Identifiability of a model for discrete frequency distributions with a multidimensional parameter space. (English) Zbl 1360.62112

The main aims of this paper are twofold. In the first part the authors shortly describe the problem of identifiability of a model for discrete frequency distributions with a multidimensional parameter space using the so-called CUB (combination of shifted binomial and discrete uniform variables) mixture models introduced recently by M. Iannario [Metron 68, No. 1, 87–94 (2010; Zbl 1301.62017)].
In the second part, being the most important contribution of this paper, a general approach described previously is applied to investigate the identifiability of nonlinear CUB (NLCUB) models. While CUB models fit rating and/or ranking data by means of a mixture of uniform and shifted binomial random variables, the main innovation of NLCUB models is that they can be used to model rating data with a non-constant intensity probabilities, i.e. probabilities moving from one rating to the next during the decision process. First, the authors discuss issues of parameter estimation of NLCUB models. Later, the conditions under which the models are identifiable are given. A short numerical study is included and several open issues are concisely discussed.


62F99 Parametric inference
62E10 Characterization and structure theory of statistical distributions
62F10 Point estimation
62-07 Data analysis (statistics) (MSC2010)


Zbl 1301.62017


CUB; catdata
Full Text: DOI


[1] H, Teicher, Identifiability of finite mixtures, Ann. Math. Stat., 34, 1265-1269, (1963) · Zbl 0137.12704
[2] Yakowitz, S.; Spragins, J., On the identifiability of finite mixtures, Ann. Math. Stat., 39, 209-214, (1968) · Zbl 0155.25703
[3] Chandra, S., On the mixtures of probability distributions, Scand. J. Stat., 4, 105-112, (1977) · Zbl 0369.60024
[4] Al-Hussaini, E.; Ahmad, K., On the identifiability of finite mixtures of distributions (corresp.), IEEE Trans. Inform Theory, 27, 5, 664-668, (1981) · Zbl 0465.60024
[5] McLachlan, G.; Peel, D., Finite mixture models, (2000), Wiley New York · Zbl 0963.62061
[6] Atienza, N.; Garcia-Heras, J.; Munoz-Pichardo, J., A new condition for identifiability of finite mixture distributions, Metrika, 63, 215-221, (2006) · Zbl 1095.62016
[7] Allman, E. S.; Matias, C.; Rhodes, J. A., Identifiability of parameters in latent structure models with many observed variables, Ann. Statist., 37, 6A, 3099-3132, (2009), URL http://dx.doi.org/10.1214/09-AOS689 · Zbl 1191.62003
[8] P. Coretto, C. Hennig, Identifiability for mixtures of distributions from a location-scale family with uniforms, Working Paper 3.186, DISES, Università di Salerno, 2007.
[9] Coretto, P.; Hennig, C., Maximum likelihood estimation of heterogeneous mixtures of Gaussian and uniform distributions, J. Statist. Plann. Inference, 141, 1, 462-473, (2011) · Zbl 1203.62017
[10] Ljung, L.; Glad, T., On global identifiability for arbitrary model parametrizations, Automatica, 30, 2, 265-276, (1994) · Zbl 0795.93026
[11] Iannario, M., On the identifiability of a mixture model for ordinal data, Metron LXVIII, 87-94, (2010) · Zbl 1301.62017
[12] Piccolo, D., On the moments of a mixture of uniform and shifted binomial random variables, Quad. Stat., 5, 85-104, (2003)
[13] D’Elia, A.; Piccolo, D., A mixture model for preference data analysis, Comput. Statist. Data Anal., 49, 917-934, (2005) · Zbl 1429.62077
[14] Agresti, A., Categorical data analysis, (2013), Wiley New York · Zbl 1281.62022
[15] G, Tutz, Regression for categorical data, (2012), Cambridge University Press Cambridge
[16] Manisera, M.; Zuccolotto, P., Modeling rating data with {\scn}onlinear {\sccub} models, Comput. Statist. Data Anal., 78, 100-118, (2014)
[17] Piccolo, D., Observed information matrix for {\scmub} models, Quad. Stat., 8, 33-78, (2006)
[18] Piccolo, D., Inferential issues on {\sccube} models with covariates, Commun. Stat. - Theory Methods, 43, (2014)
[19] Corduas, M.; Iannario, M.; Piccolo, D., A class of statistical models for evaluating services and performances, (Monari, P.; Bini, M.; Piccolo, D.; Salmaso, L., Statistical Methods for the Evaluation of Educational Services and Quality of Products, (2009), Springer), 99-117
[20] Iannario, M.; Piccolo, D., A new statistical model for the analysis of customer satisfaction, Qual Technol Quantit Manag., 7, 149-168, (2010)
[21] Iannario, M.; Piccolo, D., {\sccub} models: statistical methods and empirical evidence, (Kenett, R. S.; Salini, S., Modern Analysis of Customer Surveys, (2012), Wiley New York), 231-258
[22] Bonnini, S.; Piccolo, D.; Salmaso, L.; Solmi, F., Permutation inference for a class of mixture models, Commun. Stat. - Theory Methods, 41, 2879-2895, (2012) · Zbl 1319.62092
[23] Bonnini, S.; Salmaso, L.; Solmi, F., Nonparametric multivariate inference via permutation tests for {\sccub} models, (Giudici, P.; Ingrassia, S.; Vichi, M., Statistical Models for Data Analysis, (2013), Springer International Publishing), 45-53
[24] Iannario, M., Modelling shelter choices in a class of mixture models for ordinal responses, Stat. Method Appl., 20, 1-22, (2012) · Zbl 1333.62181
[25] Iannario, M., Hierarchical {\sccub} models for ordinal variables, Commun. Stat. - Theory Methods, 41, 3110-3125, (2012) · Zbl 1296.62067
[26] Iannario, M., {\sccube} models for interpreting ordered categorical data with overdispersion, Quad. Stat., 14, 137-140, (2012)
[27] Iannario, M., Modelling uncertainty and overdispersion in ordinal data, Commun. Stat. - Theory Methods, 43, 771-786, (2014) · Zbl 1287.62001
[28] Grilli, L.; Iannario, M.; Piccolo, D.; Rampichini, C., Latent class {\sccub} models, Adv. Data Anal. Classif., 8, 105-119, (2013)
[29] Gambacorta, R.; Iannario, M.; Vallian, R., Design-based inference in a mixture model for ordinal variables for a two stage stratied design, Aust. N.Z. J. Stat., 56, 125-143, (2014) · Zbl 1336.62010
[30] Manisera, M.; Zuccolotto, P., Modeling “don’t know” responses in rating scales, Pattern Recogn. Lett., 45, 226-234, (2014)
[31] Gambacorta, R.; Iannario, M., Measuring job satisfaction with {\sccub} models, Labour, 27, 198-224, (2013)
[32] Iannario, M.; Piccolo, D., Statistical modelling of subjective survival probabilities, GENUS LXVI, 17-42, (2010)
[33] Capecchi, S.; Piccolo, D., Modelling the latent components of personal happiness, (Perna, C.; Sibillo, M., Mathematical and Statistical Methods for Actuarial Sciences and Finance, (2014), Springer), 49-52
[34] Balirano, G.; Corduas, M., Detecting semiotically expressed humor in diasporic TV productions, Humor, 3, 227-251, (2008)
[35] Iannario, M.; Manisera, M.; Piccolo, D.; Zuccolotto, P., Sensory analysis in the food industry as a tool for marketing decisions, Adv. Data Anal. Classif., 6, 303-321, (2012) · Zbl 1257.62124
[36] D’Elia, A., A statistical modelling approach for the analysis of TMD chronic pain data, Stat. Methods Med. Res., 17, 389-403, (2008) · Zbl 1156.62366
[37] Piccolo, D.; D’Elia, A., A new approach for modelling consumers’ preferences, Food Qual. Prefer, 19, 247-259, (2008)
[38] Manisera, M.; Zuccolotto, P., {\scn}onlinear {\sccub} models: some stylized facts, QdS - J. Methodol. Appl. Statist., 1-2, (2013)
[39] M. Manisera, P. Zuccolotto, Estimation of {\scN}onlinear {\scCUB} models via numerical optimization and {\scEM} algorithm, Working Paper, 2014.
[40] McCullagh, P., What is a statistical model?, Ann. Stat., 1225-1267, (2002) · Zbl 1039.62003
[41] Ljung, L., System identification, (1987), Prentice Hall · Zbl 0639.93059
[42] Stoica, P.; Selen, Y., Model-order selection: a review of information criterion rules, IEEE Signal Process. Mag., 21, 4, 36-47, (2004)
[43] Breiman, L., Bagging predictors, Mach. Learn, 24, 123-140, (1996) · Zbl 0858.68080
[44] Gray, R. M., Probability, random processes, and ergodic properties, (1988), Springer New York · Zbl 0644.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.