Bayesian Analysis

Nonparametric estimation of the distribution function in contingent valuation models

Denzil G. Fiebig, Robert Kohn, and David S. Leslie

Full-text: Open access

Abstract

Contingent valuation models are used in Economics to value non-market goods and can be expressed as binary choice regression models with one of the regression coefficients fixed. A method for flexibly estimating the link function of such binary choice model is proposed by using a Dirichlet process mixture prior on the space of all latent variable distributions, instead of the more restricted distributions in earlier papers. The model is estimated using a novel MCMC sampling scheme that avoids the high autocorrelations in the iterates that usually arise when sampling latent variables that are mixtures. The method allows for variable selection and is illustrated using simulated and real data.

Article information

Source
Bayesian Anal. Volume 4, Number 3 (2009), 573-597.

Dates
First available in Project Euclid: 22 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.ba/1340369855

Digital Object Identifier
doi:10.1214/09-BA421

Mathematical Reviews number (MathSciNet)
MR2551046

Zentralblatt MATH identifier
1330.62428

Keywords
binary choice regression Dirichlet process latent variable mixture model variable selection

Citation

Leslie, David S.; Kohn, Robert; Fiebig, Denzil G. Nonparametric estimation of the distribution function in contingent valuation models. Bayesian Anal. 4 (2009), no. 3, 573--597. doi:10.1214/09-BA421. https://projecteuclid.org/euclid.ba/1340369855


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References

  • Albert, J. H. and S. Chib (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association 88(422), 669–679.
  • Antoniak, C. E. (1974). Mixtures of dirichlet processes with applications to bayesian nonparametric problems. The Annals of Statistics 2(6), 1152–1174.
  • Arrow, K., R. Solow, P. R. Portnoy, E. E. Leamer, R. Radner, and H. Schuman (1993). Report of the NOAA panel on contingent valuation. Federal Register 58, 4601–4614.
  • Basu, S. and S. Mukhopadhyay (2000a). Bayesian analysis of binary regression using symmetric and asymmetric links. Sankhy$\overline{\mbox{a}}$, Series B 62, 372–387.
  • Basu, S. and S. Mukhopadhyay (2000b). Binary response regression with normal scale mixture links. In D. K. Dey, S. K. Ghosh, and B. K. Mallick (Eds.), Generalized Linear Models: A Bayesian Perspective, pp. 231–253. New York: Marcel Dekker.
  • Belkar, R., D. G. Fiebig, M. Haas, and R. Viney (2006). Why worry about awareness in choice problems? Econometric analysis of screening for cervical cancer. Health Economics 15(1), 33–47.
  • Cameron, T. A. (1988). A new paradigm for valuing non-market goods using referendum data: Mximum likelihood estimation by censored logistic regression. Journal of Environmental Economics and Management 15, 355–379.
  • Cameron, T. A. and M. D. James (1987). Efficient estimation methods for `Closed-ended' contingent valuation surveys. Review of Economics and Statistics 69, 269–276.
  • Carson, R. T. and W. M. Hanemann (2005). Contingent valuation. In K.-G. Mäler and J. R. Vincent (Eds.), Handbook of Environmental Economics, Volume 2. Elsevier.
  • Chan, D., R. Kohn, D. J. Nott, and C. Kirby (2006). Locally adaptive semiparametric estimation of the mean and variance functions in regression models. Journal of Computational and Graphical Statistics 15(4), 915–936.
  • Clarke, P. M. (2000). Valuing the benefits of mobile mammographic screening units using the contingent valuation method. Applied Economics 32, 1647–1655.
  • Diener, A., B. Obrien, and A. Gafni (1998). Health care contingent valuation studies: A review and classification of the literature. Health Economics 7, 313–326.
  • Erkanli, A., D. Stangl, and P. Müller (1993). A Bayesian analysis of ordinal data using mixtures. ISDS Discussion Paper 93-A01, Duke University.
  • Escobar, M. D. and M. West (1995). Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association 90(430), 577–588.
  • Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. The Annals of Statistics 1(2), 209–230.
  • Fernández, C., C. J. León, M. F. J. Steel, and F. J. Vásquez-Polo (2004). Bayesian analysis of interval data contingent valuation models and pricing policies. Journal of Business & Economic Statistics 22(4), 431–442.
  • Gelfand, A. E., S. K. Sahu, and B. P. Carlin (1995). Efficient parametrisations for normal linear mixed models. Biometrika 82(3), 479–488.
  • Geweke, J. and M. Keane (1999). Mixture of normals probit models. In C. Hsiao, K. Lahiri, L.-F. Lee, and M. H. Pesaran (Eds.), Analysis of panels and limited dependent variables: A volume in honor of G. S. Maddala, 49–78. Cambridge University Press.
  • Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4), 711–732.
  • Green, P. J. and S. Richardson (2001). Modelling heterogeneity with and without the Dirichlet process. Scandinavian Journal of Statistics 28, 355–375.
  • Handcock, M. S., A. E. Raftery, and J. M. Tantrum (2007). Model-based clustering for social networks. Journal of the Royal Statistical Society A 170(2), 1–22.
  • Holmes, C. C. and L. Held (2006). Bayesian auxiliary variable models for binary and multinomial regression. Bayesian Analysis 1(1), 145–168.
  • Kass, R. E. and L. Wasserman (1995). Mixtures of g priors for bayesian variable selection. Journal of the American Statistical Association 90, 928–934.
  • Kohn, R., M. Smith, and D. Chan (2001). Nonparametric regression using linear combinations of basis functions. Statistics and Computing 11, 313–322.
  • Leslie, D. S., R. Kohn, and D. J. Nott (2007). A general approach to heteroscedastic linear regression. Statistics and Computing 17, 131–146.
  • Liang, F., R. Paulo, G. Molina, M. Clyde, and J. Berger (2008). Mixtures of g priors for bayesian variable selection. Journal of the American Statistical Association 103, 410–423.
  • Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates: I. Density estimates. The Annals of Statistics 12(1), 351–357.
  • MacEachern, S. N. (1994). Estimating normal means with a conjugate style Dirichlet process prior. Communications in Statistics: Simulation and Computation 7, 727–741.
  • Mallick, B. K., D. G. T. Denison, and A. F. M. Smith (2000). Semiparametric generalized linear models: Bayesian approaches. In D. K. Dey, S. K. Ghosh, and B. K. Mallick (Eds.), Generalized Linear Models: A Bayesian Perspective, pp. 217–230. New York: Marcel Dekker.
  • Marshall, E. C. and D. J. Spiegelhalter (2003). Approximate cross-validatory predictive checks in disease mapping models. Statistics in Medicine 22, 1649–1660.
  • Mukhopadhyay, S. and A. E. Gelfand (1997). Dirichlet process mixed generalised linear models. Journal of the American Statistical Association 92(438), 633–639.
  • Newton, M. A., C. Czado, and R. Chappell (1996). Bayesian inference for semiparametric binary regression. Journal of the American Statistical Association 91, 142–153.
  • Nott, D. J. and D. Leonte (2004). Sampling schemes for Bayesian variable selection in generalized linear models. Journal of Computational and Graphical Statistics 13(2), 362–382.
  • Richardson, S. and P. J. Green (1997). On Bayesian analysis of mixtures with an unknown number of components. Journal of the Royal Statistical Society, B 59, 731–792.
  • Silva, J. M. C. S. (2001). A score test for non-nested hypotheses with applications to discrete data models. Journal of Applied Econometrics 16, 577–597.
  • West, M. (1992). Hyperparameter estimation in Dirichlet process mixture models. ISDS Discussion paper 92-A03, Duke University.
  • Wood, S. and R. Kohn (1998). A Bayesian approach to robust binary nonparametric regression. Journal of the American Statistical Association 93, 203–213.