Statistical Papers

, Volume 46, Issue 3, pp 397–409 | Cite as

Divergence-based estimation and testing with misclassified data

  • E. Landaburu
  • D. Morales
  • L. Pardo


The well-known chi-squared goodness-of-fit test for a multinomial distribution is generally biased when the observations are subject to misclassification. In Pardo and Zografos (2000) the problem was considered using a double sampling scheme and ø-divergence test statistics. A new problem appears if the null hypothesis is not simple because it is necessary to give estimators for the unknown parameters. In this paper the minimum ø-divergence estimators are considered and some of their properties are established. The proposed ø-divergence test statistics are obtained by calculating ø-divergences between probability density functions and by replacing parameters by their minimum ø-divergence estimators in the derived expressions. Asymptotic distributions of the new test statistics are also obtained. The testing procedure is illustrated with an example.

Key words and phrases

Misclassification Double sampling Divergence estimators Goodness-of-fit tests Divergence statistics 

AMS Classification

62F05 62B10 


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  1. [1]
    Ali, S.M. and Silvey, S.D. (1966). A general class of coefficient of divergence of one distribution from another.J. of Royal Statistical Society, Series B 286, 131–142.MathSciNetGoogle Scholar
  2. [2]
    Birch, M. H. (1964). A new proof of the Pearson-Fisher theorem.Annals of Mathematical Statistics,35, 817–824.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Cheng, K. F., Hsueh, H. M. and Chien, T.H. (1998). Goodness of fit tests with misclassified data.Communications in Statistics—Theory and Methods,27, 1379–1393.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Csiszár, I. (1963). Eine Informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten.Publications of the Mathematical Institute of Hungarian Academy of Sciences,8,Ser. A, 85–108.zbMATHGoogle Scholar
  5. [5]
    Cressie, N. and Read, T.R.C. (1984). Multinomial Goodness-of-fit Tests.J. of the Royal Statistical Society, Series B,46, 440–464.zbMATHMathSciNetGoogle Scholar
  6. [6]
    Dik, J. J. and Gunst, M. C. M. (1985): The distribution of general quadratic forms in normal variables.Statistica Neerlandica,39, 14–26.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    Eckler, A. R. (1969): A survey of coverage problems associated with point and area targets.Tecnometrics 11, 561–589.zbMATHCrossRefGoogle Scholar
  8. [8]
    Gupta, S. S. (1963): Bibliography on the multivariate normal integrals and related topics.Annals of Mathematical Statistics 34, 829–838.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Imhof, J. P. (1961): Computing the distribution of quadratic forms in normal variables.Biometrika,48, 419–426.zbMATHMathSciNetGoogle Scholar
  10. [10]
    Jensen, D. R. and Solomon, H. (1972): A Gaussian approximation to the distribution of a definite quadratic form.J. of the American Statistical Association,67, 898–902.CrossRefGoogle Scholar
  11. [11]
    Johnson, N. L. and Kotz, S. (1968): Tables of distributions of positive definite quadratic forms in central normal variables.Sankhya,30, 303–314.MathSciNetGoogle Scholar
  12. [12]
    Morales, D., Pardo, L. and Vajda, I. (1995): Asymptotic divergence of estimates of discrete distributions.J. of Statistical Planning and Inference,48, 347–369.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Pardo, L. and Zografos K. (2000): Goodness of fit tests with misclassified data based on ø-divergences.Biometrical Journal,42, 223–237.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Rao, J. N. K. and Scott, A. J. (1981): The Analysis of categorical data from complex sample surveys: Chi-squared tests for goodness-of-fit and independence in two-way tables.J. of the American Statistical Association,76, 221–230.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Read, R. C. and Cressie, N. A. C. (1988).Goodness-of-fit Statistics for Discrete Multivariate Data, Springer Verlag, New York.zbMATHGoogle Scholar
  16. [16]
    Satterhwaite, F. E. (1946): An approximation distribution of estimates of variance components.Biometrics,2, 110–114.CrossRefGoogle Scholar
  17. [17]
    Solomon, H. (1960): Distribution of quadratic forms-tables and applications, Technical Report 45, Applied Mathematics and statistics laboratories, Stanford University, Stanford, CA.Google Scholar
  18. [18]
    Tenenbein, A. (1970): A double sampling scheme for estimating from binomial data with misclassification.J. of the American Statistical Association,65, 1350–1361.CrossRefGoogle Scholar
  19. [19]
    Tenenbein, A. (1971). A double sampling scheme for estimating from binomial data with misclassification: Sample size and determination.Biometrics,27, 935–944.CrossRefGoogle Scholar
  20. [20]
    Tenenbein, A. (1972). A double sampling scheme for estimating from misclassified multinomial data with applications to sampling inspection.Technometrics,14, 187–202.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  • E. Landaburu
    • 1
  • D. Morales
    • 2
  • L. Pardo
    • 1
  1. 1.Department of Statistics & O. R.Complutense University of MadridMadrid
  2. 2.Operations Research CenterMiguel Hernández University of ElcheElche

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