## Bayesian Analysis

### Asymptotics for Constrained Dirichlet Distributions

#### Abstract

We derive the asymptotic approximation for the posterior distribution when the data are multinomial and the prior is Dirichlet conditioned on satisfying a finite set of linear equality and inequality constraints so the posterior is also Dirichlet conditioned on satisfying these same constraints. When only equality constraints are imposed, the asymptotic approximation is normal. Otherwise it is normal conditioned on satisfying the inequality constraints. In both cases the posterior is a root-$n$-consistent estimator of the parameter vector of the multinomial distribution. As an application we consider the constrained Polya posterior which is a non-informative stepwise Bayes posterior for finite population sampling which incorporates prior information involving auxiliary variables. The constrained Polya posterior is a root-$n$-consistent estimator of the population distribution, hence yields a root-$n$-consistent estimator of the population mean or any other differentiable function of the vector of population probabilities.

#### Article information

Source
Bayesian Anal. Volume 8, Number 1 (2013), 89-110.

Dates
First available in Project Euclid: 4 March 2013

https://projecteuclid.org/euclid.ba/1362406653

Digital Object Identifier
doi:10.1214/13-BA804

Mathematical Reviews number (MathSciNet)
MR3036255

Zentralblatt MATH identifier
1329.62080

#### Citation

Geyer, Charles; Meeden, Glen. Asymptotics for Constrained Dirichlet Distributions. Bayesian Anal. 8 (2013), no. 1, 89--110. doi:10.1214/13-BA804. https://projecteuclid.org/euclid.ba/1362406653

#### References

• Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis. Hoboken: Wiley, third edition.
• Billingsley, P. (1999). Convergence of Probability Measures. New York: Wiley, second edition.
• Binder, D. (1982). “Non-parametric Bayesian models for samples from a finite population.” Journal of the Royal Statistical Society, Series B, 44: 388–393.
• Booth, J. G., Butler, R. W., and Hall, P. (1994). “Bootstrap methods for finite population sampling.” Journal of the American Statistical Association, 89: 1282–1289.
• Cramér, H. (1951). Mathematical Methods of Statistics. Princeton: Princeton University Press.
• Deville, J. and Särndal, C. (1992). “Calibration estimators in survey sampling.” Journal of the American Statistical Association, 87: 376–382.
• Fuller, W. (2009). Sampling Statistics. New York: John Wiley and Sons.
• Geyer, C. J. (1994). “On the Asymptotics of Constrained M-Estimation.” Annals of Statistics, 22: 1993–2010.
• Geyer, C. J. and Meeden, G. D. (2012). “Supplementary Material for the paper “Asymptotics for Constrained Dirichlet Distributions”.” Technical Report 691, School of Statistics, University of Minnesota. URL http://purl.umn.edu/126224
• Geyer, C. J., Meeden, G. D., and Fukuda, K. (2011). rcdd: Computational Geometry. R package version 1.1-4. URL http://CRAN.R-project.org/package=rcdd
• Ghosh, M. and Meeden, G. (1997). Bayesian Methods for Finite Population Sampling. London: Chapman and Hall.
• Gross, S. (1980). “Median estimation in survey sampling.” In Proceedings of the Section on Survey Research Methods, 181–184. American Statistical Association.
• Gupta, R. D. and Richards, D. S. P. (2001). “The history of the Dirichlet and Liouville distributions.” International Statistical Review, 69: 433–446.
• Hájek, J. (1960). “Limiting Distributions in Simple Random Sampling from a Finite Population.” Publications of the Mathematical Institute of the Hungarian Academy of Sciences, Series A, 5: 361–374.
• Hartley, H. O. and Rao, J. N. K. (1968). “A new estimation theory for sample surveys.” Biometrika, 55: 159–167.
• Lazar, R., Meeden, G., and Nelson, D. (2008). “A noninformative Bayesian approach to finite population sampling using auxiliary variables.” Survey Methodology, 34: 51–64.
• LeCam, L. (1970). “On the Assumptions Used to Prove Asymptotic Normality of Maximum Likelihood Estimates.” Annals of Mathematical Statistics, 41: 802–828.
• Lo, A. (1988). “A Bayesian Bootstrap for a finite population.” Annals of Statistics, 16: 1684–1695.
• Meeden, G. and Lazar, R. (2011). polyapost: Simulating from the Polya posterior. R package version 1.1-1. URL http://CRAN.R-project.org/package=polyapost
• R Development Core Team (2011). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/
• Rockafellar, R. T. and Wets, R. J.-B. (2004). Variational Analysis. Berlin: Springer-Verlag. Corrected 2nd printing.
• Rubin, D. (1981). “The Bayesian bootstrap.” Annals of Statistics, 9: 130–134.
• Särndal, C.-E., Swensson, B., and Wretman, J. (1992). Model Assisted Survey Sampling. New York: Springer.
• Scheffé, H. (1947). “A Useful Convergence Theorem for Probability Distributions.” Annals of Mathematical Statistics, 18: 434–438.
• Self, S. G. and Liang, K.-Y. (1987). “Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests Under Nonstandard Conditions.” Journal of the American Statistical Association, 82: 605–610.
• Weisberg, S. (2005). Applied Linear Regression. New York: Wiley, third edition.