[1] Agresti, A. (2010), Analysis of Ordinal Categorical Data, 2nd edition. J.Wiley & Sons,
Hoboken.
[2] Aitkin, M. (1996), A general maximum likelihood analysis of overdispersion in generalized
linear models. Statistics and Computing, 6(3), 251{262.
[3] Amendola, C., Faugere, J.C. and Sturmfels, B. (2016), Moment Varieties of Gaussian
Mixtures, Journal of Algebraic Statistics, 7(1), 14{28.
[4] Baudry, J.P. and Celeux, G. (2015), EM for mixtures - Initialization requires special
care. Statistics and Computing, 25(4), 713{726.
[5] Biernacki, C., Celeux, G., Govaert, G. (2003), Choosing starting values for the EM
algorithm for getting the highest likelihood in multivariate Gaussian mixture models.
Computational Statistics & Data Analysis, 41(3-4), 561 { 575.
[6] Borisevich, V.D., Potemkin, V.G. and Strunkov, S.P. (2000), Global methods for
solving systems of nonlinear algebraic equations. Computer and mathematics with
applications, 40(8-9), 1015{1025.
[7] Buchberger, B. (1976), A theoretical basis for the reduction of polynomials to canonical
forms. In ACM SIGSAM Bulletin 39, pp. 19-29.
[8] Cox, D.R. (1983), Some remarks on overdispersion. Biometrika, 70, 269{274.
[9] Cox, D., Little, J., O'Shea, D. (1997), Ideals, Varieties, and Algorithms. Springer
Verlag, New York, Second Edition.
[10] Cramer, H. (1946), Mathematical Methods of Statistics. Princeton Mathematical Series,
Princeton University Press-Princeton.
[11] D'Elia, A. (2000), A shifted binomial model for rankings, Proceedings of the 15th
International Workshop on Statistical Modelling. New Trends in Statistical Modelling
(IWSM) - Bilbao, Spain, 412{416.
REFERENCES 26
[12] D'Elia, A. (2003), Modelling ranks using the Inverse Hypergeometric distribution.
Statistical Modelling: an International Journal, 3(1), 65{78.
[13] D'Elia, A. and Piccolo, D. (2005), A mixture model for preference data analysis,
Computational Statistics & Data analysis, 49(3), 917{934.
[14] de Finetti, B. and Paciello, U. (1930), Calcolo della dierenza media, Metron, 8(3),
89{94.
[15] Fitzmaurice, G.M., Heath, A.F. and Cox, D.R. (1997), Detecting overdispersion in
large scale surveys: Application to a study of education and social class in Britain.
Applied Statistics, 46(4), 415{432.
[16] Furman, W.D. and Lindsay, B.G. (1994), Measuring the relative eectiveness of moment
estimators as starting values in maximising likelihoods, Computational Statistics
& Data Analysis, 17(5), 493{507.
[17] Iannario, M. (2012a), CUBE models for interpreting ordered categorical data with
overdispersion, Quaderni di Statistica, 14(1), 137{140.
[18] Iannario, M. (2012b), Preliminary estimators for a mixture model of ordinal data.
Advances in Data Analysis and Classication, 6(3), 163{184.
[19] Iannario, M. (2012c). Modelling shelter choices in a class of mixture models for ordinal
responses. Statistical Methods and Applications, 21(1), 1{22.
[20] Iannario, M. (2014), Modelling uncertainty and overdispersion in ordinal data. Communications
in Statistics. Theory and Methods, 43(4), 771{786.
[21] Iannario, M. (2015), Detecting latent components in ordinal data with overdispersion
by means of a mixture distribution. Quality & Quantity, 49(3), 977{987.
[22] Iannario, M. (2016), Testing the overdispersion parameter in cube models. Communications
in Statistics: Simulation and Computation, 45(5), 1621{1635.
[23] Iannario, M. and Piccolo, D. (2012), CUB models: Statistical methods and empirical
evidence, in: Kenett R. S. and Salini S. (eds.), Modern Analysis of Customer Surveys:
with applications using R. Chichester: J. Wiley & Sons, 231{258.
[24] Iannario, M., Piccolo, D. and Simone, R. (2016), CUB: A Class of Mixture Models for
Ordinal Data (R package version 1.0), http://CRAN.R-project.org/package=CUB.
[25] Karlis, D. and Xekalaki, E. (1999), Improving the EM algorithm for mixtures. Statistics
and Computing, 9(4), 303{307.
[26] Karlis, D. and Xekalaki, E. (2003), Choosing initial values for the EM algorithm for
nite mixtures, Computational Statistics & Data Analysis, 41(3-4), 577{590.
REFERENCES 27
[27] Laird, N. (1978), Nonparametric Maximum Likelihood Estimation of a Mixing Distribution.
Journal of the American Statistical Association, 73(364), 805{811.
[28] Ligget, R.E. and Delwiche, J.F. (2005), The Beta-Binomial model: variability in
overdispersion across methods and over time, Journal of Sensory Studies, 20(1), 48{
61.
[29] Mahalanobis, P.C. (1936), On the generalised distance in statistics, Proceedings of
the National Institute of Sciences of India, 2(1), 49{55.
[30] McCullagh, P. (1980), Regression models for ordinal data. Journal of the Royal Statistical
Society, Series B, 42(2), 109{142.
[31] McLachlan, G. and Krishnan, T. (1997), The EM algorithm and extensions. New
York: J.Wiley & Sons.
[32] McLachlan, G. and Peel, G.J. (2000), Finite mixture models. New York: J.Wiley &
Sons.
[33] Piccolo, D. (2003), On the moments of a mixture of uniform and shifted binomial
random variables. Quaderni di Statistica, 5(1), 85{104.
[34] Piccolo, D. (2006). Observed information matrix for MUB models. Quaderni di Statistica,
8, 33{78.
[35] Piccolo, D. (2015), Inferential issues on CUBE models with covariates. Communications
in Statistics. Theory and Methods, 44(23), 771{786.
[36] Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (2007), Numerical
Recipes: The Art of Scientic Computing (3rd ed.), New York: Cambridge University
Press.
[37] Pistone, G., Riccomagno, E. and Wynn, H.P. (2000), Algebraic Statistics: Computational
Commutative Algebra in Statistics. Chapman & Hall/CRC, Monographs on
Statistics and Applied Probability.
[38] Rao, C.R. (1973), Linear Statistical Inference and its Applications, 2nd edition. New
York: J.Wiley & Sons.
[39] Ser
ing, R.J. (1980), Approximation Theorems of Mathematical Statistics, J.Wiley &
Sons.
[40] Shacham, M., Brauner, N. and Pozin, M. (1998), Comparing software for interactive
solution of systems of nonlinear algebraic equations, Computers and Chemical
Engineering, 22(1-2), 323{331.
[41] 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, 10(2), 257{261.
REFERENCES 28
[42] Tamhane, A., Ankemanman, B. and Yang, Y. (2002), The Beta distribution as a
latent response model for ordinal data (I): Estimation of location and dispersion
parameters. Journal of Statistical Computation and Simulation, 72(6), 473{494.
[43] Tourangeau, R., Rips, L. J., Rasinski, K. (2000). The Psychology of Survey Response.
Cambridge: Cambridge University Press.
[44] Tripathi, R.C., Gupta, R.C. and Gurland, J. (1994), Estimation of parameters in Beta
Binomial models. Annals of the Institute of Statistical Mathematics, 46(2), 317{331.
[45] Wilcox, R.R. (1979), Estimating the parameters of the Beta-binomial distribution,
Educational and Psychological Measurement, 39(3), 527{535.
[46] Yamamoto, E., Yanagimoto, T. (1992), Moment estimators for the beta-binomial
distribution, Journal of Applied Statistics, 19(2), 273{283.
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