A Functional-Based Distribution Diagnostic for a Linear Model with Correlated Outcomes: Technical Report

Despite the widespread popularity of linear models for correlated outcomes (e.g. linear mixed modesl and time series models), distribution diagnostic methodology remains relatively underdeveloped in this context. In this paper we present an easy-to-implement approach that lends itself to graphical displays of model fit. Our approach involves multiplying the estimated marginal residual vector by the Cholesky decomposition of the inverse of the estimated marginal variance matrix. Linear functions or the resulting "rotated" residuals are used to construct an empirical cumulative distribution function (ECDF), whose stochastic limit is characterized. We describe a resampling technique that serves as a computationally efficient parametric bootstrap for generating representatives of the stochastic limit of the ECDF. Through functionals, such representatives are used to construct global tests for the hypothesis of normal margional errors. In addition, we demonstrate that the ECDF of the predicted random effects, as described by Lange and Ryan (1989), can be formulated as a special case of our approach. Thus, our method supports both omnibus and directed tests. Our method works well in a variety of circumstances, including models having independent units of sampling (clustered data) and models for which all observations are correlated (e.g., a single time series).

Cholesky Residuals for Assessing Normal Errors in a Linear Model with Correlated Outcomes: Technical Report

Despite the widespread popularity of linear models for correlated outcomes (e.g. linear mixed models and time series models), distribution diagnostic methodology remains relatively underdeveloped in this context. In this paper we present an easy-to-implement approach that lends itself to graphical displays of model fit. Our approach involves multiplying the estimated margional residual vector by the Cholesky decomposition of the inverse of the estimated margional variance matrix. The resulting "rotated" residuals are used to construct an empirical cumulative distribution function and pointwise standard errors. The theoretical framework, including conditions and asymptotic properties, involves technical details that are motivated by Lange and Ryan (1989), Pierce (1982), and Randles (1982). Our method appears to work well in a variety of circumstances, including models having independent units of sampling (clustered data) and models for which all observations are correlated (e.g., a single time series). Our methods can produce satisfactory results even for models that do not satisfy all of the technical conditions stated in our theory.

Bayesian change-point analysis in linear regression model with scale mixtures of normal distributions

In this thesis, we consider Bayesian inference on the detection of variance change-point models with scale mixtures of normal (for short SMN) distributions. This class of distributions is symmetric and thick-tailed and includes as special cases: Gaussian, Student-t, contaminated normal, and slash distributions. The proposed models provide greater flexibility to analyze a lot of practical data, which often show heavy-tail and may not satisfy the normal assumption. As to the Bayesian analysis, we specify some prior distributions for the unknown parameters in the variance change-point models with the SMN distributions. Due to the complexity of the joint posterior distribution, we propose an efficient Gibbs-type with Metropolis- Hastings sampling algorithm for posterior Bayesian inference. Thereafter, following the idea of [1], we consider the problems of the single and multiple change-point detections. The performance of the proposed procedures is illustrated and analyzed by simulation studies. A real application to the closing price data of U.S. stock market has been analyzed for illustrative purposes.

The evolutionary diversification of parrots supports a taxon pulse model with multiple trans-oceanic dispersal events and local radiations

Vicariance is thought to have played a major role in the evolution of modern parrots. However, as the relationships especially of the African taxa remained mostly unresolved, it has been difficult to draw firm conclusions about the roles of dispersal and vicariance. Our analyses using the broadest taxon sampling of old world parrots ever based on 3219 bp of three nuclear genes revealed well-resolved and congruent phylogenetic hypotheses. Agapornis of Africa and Madagascar was found to be the sister group to Loriculus of Australasia and Indo-Malayasia and together they clustered with the Australasian Loriinae, Cyclopsittacini and Melopsittacus. Poicephalus and Psittacus from mainland Africa formed the sister group Of the Neotropical Arini and Coracopsis from Madagascar and adjacent islands may be the closest relative of Psittrichas from New Guinea. These biogeographic relationships are best explained by independent colonization of the African continent via trans-oceanic dispersal from Australasia and Antarctica in the Paleogene following what may have been vicariance events in the late Cretaceous and/or early Paleogene. Our data support a taxon pulse model for the diversification of parrots whereby trans-oceanic dispersal played a more important role than previously thought and was the prerequisite for range expansion into new continents. (C) 2009 Elsevier Inc. All rights reserved