Abstract
There are several ways of adding one or more parameters to a distribution function. Such an addition of parameters makes the resulting distribution richer and more flexible for modeling data. A positive parameter was added to a general survival function (SF) by Marshall and Olkin (1997).
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References
Ahsanullah, M., Nevzorov, V.B.: Ordered Random Variables. Nova Publishers, New York (2001)
Ahsanullah, M., Raqab, M.R.: Recent Developments in Ordered Random Variables. Nova Publishers, New York (2007)
Ahuja, J.C., Nash, S.W.: The generalized Gompertz-Verhulst family of distributions. Sankhyā 29, 141–161 (1967)
Aicheson, J., Dunsmore, I.R.: Statistical Prediction Analysis. Cambridge University Press, Cambridge, UK (1975)
AL-Hussaini, E.K.: Prediction: advances and new research. Presented as invited topical paper in the international mathematics conference, Cairo, 15–20 Jan 2000. Proceedings of Mathematics and the 21st century. World Scientific, Singapore, pp. 223–245 (2000)
AL-Hussaini, E.K.: Generalized order statistics: prospective and applications. J. Appl. Stat. Sci. 13, 59–85 (2004)
AL-Hussaini, E.K.: A class of compound exponentiated survival functions and multivariate extention. J. Stat. Theory Appl. 10, 63–84 (2011)
AL-Hussaini, E.K., Ghitany, M.: On certain countable mixtures of absolutely continuous distrbutions. Metron LXIII(1), 39–53 (2005)
AL-Hussaini, E.K., Gharib, M.: A new family of distributions as a countable mixture with Poisson added parameters. J. Stat. Theory Appl. 8, 169–190 (2009)
Berger, J.O.: Statistical Decision Theory and Bayesian Analysis, 2nd edn. Springer, New York (1985)
Bernardo, J.M., Smith, A.F.M.: Bayesian Theory. Wiley, New York (1994)
Cowles, M.K., Carlin, B.P.: Markov chain Monte Carlo convergence diagnostics: a comparative review. J. Am. Stat. Assoc. 91, 883–904 (1996)
Cramer, E.: Contributions to Generalized Order Statistics. University of Oldenburg, Oldenburg (2002)
Cramer, E., Kamps, U.: On distributions of generalized order statistics. Statistics 35, 269–280 (2001)
Dalal, S.R., Fowlkes, E.B., Hoadley, B.: Risk analysis of the space shuttle: pre-challenger prediction failure. J. Am. Stat. Assoc. 84, 945–957 (1989)
De Groot, M.H.: Optimal Statistical Decisions. McGraw-Hill, New York (1970)
Dunsmore, I.: The Bayesian predictive distribution in life testing models. Technometrics 16, 455–460 (1974)
Everitt, B.S., Hand, D.J.: Finite Mixture Distributions. Chapman and Hall, London (1981)
Feynman, R.P.: What Do You Care What Other People Think? Further Adventures of a Curious Character. Norton, New York (1988)
Fisher, R.A.: The Mathematical Theory of Probabilities and its Applications to Frequency-Curves and Statistical Methods, vol. 1. McMillan, New York (1936)
Gamerman, D., Lopes, H.F.: Markov chain Monte Carlo: Stochastic Simulation for Bayesian inference, 2nd edn. Chapman & Hall, London (2006)
Geisser, S.: Predictive Inference: An Introduction. Chapman and Hall, London (1993)
Gelman, A., Rubin, D.B.: Inference from iterative simulation using multiple sequences. Stat. Sci. 7, 457–472 (1992)
Gompertz, B.: On the nature of the function of the law of human mortality and on a new model of determining the value of life contingencies. Philos. Trans. Roy. Soc. Lond. 115, 513–585 (1825)
Gupta, R.C., Gupta, R.D.: Proportional reversed hazard rate model and its applications. J. Stat. Plann. Inf. 137, 3525–3536 (2007)
Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)
Jefferys, H.: Theory of Probability. Oxford University Press, London (1961)
Kaminsky, K., Nelson, P.: Prediction of order statistics. In: Balakrishnan, N., Rao, C.R. (eds.) Handbook of Statistics, vol. 1, pp. 431–450. Elsevier Sciences, Amsterdam, The Netherland (1998)
Kamps, U.: A Concept of Generalized Order Statistics. Teubner, Stuttgart (1995a)
Kamps, U.: A concept of generalized order statistics. J. Stat. Plann. Inf. 48, 1–23 (1995b)
Lehmann, E.L.: The power of rank test. Ann. Math. Stat. 24, 28–43 (1953)
Lindley, D.V.: Introduction to Probability and Statistics from a Bayes viewpoint (two parts). Cambridge University Press, Cambridge (1965)
Lindley, D.V.: Approximate Bayes method. Trabajos de Estadistica 24, 28–43 (1980)
Lingappaiah, G.S.: Bayes approach to prediction and the spacings in the exponential population. Ann. Inst. Stat. Math. 31, 391–401 (1979)
Maritz, J.S., Lwin, T.: Empirical Bayes Methods. Chapman & Hall, London (1989)
Marshall, A.W., Olkin, I.: A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families. Biometrika 84(3), 641–652 (1997)
Martz, H.F., Waller, R.: Bayesian Reliability Analysis. Wiley, New York (1982)
McLachlan, G.J., Basford, K.E.: Mixture Models: Applications to Clustering. Marcel Dekker, New York (1988)
McLachlan, G.J., Peel, D.: Finite Mixture Models. Wiley, New York (2000)
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Taller, A., Teller, H.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1091 (1953)
Nagaraja, H.: Prediction problems. In: Balakrishnan, N., Basu, A.P. (eds.) The Exponential Distribution: Theory and Applications, pp. 139–163. Gordon and Breach, New York (1995)
Newcomb, S.: A generalized theory of the combination of observations so as to obtain the best result. Am. J. Math. 8, 343–366 (1886)
Patel, J.K.: Prediction intervals: a review. Commun. Stat. Theory Meth. 18, 2393–2465 (1989)
Pearson, K.: Contributions to the mathematical theory of evolution. Phil. Trans. A 185, 71–110 (1894)
Press, S.J.: Subjective and Objective Bayesian Statistics. Wiley, New York (2003)
Roberts, G.O., Gelman, A., Gilks, W.R.: Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Prob. 7, 110–120 (1997)
Savage, V.P.: The Foundation of Statistics. Wiley, New York (1954)
Savchuk, V.P., Tsokos, C.P.: Bayesian Statistical Methods with Applications to Reliability. Atlantis Press, Paris (2013)
Sinha, D., Maiti, T., Ibrahim, J., Ouyang, B.: Current methods for recurrent events data with dependent termination: A Bayesian perspective. J. Am. Stat. Assoc. 103, 866–878 (2008)
Teicher, H.: On the mixture of distributions. Ann. Math. Stat. 31, 55–73 (1960)
Tierney, L.: Markov chains for exploring posterior distributions (with discuss-ion). Ann. Stat. 22, 1701–1762 (1994)
Tierney, L., Kadane, J.B.: Accurate approximations for posterior moments and marginal densities. J. Am. Stat. Assoc. 81, 82–86 (1986)
Thompson, R.D., Basu, A.P.: Asymmetric loss function for estimating reliability. In: Berry, D.A., Chalenor, K.M., Geweke, J.K. (eds.) Bayesian Analysis in Statistics and Econometrics. Wiley, New York (1996)
Titterington, D.M., Smith, A.F.M., Makov, U.E.: Statistical Analysis of Finite Mixture Distributions. Wiley, New York (1985)
Varian, H.: A Bayesian approach to real estate assessment. In: Fienberg, S.E., Zellner, A. (eds.) Studies in Bayesian Econometrics and Statistics. Honor of Leonard J. Savage, North Holland, Amsterdam (1975)
Verhulst, P.F.: Notice sur la loi que la population pursuit dans son accroissement. Correspondance Mathématique et physique, publiée par L.A.L. Quet-elet 10, 113–121 (1838)
Verhulst, P.F.: Recherches mathématiques sur la loi d’accroissement de la population. Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Bruxelles 18, 1–42 (1845) (i.e. Mémoires, Series 2)
Verhulst, P.F.: Deuxiéme mémoire sur la loi d’accroissement de la population. Mémoires de l’Académie Royale des Sciences, des Lettres et de Beaux-Arts de Belgique, Series 2, 20, pp. 1–32 (1847)
Zellner, A.: Bayesian and non-Bayesian estimation using balanced loss functions. In: Gupta, S.S., Burger, J.O. (eds.) Statistical Decision Theory and Related Topics, pp. 371–390. Springer, New York (1994)
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AL-Hussaini, E.K., Ahsanullah, M. (2015). Class of Exponentiated Distributions Introduction. In: Exponentiated Distributions. Atlantis Studies in Probability and Statistics, vol 5. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-079-9_1
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