Dirichlet and Related Distributions: Theory, Methods and Applications

Dirichlet and Related Distributions: Theory, Methods and Applications

Volumn , Issue , 2011, Pages 1-310

  Ng, Kai Wang ^a   Tian, Guo Liang ^a   Tang, Man Lai ^b  

^a UNIVERSITY OF HONG KONG   (Hong Kong)

^b HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

Author keywords

[No Author keywords available]

Indexed keywords

MAXIMUM PRINCIPLE;

CHARACTERISTIC FUNCTIONS; DIRICHLET DISTRIBUTIONS; DIRICHLET-MULTINOMIAL DISTRIBUTION; EXPECTATION - MAXIMIZATIONS; GENERALIZED DIRICHLET DISTRIBUTION; MULTINOMIAL DISTRIBUTIONS; NONPARAMETRIC INFERENCE; UNIFORM DISTRIBUTION;

REVIEWS;

EID: 84861573300     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9781119995784     Document Type: Book

References (234)

1. Aalo, V. A., Piboongungon, T. and Efthymoglou, G. P. (2005) Another look at the performance of MRC schemes in Nakagami-m fading channel with arbitrary parameters. IEEE Transactions on Communications, 53, 2002-2005.
2. Abramowitz, M. and Stegun, I. (1965) Handbook of Mathematical Functions. Dover, New York.
3. Aitchison, J. (1963) Inverse distributions and independent gamma-distributed products of random variables. Biometrika, 50, 505-508.
4. Aitchison, J. (1986) The Statistical Analysis of Compositional Data. Chapman & Hall, London.
5. Aitchison, J. (2003) The Statistical Analysis of Compositional Data. The Blackburn Press, Caldwell, NJ, USA.
6. Albert, J. H. and Gupta, A. K. (1985) Bayesian methods for binomial data with applications to a non-response problem. Journal of the American Statistical Association, 80, 167-174.
7. Alouini, M. S., Abdi, A. and Kaveh, M. (2001) Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels. IEEE Transactions on Vehiculer Technology, 50, 1471-1480.
8. Al-Ruzaiza, A. S. and El-Gohary, A. (2008) The distributions of sums, products and ratios of inverted bivariate beta distribution. Applied Mathematical Sciences, 2 (48), 2377-2391.
9. Anderson, T. W. (1957) Maximum likelihood estimates for the multivariate normal distribution when some observations are missing. Journal of the American Statistical Association, 52, 200-203.
10. Anderson, T. W. (1984) An Introduction to Multivariate Statistical Analysis (2nd edn). John Wiley & Sons, Inc., New York.
11. Anderson, T. W. and Fang, K. T. (1982) Distributions of quadratic forms and Cochran's theorem for elliptically contoured distributions and their applications. Technical Report, No.53, Department of Statistics, Stanford University, California.
12. Anderson, T.W. and Fang, K. T. (1987) Cochran's theorem for elliptically contoured distributions. Sankhya, A, 49, 305-315.
13. Arnold, S. F. (1993) Gibbs sampling. In Handbook of Statistics: Computational Statistics (C. R. Rao, ed), Vol. 9, 599-625. Elsevier Science, New York.
14. Basu, D. and Tiwari, R. C. (1982) A note on the Dirichlet process. In Statistics and Probability: Essays in Honor of C. R. Rao (G. Kallianpur, P. R. Krishnaiah and J. K. Ghosh, eds), 89-103. North-Holland, Amsterdam.
15. Bilodeau, M. and Brenner, D. (1999) Theory of Multivariate Statistics. Springer, New York.
16. Blei, D. M., Franks, K., Jordan, M. I. and Mian, I. S. (2006) Statistical modeling of biomedical corpora: mining the Caenorhabditis genetic center bibliography for genes related to life span. BMC Bioinformatics, 7, Article 250.
17. Broemeling, L. D. (1969) Confidence regions for variance ratios of random models. Journal of the American Statistical Association, 64, 660-664.
18. Broemeling, L. D. (2007) Bayesian Biostatistics and Diagnostic Medicine. Chapman & Hall/CRC, Boca Raton, FL.
19. Cabanlit, E. A. Jr., Padua, R. N. and Alam, K. (2006) Generalization of the Dirichlet distribution. SCRA 2006-FIM XIII-Thirteenth International Conference of the Forum for Interdisciplinary Mathematics on Interdisciplinary Mathematical and Statistical Techniques, September 1-4, 2006. New University of Lisbon-Tomar Polytechnic Institute, Lisbon-Tomar, Portugal.
20. Cambanis, S., Keener, R. and Simons, G. (1983) On α-symmetric multivariate distributions. Journal of Multivariate Analysis, 13, 213-233.
21. Carlson, B. C. (1977) Special Functions of Applied Mathematics. Academic Press, New York.
22. Chen, J. J., Kodell, R. L., Howe, R. B. and Gaylor, D. W. (1991) Analysis of trinomial responses from reproductive and developmental toxicity experiments. Biometrics 47, 1049-1058.
23. Chen, T. T. and Fienberg, S. E. (1974) Two-dimensional contingency tables with both completely and partially cross-classified data. Biometrics, 30, 629-642.
24. Choi, S. C. and Stablein, D. M. (1982) Practical tests for comparing two proportions with incomplete data. Applied Statistics, 31, 256-262.
25. Choi, S. C. and Stablein, D. M. (1988) Comparing incomplete paired binomial data under non-random mechanisms. Statistics in Medicine, 7, 929-939.
26. Connor, R. J. and Mosimann, J. E. (1969) Concepts of independence for proportions with a generalization of the Dirichlet distribution. Journal of the American Statistical Association, 64, 194-206.
27. Cook, R. D. and Johnson, M. E. (1981) A family of distributions for modeling nonelliptically symmetric multivariate data. Journal of the Royal Statistical Society, Series B (Methodological), 43, 210-218.
28. Cornell, J. A. (1990) Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data (2nd edn). John Wiley & Sons, Inc., New York.
29. Cornish, E. A. (1954) The multivariate t-distribution associated with a set of normal sample deviates. Australian Journal of Physics, 7, 531-542.
30. Cox, D. R. (1959) The analysis of exponentially distributed lifetimes with two types of failures. Journal of the Royal Statistical Society, Series B (Methodological), 21, 411-421.
31. Crosier, R. B. (1984) Mixture experiments: geometry and pseudocomponents. Technometrics, 26, 209-216.
32. Crosier, R. B. (1986) The geometry of constrained mixture experiments. Technometrics, 28, 95-102.
33. Darroch, J. N. and Ratcliff, D. (1971) Acharacterization of the Dirichlet distribution. Journal of the American Statistical Association, 66, 641-643.
34. David, H. A. (1981) Order Statistics (2nd edn). JohnWiley&Sons, Inc., NewYork.
35. Davis, T. P. and Lawrence, A. J. (1989) The likelihood for competing risks survival analysis. Scandinavian Journal of Statistics, 16, 23-30.
36. Dawid, A. P. and Lauritzen, S. L. (1993) Hyper Markov laws in the statistical analysis of decomposable graphical models. Annals of Statistics, 21 (3), 1272-1317.
37. Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977) Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B (Methodological), 39, 1-38.
38. Devroye, L. (1986) Non-uniform Random Variate Generation. Springer, New York.
39. Dickey, J. M. (1983) Multiple hypergeometric functions: probabilistic interpretations and statistical uses. Journal of the American Statistical Association, 78, 628-637.
40. Dickey, J. M., Jiang J. M. and Kadane, J. B. (1987) Bayesian methods for censored categorical data. Journal of the American Statistical Association, 82, 773-781.
41. Didelez, V. and Pigeot, I. (1998) Maximum likelihood estimation in graphical models with missing values. Biomatrika, 85, 960-966.
42. Dirichlet, P. G. L. (1839) Sur une nouvelle méthode pour la détermination des intégrales multiple. Liouville Journal des Mathématiques, Series I, 4, 164-168. Reprinted in Dirichlet's Werke, Vol. 1, 377-380.
43. Dishon, M. andWeiss, G. (1980) Small sample comparison of estimation methods for the beta distribution. Journal of Statistical Computation and Simulation, 11, 1-11.
44. Doksum, K. A. (1974) Tailfree and neutral random probabilities and their posterior distributions. Annals of Probability, 2, 183-201.
45. Dunnett, C. W. and Sobel, M. (1954) A bivariate generalization of Student's t-distribution, with tables for certain special cases. Biometrika, 41, 153-169.
46. Dykstra, R., Kochar, S. and Robertson, T. (1991) Statistical inference for uniform stochastic ordering in several populations. Annals of Statistics, 19, 870-888.
47. Dykstra, R., Kochar, S. and Robertson, T. (1998) Restricted tests for testing independence of time to failure and cause of failure in a competing-risks model. Canadian Journal of Statistics, 26 (1), 57-68.
48. Edwards, J. (1922) A Treatise on the Integral Calculus (Vol. 2). Macmillan, New York.
49. Efron, B. (1979) Bootstrap methods: another look at the jackknife. Annals of Statistics, 7, 1-26.
50. Efron, B. and Tibshirani, R. J. (1993) An Introduction to the Bootstrap. Chapman & Hall/CRC, Boca Raton, FL.
51. Efthymoglou, G. P. and Aalo, V. A. (1995) Performance of RAKE receivers in Nakagami fading channel with arbitrary fading parameters. Electronics Letters, 31, 1610-1612.
52. Erdéyli, A. (1939) Integration of certain systems of linear partial differential equations of hypergeometric type. Proceedings of the Royal Society of Edinburgh, Section A, 59, 224-241.
53. Erdéyli, A. (1953) Higher Transcendental Functions. McGraw-Hill, New York.
54. Evans, M. and Swartz, T. (1995) Methods for approximating integrals in statistics with special emphasis on Bayesian integration problems (with discussion). Statistical Science, 10, 254-272.
55. Evans, M. and Swartz, T. (2000) Approximating Integrals via Monte Carlo and Deterministic Methods. Oxford University Press, Oxford.
56. Evans, M., Guttman, I., Haitovsky, Y. and Swartz, T. (1996) Bayesian analysis of binary data subject to misclassification. In Bayesian Ana-lysis in Statistics and Econometrics: Essays in Honor of Arnold Zellner (D. A. Berry, K. M. Chaloner and J. K. Geweke, eds), 67-77. John Wiley & Sons, Inc., New York.
57. Exton, H. (1976) Multiple Hypergeometric Functions and Applications. JohnWiley & Sons, Inc., New York.
58. Fabius, J. (1973) Two characterizations of the Dirichlet distribution. Annals of Statistics, 1 (3), 583-587.
59. Fang, K. T. and Chen, Q. Y. (1986) Probabilistic methods in statistics. Chinese Journal of Mathematical Statistics and Applied Probability, 2 (3), 355-367.
60. Fang, K. T. and Fang, B. Q. (1988) Some families of multivariate symmetric distributions related to exponential distribution. Journal of Multivariate Analysis, 24, 109-122.
61. Fang, K. T. andWang, Y. (1994) Number-theoretic Methods in Statistics. Chapman & Hall, London.
62. Fang, K. T. andYang, Z. H. (2000) On uniform design of experiments with restricted mixtures and generation of uniform distribution on some domains. Statistics and Probability Letters, 46, 113-120.
63. Fang, K. T. and Zhang, Y. T. (1990) Generalized Multivariate Analysis. Springer/Science Press, Berlin/Beijing.
64. Fang, K. T., Kotz, S. and Ng, K. W. (1990) Symmetric Multivariate and Related Distributions. Chapman & Hall, London.
65. Fang, K. T., Geng, Z. and Tian, G. L. (2000) Statistical inference for truncated Dirichlet distribution and its application in misclassification. Biometrical Journal, 42 (8), 1053-1068.
66. Feller, W. (1966) An Introduction to Probability Theory and Its Applications (Vol. 2). John Wiley & Sons, Inc., New York.
67. Geiger,D. and Heckerman, D. (1997) Acharacterization of the Dirichlet distribution through global and local parameter independence. Annals of Statistics, 25 (3), 1344-1369.
68. Gelfand, A. E. (2002) Gibbs sampling. In Statistics in the 21-st Century (A. E. Raftery, M. A. Tanner and M. T. Wells, eds), 341-349. Chapman & Hall/CRC, Boca Raton, FL.
69. Gelfand, A. E. and Smith, A. F. M. (1990) Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association, 85, 398-409.
70. Gelfand, A. E. and Smith, A. F. M. (1991) Gibbs sampling for marginal posterior expectation. Communications in Statistics: Theory and Methods, 20, 1747-1766.
71. Geman, D. and Geman, S. (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721-741.
72. Geng, Z., Wan, K. and Tao, F. (2000) Mixed graphical models with missing data and the partial imputation EM algorithm. Scandinavian Journal of Statistics, 27, 433-444.
73. Geweke, J. (1989) Bayesian inference in econometric models using Monte Carlo integration. Econometrica, 57, 1317-1339.
74. Gilks, W. R. and Wild, P. (1992) Adaptive rejection sampling for Gibbs sampling. Applied Statistics, 41, 337-348.
75. Gouda, A. A. and Szántai, T. (2004) New sampling techniques for calculation of Dirichlet probabilities. The Central European Journal of Operations Research, 12, 389-403.
76. Gouda, A. A. and Szántai, T. (2010) On numerical calculation of probabilities according to Dirichlet distribution. Annals of Operations Research, 177, 185-200.
77. Graybill, F. A. (1961) An Introduction to Linear Models (Vol. 1). McGraw Hill, New York.
78. Greenland, S. (1989) On correcting for misclassification in twin studies and other matched-pair studies. Statistics in Medicine, 8, 825-829.
79. Gupta, A. K. and Nadarajah, S. (2006) Exact and approximate distributions for the linear combination of inverted Dirichlet components. Journal of the Japan Statistical Society, 36 (2), 225-236.
80. Gupta, R. D. and Richards, D. St. P. (1987) Multivariate Liouville distributions. Journal of Multivariate Analysis, 23, 233-256.
81. Gupta, R. D. and Richards, D. St. P. (1992a) Multivariate Liouville distributions, II. Probability and Mathematical Statistics, 12, 291-309.
82. Gupta, R. D. and Richards, D. St. P. (1992b) Multivariate Liouville distributions, III. Journal of Multivariate Analysis 43, 29-57.
83. Gupta, R. D. and Richards, D. St. P. (2001a) The covariance structure of the multivariate Liouville distributions. Contemporary Mathematics, 287, 125-138.
84. Gupta, R. D. and Richards, D. St. P. (2001b) The history of the Dirichlet and Liouville distributions. International Statistical Review, 69(3), 433-446.
85. Hankin, R. K. S. (2010) A generalization of the Dirichlet distribution. Journal of Statistics Software, 33 (11), 1-18.
86. Hanson, L. P. (1982) Large sample properties of generalized method of moments estimators. Econometrica, 50, 1029-1054.
87. Hartley, H. O. (1967) Expectations, variances, and covariances of ANOVA mean squares by 'synthesis'. Biometrics, 23 (1), 105-114.
88. Hazewinkel, M. (1990) Encyclopedia of Mathematics (Vol. 9). Kluwer, Dordrecht.
89. Hellmich, M., Abrams, K. R., Jones, D. R. and Lambert, P. C. (1998) ABayesian approach to a general regression model for ROC curves. Medical Decision Making, 18, 436-443.
90. Hille, E. (1959) Analytic Function Theory (Vol. 1). Blaisdell Ginn, New York.
91. Hobert, J. P. and Casella, G. (1996) The effect of improper priors on Gibbs sampling in hierarchical linear models. Journal of the American Statistical Association, 91, 1461-1473.
92. Hocking, R. R. and Oxspring, H. H. (1974) The analysis of partially categorized contingency data. Biometrics, 30, 469-483.
93. Hsu, P. L. (1949) The limiting distribution of functions of sample means and application to testing hypotheses. In Proceedings of the Berkeley Symposium on Mathematical Statics and Probability (J. Neymen, ed) 359-402.
94. Humbert, P. (1920-1921) The confluent hypergeometric function of two variables. Proceedings of the Royal Society of Edinburgh, Section A, 41, 73-96.
95. Imhof, J. P. (1961) Computing the distribution of quadratic forms in normal variables. Biometrika, 48, 419-426.
96. Ishii, G. and Hayakawa, R. (1960) On the compound binomial distribution. Annals of the Institute of Statistical Mathematics, 12, 69-80. (Errata, 12, 208).
97. James, I. R. (1972) Products of independent beta variables with application to Connor and Mosimann's generalized Dirichlet distribution. Journal of the American Statistical Association, 67 (340), 910-912.
98. James, I. R. (1975) Multivariate distributions which have beta conditional distributions. Journal of the American Statistical Association, 70 (351), 681-684.
99. James, I. R. and Mosimann, J. E. (1980) A new characterization of the Dirichlet distribution through neutrality. Annals of Statistics, 8 (1), 183-189.
100. Jech, T. (1983) The ranking of incomplete tournaments: a mathematician's guide to popular sports. American Mathematical Monthly, 90 (4), 246-266.
101. Johnson,W. O. and Gastwirth, J. L. (1991) Bayesian inference for medical screening tests: approximations useful for the analysis of AIDS. Journal of the Royal Statistical Society, Series B (Methodological), 53(2), 427-439.
102. Johnson, N. L. and Kotz, S. (1969) Discrete Distributions. JohnWiley&Sons, Inc., New York.
103. Johnson, N. L. and Kotz, S. (1970a) Continuous Univariate Distributions-1. John Wiley & Sons, Inc., New York.
104. Johnson, N. L. and Kotz, S. (1970b) Continuous Univariate Distributions-2. John Wiley & Sons, Inc., New York.
105. Johnson, N. L. and Kotz, S. (1972) Distributions in Statistics - Continuous Multivariate Distributions. John Wiley & Sons, Inc., New York.
106. Kadane, J. B. (1983) Juries hearing death penalty cases: statistical analysis of a legal procedure. Journal of the American Statistical Association, 78, 544-552.
107. Kadane, J. B. (1985) Is victimization chronic? A Bayesian analysis of multinomial missing data. Journal of Econometrics, 29, 47-67.
108. Karlin, S., Micchelli, C. A. and Rinott, Y. (1986) Multivariate splines: A probabilistic perspective. Journal of Multivariate Analysis, 20, 69-90.
109. Knezek, G., Wallace, S. and Dunn-Rankin, P. (1998) Accuracy of Kendall's chisquare approximation to circular triad distributions. Psychometrika, 63 (1), 23-34.
110. Kotz, S. and Johnson, N. L. (1982) Encyclopedia of Statistical Sciences. JohnWiley & Sons, Inc., New York.
111. Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000) Continuous Multivariate Distributions: Models and Applications (Vol. 1, 2nd edn). John Wiley & Sons, Inc., New York.
112. Kupper, L. L. and Haseman, J. K. (1978) The use of a correlated-binomial model for the analysis of certain toxicological experiments. Biometrics, 34, 69-76.
113. Laval, G., SanCristobal, M. and Chevalet, C. (2003) Maximum-likelihood and Markov chain Monte Carlo approaches to estimate inbreeding and effective size from allele frequency changes. Genetics, 164 (3), 1189-1204.
114. Lange, K. (1995a) A gradient algorithm locally equivalent to the EM algorithm. Journal of the Royal Statistical Society, Series B (Methodological), 57(2), 425-437.
115. Lange, K. (1995b) Applications of the Dirichlet distribution to forensic match probabilities. Genetica, 96, 107-117.
116. Lee, P. A. (1971) Adiagonal expansion for the 2-variate Dirichlet probability density function. SIAM Journal of Applied Mathematics, 21, 155-165.
117. Lehmann, E. L. (1983) Theory of Point Estimation. John Wiley & Sons, Inc., New York.
118. Lew, R. A. and Levy, P. S. (1989) Estimation of prevalence on the basis of screening tests. Statistics in Medicine, 8, 1225-1230.
119. Liouville, J. (1839) Note sur quelques intégrales définies. Journal de Mathématiques Pures et Appliquées, 4, 225-235.
120. Lipsitz, S. R. and Fitzmaurice, G. M. (1996) The score test for independence in R × C contingency tables with missing data. Biometrics, 52, 751-762.
121. Little, R. J. A. and Rubin, D. B. (2002) Statistical Analysis with Missing Data (2nd edn). John Wiley & Sons, Inc., New York.
122. Liu, C. H. (1999) Efficient ML estimation of the multivariate normal distribution from incomplete data. Journal of Multivariate Analysis, 69, 206-217.
123. Liu, J. S. (1994) Fraction of missing information and convergence rate of data augmentation. In Computationally Intensive Statistical Methods: Proceedings of the 26th Symposium Interface (J. Sall and A. Lehmann, eds), 490-497. Interface Foundation of North American, Fairfax Station, VA.
124. Liu, J. S., Wong, W. H. and Kong, A. (1994) Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes. Biometrika, 81, 27-40.
125. Liu, J. S.,Wong,W. H. and Kong, A. (1995) Covariance structure and convergence rate of the Gibbs sampler with various scans. Journal of the Royal Statistical Society, Series B (Methodological), 57, 157-169.
126. Louis, T. A. (1982) Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, Series B (Methodological), 44, 226-233.
127. Lukacs, E. (1955) A characterization of the gamma distribution. Annals of Mathematical Statistics, 26, 319-324.
128. Malik, H. J. and Abraham, B. (1973) Multivariate logistic distribution. Annals of Statistics, 1, 588-590.
129. Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
130. Mathai, A. M. (1993) A Handbook of Generalized Special Functions for Statistical and Physical Sciences. Oxford University Press, New York.
131. Mathai, A. M. (2002) On a general multivariate Dirichlet density and its applications. In Statistical Methods and Practice: Recent Advances (N. Balakrishnan, N. Kannan and M. R. Sreenivasan, eds), 57-67. Narosa Publishing House, Chennai.
132. Mathai, A. M. and Saxena, R. K. (1978) The H-function with Applications in Statistics and Other Disciplines. John Wiley & Sons, Inc., New York.
133. Mauchly, J. W. (1940) Significance test for sphericity of a normal n-variate distribution. Annals of Mathematical Statistics, 11, 204-209.
134. Medenhall, W. and Hader, R. J. (1958) Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika, 45, 504-520.
135. Meng, X. L. (1994) On the rate of convergence of the ECM algorithm. Annals of Statistics, 22, 326-339.
136. Meng, X. L. and Rubin, D. B. (1993) Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika, 80, 267-278.
137. Meng, X. L. and van Dyk, D. (1997) The EM algorithm - an old folk-song sung to a fast newtune. Journal of the Royal Statistical Society, Series B (Methodological), 59, 511-567.
138. Miller, K. S. (1987) Some Eclectic Matrix Theory. Robert E. Krieger Publishing, Malabar, FL.
139. Miller,R.G. Jr. (1966) Simultaneous Statistical Inference. McGraw-Hill, NewYork.
140. Minka, T. P. (2003) Estimating a Dirichlet distribution. Unpublished manuscript available at http://research.microsoft.com/minka.
141. Moschopoulos, P. G. (1985) The distribution of the sum of independent gamma random variables. Annals of the Institute of Statistical Mathematics, 37, 541-544.
142. Mosimann, J. E. (1962) On the compound multinomial distribution, the multivariate β-distribution, and correlations among proportions. Biometrika, 49, 65-82.
143. Mosimann, J. E. (1975) Statistical problems of size and shape (II): characterizations of the lognormal, gamma and Dirichlet distributions. In Statistical Distributions in Scientific Work (G. P. Patil, S. Kotz and J. K. Ord, eds), Vol. 2, 219-239. D. Reidel, Dordrecht.
144. Nadarajah, S. (2008) A review of results on sums of random variables. Acta Applicandae Mathematicae, 103, 131-140.
145. Narayanan, A. (1991) Algorithm AS266: maximum likelihood estimation of the parameters of the Dirichlet distribution. Applied Statistics, 40 (2), 365-374.
146. Neerchal, N. K. and Morel, J. G. (2005) An improved method for the computation of maximum likelihood estimates for multinomial overdispersion models. Computational Statistics of Data Analysis, 49, 33-43.
147. Neyman, J. (1959) Optimal asymptotic tests of composite hypothesis. In Probability and Statistics: The Harold Cramér Volume (U. Grenander, ed). John Wiley & Sons, Inc., New York.
148. Ng, K. W. (1995a) Explicit formulas for unconditional pdf. Research Report No. 82 (March 1995), Department of Statistics, the University of Hong Kong, Hong Kong.
149. Ng, K. W. (1995b) On the inversion of Bayes theorem. Presentation in The 3rd ICSA Statistical Conference, August 17-20, Beijing, P.R. China.
150. Ng, K. W. (1996) Inversion of Bayes formula without positivity assumption. Presentation in Sydney International Statistical Congress 1996, ASC Contributed Session: Topics in Statistical Inference III, July 12, 8:30-10:20 am, Sydney, Australia; Abstract #438 in Final Programme.
151. Ng, K.W. (1997a) Inversion of Bayes formula: explicit formulas for unconditional pdf. In Advances in the Theory and Practice in Statistics - A Volume in Honor of Samuel Kotz (N. L. Johnson and N. Balakrishnan, eds), 571-584. John Wiley & Sons, Inc., New York.
152. Ng, K. W. (1997b) Applications of inversion Bayes formula. Presentation in The 1997 International Symposium on Contemporary Multivariate Analysis and Its Applications, under the I. Theme: Bayesian Computation organized by A. Lo and W. H. Wong, Preprint pp. I.1-10, May 19-22, 1997, Hong Kong.
153. Ng, K.W. (2010a) Inversion of Bayes formula for events. Research Report No. 466 (January 2010), Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong.
154. Ng, K. W. (2010b) Inversion of Bayes' formula for events, International Encyclopedia of Statistical Science (M. Lovric, ed), 690-695. Springer, New York.
155. Ng, K. W. and Tian, G. L. (2001) Characteristic functions of L1-spherical and L1-norm symmetric distributions and their applications. Journal of Multivariate Analysis, 76(2), 192-213.
156. Ng, K.W., Tang, M. L., Tan, M. and Tian, G. L. (2008) Grouped Dirichlet distribution:a new tool for incomplete categorical data analysis. Journal of Multivariate Analysis, 99 (3), 490-509.
157. Ng, K. W., Tang, M. L., Tian, G. L. and Tan, M. (2009) The nested Dirichlet distribution and incomplete categorical data analysis. Statistica Sinica, 19 (1), 251-271.
158. Nordheim, E. V. (1984) Inference from non-randomly missing categorical data:an example from a genetic study on Turner's syndrome. Journal of American Statistical Association, 79, 772-780.
159. Ongaro, A., Migliorati, S. and Monti, G. S. (2008) Anewdistribution on the simplex containing the Dirichlet family. In Proceedings of the Third Compositional Data Analysis Workshop, May 27-30, University of Girona, Girona, Spain.
160. Pan, J. J. (1968) Distribution of the serial correlation coefficients with non-circular statistics. In Selected Translations in Mathematical Statistics and Probability (Vol. 7), 281-291. American Mathematics Society, Providence, RI.
161. Paul, S. R., Liang, K. Y. and Self, S. G. (1989) On testing departure from the binomial and multinomial assumptions. Biometrics, 45, 231-236.
162. Paul, S. R., Balasooriya, U. and Banerjee, T. (2005) Fisher information matrix of the Dirichlet-multinomial distribution. Biometrical Journal, 47 (2), 230-236.
163. Paulino, C. D. M. and de Bragarça Pereira, C. A. (1992) Bayesian analysis of categorical data informatively censored. Commununications in Statistics: Theory and Methods, 21, 2689-2705.
164. Paulino, C. D. M. and de Bragarça Pereira, C. A. (1995) Bayesian methods for categorical data under informative general censoring. Biometrika, 82, 439-446.
165. Peng, F. and Hall, W. J. (1996) Bayesian analysis of ROC curves using Markov Chain Monte Carlo methods. Medical Decision Making, 16, 404-411.
166. Pepe, M. S. (2003) The Statistical Evaluation of Medical Tests for Classification and Prediction. Oxford University Press, New York.
167. Phillips, P. C. B. (1988) The characteristic function of the Dirichlet and multivariate F distributions. CowlesFoundation for Research in Economics (Discussion Paper 865), 1-17.
168. Piepel, G. F. (1983) Defining consistent constraint regions in mixture experiments. Technometrics, 25, 97-101.
169. Potthoff, R. F. and Whittinghill, M. (1966) Testing for homogeneity: I. The binomial and multinomial distributions. Biometrika, 53, 167-182.
170. Prékopa, A. (1995) Stochastic Programming. Kluwer Academic, Dordrecht.
171. Prentice, R. L., Kalbfleisch, J. D., Peterson, A. V. et al. (1978) The analysis of failure times in the presence of competing risks. Biometrics, 34, 541-554.
172. Press, S. J. (1982) Applied Multivariate Analysis (2nd edn). Robert E. Krieger, Malabar, FL.
173. Provost, S. B. (1988) The exact density of a general linear combination of gamma variates. Metron, 46, 61-69.
174. Provost, S. B. (1989) On sums of independent gamma variables. Statistics, 20, 583-591.
175. Provost, S. B. and Cheong, Y. H. (2000) On the distribution of linear combinations of the components of a Dirichlet random vector. Canadian Journal of Statistics, 28 (2), 417-425.
176. Raiffa, H. and Schlaifer, R. (1961) Applied Statistical Decision Theory. Harvard University, Boston, MA.
177. Rao, C. R. (1973) Linear Statistical Inference and Its Applications. John Wiley & Sons, Inc., New York.
178. Rao, J. S. and Sobel, M. (1980) Incomplete Dirichlet integrals with applications to ordered uniform spacings. Journal of Multivariate Analysis, 10, 603-610.
179. Rayens, W. S. and Srinivasan, C. (1994) Dependence properties of generalized Liouville distributions on the simplex. Journal of the American Statistical Association 89, 1465-1470.
180. Ronning, G. (1989) Maximumlikelihood estimation of Dirichlet distributions. Journal of Statistical Computing and Simulation, 32, 215-221.
181. Rubin, D. B. (1974) Characterizing the estimation of parameters in incompletedata problems. Journal of the American Statistical Association, 69, 467-474.
182. Rubin, D. R. (1976) Inference and missing data. Biometrika, 63, 581-592.
183. Rubin, D. B. (1987) Multiple Imputation for Non-response in Surveys. John Wiley & Sons, Inc., New York.
184. Rubin, D. B. (1988) Using the SIR algorithm to simulate posterior distributions (with discussion). In Bayesian Statistics, Vol. 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds), 395-402. Oxford University Press, Oxford.
185. Sahai, H. and Anderson, R. L. (1973) Confidence regions for variance ratios of random models for balanced data. Journal of the American Statistical Association, 68, 951-952.
186. Schafer, J. L. (1997) Analysis of Incomplete Multivariate Data. Chapman & Hall, London.
187. Schervish, M. J. and Carlin, B. P. (1992) On the convergence of successive substitution sampling. Journal of Computational and Graphical Statistics, 1, 111-127.
188. Self, S. G. and Liang, K. Y. (1987) Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under non-standard conditions. Journal of the American Statistical Association, 82, 605-610.
189. Sivazlian, B. D. (1981a) On a multivariate extension of gamma and beta distributions. SIAM Journal of Applied Mathematics, 41, 205-209.
190. Sivazlian, B. D. (1981b) A class of multivariate distributions. Australian Journal of Statistics, 23, 251-255.
191. Sjölander, K., Karplus, K., Brown, M. et al. (1996) Dirichlet mixtures: a method for improved detection of weak but significant protein sequence homology. Computer Applications in the Biosciences, 12, 327-345.
192. Skellam, J. G. (1948) A probability distribution derived from the binomial distribution by regarding the probability of success as variable between the sets of trials. Journal of the Royal Statistical Society, Series B (Methodological), 10 (2), 257-261.
193. Spiegelhalter, D. J., Harris, N. L., Bull, K. and Franklin, R. C. G. (1994) Empiricalevaluation of prior beliefs about frequencies: methodology and a case-study in congenital heart disease. Journal of the American Statistical Society, 89, 435-443.
194. Springer,M. D. (1979) The Algebra of Random Variables. JohnWiley&Sons, Inc., New York.
195. Srivastava, M. S. (2002) Methods of Multivariate Statistics. John Wiley & Sons, Inc., New York.
196. Stuart, A. and Ord, J. K. (1987) Kendall's Advanced Theory of Statistics: Distribution Theory (Vol. 1, 5th edn) Oxford University Press, Oxford.
197. Szántai, T. (1985) Numerical evaluation of probabilities concerning multidimensional probability distributions. Thesis, Hungarian Academy of Sciences, Budapest.
198. Tan, M., Tian, G. L. and Ng, K. W. (2003) A non-iterative sampling method for computing posteriors in the structure of EM-type algorithms. Statistica Sinica, 13 (3), 625-639.
199. Tan, M., Tian, G. L. and Ng, K. W. (2010) Bayesian Missing Data Problems:EM, Data Augmentation and Non-iterative Computation. Chapman&Hall/CRC, Boca Raton, FL.
200. Tang, M. L., Ng, K.W., Tian, G. L. and TanM. (2007) On improved EMalgorithm and confidence interval construction for incomplete r × c tables. Computational Statistics and Data Analysis, 51, 2919-2933.
201. Tanner, M. A. (1993) Tools for Statistical Inference: Observed Data and Data Augmentation Methods (3rd printing of the 1st edn). Lecture Notes in Statistcs 67. Springer, New York.
202. Tanner,M. A. (1996) Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions (3rd edn) Springer, New York.
203. Tanner, M. A. and Wong, W. H. (1987) The calculation of posterior distributions by data augmentation (with discussion) Journal of the American Statistical Association, 82, 528-540.
204. Tarone, R. E. (1979) Testing the goodness of fit of the binomial distribution. Biometrika, 66 (3), 585-590.
205. Theil, H. (1975) Theory and Measurement of Consumer Demand (Vol. 1). North-Holland, Amsterdam.
206. Thomas, S. and George, S. (2004) A review of Dirichlet distribution and its generalizations. Journal of the Indian Society for Probability and Statistics, 8, 72-91.
207. Thomas, S. and Thannippara, A. (2008) Distribution of theΛ-criterion for sphericity test and its connection to a generalized Dirichlet model. Communications in Statistics: Simulation and Computation, 37, 1385-1395.
208. Tian, G. L., Ng, K.W. and Geng, Z. (2003) Bayesian computation for contingency tables with incomplete cell-counts. Statistica Sinica, 13, 189-206.
209. Tian, G. L.,Tan, M. and Ng, K.W. (2007) Anexact non-iterative sampling procedure for discrete missing data problems. Statistica Neerlandica, 61 (2), 232-242.
210. Tian, G. L., Tang, M. L., Yuen, K. C. and Ng, K.W. (2010). Further properties and new applications for the nested Dirichlet distribution. Computational Statistics and Data Analysis, 54, 394-405.
211. Tiao, G. G. and Guttman, I. (1965) The inverted Dirichlet distribution with applications. Journal of the American Statistical Association, 60, 793-805. With corrigendum, 1251-1252.
212. Tierney, L. and Kadane, J. B. (1986) Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association, 81, 82-86.
213. van Dyk, D. and Meng, X. L. (2001) The art of data augmentation (with discussion) Journal of Computational and Graphical Statistics, 10, 1-50.
214. Viana, M. A. G. (1994) Bayesian small-sample estimation of misclassified multinomial data. Biometrics, 50, 237-243.
215. Viana, M. A. G., Ramakrishnan, V. and Levy, P. S. (1993) Bayesian analysis of prevalence from the results of small screening samples. Communications in Statistics: Theory and Methods, 22 (2), 575-585.
216. Wang, Y. and Fang, K. T. (1996) Uniform design of experiments with mixtures. Science in China, Series A, 26, 1-10.
217. Ware, J. H., Dockery, D. W., Spiro III, A. et al. (1984) Passive smoking, gas cooking, and respiratory health of children living in six cities. American Reviewof Respiratory Diseases, 129, 366-374.
218. Wei, G. C. G. and Tanner, M. A. (1990a) A Monte Carlo implementation of the EM algorithm and the poor man's data augmentation algorithm. Journal of the American Statistical Association, 85, 699-704.
219. Wei, G. C. G. and Tanner, M. A. (1990b) Calculating the content and boundary of the HPD region via data augmentation. Biometrika, 77, 649-652.
220. Wei, G. C. G. and Tanner, M. A. (1990c) Posterior computations for censored regression data. Journal of the American Statistical Association, 85, 829-839.
221. Wicker, N., Muller, J., Kalathur, R. K. R. and Poch, O. (2008) A maximum likelihood approximation method for Dirichlet's parameter estimation. Computational Statistics and Data Analysis, 52, 1315-1322.
222. Wilks, S. S. (1962) Mathematical Statistics. John Wiley & Sons, Inc., New York.
223. Williamson, G. D. and Haber,M. (1994) Models for three-dimensional contingency tables with completely and partially cross-classified data. Biometrics, 50, 194-203.
224. Wong, T. T. (1998) Generalized Dirichlet distribution in Bayesian analysis. Applied Mathematics and Computation, 97, 165-181.
225. Wong,W. H. and Li, B. (1992) Laplace expansion for posterior densities of nonlinear function of parameters. Biometrika, 79, 393-398.
226. Wu, C. F. J. (1983) On the convergence properties of the EM algorithm. Annals of Statistics, 11, 95-103.
227. Yassaee, H. (1974) Inverted Dirichlet distribution and multivariate logistic distribution. Canadian Journal of Statistics, 2 (1), 99-105.
228. Yassaee, H. (1976) Probability integral of Dirichlet distribution and its application. In COMPSTAT: Proceedings of the Second Symposium on Computational Statistics (J. Gordesch and P. Naeve, eds), 64-71. Physica-Verlag, Wien, Germany.
229. Yassaee, H. (1981) On integral of Dirichlet distributions and their applications. Communications in Statistics: Theory and Methods, 10 (9), 897-906.
230. Zhang, Y. T. (2000) An Introduction to the Statistical Analysis of Compositional Data. Science Press, Beijing, P.R. China.
231. Zheng, Z. G. (1987) Random weighting methods. Acta Mathematical Applicatae Sinica 10, 247-253.
232. Zheng, Z. G. (1992) Random weighting method: another approach to approximate the error distribution of estimators. In The Development of Statistics: Recent Contributions from China (X. R. Chen, K. T. Fang and C. C. Yang, eds), 219-235. Longman Scientific and Technical, UK.
233. Zhou, X. H., Obuchowski, N. A. and McClish, D. K. (2002) Statistical Methods in Diagnostic Medicine. John Wiley & Sons, Inc., New York.
234. Zolotarev, V. M. (1985) One-Dimensional Stable Distributions. Translations of Mathematical Monographs, Vol. 65. American Mathematical Society, Providence, RI.