Finite mixtures of quantile and M-quantile regression models

Abstract

In this paper we define a finite mixture of quan- tile 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 ran- dom 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 empiri- cal performance of the proposed standard error estimates for model parameters. We considered a variety of empirical set- tings 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

M-Quantile Regression for Multivariate Longitudinal Data with an Application to the Millennium Cohort Study

“…Alfò et al. (2017) consider finite mixtures of quantile regression models where the discrete distribution represents a nonparametric estimate of an unspecified, possibly continuous, distribution for the random effects. The time constant random effect model has been extended to time‐varying subject‐speficic intercepts by Farcomeni (2012); Marino et al.…”

Robust finite mixture regression for heterogeneous targets

“…Weruaga and Vía (2015) proposed multivariate Gaussian mixture regression and used 1 penalty for sparseness of parameters. Besides mixture of GLMs, there have been many works on mixture of other continuous distributions such as t and Laplace distributions, mainly motivated by the needs of robust estimation for handling heavy tailed or skewed continuous distribution; see, e.g., Xian Wang et al (2004); Do˘ru and Arslan (2017); Alfò et al (2016); Dogru and Arslan (2016); Kyung Lim et al (2016); Bai et al (2016). However, these methods assume that the targets are of the same type, and only consider interrelationship among tasks with continuous outcomes.…”

Mixture Markov regression model with application to mosquito surveillance data analysis

“…To model the effect of exogenous variables, one may consider mixtures of regression models on time series. Existing mixture regression models include mixtures of generalized linear models (McLachlan and Peel, 2004), mixtures of nonparametric regression models (Gaffney and Smyth, 1999), or finite mixtures of quantile and M-quantile regression models (Alfò et al, 2016). However, these methods cannot be directly applied on mixture time series with exogenous covariates.…”