Brazilian Journal of Probability and Statistics

A note on extendibility and predictivistic inference in finite populations

Pilar L. Iglesias, Rosangela H. Loschi, Carlos A. B. Pereira, and Sergio Wechsler

Full-text: Open access


The usual finite population model—where information provided by a subset of units is used to reduce uncertainty about functions of the complete population list of values—is explored from a predictivistic point of view. Under this approach, only operationally meaningful quantities (operational parameters) are considered and therefore no superpopulation parameters are involved. This paper addresses the estimation of both population total and maximum based on uniformity and/or exchangeability judgments on finite sequences of random variables. A central point of this paper is that there are contexts in which the superpopulation approach cannot be employed in inferential problems in finite populations. There are circumstances in which the prior distributions for the operational parameters cannot be obtained from any superpopulation model. Conditions for the extendibility to infinite populations are also established for some models, as this approach may ease the inferential problem.

Article information

Braz. J. Probab. Stat., Volume 23, Number 2 (2009), 216-226.

First available in Project Euclid: 26 October 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

De Finetti-style theorems exchangeability extendibility operational parameters superpopulation models


Iglesias, Pilar L.; Loschi, Rosangela H.; Pereira, Carlos A. B.; Wechsler, Sergio. A note on extendibility and predictivistic inference in finite populations. Braz. J. Probab. Stat. 23 (2009), no. 2, 216--226. doi:10.1214/08-BJPS011.

Export citation


  • Barlow, R. E. and Mendel, M. (1992). De Finetti-type representations for life distributions. Journal of the American Statistical Association 87 1116–1122.
  • Bernardo, J. and Smith, A. F. M. (1994). Bayesian Theory. Wiley, New York.
  • Bolfarine, H. and Zacks, S. (1992). Prediction Theory for Finite Populations. Springer, Berlin.
  • Bolfarine, H., Gasco, L. B. and Iglesias, P. (2003). Inference under representable priors for Pearson type II models in finite populations. Journal of Statistical Planning and Inference 111 23–26.
  • Cochran, W. E. (1977). Sampling Techniques. Wiley, New York.
  • Daboni, L. and Wedlin, A. (1982). Statistica-Un’introduzione all’impostazione neo-bayesiana. Unione Tipografico Editrice Torinese, Torino.
  • de Finetti, B. (1937). La prévision: ses lois logiques, ses sources subjectives. Annales de l’Institut Henry Poincaré (Translated in Kyburg, Jr., H. E.; Smokler, H. E., 1964). Studies in subjective probability 7 1–68.
  • de Finetti, B. (1975). Theory of Probability 1. Wiley, New York.
  • Diaconis, P. and Freedman, D. A. (1990). Cauchy’s equation and de Finetti’s theorem. Scandinavian Journal of Statistics 17 235–250.
  • Diaconis, P. and Ylvisaker, D. (1979). Conjugate priors for exponential families. Annals of Statistics 7 269–281.
  • Diaconis, P., Eaton, M. L. and Lauritzen, S. L. (1992). Finite de Finetti theorems in linear models and multivariate analysis. Scandinavian Journal of Statistics 19 289–316.
  • Ericson, W. A. (1969). A note on the posterior mean of a population mean. Journal of the Royal Statistical Society 31 332–334.
  • Esteves, L. G., Wechsler, S. and Iglesias, P. (2004). Some characterizations of uniform models. Statistics & Probability Letters 69 397–404.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications 1. Wiley, New York.
  • Feller, W. (1991). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
  • Fossaluza, V., Diniz, J. B., Pereira, B. B., Miguel, E. and Pereira, C. A. B. (2009). Sequential allocation and balancing prognostic factors in psychiatric clinical trial. Clinics. To appear.
  • Iglesias, P. L. (1993). Finite forms of de Finetti’s theorem: A predictivistic approach to statistical inference in finite populations (in Portuguese). Ph.D. thesis, Instituto de Matemática e Estatística, Universidade de São Paulo.
  • Iglesias, P. L., Pereira, C. A. B. and Tanaka, N. I. (1998). Characterizations of multivariate spherical distributions in l-norm. Test 7 307–324.
  • Irony, T. Z. and Pereira, C. A. B. (1994). Motivation for the use of discrete distributions in quality assurance. Test 3 181–193.
  • Kadane, J. (1996). Bayesian Methods and Ethics in a Clinical Trial Design. Wiley, New York.
  • Loschi, R. H. and Wechsler, S. (2002). Coherence, Bayes’s theorem and posterior distributions. Brazilian Journal of Probability and Statistics 16 169–185.
  • Mendel, M. (1994). Operational parameters in Bayesian models. Test 3 195–206.
  • Mendel, M. and Kempthorne, P. J. (1996). Operational parameters for the use of Bayesian methods in Engineering. Brazilian Journal of Probability and Statistics 10 1–13.
  • Nagae, C. Y. (2007). Amostragem Intencional (in Portuguese). Master thesis, Instituto de Matemática e Estatística, Universidade de São Paulo.
  • Shohat, J. A. and Tamarkin, J. D. (1943). The problems of moments 1. Amer. Math. Soc., New York.
  • Wechsler, S. (1993). Exchangeability and predictivism. Erkenntnis—International Journal of Analytic Philosophy 38 343–350.
  • Zabell, S. L. (1989). The rule of succession. Erkenntnis—International Journal of Analytic Philosophy 31 283–321.