Noninformative Priors for the Ratio of the Lognormal Means with Equal Variances

- Journal title : Communications for Statistical Applications and Methods
- Volume 14, Issue 3, 2007, pp.633-640
- Publisher : The Korean Statistical Society
- DOI : 10.5351/CKSS.2007.14.3.633

Title & Authors

Noninformative Priors for the Ratio of the Lognormal Means with Equal Variances

Lee, Seung-A; Kim, Dal-Ho;

Lee, Seung-A; Kim, Dal-Ho;

Abstract

We develop noninformative priors for the ratio of the lognormal means in equal variances case. The Jeffreys' prior and reference priors are derived. We find a first order matching prior and a second order matching prior. It turns out that Jeffreys' prior and all of the reference priors are first order matching priors and in particular, one-at-a-time reference prior is a second order matching prior. One-at-a-time reference prior meets very well the target coverage probabilities. We consider the bioequivalence problem. We calculate the posterior probabilities of the hypotheses and Bayes factors under Jeffreys' prior, reference prior and matching prior using a real-life example.

Keywords

Bioequivalence problem;equal variance;Jeffreys' prior;matching priors;ratio of the lognormal means;reference priors;

Language

English

References

1.

Alkalay, D., Wagner, W. E., Carlsen, S., Khemani, L., Yolk, J., Bartlett, M. F. and LeSher, A. (1980). Bioavailability and kinetics of maprotiline. Clinical Pharmacology and Therapeutics, 27, 697-703

2.

Berger, J. O. and Bernardo, J. M. (1989). Estimating a product of means: Bayesian analysis with reference priors. Journal of the American Statistical Association, 84, 200-207

3.

Berger, J. O. and Bernardo, J. M. (1992). On the development of reference priors (with discussion). In Bayesian Statistics 4 (J. M. Bernardo, et al., eds), 35-60, Oxford University Press, Oxford

4.

Berger, R. L. and Hsu, J. C. (1996). Bioequivalence trials, intersection-union tests and equivalence confidence sets. Statistical Science, 11, 283-315

5.

Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion). Journal of the Royal Statistical Society, Ser. B, 41, 113-147

6.

Chow, S. C. and Liu, J. P. (2000). Design and Analysis of Bioavailability and Bioequivalence Studies, 2nd ed., Marcel Dekker, New York

7.

Cox, D. R. and Reid, N. (1987). Orthogonal parameters and approximate conditional inference (with discussion). Journal of Royal Statistical Society, Ser. B, 49, 1-39

8.

Datta, G. S. and Ghosh, J. K. (1995a). On priors providing frequentist validity for Bayesian !inference. Biometrika, 82, 37-45

9.

Datta, G. S. and Ghosh, J. K. (1995b). Noninformative priors for maximal invariant parameter in group models. Test, 4, 95-114

10.

Datta, G. S. and Ghosh, M. (1995c). Some remarks on noninformative priors. Journal of the American Statistical Association, 90, 1357-1363

11.

Datta, G. S. and Ghosh, M. (1996). On the invariance of noninformative priors. The Annals of Statistics, 24, 141-159

12.

Ghosh, J. K. and Mukerjee, R. (1992). Noninformative Ppriors (with discussion). In Bayesian Statistics 4 (J. M. Bernardo, et al., eds.) 195-210, Oxford University Press, Oxford

13.

Moon, K. A. and Kim, D. H. (2001). Bayesian testing for the equality of two lognormal populations with the fractional Bayes factor. Journal of the Korean Data & Information Science Society, 12, 51-59

14.

Moon, K. A., Shin, I. H. and Kim, D. H. (2000). Bayesian testing for the equality of two lognormal populations. Journal of the Korean Data & Information Science Society, 11, 269-277

15.

Mukerjee, R. and Dey, D. K. (1993). Frequentist validity of posterior quantiles in the presence of a nuisance parameter: higher order asymptotics. Biometrika, 80, 499-505

16.

Mukerjee, R. and Ghosh, M. (1997). Second order probability matching priors. Biometrika, 84, 970-975

17.

Stein, C. (1985). On the coverage probability of confidence sets based on a prior distribution. Sequential Methods in Statistics, 16, 485-514

19.

Welch, B. L. and Peers, H. W. (1963). On formula for confidence points based on integrals of weighted likelihood. Journal of the Royal Statistical Society, Ser, B, 25, 318-329