Choice of Suitable Informative Prior for the Scale Parameter of Mixture of Laplace Distribution using Type-I Censoring Scheme under Different Loss Function


A comprehensive simulation scheme including a large number of parameter points is followed to highlight the properties and performance of the Bayes estimates and their posterior risk in terms of sample size, censoring rate and proportion of the component of the mixture using Levy and Gumbel Type-II informative priors. Limiting expressions for complete sample are also derived. The system of three non-linear equations, required to be solved iteratively for the computations of maximum likelihood (ML) estimates and predictive intervals are derived. A real-life mixture data application has been discussed. The Elicitation of hyperparameters of mixture through prior predictive approach has also argued. The Bayes estimates are evaluated under square error loss function, precautionary loss function, weighted squared error loss function and modified (quadratic) squared error loss function.

DOI Code: 10.1285/i20705948v6n1p32

Keywords: Information matrix; Censored sampling; Inverse transformation method; Squared error loss function; Precautionary loss function; Weighted squared error loss function; Modified (quadratic) squared error loss function; Hyperparameters; Elicitation; Fixed tes


Ali, M. M. and Nadarajah, S. (2007) Information matrices for normal and Laplace mixtures. Information Sciences, Vol. 177, PP. 947–955.

Amin, T. and Guan, L., (2004), Interactive content-based image retrieval using Laplacian mixture model in wavelet domain. In: Proc. IEEE Internat. Symp. On Circuits and Systems (ISCAS), Vancouver, Canada. Vol. 2, N (23-26 May 2004), PP. II - 45-8. Digital Object Identifier.

Aryal, G. and Rao, V. N. A., (2005), Reliability model using truncated skew-Laplace distribution, Nonlinear Analysis, Vol. 63, N (5-7), PP. e639-e646.

Aslam, M. (2003). An application of prior predictive distribution to elicit the prior density. Journal of Statistical Theory and Applications. Vol. 2, N (1),PP. 70-83.

Ayyub, B. M., (2001), Elicitation of Expert Opinions for Uncertainty and Risks, CRC Press, ISBN 0-8493-1087-3.

Balakrishnan, N. and Chandramouleeswaran, P. M., (1996), Reliability estimation and tolerance limits for Laplace distribution based on censored samples. Microelectron Reliab. Vol. 36, N (3), PP. 375-378.

Bansal, A. K., (2007), Bayesian Parametric Inference, Narosa Publishing House Pvt. Ltd., New Delhi.

Berger O. James “Statistical Decision Theory and Bayesian Analysis”, 2nd edition, Springer Series in Statistics, ISBN-10: 0-387-96098-8 and -13: 978-0387-96098-2.

Bishop, M. C., and Svensén, M. (2004), Robust Bayesian Mixture Modelling. Proceedings of ESANN 2004 and also available at{cmbishop,markussv}

Bhowmick, D., Davison, A. C., Goldstein, R.D., Ruffieux, y. (2006), A Laplace mixture model for identification of differential expression in microarray experiments. Journal of Biostatistics, Vol. 4, N (4), PP.630-641.

Burmaster, D.E. and K.M. Thompson (1998), Fitted second-order parametric distributions to data using maximum likelihood estimation. Human and Ecological Risk Assessment, Vol. 3, No. 2, PP. 235-255.

Childs, A. and Balakrishnan, N, (2000), Conditional inference procedures for the Laplace distribution when the observed samples are progressively censored. Metrika, Vol. 52, N (3), PP. 253-265.

Childs, A. and Balakrishnan, N. (1997), Maximum likelihood estimation of Laplace parameters based on general Type-II censored samples. Statistical Papers, Vol. 38, N (3), PP. 343-349.

Childs, A. and Balakrishnan, N., (1996), Conditional inference procedures for the Laplace distribution based on Type-II right censored samples. Statistics & Probability Letters, Vol. 31, N (1), PP. 31-39.

Choi, D. and Nadarajah, S. (2009) Information matrix for a mixture of two Laplace distributions. Stat Papers, Vol. 50, PP.1–12, DOI 10.1007/s00362-007-0053-8

Chuan-Chong, C. and Mhee-Meng, K. (1992). Principles and Techniques in Combinatorics, World Scientific Publishing Co. Pvt. Ltd. Singapore.

Dey, K. D., (2007), Prior Elicitation from Expert Opinion, (Lecture Notes), University of Connecticut and Some parts joint with: Junfeng Liu Case Western Reserve University.

Gaioni, E., Dey, K. D., and Grigoriu, M. (2008), Semiparametric Functional Estimation Using

Quantile-Based Prior Elicitation, samsi Technical Report #2008-6, June 23, 2008, Statistical and Applied Mathematical Sciences Institute PO Box 14006 ,Research Triangle Park, NC 27709-4006,

Gajewski, J. B., Simon D. S. and Carlson, E. S., (2007), Predicting accrual in clinical trials with Bayesian posterior predictive distributions. Statist. Med. Published online in Wiley InterScience ( DOI: 10.1002/sim.3128

Gijbels, I. (2010), Censored Data, WIREs Comp Stat, Vol. 2, March/April 2010, PP. 178-188.

Gauss, C. F. (1810), Method des Moindres Carres Memoire sur la Combination des Observations. Translated by J. Bertrand (1955). Mallet-Bachelier, Paris.

Harris, C.M., (1983), On finite mixtures of geometric and negative binomial distributions. Commun, Statist.-Ther. Meth., Vol.12, PP. 987-1007.

Inusah, S. and Kozubowski, J. T., (2006) A discrete analogue of the Laplace distribution. Journal of Statistical Planning and Inference 136 (2006) 1090 – 1102.

Jeffreys, H. (1961). Theory of Probability (3rd edt.) Oxford University Press.

Jenkinson, D. (2005), The Elicitation of Probabilities: A Review of the Statistical Literature. Department of Probability and Statistics, University of Sheffield.

Jones, P. N. and McLachlan, J. G., (1990), Laplace-Normal mixtures fitted to wind shear data. Journal of Applied Statistics, Vol. 17, No. 2, PP. 271-276.

Kappenman, Russell F. (1975), Conditional confidence intervals for Double Exponential distribution parameters. Technometrics, Vol. 17, N (2), PP. 233-235.

Kappenman, Russell F. (1977), Tolerance intervals for the Double Exponential distribution. Journal of the American Statistical Association (JASA), Vol. 72, N (360), PP. 908-909.

Kanji, K. G. (1985), A mixture model for wind shear data. Journal of Applied Statistics, Vol. 12, No. 1, PP. 49-58.

Kozubowski, J. T. and Nadarajah, S., (2010), Multitude of Laplace distributions. Stat Papers, Vol.51:127–148,DOI 10.1007/s00362-008-0127-2

Legendre, A. (1805), Nouvelles Methodes Pour la Determination des Orbites des Cometes, Courcier, Paris.

Leόn, J. C., Vázquez -Polo ,J. F. and González, L.R.,(2003), Environmental and Resource Economics Vol. 26 PP. 199–210.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

Mendenhall, W. and Hader, R. A. (1958), Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data, Biometrika, Vol. 45, PP. 1207-1212.

Mitianoudis, N. and Stathaki, T., (2005), Overcomplete source separation using Laplacian mixture models. IEEE Signal Process. Letters, Vol. 12, N (4), PP. 277-280. Digital Object Identifier 10.1109/LSP.2005.843759.

Nadarajah, S., (2009) Laplace random variables with application to price indices. AStA Adv Stat Anal (2009) 93: 345–369 DOI 10.1007/s10182-009-0108-3

Nadarajah, S. (2004), Reliability for Laplace Distribution. Mathematical Problems in Engineering, Vol. 2, PP. 169–183.

Norstrom, J. G. (1996), The use of precautionary loss functions in risk analysis, IEEE Trans. Reliab. Vol. 45, N (1), PP. 400-403.

Oakley, J., and O'Hagan, A. (2005). Uncertainty in prior elicitations: a non-parametric approach. Revised version of research report No. 521/02 Department of Probability and Statistics, University of Sheffield.

O’ Hagan, A., Buck, E. C., Daneshkhah, A., Eiser, J. E., Garthwaite, H.P., Jenkinson, J. D., Jeremy E. Oakley, E. J., and Rakow, T. (2006), Uncertain Judgments: Eliciting Experts’ Probabilities. John Wiley & Sons, Ltd. ISBN-13: 978-0-470-02999-2 (HB),ISBN-10: 0-470-02999-4 (HB)

Rabbani, H. and Vafadoost, M. (2006), Wavelet based Image denoising based on a mixture of Laplace distribution. Iranian Journal of Science & Technology, Transaction B, Engineering, Vol. 30, No. B6, PP 711-733

Rabbani, H., and Vafadust, M. (2008), Image/ video denoising based on a mixture of Laplace distributions with local parameters in multidimensional complex wavelet domain. Signal Processing, Vol. 88, N (1), PP. 158-173.

Razali, M. A., and Salih, A. A., (2009), Combining Two Weibull Distributions Using a Mixing Parameter. European Journal of Scientific Research ISSN 1450-216X Vol.31 No.2 (2009), pp.296-305

Romeu, L. J. (2004), Censored Data. START, Vol. 11, N (3), PP. 1-8.

Sabarinath, A. and Anilkumar, A. K., (2008), Modeling of Sunspot numbers by a modified binary mixture of Laplace distribution functions. Solar Phys, 250, N (1), PP. 183-197.

Saleem, M. and Aslam, M. (2008), Bayesian Analysis of the Two Component Mixture of the Rayleigh Distribution assuming the Uniform and the Jeffreys Priors. Journal of Applied Statistical Science, ISSN 1067-5817 ,Vol. 16, No. 4, pp. 105-113

Saleem, M. and Aslam, M. and Economus, P. (2010), On the Bayesian analysis of the mixture of the power distribution using the complete and censored sample. Journal of Applied Statistics, Vol. 37, No. 1, pp. 25-40.

Scallan, A. J. (1992), Maximum likelihood estimation for a Normal/Laplace mixture distribution. The Statistician, Vol. 41, PP. 227-231.

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