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Finite mixtures of quantile and M-quantile regression models. (English) Zbl 06697673
Summary: In this paper we define a finite mixture of quantile and M-quantile regression models for heterogeneous and /or for dependent/clustered data. Components of the finite mixture represent clusters of individuals with homogeneous values of model parameters. For its flexibility and ease of estimation, the proposed approaches can be extended to random coefficients with a higher dimension than the simple random intercept case. Estimation of model parameters is obtained through maximum likelihood, by implementing an EM-type algorithm. The standard error estimates for model parameters are obtained using the inverse of the observed information matrix, derived through the Oakes (J R Stat Soc Ser B 61:479-482, 1999) formula in the M-quantile setting, and through nonparametric bootstrap in the quantile case. We present a large scale simulation study to analyse the practical behaviour of the proposed model and to evaluate the empirical performance of the proposed standard error estimates for model parameters. We considered a variety of empirical settings in both the random intercept and the random coefficient case. The proposed modelling approaches are also applied to two well-known datasets which give further insights on their empirical behaviour.

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[1] Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. National Bureau of standards, Washington, DC (1964) · Zbl 0171.38503
[2] Aitkin, M, A general maximum likelihood analysis of overdispersion in generalized linear models, Stat. Comput., 6, 127-130, (1996)
[3] Aitkin, M, A general maximum likelihood analysis of variance components in generalized linear models, Biometrics, 55, 117-128, (1999) · Zbl 1059.62564
[4] Aitkin, M., Francis, B., Hinde, J.: Statistical Modelling in GLIM, 2nd edn. Oxford University Press, Oxford (2005) · Zbl 1078.62001
[5] Alfó, M; Trovato, G, Semiparametric mixture models for multivariate count data, with application, Econom. J., 7, 426-454, (2004) · Zbl 1064.62031
[6] Bianchi, A., Fabrizi, E., Salvati, N., Tzavidis, N.: M-quantile regression: diagnostics and parametric representation of the model. Working paper. http://www.sp.unipg.it/surwey/dowload/publications/24-mq-diagn.html (2015) · Zbl 0373.62038
[7] Bianchi, A; Salvati, N, Asymptotic properties and variance estimators of the M-quantile regression coefficients estimators, Commun. Stat., 44, 2416-2429, (2015) · Zbl 1329.62242
[8] Breckling, J; Chambers, R, \({M}\)-quantiles, Biometrika, 75, 761-771, (1988) · Zbl 0653.62024
[9] Cantoni, E; Ronchetti, E, Robust inference for generalized linear models, J. Am. Stat. Assoc., 96, 1022-1030, (2001) · Zbl 1072.62610
[10] Davis, C, Semi-parametric and non-parametric methods for the analysis of repeated measurements with applications to clinical trials, Stat. Med., 10, 1959-1980, (1991)
[11] DeSarbo, W; Cron, W, A maximum likelihood methodology for clusterwise regression, J. Classif., 5, 249-282, (1988) · Zbl 0692.62052
[12] Farcomeni, A, Quantile regression for longitudinal data based on latent Markov subject-specific parameters, Stat. Comput., 22, 141-152, (2012) · Zbl 1322.62206
[13] Follmann, D; Lambert, D, Generalizing logistic regression by nonparametric mixing, J. Am. Stat. Assoc., 84, 295-300, (1989)
[14] Friedl, H; Kauermann, G, Standard errors for EM estimates in generalized linear models with random effects, Biometrics, 56, 761-767, (2000) · Zbl 1060.62534
[15] Geraci, M; Bottai, M, Quantile regression for longitudinal data using the asymmetric Laplace distribution, Biostatistics, 8, 140-154, (2007) · Zbl 1170.62380
[16] Geraci, M; Bottai, M, Linear quantile mixed models, Stat. Comput., 24, 461-479, (2014) · Zbl 1325.62010
[17] Geyer, C; Thompson, E, Constrained Monte Carlo maximum likelihood for dependent data, J. R. Stat. Soc. B, 54, 657-699, (1992)
[18] Gueorguieva, R, A multivariate generalized linear mixed model for joint modelling of clustered outcomes in the exponential family, Stat. Model., 1, 177-193, (2001) · Zbl 1106.62060
[19] Hennig, C, Identifiability of models for clusterwise linear regression, J. Classif., 17, 273-296, (2000) · Zbl 1017.62058
[20] Huber, P, Robust estimation of a location parameter, Ann. Math. Stat., 35, 73-101, (1964) · Zbl 0136.39805
[21] Huber, P, Robust regression: asymptotics, conjectures and Monte Carlo, Ann. Stat., 1, 799-821, (1973) · Zbl 0289.62033
[22] Huber, P. J.: The behavior of maximum likelihood estimates under nonstandard conditions. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, pp. 221-233. Wiley, Amsterdam (1967) · Zbl 0212.21504
[23] Huber, P.J.: Robust Statistics. Wiley, Hoboken (1981) · Zbl 0536.62025
[24] Jank, W., Booth, J.: Efficiency of Monte Carlo EM and simulated maximum likelihood in two-stage hierarchical models. J. Comput. Graph. Stat. 12, 214-229 (2003) · Zbl 1017.62058
[25] Jones, MC, Expectiles and m-quantiles are quantiles, Stat. Probab. Lett., 20, 149-153, (1994) · Zbl 0801.62012
[26] Jung, S, Quasi-likelihood for Median regression models, J. Am. Stat. Assoc., 91, 251-257, (1996) · Zbl 0871.62060
[27] Koenker, R; Bassett, G, Regression quantiles, Econometrica, 46, 33-50, (1978) · Zbl 0373.62038
[28] Koenker, R; D’Orey, V, Computing regression quantiles, Biometrika, 93, 255-268, (1987)
[29] Kokic, P; Chambers, R; Breckling, J; Beare, S, A measure of production performance, J. Bus. Econ. Stat., 10, 419-435, (1997)
[30] Laird, NM, Nonparametric maximum likelihood estimation of a mixing distribution, J. Am. Stat. Assoc., 73, 805-811, (1978) · Zbl 0391.62029
[31] Liu, Q; Pierce, D, A note on Gaussian-Hermite quadrature, Biometrika, 81, 624-629, (1994) · Zbl 0813.65053
[32] Liu, Y; Bottai, M, Mixed-effects models for conditional quantiles with longitudinal data, Int. J. Biostat., 5, 1-22, (2009)
[33] Louis, T, Finding the observed information matrix when using the EM algorithm, J. R. Stat. Soc. Ser. B, 44, 226-233, (1982) · Zbl 0488.62018
[34] McCulloch, C, Maximum likelihood estimation of variance components for binary data, J. Am. Stat. Assoc., 89, 330-335, (1994) · Zbl 0800.62139
[35] Munkin, MK; Trivedi, PK, Simulated maximum likelihood estimation of multivariate mixed-Poisson regression models, with application, Econom. J., 2, 29-48, (1999) · Zbl 0935.91036
[36] Newey, W; Powell, J, Asymmetric least squares estimation and testing, Econometrica, 55, 819-847, (1987) · Zbl 0625.62047
[37] Oakes, D, Direct calculation of the information matrix via the EM algorithm, J. R. Stat. Soc. Ser. B, 61, 479-482, (1999) · Zbl 0913.62036
[38] Pinheiro, J; Bates, D, Approximations to the log-likelihood function in the nonlinear mixed-effects model, J. Comput. Graph. Stat., 4, 12-35, (1995)
[39] Press, W., Teukolsky, S., Vetterling, W., Flannery, B.: Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, New York (2007) · Zbl 1132.65001
[40] Street, J; Carroll, R; Ruppert, D, A note on computing robust regression estimates via iteratively reweighed least squares, Am. Stat., 42, 152-154, (1988)
[41] Treatment of Lead-Exposed Children (TLC) Trial Group: Safety and efficacy of succimer in toddlers with blood lead levels of 20-44 \(μ {\rm g/dl}\). Pediatr. Res. 48, 593-599 (2000) · Zbl 0459.62051
[42] Tzavidis, N., Salvati, N., Schmid, T., Flouri, E., Midouhas, E.: Longitudinal analysis of the Strengths and Difficulties Questionnaire scores of the Millennium Cohort Study children in England using M-quantile random effects regression. J. R. Stat. Soc. A. 179, 427-452 (2016) · Zbl 0825.62611
[43] Wang, P; Puterman, M; Cockburn, I; Le, N, Mixed Poisson regression models with covariate dependent rates, Biometrics, 52, 381-400, (1996) · Zbl 0875.62407
[44] Wang, Y; Lin, X; Zhu, M; Bai, Z, Robust estimation using the huber funtion with a data-dependent tuning constant, J. Comput. Graph. Stat., 16, 468-481, (2007)
[45] Wedel, M; DeSarbo, W, A mixture likelihood approach for generalized linear models, J. Classif., 12, 21-55, (1995) · Zbl 0825.62611
[46] White, H, A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity, Econometrica, 48, 817-838, (1980) · Zbl 0459.62051
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