Abstract
In this paper, we consider noninformative priors for the ratio of variances in two normal populations. We develop first and second order matching priors. We find that the second order matching prior matches alternative coverage probabilities up to the second order and is also a HPD matching prior. It turns out that among the reference priors, only one-at-a-time reference prior satisfies a second order matching criterion. Our simulation study indicates that the one-at-a-time reference prior performs better than other reference priors in terms of matching the target coverage probabilities in a frequentist sense.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berger JO, Bernardo JM (1989) Estimating a product of means: Bayesian analysis with reference priors. J Am Stat Assoc 84:200–207
Berger JO, Bernardo JM (1992) On the development of reference priors (with discussion). In: Bernardo JM et al (eds) Bayesian statistics IV. Oxford University Press, Oxford, pp 35–60
Bernardo JM (1979) Reference posterior distributions for Bayesian inference (with discussion). J R Stat Soc B 41:113–147
Box GEP, Tiao GC (1973) Bayesian inference in statistical analysis. Addison–Wesley, Reading
Brown LD (1968) Inadmissibility of the usual estimators of scale parameters. Ann Math Stat 39:29–48
Cox DR, Reid N (1987) Orthogonal parameters and approximate conditional inference (with discussion). J R Stat Soc B 49:1–39
Datta GS (1996) On priors providing frequentist validity for Bayesian inference. Biometrika 83:287–298
Datta GS, Ghosh JK (1995a) On priors providing frequentist validity for Bayesian inference. Biometrika 82:37–45
Datta GS, Ghosh M (1995b) Some remarks on noninformative priors. J Am Stat Assoc 90:1357–1363
Datta GS, Ghosh M (1996) On the invariance of noninformative priors. Ann Stat 24:141–159
Datta GS, Ghosh M, Mukerjee R (2000) Some new results on probability matching priors. Calcutta Stat Assoc Bull 50:179–192
Gelfand AE, Dey DK (1988) On estimation of a variance ratio. J Stat Plann Inf 19:121–131
Ghosh M, Kundu S (1996) Decision theoretic estimation of the variance ratio. Stat Decis 14:161–175
Lee P (1989) Bayesian statistics: an introduction. Edward Arnold, London
Madi TM (1995) On invariant estimation of a normal variance ratio. J Stat Plann Inf 44:349–357
Mukerjee R, Dey DK (1993) Frequentist validity of posterior quantiles in the presence of a muisance parameter: higher order asymptotics. Biometrika 80:499–505
Mukerjee R, Ghosh M (1997) Second order probability matching priors. Biometrika 84:970–975
Mukerjee R, Reid N (1999) On a property of probability matching priors: matching the alternative coverage probabilities. Biometrika 86:333–340
Stein C (1964) Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean. Ann Inst Stat Math 42:385–388
Stein C (1985) On the coverage probability of confidence sets based on a prior distribution. In: Sequential methods in statistics, vol 16. Polish Scientific Publishers, Warsaw, pp 485–514
Tibshirani R (1989) Noninformative priors for one parameter of many. Biometrika 76:604–608
Welch BL, Peers HW (1963) On formulae for confidence points based on integrals of weighted likelihood. J R Stat Soc B 35:318–329
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by Korea Research Foundation Grant (KRF-2004-002-C00041).
Rights and permissions
About this article
Cite this article
Kim, D.H., Kang, S.G. & Lee, W.D. Noninformative priors for the normal variance ratio. Stat Papers 50, 393–402 (2009). https://doi.org/10.1007/s00362-007-0065-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-007-0065-4