Browsing by Author "Hammujuddy, Mohammad Jahvaid."

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A classical approach for the analysis of generalized linear mixed models.
(2004) Hammujuddy, Mohammad Jahvaid. ; Matthews, Glenda Beverley.
Generalized linear mixed models (GLMMs) accommodate the study of overdispersion and correlation inherent in hierarchically structured data. These models are an extension of generalized linear models (GLMs) and linear mixed models (LMMs). The linear predictor of a GLM is extended to include an unobserved, albeit realized, vector of Gaussian distributed random effects. Conditional on these random effects, responses are assumed to be independent. The objective function for parameter estimation is an integrated quasi-likelihood (IQL) function which is often intractable since it may consist of high-dimensional integrals. Therefore, an exact maximum likelihood analysis is not feasible. The penalized quasi-likelihood (PQL) function, derived from a first-order Laplace expansion to the IQL about the optimum value of the random effects and under the assumption of slowly varying weights, is an approximate technique for statistical inference in GLMMs. Replacing the conditional weighted quasi-deviance function in the Laplace-approximated IQL by the generalized chi-squared statistic leads to a corrected profile quasilikelihood function for the restricted maximum likelihood (REML) estimation of dispersion components by Fisher scoring. Evaluation of mean parameters, for fixed dispersion components, by iterative weighted least squares (IWLS) yields joint estimates of fixed effects and random effects. Thus, the PQL criterion involves repeated fitting of a Gaussian LMM with a linked response vector and a conditional iterated weight matrix. In some instances, PQL estimates fail to converge to a neighbourhood of their true values. Bias-corrected PQL estimators (CPQL) have hence been proposed, using asymptotic analysis and simulation. The pseudo-likelihood algorithm is an alternative estimation procedure for GLMMs. Global score statistics for hypothesis testing of overdispersion, correlation and heterogeneity in GLMMs has been developed as well as individual score statistics for testing null dispersion components separately. A conditional mean squared error of prediction (CMSEP) has also been considered as a general measure of predictive uncertainty. Local influence measures for testing the robustness of parameter estimates, by inducing minor perturbations into GLMMs, are recent advances in the study of these models. Commercial statistical software is available for the analysis of GLMMs.
Modelling obesity risk factors among adult females in South Africa via a GLMM: classical and bayesian approaches.
(2016) Pillay, Telissa.; Hammujuddy, Mohammad Jahvaid.
Abstract available in PDF file.
Multivariate elliptically contoured stable distributions with applications to BRICS financial data.
(2016) Naradh, Kimera.; Chinhamu, Knowledge.; Hammujuddy, Mohammad Jahvaid.; Chifurira, Retius.
Brazil, Russia, India, China and South Africa (BRICS) are regarded as the ve major emerging economies where all members are a part of a select group of developing industrialized countries. In the nancial industry, various models are used for the description and analysis of nancial trends. One of these models is the family of stable distributions which takes into account the skewness and heavy tails that are frequent in nancial data. The main objective of this study is to investigate the t of stable distributions for exchange rates of each of the BRICS countries against the U.S. Dollar in both the univariate and multivariate cases. The data set consists of exchange rate data from the period January 2011 to January 2016. Nolan's S0 -parameterization stable distribution was tted using the maximum likelihood method in the univariate case and in a tted stable model where a GARCH (1,1) lter was applied to the returns (Stable-GARCH(1,1)). The Kolmogorov-Smirnov test and the Anderson-Darling test show that stable distributions adequately t the returns of BRICS nancial data. Value-at-Risk (VaR) calculations and VaR in-sample backtesting using the Kupiec likelihood ratio test and the Christo ersen's conditional coverage test were applied as per the International Basel Regulatory where the robustness of each model describing the nancial data was evaluated. Thereafter, we proceeded to t bivariate elliptical stable models using the Rachev-Xin-Cheng method after visualizing the scatterplot matrix of BRICS countries. This study validates the usefulness of stable distributions for modelling BRICS nancial data.