Bayesian analysis in the \(L^{1}\)-norm of the mixing proportion using discriminant analysis.(English)Zbl 1099.62026
Summary: We consider the mixing proportion \(\pi\) 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.
MSC:
| 62F15 | Bayesian inference |
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
Keywords:
Overlapping coefficient;Discriminant analysis;Misclassification;Hypergeometric functions;Highest posterior density;Beta distributionCite
Full Text:DOI
References:
| [1] | Bolotin VV (1969) Statistical methods in structural mechanics. Holden-Day, San Francisco ·Zbl 0194.57001 |
| [2] | Clemons TE, Bradley EL Jr (2000) A non-parametric measure of the overlapping coefficient. Comput Stat Data Anal 34:51–61 ·Zbl 1052.62514 ·doi:10.1016/S0167-9473(99)00074-2 |
| [3] | Devroye L, Gyorfi L (1985) Non-parametric density estimation, the L1 View. Wiley, New York |
| [4] | Exton H (1976) Multiple hypergeometric functions and applications. Chichester-Ellis Howard, London ·Zbl 0337.33001 |
| [5] | Everitt BS (1985) Mixture distributions. In: Johnson N, Kotz S (eds) Encyclopedia of statistical sciences, vol 5, pp 559–569 |
| [6] | Gastwirth JL (1975) Statistical measures of earnings differentials. Am Stat 29:32–35 ·doi:10.2307/2683677 |
| [7] | 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 ·Zbl 0696.62131 ·doi:10.1080/03610928908830127 |
| [8] | James IR (1978) Estimation of the mixing proportion in a mixture of two normal distributions from simple, rapid measurements. Biometrics 34:265–278 ·Zbl 0384.62027 ·doi:10.2307/2530016 |
| [9] | Johnson N, Kotz S, Balakhrishnan N (1995) Continuous univariate distributions, vol 2. Wiley, New York |
| [10] | 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 |
| [11] | McLachlan GJ, Basford K (1988) Mixture Models. Marcel Dekker, New York ·Zbl 0697.62050 |
| [12] | Mulekar MS, Mishra SN (1994) Overlap coefficients of two normal densities: equal means case. J Jpn Stat Soc 34:169–180 ·Zbl 0818.62100 |
| [13] | Pham-Gia T (2004) Bayes Inference. In: Gupta AK, Nadarajah S (eds) Handbook of the beta distribution. Marcel Dekker, New York, pp 361–422 |
| [14] | Pham-Gia T, Tran Loc H (2001) The mean and median absolute deviations. Math Comput Model 34:921–936 ·Zbl 0996.60008 ·doi:10.1016/S0895-7177(01)00109-1 |
| [15] | Pham-Gia T, Turkkan N (1992a) Bayes binomial sampling by attributes with a general beta prior. IEEE Trans Reliab 4:310–316 ·Zbl 0850.62751 |
| [16] | Pham-Gia T, Turkkan N (1992b) Sample size determination in bayesian analysis. The Statistician 41(4):389–397 ·Zbl 0756.90024 ·doi:10.2307/2349003 |
| [17] | 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 ·Zbl 0743.62023 ·doi:10.1016/0167-7152(92)90110-Q |
| [18] | 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 ·Zbl 0944.62032 |
| [19] | Rao CR (1988) Methodology Based on the L1-norm in Statistical Inference. Sankhya 50:289–313 ·Zbl 0677.62058 |
| [20] | Reiser B, Faraggi D (1999) Confidence Intervals for the overlapping coefficient: the normal equal variance case. Statistician 48:413–418 |
| [21] | 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 ·doi:10.1007/BF02312508 |
| [22] | Styne RA, Heyse JF (2001) Non-parametric estimates of overlap. Stat Med 20:215–236 ·doi:10.1002/1097-0258(20010130)20:2<215::AID-SIM642>3.0.CO;2-X |
| [23] | Titterington DM (1997) Mixture distributions. In: Johnson N, Kotz S (eds) Encyclopedia of statistical sciences, vol 1 , pp 399–407 ·Zbl 1007.62505 |
| [24] | Titterington DM, Smith AFM, Makov UE (1985) Statistical analysis of mixture distributions. Wiley, Chichester, London |
| [25] | Turkkan N, Pham-Gia T (1993) Computation of the highest posterior density interval in bayesian analysis. J Stat Comput Simul 44:243–250 ·Zbl 0784.62024 ·doi:10.1080/00949659308811461 |
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