Abstract
Ordered parameter problems arise in a wide variety of real world situations and are dealt with extensively in the literature. Traditional frequentist methods for dealing with these problems are rather complicated theoretically, especially when sample sizes are small. Bayesian methods are not widely used because high dimensional numerical integration is often required. However, Markov chain Monte Carlo methods provide alternatives to such numerical integration and also deal with ordered parameter problems in a straightforward manner. Little is known about the situation where functions of parameters are ordered. Such problems may seem to be of little practical concern initially, but one can readily see their importance in situations where ordering is placed on the means and variances of several normal or Gamma populations. For the Gamma distribution we will present real examples where we will analyze monthly precipitation data from San Francisco, California and Oakland Mills, Iowa. For the San Francisco data we will simultaneously order both monthly precipitation means and variances. For the Iowa data we will place ordering on seasonal average while still estimating monthly means. Our results show that we would obtain sharper, more accurate inference when order restrictions are employed.
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References
Devroye, L. (1986) Non-Uniform Random Variate Generation, Springer-Verlag, New York.
Gelfand, A.E., Hills, S.E., Racine-Poon, A., and Smith, A.F.M. (1990), Illustration of bayesian inference in normal data models using gibbs sampling. Journal of the American Statistical Association, 85, 972–85.
Gelfand, A.E., Smith, A.F.M., and Lee, T.-M. (1992) Bayesian analysis of constrained parameter and truncated data problems using gibbs sampling. Journal of the American Statistical Association, 87, 523–7.
Gilks, W.R. (1992) Derivative-Free Adaptive Rejection Sampling for Gibbs Sampling, Bayesian Statistics, Oxford University Press.
Gilks, W.R., Best, N.G., and Tan, K.K.C. (1995) Adaptive rejection metropolis sampling. Applied Statistics, 44, 455–72.
Gilks, W.R. and Wild, P. (1992) Adaptive rejection sampling for gibbs sampling. Applied Statistics, 41, 337–48.
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., and Teller, A.H. (1953) Equations of state calculations by fast computing machines. J. Chem. Phys., 21, 1087–92.
National Oceanic and Atmospheric Administration (1981) Climatological Data, 85, California.
Robertson, T., Wright, F.T., and Dykstra, R.L. (1988) Order Restricted Statistical Inference, Wiley Series in Probability and Mathematical Statistics, Wiley.
Sun, D. and Ye, K. (1996) Frequentist validity of posterior quantiles for a two-parameter exponential family. Biometrika, 83, 55–65.
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Molitor, J., Sun, D. Bayesian analysis under ordered functions of parameters . Environmental and Ecological Statistics 9, 179–193 (2002). https://doi.org/10.1023/A:1015122221315
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DOI: https://doi.org/10.1023/A:1015122221315