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The multilevel models are used in applied sciences including social sciences, sociology, medicine, economic for analysing correlated data. There are various approaches to estimate the model parameters when the responses are normally distributed. To implement the Bayesian approach, a generalized version of the Markov Chain Monte Carlo algorithm, which has a simple structure and removes the correlations among the simulated samples for the fixed parameters and the errors in higher levels, is used in this article. Because the dimension of the covariance matrix for the new error vector is increased, based upon the Cholesky decomposition of the covariance matrix, two methods are proposed to speed the convergence of this approach. Then, the performances of these methods are evaluated in a simulation study and real life data.

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Introducing some efficient model selection criteria for mixed models is a substantial challenge; Its source is indeed fitting the model and computing the maximum likelihood estimates of the parameters. Data cloning is a new method to fit mixed models efficiently in a likelihood-based approach. This method has been popular recently and avoids the main problems of other likelihood-based methods in mixed models. A disadvantage of data cloning is its inability of computing the maximum of likelihood function of the model. This value is a key quantity in proposing and calculating information criteria. Therefore, it seems that we can not, directly, define an appropriate information criterion by data cloning approach. In this paper, this believe is broken and a criterion based on data cloning is introduced. The performance of the proposed model selection criterion is also evaluated by a simulation study.

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In this paper, we present a comprehensive review and comparative analysis of estimation methods for periodic autoregressive (PAR) models driven by scale mixture of skew-normal (SMSN) innovations, a flexible class suitable for modeling both symmetric and asymmetric data. Expectation-conditional maximization algorithms are employed to develop maximum likelihood, maximum a posteriori, and Bayesian estimation procedures. A thorough evaluation of these methods is conducted using simulation studies, with particular attention to asymptotic properties and robustness against outliers, high peaks, and heavy tails. To demonstrate their practical utility, these methods are applied to monthly Google stock price data.