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Bayesian Analysis in the L 1-Norm of the Mixing Proportion Using Discriminant Analysis

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Abstract

We consider the mixing proportion π in a mixture of two independent distributions, and establish the expression of its posterior density, in closed form and in terms of L 1-norms of various related functions, using a prior beta and the optimal classification rule for the two populations provided by Discriminant analysis. A numerical example fully illustrates the concepts presented.

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References

  • Bolotin VV (1969) Statistical methods in structural mechanics. Holden-Day, San Francisco

    MATH  Google Scholar 

  • Clemons TE, Bradley EL Jr (2000) A non-parametric measure of the overlapping coefficient. Comput Stat Data Anal 34:51–61

    Article  Google Scholar 

  • Devroye L, Gyorfi L (1985) Non-parametric density estimation, the L1 View. Wiley, New York

    Google Scholar 

  • Exton H (1976) Multiple hypergeometric functions and applications. Chichester-Ellis Howard, London

    MATH  Google Scholar 

  • Everitt BS (1985) Mixture distributions. In: Johnson N, Kotz S (eds) Encyclopedia of statistical sciences, vol 5, pp 559–569

  • Gastwirth JL (1975) Statistical measures of earnings differentials. Am Stat 29:32–35

    Article  Google Scholar 

  • Inman HF, Bradley EL (1989) The overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities. Commun Stat Theory Methods 18:3851–3874

    Article  MATH  MathSciNet  Google Scholar 

  • James IR (1978) Estimation of the mixing proportion in a mixture of two normal distributions from simple, rapid measurements. Biometrics 34:265–278

    Article  PubMed  MATH  Google Scholar 

  • Johnson N, Kotz S, Balakhrishnan N (1995) Continuous univariate distributions, vol 2. Wiley, New York

    MATH  Google Scholar 

  • Krishnan T (2001) Imperfect supervision in statistical pattern recognition. In: Pal SK, Pal A (eds) Pattern Recognition, from Classical to Modern Approaches. World Scientific, Singapore, pp 25–65

    Google Scholar 

  • McLachlan GJ, Basford K (1988) Mixture Models. Marcel Dekker, New York

    MATH  Google Scholar 

  • Mulekar MS, Mishra SN (1994) Overlap coefficients of two normal densities: equal means case. J Jpn Stat Soc 34:169–180

    MathSciNet  Google Scholar 

  • Pham-Gia T (2004) Bayes Inference. In: Gupta AK, Nadarajah S (eds) Handbook of the beta distribution. Marcel Dekker, New York, pp 361–422

    Google Scholar 

  • Pham-Gia T, Tran Loc H (2001) The mean and median absolute deviations. Math Comput Model 34:921–936

    Article  MATH  Google Scholar 

  • Pham-Gia T, Turkkan N (1992a) Bayes binomial sampling by attributes with a general beta prior. IEEE Trans Reliab 4:310–316

    Article  Google Scholar 

  • Pham-Gia T, Turkkan N (1992b) Sample size determination in bayesian analysis. The Statistician 41(4):389–397

    Article  Google Scholar 

  • Pham-Gia T, Turkkan N, Duong QP (1992c) Using the mean deviation in the determination of the prior distribution. Stat Prob Lett 13:373–381

    Article  MATH  MathSciNet  Google Scholar 

  • Rahme E, Joseph L, Gyorkos TW (2000) Bayesian sample size determination for estimating binomial parameters from data subject to misclassification. Appl Stat 49:119–128

    MATH  MathSciNet  Google Scholar 

  • Rao CR (1988) Methodology Based on the L1-norm in Statistical Inference. Sankhya 50:289–313

    MATH  Google Scholar 

  • Reiser B, Faraggi D (1999) Confidence Intervals for the overlapping coefficient: the normal equal variance case. Statistician 48:413–418

    Google Scholar 

  • Sneath PHA (1977) A method for testing the distinctness of clusters: a test of the disjunction of two clusters in euclidean space, as measured by their overlap. Math Geol 9:123–143

    Article  Google Scholar 

  • Styne RA, Heyse JF (2001) Non-parametric estimates of overlap. Stat Med 20:215–236

    Article  PubMed  Google Scholar 

  • Titterington DM (1997) Mixture distributions. In: Johnson N, Kotz S (eds) Encyclopedia of statistical sciences, vol 1 , pp 399–407

  • Titterington DM, Smith AFM, Makov UE (1985) Statistical analysis of mixture distributions. Wiley, Chichester, London

    MATH  Google Scholar 

  • Turkkan N, Pham-Gia T (1993) Computation of the highest posterior density interval in bayesian analysis. J Stat Comput Simul 44:243–250

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Université de Moncton, Moncton, EIA 3E9, Canada

    T. Pham-Gia & N. Turkkan

  2. Department of Statistics, University of South Africa (UNISA), Medunsa, South Africa

    A. Bekker

Authors
  1. T. Pham-Gia
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  2. N. Turkkan
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  3. A. Bekker
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Corresponding author

Correspondence to T. Pham-Gia.

Additional information

Research partially supported by CRSNG 9249 (Canada). The authors wish to thank the Faculty of Science and the Department of Statistics of UNISA for their generous support that has led to this joint work. Also, thanks to Ms. Jeannette LeBlanc for her excellent technical support, and to an anonymous referee for very helpful comments that have helped to improve the presentation of the paper.

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Pham-Gia, T., Turkkan, N. & Bekker, A. Bayesian Analysis in the L 1-Norm of the Mixing Proportion Using Discriminant Analysis. Metrika 64, 1–22 (2006). https://doi.org/10.1007/s00184-006-0027-1

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  • DOI: https://doi.org/10.1007/s00184-006-0027-1

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