Robust estimation and forecasting for beta-mixed hierarchical models of grouped binaria data

• Autores: Yurij S. Kharin, Maxim Pashkevich
• Localización: Sort: Statistics and Operations Research Transactions, ISSN 1696-2281, Vol. 28, Nº. 2, 2004, págs. 125-160
• Idioma: inglés
• Títulos paralelos:
  • Estimación robusta y pronóstico para modelos jerárquicos beta-mezclados de datos binarios agrupados.
• Enlaces
  • Texto completo (pdf)
• Resumen
  • català

    L'article desenvolupa t`ecniques d'estimaci¿o i predicci¿o robusta per a dades bin`aries agrupades quan hi ha respostes classificades malament. Les dades es descriuen mitjan¿cant un model jer`arquic, b¿e el beta-binomial o b¿e el beta-log¿istic, mentre que les males classificacions estan causades per les distorsions estoc`astiques additives de les observacions bin`aries. Per a aquests models, s'avalua l'efecte d'ignorar la mala classificaci¿o i es donen expressions per als biaixos dels estimadors obtinguts pel m`etode dels moments i per la m`axima versemblan¿ca. Tamb¿e es donen expressions per a l'augment de l'error de predicci¿o, en mitjana quadr`atica, dels predictors de Bayes. Per compensar els efectes de la mala classificaci¿o, es construeixen nous estimadors consistents i un nou predictor de Bayes que tenen en compte la distorsi¿o en el model. La robustesa de les t`ecniques desenvolupades es demostra mitjan¿cant simulacions i l'estudi d'un cas real.

  • English

    The paper focuses on robust estimation and forecasting techniques for grouped binary data with misclassified responses. It is assumed that the data are described by the beta-mixed hierarchical model (the beta-binomial or the beta-logistic), while the misclassifications are caused by the stochastic additive distortions of binary observations. For these models, the effect of ignoring the misclassifications is evaluated and expressions for the biases of the method-of-moments estimators and maximum likelihood estimators, as well as expressions for the increase in the mean square error of forecasting for the Bayes predictor are given. To compensate the misclassification effects, new consistent estimators and a new Bayes predictor, which take into account the distortion model, are constructed. The robustness of the developed techniques is demonstrated via computer simulations and a real-life case study.

• Referencias bibliográficas
  • Agresti, A., Booth, J. G., Hobert, J. P. and Caffo, B. (2000). Random effects modeling of categorical response data.Sociological Methodology,...
  • Brooks, S. P. (2001). On Bayesian analysis and finite mixtures for proportions.Statistics and Computing, 11, 179-190.
  • Collet, D. (2002).Modeling Binary Data. London: Champton and Hall/CRC.
  • Coull, B. A. and Agresti, A. (2000). Random effects modeling of multiple binomial responses using the multivariate binomial logit-normal distribution.Biometrics,...
  • Copas, J. B. (1988). Binary regression models for contaminated data.Journal of Royal Statistical Society, 50B, 225-265.
  • Danaher, P. J. (1992). Some statistical modeling problems in the advertising industry: a look at media exposure distributions.The American...
  • Demidovich B. P. and Maron, I. A. (1970).Basics of Computational Mathematics. Moscow (in Russian).
  • Diggle, P. J., Heagerty, P., Liang, K.-Y. and Zeger, S. L. (2002). Analysis of Longitudinal Data. Oxford : University Press.
  • Gaba, A. and Winkler, R. L. (1992). Implication of errors in survey data: a Bayesian model.Management Science., 38 (7), 913-925.
  • Gill, P. S. (2001). A robust mixed linear model analysis for longitudinal data.Statistics in Medicine, 19, 975-987.
  • Hampel, F. R., Rousseeuw, P. J., Ronchetti, E. M. and Stahel,W. A. (1986). Robust Statistics. New York: John Wiley and Sons.
  • Heckman, J. J. and Willis, R. J. (1977). A beta logistic modelfor the analysis of sequential labor force participation by married women.Journal...
  • Huber, P. J. (1981).Robust Statistics. New York: Wiley.
  • Ivchenko, G. I. and Medvedev, U. I. (1984).Mathematical Statistics. Moscow (in Russian).
  • Johnson, N. L., Kotz, S. and Kemp, A. W. (1996).Univariate Discrete Distributions. Wiley-Interscience, New York.
  • Kharin, Yu. (1996).Robustness in Statistical Pattern Recognition. Kluwer Academic Publishers, Dordrecht.
  • Kordzakhia, N., Mishra, G. D. and Reiersolmoen, L. (2001). Robust estimation in the logistic regression model.Journal of Statistical Planning...
  • Liang, K.-Y. and Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73, 13-22.
  • Nathan, D. (1999). A beta-logistic model of presidential influence on voting on civil rights issues in the house of representatives, 1960-1988.Midwest...
  • Neuhaus, J. M. (1999). Bias and efficiency loss due to misclassified responses in binary regression. Biometrika, 86, 843-855.
  • Neuhaus, J. M. (2002). Analysis of clustered and longitudinal binary data subject to response misclassification.Biometrics, 58, 675-683.
  • Pearson, E. S. (1925). Bayes’ theorem in the light of experimental sampling.Biometrika, 17, 338-442.
  • Pfeifer, P. E. (1998). On using the beta-logistic model to update response probabilities given nonresponse. Journal of Interactive Marketing,...
  • Prentice, R. L. (1988). Correlated binary regression with covariates specific to each binary observation. Biometrics, 44, 1033-1048.
  • Prentice, R. L. (1986). Binary Regression Using an ExtendedBeta-Binomial Distribution, With Discussion of Correlation Induced by Covariate...
  • Ruckstuhl, A. F. and Welsh, A. H. (2001). Robust fitting of thebinomial model.The Annals of Statistics, 29, 1117-1136.
  • Sissors, J. and Lincoln, B. (1994).Advertising Media Planning. NTC Business Books. Lincolnwood.
  • Skellam, J. G. (1948). A probability distribution derived from the binomial distribution by regarding the probability of success as a variable...
  • Slaton, T. L., Piegorsch, W. W. and Durham, S. D. (2000). Estimation and testing with overdispersed proportions using the beta-logistic regression...
  • Swartz, T., Haitovsku, Y., Vexler, A. and Yang, T. Bayesian identifiability and misclassification in multinomial data.The Canadian Journal...
  • Tripathi, R. C., Gupta, R. C. and Gurland, J. (1994). Estimation of parameters in the beta binomial model. Ann. Inst. Statist. Math., 46,...
  • White, H. (1982). Maximum Likelihood Estimation of Misspecified Model.Econometrica, 50(1), 1-26.
  • Zeger, S. L. and Karim, M. R. (1991). Generalized linear models with random effects: A Gibbs sampling approach.Journal of the American Statistical...