International Statistical Review

A Measure of Representativeness of a Sample for Inferential Purposes

Salvatore Bertino

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Abstract

After defining the concept of representativeness of a random sample, the author proposes a measure of how much the observed sample represents its parent distribution. This measure is called Representativeness Index. The same measure, seen as a function of a sample and of a distribution, will be called Representativeness Function. For a given sample it provides the value of the index for the different distributions under examination, and for a given distribution it provides a measure of the representativeness of its possible samples. Such Representativeness Function can be used in an inferential framework just as the likelihood function, since it gives to any distribution the ''experimental support'' provided by the observed sample. This measure is distribution-free and it is shown to be a transformation of the well-known Cramér-von Mises statistic. By using the properties of that statistic, criteria for providing set estimators and tests of hypotheses are introduced. The utilization of the representativeness function in many standard statistical problems is outlined through examples. The quality of the inferential decisions can be assessed with the usual techniques (MSE, power function, coverage probabilities). The most interesting examples turn out to be those of situations that are ''non-regular'', as for instance the estimation of parameters involved in the support of the parent distribution, or less explored (model choice).

Article information

Source
Internat. Statist. Rev., Volume 74, Number 2 (2006), 149-159.

Dates
First available in Project Euclid: 24 July 2006

Permanent link to this document
https://projecteuclid.org/euclid.isr/1153748789

Keywords
Representativeness function Parameter estimation Test of hypotheses Prediction distribution Cramér-von Mises statistic Model choice

Citation

Bertino, Salvatore. A Measure of Representativeness of a Sample for Inferential Purposes. Internat. Statist. Rev. 74 (2006), no. 2, 149--159. https://projecteuclid.org/euclid.isr/1153748789


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References

  • [1] Anderson, T.W. & Darling, D.A. (1952). Asymptotic theory of certain ''goodness of fit'' criteria based on stochastic processes. Ann. Math. Stat., 23, 193-212.
  • [2] Bertino, S. (1971). Gli indici di dissomiglianza e la stima dei parametri. In Studi di probabilità, statistica e ricerca operativa in onore di Giuseppe Pompilj. Pubblicazioni dell'Istituto di Calcolo delle Probabilitàdell'Universitàdi Roma, 1971, pp. 187-202.
  • [3] Bertino, S. (1998). Un nuovo strumento per le decisioni statistiche: la funzione di rappresentatività. Atti della XXXIX Riunione scientifica della SIS, vol. II, Sorrento, 311-318.
  • [4] Csörg\H{o}, S. & Faraway, J.J. (1996). The Exact and Asymptotic Distributions of Cramér-von Mises Statistics. J. R. Statist. Soc. B, 58, 221-234.
  • [5] Dall'Aglio, G. (1956). Sugli estremi dei momenti delle funzioni di ripartizione doppia. Annali della Scuola Normale di Pisa, ser. III, vol. X, I-II, 35-74.
  • [6] Fréchet, M. (1951). Sur les tableaux de corrélation dont les marges sont données. \textsl {Annales de l'Universitéde Lyon. Section A, Series 3}, 14, 53-77.
  • [7] Gini, C. (1914). Di una misura della dissomiglianza tra due gruppi di quantitàe delle sue applicazioni allo studio delle relazioni statistiche. Atti del Regio Istituto Veneto di Scienze, Lettere ed Arti, vol. LXXIV, parte II, 185-213.
  • [8] Landenna, G. (1957). Osservazioni sulla connessione. Statistica, vol. XVII, 4, 351-391.
  • [9] Marshall, A.W. (1958). The small sample distribution of nω^2. Ann. Math. Stat., 29, 307-309.