Estimating the parameters of a BINMA Poisson model for a non-stationary bivariate time series

This article proposes a novel non-stationary BINMA time series model by extending two INMA processes where their innovation series follow the bivariate Poisson under time-varying moment assumptions. This article also demonstrates, through simulation studies, the use and superiority of the generalized quasi-likelihood (GQL) approach to estimate the regression effects, which is computationally less complicated as compared to conditional maximum likelihood estimation (CMLE) and the feasible generalized least squares (FGLS). The serial and bivariate dependence correlations are estimated by a robust method of moments.     

Comparative GMM and GQL logistic regression models on hierarchical data

We often rely on the likelihood to obtain estimates of regression parameters but it is not readily available for generalized linear mixed models (GLMMs). Inferences for the regression coefficients and the covariance parameters are key in these models. We presented alternative approaches for analyzing binary data from a hierarchical structure that do not rely on any distributional assumptions: a generalized quasi-likelihood (GQL) approach and a generalized method of moments (GMM) approach. These are alternative approaches to the typical maximum-likelihood approximation approach in Statistical Analysis System (SAS) such as Laplace approximation (LAP). We examined and compared the performance of GQL and GMM approaches with multiple random effects to the LAP approach as used in PROC GLIMMIX, SAS. The GQL approach tends to produce unbiased estimates, whereas the LAP approach can lead to highly biased estimates for certain scenarios. The GQL approach produces more accurate estimates on both the regression coefficients and the covariance parameters with smaller standard errors as compared to the GMM approach. We found that both GQL and GMM approaches are less likely to result in non-convergence as opposed to the LAP approach. A simulation study was conducted and a numerical example was presented for illustrative purposes.     

A case study of MCB and SBMH stock transaction using a novel BINMA(1) with non-stationary NB correlated innovations

This paper focuses on the modeling of the intra-day transactions at the Stock Exchange Mauritius (SEM) of the two major banking companies: Mauritius Commercial Bank Group Limited (MCB) and State Bank of Mauritius Holdings Ltd (SBMH) in Mauritius using a flexible non-stationary bivariate integer-valued moving average of order 1 (BINMA(1)) process with negative binomial (NB) innovations that may cater for different levels of over-dispersion. The generalized quasi-likelihood (GQL) approach is used to estimate the regression, dependence and over-dispersion effects. However, for the over-dispersion parameters, the auto-covariance structure in the GQL is constructed using some higher order moments. This new model is tested over some Monte-Carlo experiments and is applied to analyze the inter-related intra-day series of volume of stocks for the two banking institutions using data collected from 3 August to 16 October 2015 in the presence of some time-varying covariates such as the news effect, Friday effect and time of the day effect.     

A BINAR(1) time-series model with cross-correlated COM–Poisson innovations

This article proposes a bivariate integer-valued autoregressive time-series model of order 1 (BINAR(1) with COM–Poisson marginals to analyze a pair of non stationary time series of counts. The interrelation between the series is induced by the correlated innovations, while the non stationarity is captured through a common set of time-dependent covariates that influence the count responses. The regression and dependence effects are estimated using generalized quasi-likelihood (GQL) approach. Simulation experiments are performed to assess the performance of the estimation algorithms. The proposed BINAR(1) process is applied to analyze a real-life series of day and night accidents in Mauritius.     

A non‐stationary bivariate INAR(1) process with a simple cross‐dependence: Estimation with some properties

This paper considers modelling of a non‐stationary bivariate integer‐valued autoregressive process of order 1 (BINAR(1)) where the cross‐dependence between the counting series is formed through the relationship of the current series with the previous‐lagged count series observations while the pair of innovations is independent and marginally Poisson. In addition, this paper proposes a generalised quasi‐likelihood (GQL) estimating equation based on the exact specification of the mean score and the auto‐covariance structure. The proposed approach is also compared with other popular techniques such as conditional maximum likelihood (CML), generalised least squares (GLS) and generalised method of moment (GMM) based on simulated data from the proposed BINAR(1). Moreover, the model is applied to weekly series of day and night road accidents arising in some regions of Mauritius and is compared with other existing BINAR(1) models.     

On probit versus logit dynamic mixed models for binary panel data

In this paper, we consider inferences in a binary dynamic mixed model. The existing estimation approaches mainly estimate the regression effects and the dynamic dependence parameters either through the estimation of the random effects or by avoiding the random effects technically. Under the assumption that the random effects follow a Gaussian distribution, we propose a generalized quasilikelihood (GQL) approach for the estimation of the parameters of the dynamic mixed models. The proposed approach is computationally less cumbersome than the exact maximum likelihood (ML) approach. We also carry out the GQL estimation under two competitive, namely, probit and logit mixed models, and discuss both the asymptotic and small-sample behaviour of their estimators.     

Composite likelihood estimation in multivariate data analysis

The authors propose two composite likelihood estimation procedures for multivariate models with regression/univariate and dependence parameters. One is a two‐stage method based on both univariate and bivariate margins. The other estimates all the parameters simultaneously based on bivariate margins. For some special cases, the authors compare their asymptotic efficiencies with the maximum likelihood method. The performance of the two methods is reasonable, except that the first procedure is inefficient for the regression parameters under strong dependence. The second approach is generally better for the regression parameters, but less efficient for the dependence parameters under weak dependence.     

GEE estimation of the covariance structure of a bivariate panel data model with an application to wage dynamics and the incidence of profit-sharing in West Germany

We propose a generalized estimating equations (GEE) approach to the estimation of the mean and covariance structure of bivariate time series processes of panel data. The one-step approach allows for mixed continuous and discrete dependent variables. A Monte Carlo Study is presented to compare our particular GEE estimator with more standard GEE-estimators. In the empirical illustration, we apply our estimator to the analysis of individual wage dynamics and the incidence of profit-sharing in West Germany. Our findings show that time-invariant unobserved individual ability jointly influences individual wages and participation in profit sharing schemes.     

A GQL estimation approach for analysing non-stationary over-dispersed BINAR(1) time series

This paper proposes a generalized quasi-likelihood (GQL) function for estimating the vector of regression and over-dispersion effects for the respective series in the bivariate integer-valued autoregressive process of order 1 (BINAR(1)) with Negative Binomial (NB) marginals. The auto-covariance function in the proposed GQL is computed using some ‘robust’ working structures. As for the BINAR(1) process, the inter-relation between the series is induced mainly by the correlated NB innovations that are subject to different levels of over-dispersion. The performance of the GQL approach is tested via some Monte-Carlo simulations under different combination of over-dispersion together with low and high serial- and cross-correlation parameters. The model is also applied to analyse a real-life series of day and night accidents in Mauritius.     

Robust quasi-likelihood inference in generalized linear mixed models with outliers

It is well known that in a traditional outlier-free situation, the generalized quasi-likelihood (GQL) approach [B.C. Sutradhar, On exact quasilikelihood inference in generalized linear mixed models, Sankhya: Indian J. Statist. 66 (2004), pp. 261–289] performs very well to obtain the consistent as well as the efficient estimates for the parameters involved in the generalized linear mixed models (GLMMs). In this paper, we first examine the effect of the presence of one or more outliers on the GQL estimation for the parameters in such GLMMs, especially in two important models such as count and binary mixed models. The outliers appear to cause serious biases and hence inconsistency in the estimation. As a remedy, we then propose a robust GQL (RGQL) approach in order to obtain the consistent estimates for the parameters in the GLMMs in the presence of one or more outliers. An extensive simulation study is conducted to examine the consistency performance of the proposed RGQL approach.     

Marginal semiparametric multivariate accelerated failure time model with generalized estimating equations

The semiparametric accelerated failure time (AFT) model is not as widely used as the Cox relative risk model due to computational difficulties. Recent developments in least squares estimation and induced smoothing estimating equations for censored data provide promising tools to make the AFT models more attractive in practice. For multivariate AFT models, we propose a generalized estimating equations (GEE) approach, extending the GEE to censored data. The consistency of the regression coefficient estimator is robust to misspecification of working covariance, and the efficiency is higher when the working covariance structure is closer to the truth. The marginal error distributions and regression coefficients are allowed to be unique for each margin or partially shared across margins as needed. The initial estimator is a rank-based estimator with Gehan’s weight, but obtained from an induced smoothing approach with computational ease. The resulting estimator is consistent and asymptotically normal, with variance estimated through a multiplier resampling method. In a large scale simulation study, our estimator was up to three times as efficient as the estimateor that ignores the within-cluster dependence, especially when the within-cluster dependence was strong. The methods were applied to the bivariate failure times data from a diabetic retinopathy study.     

Communication in Statistics-Theory and methods improved GQL estimation method for the generalised BINMA(1) model

In a recent research, the quasi-likelihood estimation methodology was developed to estimate the regression effects in the Generalized BINMA(1) (GBINMA(1)) process. The method provides consistent parameter estimates but, in the intermediate computations, moment estimating equations were used to estimate the serial- and cross-correlation parameters. This procedure may not result optimal parameter estimates, in particular, for the regression effects. This paper provides an alternative simpler GBINMA(1) process based on multivariate thinning properties where the main effects are estimated via a robust generalized quasi-likelihood (GQL) estimation approach. The two techniques are compared through some simulation experiments. A real-life data application is studied.     

Dependence Estimation for Marginal Models of Multivariate Survival Data

We have previously(Segal and Neuhaus, 1993) devised methods for obtaining marginal regression coefficients and associated variance estimates for multivariate survival data, using a synthesis of the Poisson regression formulation for univariate censored survival analysis and generalized estimating equations (GEE's). The method is parametric in that a baseline survival distribution is specified. Analogous semiparametric models, with unspecified baseline survival, have also been developed (Wei, Lin and Weissfeld, 1989; Lin, 1994).Common to both these approaches is the provision of robust variances for the regression parameters. However, none of this work has addressed the more difficult area of dependence estimation. While GEE approaches ostensibly provide such estimates, we show that there are problems adopting these with multivariate survival data. Further, we demonstrate that these problems can affect estimation of the regression coefficients themselves. An alternate, ad hoc approach to dependence estimation, based on design effects, is proposed and evaluated via simulation and illustrative examples. This revised version was published online in July 2006 with corrections to the Cover Date.     

On efficient inferences in familial-longitudinal binary models with two variance components

When a generalized linear mixed model (GLMM) with multiple (two or more) sources of random effects is considered, the inferences may vary depending on the nature of the random effects. For example, the inference in GLMMs with two independent random effects with two distinct components of dispersion will be different from the inference in GLMMs with two random effects in a two factor factorial design set-up. In this paper, we consider a familial-longitudinal model for repeated binary data where the binary response of an individual member of a family at a given time point is assumed to be influenced by the past responses of the member as well as two but independent sources of random family effects. For the estimation of the parameters of the proposed model, we discuss the well-known maximum-likelihood (ML) method as well as a generalized quasi-likelihood (GQL) approach. The main objective of the paper is to examine the relative asymptotic efficiency performance of the ML and GQL estimators for the regression effects, dynamic (longitudinal) dependence and variance parameters of the random family effects from two sources.     

Estimation Procedure for a Multiple Time Series Model

Given a multiple time series sharing common autoregressive patterns, we estimate an additive model. The autoregressive component and the individual random effects are estimated by integrating maximum likelihood estimation and best linear unbiased predictions in a backfitting algorithm. The simulation study illustrated that the estimation procedure provides an alternative to the Arellano–Bond generalized method of moments (GMM) estimator of the panel model when T > N and the Arellano–Bond generally diverges. The estimator has high predictive ability. In cases where T ≤ N, the backfitting estimator is at least comparable to Arellano–Bond estimator.     

An alternative approach to the analysis of longitudinal data via generalized estimating equations

The generalized estimating equations (GEE) introduced by Liang and Zeger (Biometrika 73 (1986) 13–22) have been widely used over the past decade to analyze longitudinal data. The method uses a generalized quasi-score function estimate for the regression coefficients, and moment estimates for the correlation parameters. Recently, Crowder (Biometrika 82 (1995) 407–410) has pointed out some pitfalls with the estimation of the correlation parameters in the GEE method. In this paper we present a new method for estimating the correlation parameters which overcomes those pitfalls. For some commonly assumed correlation structures, we obtain unique feasible estimates for the correlation parameters. Large sample properties of our estimates are also established.     

On familial Poisson mixed models with multi-dimensional random effects

When a generalized linear mixed model with multiple (two or more) sources of random effects is considered, the inferences may vary depending on the nature of the random effects. In this paper, we consider a familial Poisson mixed model where each of the count responses of a family are influenced by two independent unobservable familial random effects with two distinct components of dispersion. A generalized quasilikelihood (GQL) approach is discussed for the estimation of the dispersion components as well as the regression effects of the model. A simulation study is conducted to examine the relative performance of the GQL approach as opposed to a simpler method of moments. Furthermore, the GQL estimation methodology is illustrated by using health care utilization data that follow a Poisson mixed model with one component of dispersion and by using simulated asthma data that follow a Poisson mixed model with two sources of random effects with two distinct components of dispersion.     

A Monte Carlo comparison of GMM and QMLE estimators for short dynamic panel data models with spatial errors

We suggest a generalized spatial system GMM (SGMM) estimation for short dynamic panel data models with spatial errors and fixed effects when n is large and T is fixed (usually small). Monte Carlo studies are conducted to evaluate the finite sample properties with the quasi-maximum likelihood estimation (QMLE). The results show that, QMLE, with a proper approximation for initial observation, performs better than SGMM in general cases. However, it performs poorly when spatial dependence is large. QMLE and SGMM perform better for different parameters when there is unknown heteroscedasticity in the disturbances and the data are highly persistent. Both estimates are not sensitive to the treatment of initial values. Estimation of the spatial autoregressive parameter is generally biased when either the data are highly persistent or spatial dependence is large. Choices of spatial weights matrices and the sign of spatial dependence do affect the performance of the estimates, especially in the case of the heteroscedastic disturbance. We also give empirical guidelines for the model.     

Modification of GEE1 and linear mixed-effects models for heteroscedastic longitudinal Gaussian data

A characterization of GLMs is given. Modification of the Gaussian GEE1, modified GEE1, was applied to heteroscedastic longitudinal data, to which linear mixed-effects models are usually applied. The modified GEE1 models scale multivariate data to homoscedastic data maintaining the correlation structure and apply usual GEE1 to homoscedastic data, which needs no-diagnostics for diagonal variances. Relationships among multivariate linear regression methods, ordinary/generalized LS, naïve/modified GEE1, and linear mixed-effects models were discussed. An application showed modified GEE1 gave most efficient parameter estimation. Correct specification of the main diagonals of heteroscedastic data variance appears to be more important for efficient mean parameter estimation.