International Statistical Review

Mixed Poisson Distributions

Dimitris Karlis and Evdokia Xekalaki

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Mixed Poisson distributions have been used in a wide range of scientific fields for modeling non-homogeneous populations. This paper aims at reviewing the existing literature on Poisson mixtures by bringing together a great number of properties, while, at the same time, providing tangential information on general mixtures. A selective presentation of some of the most prominent members of the family of Poisson mixtures is made.

Article information

Source
Internat. Statist. Rev., Volume 73, Number 1 (2005), 35-58.

Dates
First available in Project Euclid: 31 March 2005

Permanent link to this document
https://projecteuclid.org/euclid.isr/1112304811

Mathematical Reviews number (MathSciNet)
MR2142681

Zentralblatt MATH identifier
1104.62010

Keywords
Mixtures Discrete distributions Distribution theory Mixing distribution Overdispersion

Citation

Karlis, Dimitris; Xekalaki, Evdokia. Mixed Poisson Distributions. Internat. Statist. Rev. 73 (2005), no. 1, 35--58. https://projecteuclid.org/euclid.isr/1112304811


Export citation

References

  • [1] Adell, J. & de la Cal, J. (1993). On the Uniform Convergence of Normalised Poisson Mixtures to their Mixing Distribution. Statistics and Probability Letters, 18, 227-232.
  • [2] Aitchinson, J. & Ho, C.H. (1989). The Multivariate Poisson-Log Normal Distribution. Biometrika, 75, 621-629.
  • [3] Aitkin, M. (1996). A General Maximum Likelihood Analysis of Overdispersion in Generalized Linear Models. Statistics and Computing, 6, 251-262.
  • [4] Albrecht, P. (1982). On Some Statistical Methods Connected with the Mixed Poisson Process. Scandinavian Actuarial Journal, 9, 1-14.
  • [5] Albrecht, P. (1984). Laplace Transforms, Mellin Transforms and Mixed Poisson Processes. Scandinavian Actuarial Journal, 11, 58-64.
  • [6] Al-Awadhi, S.A. & Ghitany, M.E. (2001). Statistical Properties of Poisson-Lomax Distribution and its Application to Repeated Accidents Data. Journal of Applied Statistical Science, 10, 365-372.
  • [7] Al-Hussaini, E.K. & El-Dab, A.K. (1981). On the Identifiability of Finite Mixtures of Distributions. IEEE Transactions on Information Theory, 27, 664-668.
  • [8] Al-Zaid, A.A. (1989). On the Unimodality of Mixtures. Pakistan Journal of Statistics, 5, 205-209.
  • [9] Arbous, A.G. & Kerrich, J.E. (1951). Accident Statistics and the Concept of Accident-Proneness. Biometrics, 7, 340-432.
  • [10] Ashford J.R. & Hunt R.G. (1974). The Distribution of Doctor-Patient Contacts in the National Health Service. Journal of the Royal Statistical Society A, 137, 347-383.
  • [11] Barndorff-Nielsen, O.E. (1965). Identifiability of Mixtures of Exponential Families. Journal of Mathematical Analysis and Applications, 12, 115-121.
  • [12] Barndorff-Nielsen, O.E., Blaesild, P. & Seshardi, V. (1992). Multivariate Distributions with Generalized Inverse Gaussian Marginals and Associated Poisson Mixtures. Canadian Journal of Statistics, 20, 109-120.
  • [13] Bates, G. & Neyman, J. (1952a). Contributions to the Theory of Accident Proneness Part II: True Or False Contagion? University of California Publications in Statistics, pp. 255-275.
  • [14] Bates, G. & Neyman, J. (1952b). Contributions to the Theory of Accident Proneness Part I: An Optimistic Model of the Correlation Between Light and Severe Accidents. University of California Publications in Statistics, pp. 215-253.
  • [15] Bertin, E. & Theodorescu, R. (1995). Preserving Unimodality by Mixing. Statistics and Probability Letters, 25, 281-288.
  • [16] Best, A. & Gipps, B. (1974). An Improved Gamma Approximation to the Negative Binomial. Technometrics, 16, 621-624.
  • [17] Bhattacharya, S.K. (1966). Confluent Hypergeometric Distributions of Discrete and Continuous Type with Application to Accident Proneness. Bulletin of the Calcutta Statistical Association, 15, 20-31.
  • [18] Bhattacharya, S.K. (1967). A Result in Accident Proneness. Biometrika, 54, 324-325.
  • [19] Böhning, D. (1999). Computer Assisted Analysis of Mixtures (C.A.M.AN). New York: Marcel Dekker Inc.
  • [20] Buchmann, B. & Grubel, R. (2003). Decompounding: an Estimation Problem for Poisson Random Sums. Annals of Statistics, 31, 1054-1074.
  • [21] Bulmer, M.G. (1974). On Fitting the Poisson Lognormal Distribution to Species Abundance Data. Biometrics, 30, 101-110.
  • [22] Burrel, Q. & Cane, V. (1982). The Analysis of Library Data. Journal of the Royal Statistical Society A, 145, 439-471.
  • [23] Cane, V. (1977). A Class of Non-Identifiable Stochastic Models. Journal of Applied Probability, 14, 475-482.
  • [24] Carriere, J. (1993). Nonparametric Tests for Mixed Poisson Distributions. Insurance Mathematics and Economics, 12, 3-8.
  • [25] Cassie, M. (1964). Frequency Distributions Models in the Ecology of Plankton and Other Organisms. Journal of Animal Ecology, 31, 65-92.
  • [26] Chadjiconstantinidis, S & Antzoulakos, D.L. (2002). Moments of Compound Mixed Poisson Distributions. Scandinavian Actuarial Journal, (3) 138-161.
  • [27] Chen, J. (1995). Optimal Rate of Convergence for Finite Mixture Models. Annals of Statistics, 23, 221-233.
  • [28] Chen, J. & Ahn, H. (1996). Fitting Mixed Poisson Regression Models Using Quasi-Likelihood Methods. Biometrical Journal, 38, 81-96.
  • [29] Chib, S. & Winkelmann, R. (2001). Markov Chain Monte Carlo Analysis of Correlated Count Data. Journal of Business and Economic Statistics, 19, 428-435.
  • [30] Cox, D. (1983). Some Remarks on Overdispersion. Biometrika, 70, 269-274.
  • [31] Cressie, N. (1982). A Useful Empirical Bayes Identity. Annals of Statistics, 10, 625-629.
  • [32] De Vylder, F. (1989). Compound and Mixed Distributions. Insurance Mathematics and Economics, 8, 57-62.
  • [33] Dean, C.B., Lawless, J. & Willmot, G.E. (1989). A Mixed Poisson-Inverse Gaussian Regression Model. Canadian Journal of Statistics, 17, 171-182.
  • [34] Denuit, M., Lefevre, C. & Shaked, M. (2001). Stochastic Convexity of the Poisson Mixture Model. Methodology and Computing in Applied Probability, 2, 231-254.
  • [35] Devroye, L. (1993). A Triptych of Discrete Distributions Related to the Stable Law. Statistics and Probability Letters, 18, 349-351.
  • [36] Douglas, J.B. (1980). Analysis with Standard Contagious Distributions. Statistical Distributions in Scientific Work Series 4. Fairland, MD, USA: International Cooperative Publishing House.
  • [37] Edwards, C.B. & Gurland, J. (1961). A Class of Distributions Applicable to Accidents. Journal of the American Statistical Association, 56, 503-517.
  • [38] Everitt, B.S. & Hand, D.J. (1981). Finite Mixtures Distributions. Chapman and Hall.
  • [39] Feller, W. (1943). On a Generalized Class of Contagious Distributions. Annals of Mathematical Statistics, 14, 389-400.
  • [40] Feller, W. (1968). An Introduction to Probability Theory and its Applications. Vol I, 3rd Edition. New York: John Wiley and Sons.
  • [41] Gaver, D. & O'Muircheartaigh, I.G. (1987). Robust Empirical Bayes Analyses of Event Rates. Technometrics, 29, 1-15.
  • [42] Gelfand, A. & Dalal, S. (1990). A Note on Overdispersed Exponential Families. Biometrika, 77, 55-64.
  • [43] George, E., Makov, U. & Smith, A.F.M. (1993). Conjugate Likelihood Distributions. Scandinavian Journal of Statistics, 20, 147-156.
  • [44] Gerber, H.U. (1991). From the Generalized Gamma to the Generalized Negative Binomial Distribution. Insurance Mathematics and Economics, 10, 303-309.
  • [45] Ghitany, M.E. & Al-Awadhi, M.E. (2001). A Unified Approach to Some Mixed Poisson Distributions. Tamsui Oxford Journal of Mathematical Sciences, 17, 147-161.
  • [46] Goutis, C. & Galbraith, R.F. (1996). A Parametric Model for Heterogeneity in Paired Poisson Counts. Canadian Journal of Statistics, 24, 569-581.
  • [47] Grandell, J. (1997). Mixed Poisson Processes. Chapman and Hall.
  • [48] Greenwood, M. & Yule, G. (1920). An Inquiry into the Nature of Frequency Distributions Representative of Multiple Happenings with Particular Reference to the Occurrence of Multiple Attacks of Disease or of Repeated Accidents. Journal of the Royal Statistical Society A, 83, 255-279.
  • [49] Gupta, S. & Huang, W.T. (1981). On Mixtures of Distributions: A Survey and Some New Results on Ranking and Selection. Sankhya B, 43, 245-290.
  • [50] Gupta, R.C. & Ong, S.H. (2004). A New Generalization of the Negative Binomial Distribution. Computational Statistics and Data Analysis, 45, 287-300.
  • [51] Gurland, J. (1957). Some Interrelations among Compound and Generalized Distributions. Biometrika, 44, 263-268.
  • [52] Gurland, J. (1958). A Generalized Class of Contagious Distributions. Biometrics, 14, 229-249.
  • [53] Gurmu, S. & Elder, J. (2000). Generalized Bivariate Count Data Regression Models. Economics Letters, 68, 31-36.
  • [54] Haight, F.A. (1965). On the Effect of Removing Persons with n or More Accidents from an Accident Prone Population. Biometrika, 52, 298-300.
  • [55] Haight, F.A. (1967). Handbook of Poisson Distributions. New York: John Wiley and Sons.
  • [56] Hall, P. (1979). On Measures of the Distance of a Mixture from its Parent Distribution. Stochastic Processes and Applications, 8, 357-365.
  • [57] Hesselager, O. (1994). A Recursive Procedure for Calculation of Some Compound Distributions. ASTIN Bulletin, 24, 19-32.
  • [58] Hesselager, O. (1996). A Recursive Procedure for Calculation of Some Mixed Compound Poisson Distributions. Scandinavian Actuarial Journal, (1), 54-63.
  • [59] Hinde, J. & Demetrio, C.G.B. (1998). Overdispersion: Models and Estimation. Computational Statistics and Data Analysis, 27, 151-170.
  • [60] Holgate, P. (1970). The Modality of Some Compound Poisson Distributions. Biometrika, 57, 666-667.
  • [61] Holla, M.S. & Bhattacharya, S.K. (1965). On a Discrete Compound Distribution. Annals of the Institute of Statistical Mathematics, 15, 377-384.
  • [62] Hougaard, P., Lee, M.L.T. & Whitmore, G.A. (1997). Analysis of Overdispersed Count Data by Mixtures of Poisson Variables and Poisson Processes. Biometrics, 53, 1225-1238.
  • [63] Irwin, J. (1975). The Generalized Waring Distribution Parts I, II, III. Journal of the Royal Statistical Society A, 138, 18-31 (Part I), 204-227 (Part II), 374-384 (Part III).
  • [64] Johnson, N.L. (1957). Uniqueness of a Result in the Theory of Accident Proneness. Biometrika, 44, 530-531.
  • [65] Johnson, N.L. (1967). Note on a Uniqueness of a Result in the Theory of Accident Proneness. Journal of the American Statistical Association, 62, 288-289.
  • [66] Johnson, N.L., Kotz, S. & Balakrishnan, N. (1997). Discrete Multivariate Distributions. New York: John Wiley and Sons.
  • [67] Johnson, N.L., Kotz, S. & Kemp, A.W. (1992). Univariate Discrete Distributions. 2nd Edition. New York: John Wiley and Sons.
  • [68] Karlis, D. (1998). Estimation and Testing Problems in Poisson Mixtures. Ph.D. Thesis, Department of Statistics, Athens University of Economics. ISBN: 960-7929-19-5.
  • [69] Karlis, D. & Xekalaki, E. (2000). On some Distributions Arising from the Triangular Distribution. Technical Report 111, Department of Statistics, Athens University of Economics and Business, August 2000.
  • [70] Katti, S. (1966). Interrelations among Generalized Distributions and their Components. Biometrics, 22, 44-52.
  • [71] Kemp, A.W. (1995). Splitters, Lumpers and Species per Genus. Mathematical Scientist, 20, 107-118.
  • [72] Kemp, C.D. (1967). Stuttering Poisson Distributions. Journal of the Statistical and Social Enquiry Society of Ireland, 21, 151-157.
  • [73] Kemperman, J.H.B. (1991). Mixtures with a Limited Number of Modal Intervals. Annals of Statistics, 19, 2120-2144.
  • [74] Kempton, R.A. (1975). A Generalized Form of Fisher's Logarithmic Series. Biometrika, 62, 29-38.
  • [75] Khaledi, B.E. & Shaked, M. (2003). Bounds on the Kolmogorov Distance of a Mixture from its Parent Distribution. Sankhya, A, 65, 317-332.
  • [76] Kling, B. & Goovaerts, M. (1993). A Note on Compound Generalized Distributions. Scandinavian Actuarial Journal, 20, 60-72.
  • [77] Kocherlakota, S. (1988). On the Compounded Bivariate Poisson Distribution: A Unified Treatment. Annals of the Institute of Statistical Mathematics, 40, 61-76.
  • [78] Kocherlakota, S. & Kocherlakota, K. (1992). Bivariate Discrete Distributions. New York: Marcel Dekker Inc.
  • [79] Lawless, J. (1987). Negative Binomial and Mixed Poisson Regression. Canadian Journal of Statistics, 15, 209-225.
  • [80] Lindsay, B. (1995). Mixture Models: Theory, Geometry and Applications. Regional Conference Series in Probability and Statistics, Vol 5. Institute of Mathematical Statistics and American Statistical Association.
  • [81] Lindsay, B. & Roeder, K. (1993). Uniqueness of Estimation and Identifiability in Mixture Models. Canadian Journal of Statistics, 21, 139-147.
  • [82] Lynch, J. (1988). Mixtures, Generalized Convexity and Balayages. Scandinavian Journal of Statistics, 15, 203-210.
  • [83] Maceda, E.C. (1948). On the Compound and Generalized Poisson Distributions. Annals of Mathematical Statistics, 19, 414-416.
  • [84] Maiti, T. (1998). Hierarchical Bayes Estimation of Mortality Rates for Disease Mapping. Journal of Statistical Planning and Inference, 69, 339-348.
  • [85] McFadden, J.A. (1965). The Mixed Poisson Process. Sankhya, A, 27, 83-92.
  • [86] McLachlan, G. & Basford, K. (1988). Mixture Models: Inference and Application to Clustering. New York: Marcel Dekker Inc.
  • [87] McLachlan, J.A. & Peel, D. (2000). Finite Mixture Models. New York: John Wiley and Sons.
  • [88] McNeney, B. & Petkau, J. (1994). Overdispersed Poisson Regression Models for Studies of Air Pollution and Human Health. Canadian Journal of Statistics, 22, 421-440.
  • [89] Mellinger, G.D., Sylwester, D.L., Gaffey, W.R. & Manheimer, D.I. (1965). A Mathematical Model with Applications to a Study of Accident Repeatedness among Children. Journal of the American Statistical Association, 60, 1046-1059.
  • [90] Misra, N., Singh, H. & Harner, E.J (2003). Stochastic Comparisons of Poisson and Binomial Random Variables with their Mixtures. Statistics and Probability Letters, 65, 279-290.
  • [91] Molenaar, W. & Van Zwet, W. (1966). On Mixtures of Distributions. Annals of Mathematical Statistics, 37, 201-203.
  • [92] Munkin, M.K. & Trivedi, P.K. (1999). Simulated Maximum Likelihood Estimation of Multivariate Mixed-Poisson Regression Models, with Applications. Econometrics Journal, 2, 29-48.
  • [93] Nelson J. (1985). Multivariate Gamma-Poisson Models. Journal of the American Statistical Association, 80, 828-834.
  • [94] Neuts, M.F. & Ramalhoto, M.F. (1984). A Service Model in Which the Server is Required to Search for Customers. Journal of Applied Probability, 21, 157-166.
  • [95] Nichols, W.G. & Tsokos, C. (1972). Empirical Bayes Point Estimation in a Family of Probability Distributions. International Statistical Review, 40, 147-151.
  • [96] Ong, S.H. (1995). Computation of Probabilities of a Generalized Log-Series and Related Distributions. Communication in Statistics-Theory and Methods, 24, 253-271.
  • [97] Ong, S.H. (1996). On a Class of Discrete Distributions Arising from the Birth-Death with Immigration Process. Metrika, 43, 221-235.
  • [98] Ong, S.H. & Muthaloo, S. (1995). A Class of Discrete Distributions Suited to Fitting Very Long Tailed Data. Communication in Statistics-Simulation and Computation, 24, 929-945.
  • [99] Ord, K. (1967). Graphical Methods for a Class of Discrete Distributions. Journal of the Royal Statistical Society A, 130, 232-238.
  • [100] Ord, K. (1972). Families of Frequency Distributions. London: Griffin.
  • [101] Ospina, V. & Gerber, H.U. (1987). A Simple Proof of Feller's Characterization of the Compound Poisson Distribution. Insurance Mathematics and Economics, 6, 63-64.
  • [102] Panjer, H. (1981). Recursive Evaluation of a Family of Compound Distributions. ASTIN Bulletin, 18, 57-68.
  • [103] Papageorgiou, H. & Wesolowski, J. (1997). Posterior Mean Identifies the Prior Distribution in NB and Related Models. Statistics and Probability Letters, 36, 127-134.
  • [104] Patil, G.P. (1964). On Certain Compound Poisson and Compound Binomial Distributions. Sankha A, 27, 293-294.
  • [105] Patil, G.P. & Rao, C.R. (1978). Weighted Distributions and Size-Biased Sampling with Applications to Wildlife Populations and Human Families. Biometrics, 34, 179-189.
  • [106] Patil, G.P., Rao, C.R. & Ratnaparkhi, M.V. (1986). On Discrete Weighted Distributions and Their Use in Model Choice for Observed Data. Communication in Statistics-Theory and Methods, 15, 907-918.
  • [107] Perline, R. (1998). Mixed Poisson Distributions Tail Equivalent to their Mixing Distributions. Statistics and Probability Letters, 38, 229-233.
  • [108] Pfeifer, D. (1987). On the Distance between Mixed Poisson and Poisson Distributions. Statistics and Decision, 5, 367-379.
  • [109] Pielou, E. (1962). Run of One Species with Respect to Another in Transects through Plant Populations. Biometrics, 18, 579-593.
  • [110] Press, W., Teukolsky, S., Vetterling, W., & Flannery, B. (1992). Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd Edition. Cambridge University Press.
  • [111] Quenouille, M.H. (1949). A Relation between the Logarithmic, Poisson and Negative Binomial Series. Biometrics, 5, 162-164.
  • [112] Rai, G. (1971). A Mathematical Model for Accident Proneness. Trabajos Estad\'{\istica}, 22, 207-212.
  • [113] Rao, C.R. (1965). On Discrete Distributions Arising out of Methods of Ascertainment. In Classical and Contagious Discrete Distributions, Ed. G.P. Patil, pp. 320-332. Pergamon Press and Statistical Publishing Society, Calcutta.
  • [114] Redner, R. & Walker, H. (1984). Mixture Densities, Maximum Likelihood and the EM Algorithm. SIAM Review, 26, 195-230.
  • [115] Remillard, B. & Theodorescu, R. (2000). Inference Based on the Empirical Probability Generating Function for Mixtures of Poisson Distributions. Statistics and Decisions, 18, 349-266.
  • [116] Roos, B. (2003). Improvements in the Poisson Approximations of Mixed Poisson Distributions. Journal of Statistical Planning and Inference, 113, 467-483.
  • [117] Ruohonen, M. (1988). A Model for the Claim Number Process. ASTIN Bulletin, 18, 57-68.
  • [118] Sankaran, M. (1969). On Certain Properties of a Class of Compound Poisson Distributions. Sankha B, 32, 353-362.
  • [119] Sankaran, M. (1970). The Discrete Poisson-Lindley Distribution. Biometrics, 26, 145-149.
  • [120] Sapatinas, T. (1995). Identifiability of Mixtures of Power Series Distributions and Related Characterizations. Annals of the Institute of Statistical Mathematics, 47, 447-459.
  • [121] Schweder, T. (1982). On the Dispersion of Mixtures. Scandinavian Journal of Statistics, 9, 165-169.
  • [122] Seshadri, V. (1991). Finite Mixtures of Natural Exponential Families. Canadian Journal of Statistics, 19, 437-445.
  • [123] Shaked, M. (1980). On Mixtures from Exponential Families. Journal of the Royal Statistical Society B, 42, 192-198.
  • [124] Sibuya, M. (1979). Generalized Hypergeometric, Digamma and Trigamma Distributions. Annals of the Institute of Statistical Mathematics, 31, 373-390.
  • [125] Sichel, H.S. (1974). On a Distribution Representing Sentence-Length in Written Prose. Journal of the Royal Statistical Society A, 137, 25-34.
  • [126] Sichel, H.S. (1975). On a Distribution Law for Word Frequencies. Journal of the American Statistical Association, 70, 542-547.
  • [127] Simon, P. (1955). On a Class of Skew Distributions. Biometrika, 42, 425-440.
  • [128] Sivaganesan, S. & Berger, J.O. (1993). Robust Bayesian Analysis of the Binomial Empirical Bayes Problem. The Canadian Journal of Statistics, 21, 107-119.
  • [129] Stein, G.Z., Zucchini, W. & Juritz, J.M. (1987). Parameter Estimation for the Sichel Distribution and its Multivariate Extension. Journal of the American Statistical Association, 82, 938-944.
  • [130] Stein, G. & Juritz, J.M. (1987). Bivariate Compound Poisson Distributions. Communications in Statistics-Theory and Methods, 16, 3591-3607.
  • [131] Steyn, H. (1976). On the Multivariate Poisson Normal Distribution. Journal of the American Statistical Association, 71, 233-236.
  • [132] Subrahmaniam, K. (1966). A Test for Intrinsic Correlation in the Theory of Accident Proneness. Journal of the Royal Statistical Society B, 28, 180-189.
  • [133] Sundt, B. (1999). An Introduction to Non-Life Insurance Mathematics, 4th Edition. Karlsruhe: University of Mannheim Press.
  • [134] Tallis, G.M. (1969). The Identifiability of Mixtures of Distributions. Journal of Applied Probability, 6, 389-398.
  • [135] Teicher, H. (1961). Identifiability of Mixtures. Annals of Mathematical Statistics, 26, 244-248.
  • [136] Teicher, H. (1963). Identifiability of Finite Mixtures. Annals of Mathematical Statistics, 28, 75-88.
  • [137] Titterington, D.M., Smith, A.F.M. & Makov, U.E. (1985). Statistical Analysis of Finite Mixture Distributions. New York: John Wiley and Sons.
  • [138] Titterington, D.M. (1990). Some Recent Research in the Analysis of Mixture Distributions. Statistics, 21, 619-641.
  • [139] Wahlin, J.F. & Paris, J. (1999). Using Mixed Poisson Distributions in Connection with Bonus-Malus Systems. ASTIN Bulletin, 29, 81-99.
  • [140] Wang, S. & Panjer, H. (1993). Critical Starting Points for Stable Evaluation of Mixed Poisson Probabilities. Insurance Mathematics and Economics, 13, 287-297.
  • [141] Wang, S. & Sobrero, M. (1994). Further Results on Hesselager's Recursive Procedure for Calculation of Some Compound Distributions. ASTIN Bulletin, 24, 160-166.
  • [142] Wang, P., Puterman, M., Cockburn, I. & Le, N. (1996). Mixed Poisson Regression Models with Covariate Dependent Rates. Biometrics, 52, 381-400.
  • [143] Willmot, G.E. (1986). Mixed Compound Poisson Distribution. ASTIN Bulletin Supplement, 16, 59-79.
  • [144] Willmot, G.E. (1990). Asymptotic Tail Behaviour of Poisson Mixtures with Applications. Advances in Applied Probability, 22, 147-159.
  • [145] Willmot, G.E. (1993). On Recursive Evaluation of Mixed Poisson Probabilities and Related Quantities. Scandinavian Actuarial Journal, 18, 114-133.
  • [146] Willmot, G.E. & Sundt, B. (1989). On Posterior Probabilities and Moments in Mixed Poisson Processes. Scandinavian Actuarial Journal, 14, 139-146.
  • [147] Xekalaki, E. (1981). Chance Mechanisms for the Univariate Generalized Waring Distribution and Related Characterizations. In Statistical Distributions in Scientific Work, Eds. C. Taillie, G.P. Patil and B. Baldessari, 4, 157-171. The Netherlands: D. Reidel Publishing Co.
  • [148] Xekalaki, E. (1983a). The Univariate Generalized Waring Distribution in Relation to Accident Theory: Proneness, Spells Or Contagion? Biometrics, 39, 887-895.
  • [149] Xekalaki, E. (1983b). Infinite Divisibility, Completeness and Regression Properties of the Univariate Generalized Waring Distribution. Annals of the Institute of Statistical Mathematics, 32, 279-289.
  • [150] Xekalaki, E. (1984a). The Bivariate Generalized Waring Distribution and its Application to Accident Theory. Journal of the Royal Statistical Society A, 147, 488-498.
  • [151] Xekalaki, E. (1984b). Models Leading to the Bivariate Generalized Waring Distribution. Utilitas Mathematica, 35, 263-290.
  • [152] Xekalaki, E. (1986). The Multivariate Generalized Waring Distribution. Communications in Statistics, 15, 1047-1064.
  • [153] Xekalaki, E. & Panaretos, J. (1983). Identifiability of Compound Poisson Distributions. Scandinavian Actuarial Journal, 39-45.
  • [154] Xue, D. & Deddens, J. (1992). Overdispersed Negative Binomial Models. Communications in Statistics-Theory and Methods, 21, 2215-2226.
  • [155] Yakowitz, S. & Spragins, J. (1969). On the Identifiability of Finite Mixtures. Annals of Mathematical Statistics, 39, 209-214.