian nonparametric inference on stochastic ordering. Biometrika 95: 859–874.
  • Erkanli, A., M. Sung, E.J. Costello and A. Angold 2006. Bayesian semi-parametric ROC analysis. Statistics in Medicine 25: 3905–3928.
  • Fraser, D.A.S. and N. Reid 1996. Bayes posteriors for scalar interest parameters. Bayesian Statistics 5: 581–585.
  • ––. 2002. Strong matching of frequentist and Bayesian parametric inference. Journal of Statistical Planning and Inference 103: 263–285.
  • Ghosh, M. and M.C. Yang 1996. Noninformative priors for the two sample normal problem. Test 5: 145–157.
  • Ghosh, M. and D. Sun 1998. Recent development of Bayesian inference for stress-strength models. In: Frontier of Reliability, 143–158. New Jersey: World Scientific.
  • Giummolé, F. and L. Ventura 2002. Practical point estimation from higher-order pivots. Journal of Statistical Computation and Simulation 72: 419–430.
  • Guttman, I., R.A. Johnson, G.K. Bhattacharyya and B. Reiser 1988. Confidence limits for stress-strength models with explanatory variables. Technometrics 30: 161–168.
  • Guttman, I. and G.D. Papandonatos 1997. A Bayesian approach to a reliability problem: theory, analysis and interesting numerics. Canadian Journal of Statistics 25: 143–158.
  • Hanson, T.E., A. Kottas and A.J. Branscum 2008. Modeling stochastic order in the analysis of receiver operating characteristic data: Bayesian non-parametric approaches. Applied Statistics 57: 207–225.
  • Jiang, L. and A.C.M. Wong 2008. A note on inference for $P(X<Y)$ for right truncated exponentially distributed data. Statistical Papers 49: 637–651.
  • Johnson, R.A. 1988. Stress-strength model for reliability. Handbook of Statistics, Krishnaiah P.R. and Rao C.R. eds., 7: 27–54. Amsterdam: Elsevier.
  • Kotz S., Y. Lumelskii and M. Pensky 2003. The Stress-Strength Model and its Generalizations. Theory and Applications. Singapore: World Scientific.
  • Kundu, D. and R.D. Gupta 2006. Estimation of $P(Y<X)$ for Weibull distribution. IEEE Transactions in Reliability 55: 270–280.
  • Mayer, M.P. and B. Bukau 2005. Hsp70 chaperones: cellular functions and molecular mechanism. Cellular and Molecular Life Sciences 62: 670–684.
  • Monahan, J.F. and D.D. Boos 1992. Proper likelihoods for Bayesian analysis. Biometrika 79: 271–278.
  • Nyhsen, C.M. and S.T. Elliott 2007. Rapid assessment of abdominal aortic aneurysms in 3-dimensional ultrasonography. Journal of Ultrasound in Medicine 26: 223–226.
  • Pace, L. and A. Salvan 1997. Principles of Statistical Inference from a Neo-Fisherian Perspective. Singapore: World Scientific.
  • ––. 1999. Point estimation based on confidence intervals: Exponential families. Journal of Statistical Computation and Simulation 64: 1–21.
  • ––. 2006. Adjustments of the profile likelihood from a new perspective. Journal of Statistical Planning and Inference 136: 3554–3564.
  • Pauli, F., W. Racugno and L. Ventura 2011. Bayesian composite marginal likelihoods. Statistica Sinica, 21: 149–164.
  • Racugno, W., A. Salvan and L. Ventura 2010. Bayesian analysis in regression models using pseudo-likelihoods. Communications in Statistics - Theory and Methods 39: 3444–3455.
  • Reid, N. 1995. Likelihood and Bayesian approximation methods. Bayesian Statistics 5: 351–368.
  • ––. 2003. The 2000 Wald memorial lectures: Asymptotics and the theory of inference. Annals of Statistics 31: 1695–1731.
  • Reiser, B. and I. Guttman 1986. Statistical inference for $Pr(Y<X)$: The normal case. Technometrics 28: 253–257.
  • ––. 1987. A comparison of three point estimators for $P(Y<X)$ in the normal case. Computational Statistics and Data Analysis 5: 59–66.
  • Severini, T.A. 1999. On the relationship between Bayesian and non-Bayesian elimination of nuisance parameters. Statistica Sinica 9: 713–724.
  • ––. 2000. Likelihood Methods in Statistics. Oxford: Oxford University Press.
  • Skovgaard, I.M. 1989. A review of higher order likelihood inference. Bulletin of the International Statistical Institute 53: 331–351.
  • Tierney, L.J. and J.B. Kadane 1986. Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association 81: 82–86.
  • Ventura, L., S. Cabras and W. Racugno 2009. Prior distributions from pseudo-likelihoods in the presence of nuisance parameters. Journal of the American Statistical Association 104: 768–774.
  • ––. 2010. Default prior distributions from quasi- and quasi-profile likelihoods. Journal of Statistical Planning and Inference 140: 2937–2942.