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.     

Alternative GMM estimators for first-order autoregressive panel model: An improving efficiency approach

This article considers first-order autoregressive panel model that is a simple model for dynamic panel data (DPD) models. The generalized method of moments (GMM) gives efficient estimators for these models. This efficiency is affected by the choice of the weighting matrix that has been used in GMM estimation. The non-optimal weighting matrices have been used in the conventional GMM estimators. This led to a loss of efficiency. Therefore, we present new GMM estimators based on optimal or suboptimal weighting matrices. Monte Carlo study indicates that the bias and efficiency of the new estimators are more reliable than the conventional estimators.     

GMM nonparametric correction methods for logistic regression with error‐contaminated covariates and partially observed instrumental variables

We consider logistic regression with covariate measurement error. Most existing approaches require certain replicates of the error‐contaminated covariates, which may not be available in the data. We propose generalized method of moments (GMM) nonparametric correction approaches that use instrumental variables observed in a calibration subsample. The instrumental variable is related to the underlying true covariates through a general nonparametric model, and the probability of being in the calibration subsample may depend on the observed variables. We first take a simple approach adopting the inverse selection probability weighting technique using the calibration subsample. We then improve the approach based on the GMM using the whole sample. The asymptotic properties are derived, and the finite sample performance is evaluated through simulation studies and an application to a real data set.     

Efficient M-estimators with auxiliary information

This paper introduces a new class of M-estimators based on generalised empirical likelihood (GEL) estimation with some auxiliary information available in the sample. The resulting class of estimators is efficient in the sense that it achieves the same asymptotic lower bound as that of the efficient generalised method of moment (GMM) estimator with the same auxiliary information. The paper also shows that in case of smooth estimating equations the proposed estimators enjoy a small second order bias property compared to both efficient GMM and full GEL estimators. Analytical formulae to obtain bias corrected estimators are also provided. Simulations show that with correctly specified auxiliary information the proposed estimators and in particular those based on empirical likelihood outperform standard M and efficient GMM estimators both in terms of finite sample bias and efficiency. On the other hand with moderately misspecified auxiliary information estimators based on the nonparametric tilting method are typically characterised by the best finite sample properties.     

Probability density estimation via an infinite Gaussian mixture model: application to statistical process monitoring

Summary.  The primary goal of multivariate statistical process performance monitoring is to identify deviations from normal operation within a manufacturing process. The basis of the monitoring schemes is historical data that have been collected when the process is running under normal operating conditions. These data are then used to establish confidence bounds to detect the onset of process deviations. In contrast with the traditional approaches that are based on the Gaussian assumption, this paper proposes the application of the infinite Gaussian mixture model (GMM) for the calculation of the confidence bounds, thereby relaxing the previous restrictive assumption. The infinite GMM is a special case of Dirichlet process mixtures and is introduced as the limit of the finite GMM, i.e. when the number of mixtures tends to ∞. On the basis of the estimation of the probability density function, via the infinite GMM, the confidence bounds are calculated by using the bootstrap algorithm. The methodology proposed is demonstrated through its application to a simulated continuous chemical process, and a batch semiconductor manufacturing process.     

Multiscale Gaussian convolution algorithm for estimate of Gaussian mixture model

Abstract

This paper introduces a multiscale Gaussian convolution model of Gaussian mixture (MGC-GMM) via the convolution of the GMM and a multiscale Gaussian window function. It is found that the MGC-GMM is still a Gaussian mixture model, and its parameters can be mapped back to the parameters of the GMM. Meanwhile, the multiscale probability density function (MPDF) of the MGC-GMM can be viewed as the mathematical expectation of a random process induced by the Gaussian window function and the GMM, which can be directly estimated by the use of sample data. Based on the estimated MPDF, a novel algorithm denoted by the MGC is proposed for the selection of model and the parameter estimates of the GMM, where the component number and the means of the GMM are respectively determined by the number and the locations of the maximum points of the MPDF, and the numerical algorithms for the weight and variance parameters of the GMM are derived. The MGC is suitable for the GMM with diagonal covariance matrices. A MGC-EM algorithm is also presented for the generalized GMM, where the GMM is estimated using the EM algorithm by taking the estimates from the MGC as initial parameters of the GMM model. The proposed algorithms are tested via a series of simulated sample sets from the given GMM models, and the results show that the proposed algorithms can effectively estimate the GMM model.     

Estimation with unbalanced panel data having covariate measurement error

Panel data with covariate measurement error appear frequently in various studies. Due to the sampling design and/or missing data, panel data are often unbalanced in the sense that panels have different sizes. For balanced panel data (i.e., panels having the same size), there exists a generalized method of moments (GMM) approach for adjusting covariate measurement error, which does not require additional validation data. This paper extends the GMM approach of adjusting covariate measurement error to unbalanced panel data. Two health related longitudinal surveys are used to illustrate the implementation of the proposed method.     

GMM tests for the Katz family of distributions

Generalized method of moments (GMM) is used to develop tests for discriminating discrete distributions among the two-parameter family of Katz distributions. Relationships involving moments are exploited to obtain identifying and over-identifying restrictions. The asymptotic relative efficiencies of tests based on GMM are analyzed using the local power approach and the approximate Bahadur efficiency. The paper also gives results of Monte Carlo experiments designed to check the validity of the theoretical findings and to shed light on the small sample properties of the proposed tests. Extensions of the results to compound Poisson alternative hypotheses are discussed.     

On the asymptotic non‐equivalence of efficient‐GMM and MEL estimators in models with missing data

The generalized method of moments (GMM) and empirical likelihood (EL) are popular methods for combining sample and auxiliary information. These methods are used in very diverse fields of research, where competing theories often suggest variables satisfying different moment conditions. Results in the literature have shown that the efficient‐GMM (GMM_E) and maximum empirical likelihood (MEL) estimators have the same asymptotic distribution to order n^?1/2 and that both estimators are asymptotically semiparametric efficient. In this paper, we demonstrate that when data are missing at random from the sample, the utilization of some well‐known missing‐data handling approaches proposed in the literature can yield GMM_E and MEL estimators with nonidentical properties; in particular, it is shown that the GMM_E estimator is semiparametric efficient under all the missing‐data handling approaches considered but that the MEL estimator is not always efficient. A thorough examination of the reason for the nonequivalence of the two estimators is presented. A particularly strong feature of our analysis is that we do not assume smoothness in the underlying moment conditions. Our results are thus relevant to situations involving nonsmooth estimating functions, including quantile and rank regressions, robust estimation, the estimation of receiver operating characteristic (ROC) curves, and so on.     

Oracle GMM estimation for misspecified models via thresholding

We propose a thresholding generalized method of moments (GMM) estimator for misspecified time series moment condition models. This estimator has the following oracle property: its asymptotic behavior is the same as of any efficient GMM estimator obtained under the a priori information that the true model were known. We propose data adaptive selection methods for thresholding parameter using multiple testing procedures. We determine the limiting null distributions of classical parameter tests and show the consistency of the corresponding block-bootstrap tests used in conjunction with thresholding GMM inference. We present the results of a simulation study for a misspecified instrumental variable regression model and for a vector autoregressive model with measurement error. We illustrate an application of the proposed methodology to data analysis of a real-world dataset.     

GMM estimation of spatial autoregressive models in a system of simultaneous equations with heteroskedasticity

This paper proposes a GMM estimation framework for the SAR model in a system of simultaneous equations with heteroskedastic disturbances. Besides linear moment conditions, the proposed GMM estimator also utilizes quadratic moment conditions based on the covariance structure of model disturbances within and across equations. Compared with the QML approach, the GMM estimator is easier to implement and robust under heteroskedasticity of unknown form. We derive the heteroskedasticity-robust standard error for the GMM estimator. Monte Carlo experiments show that the proposed GMM estimator performs well in finite samples.     

A Time-Series Study of the Formation and Predictive Performance of EEC Production Survey Expectations

Generalized method of moments (GMM) has been an important innovation in econometrics. Its usefulness has motivated a search for good inference procedures based on GMM. This article presents a novel method of bootstrapping for GMM based on resampling from the empirical likelihood distribution that imposes the moment restrictions. We show that this approach yields a large-sample improvement and is efficient, and give examples. We also discuss the development of GMM and other recent work on improved inference.     

Some non-parametric tests for duration dependence: an application to UK business cycle data

A distinction between Fisher's implied data-generating process for Monte Carlo cycles and the more general Markov process leads to non-parametric tests for duration dependence. Tests are based on the method of moments, Tauchen's generalized method of moments (GMM) procedure, and a statistic whose null distribution probability limit is zero. Using finite-sample critical values obtained by Monte Carlo methods, our test results are remarkably consistent. The null distribution of the GMM test statistic for samples of the size considered is distinctly non-normal, so that asymptotic critical values give erroneous results. The tests are applied to UK business cycle data for 1854-1992. There is evidence for duration dependence in expansions but not in contractions.     

Robust inference for estimating equations with nonignorably missing data based on SIR algorithm

Nonresponse is a very common phenomenon in survey sampling. Nonignorable nonresponse – that is, a response mechanism that depends on the values of the variable having nonresponse – is the most difficult type of nonresponse to handle. This article develops a robust estimation approach to estimating equations (EEs) by incorporating the modelling of nonignorably missing data, the generalized method of moments (GMM) method and the imputation of EEs via the observed data rather than the imputed missing values when some responses are subject to nonignorably missingness. Based on a particular semiparametric logistic model for nonignorable missing response, this paper proposes the modified EEs to calculate the conditional expectation under nonignorably missing data. We can apply the GMM to infer the parameters. The advantage of our method is that it replaces the non-parametric kernel-smoothing with a parametric sampling importance resampling (SIR) procedure to avoid nonparametric kernel-smoothing problems with high dimensional covariates. The proposed method is shown to be more robust than some current approaches by the simulations.     

Double filter instrumental variable estimation of panel data models with weakly exogenous variables

In this article, we propose instrumental variables (IV) and generalized method of moments (GMM) estimators for panel data models with weakly exogenous variables. The model is allowed to include heterogeneous time trends besides the standard fixed effects (FE). The proposed IV and GMM estimators are obtained by applying a forward filter to the model and a backward filter to the instruments in order to remove FE, thereby called the double filter IV and GMM estimators. We derive the asymptotic properties of the proposed estimators under fixed T and large N, and large T and large N asymptotics where N and T denote the dimensions of cross section and time series, respectively. It is shown that the proposed IV estimator has the same asymptotic distribution as the bias corrected FE estimator when both N and T are large. Monte Carlo simulation results reveal that the proposed estimator performs well in finite samples and outperforms the conventional IV/GMM estimators using instruments in levels in many cases.     

Structural change tests for GEL criteria

This article examines structural change tests based on generalized empirical likelihood methods in the time series context, allowing for dependent data. Standard structural change tests for the Generalized method of moments (GMM) are adapted to the generalized empirical likelihood (GEL) context. We show that when moment conditions are properly smoothed, these test statistics converge to the same asymptotic distribution as in the GMM, in cases with known and unknown breakpoints. New test statistics specific to GEL methods, and that are robust to weak identification, are also introduced. A simulation study examines the small sample properties of the tests and reveals that GEL-based robust tests performed well, both in terms of the presence and location of a structural change and in terms of the nature of identification.     

Combining least-squares and quantile regressions

Least-squares and quantile regressions are method of moments techniques that are typically used in isolation. A leading example where efficiency may be gained by combining least-squares and quantile regressions is one where some information on the error quantiles is available but the error distribution cannot be fully specified. This estimation problem may be cast in terms of solving an over-determined estimating equation (EE) system for which the generalized method of moments (GMM) and empirical likelihood (EL) are approaches of recognized importance. The major difficulty with implementing these techniques here is that the EEs associated with the quantiles are non-differentiable. In this paper, we develop a kernel-based smoothing technique for non-smooth EEs, and derive the asymptotic properties of the GMM and maximum smoothed EL (MSEL) estimators based on the smoothed EEs. Via a simulation study, we investigate the finite sample properties of the GMM and MSEL estimators that combine least-squares and quantile moment relationships. Applications to real datasets are also considered.     

Prediction of accrual closure date in multi-center clinical trials with discrete-time Poisson process models

In a phase III multi‐center cancer clinical trial or a large public health study, sample size is predetermined to achieve desired power, and study participants are enrolled from tens or hundreds of participating institutions. As the accrual is closing to the target size, the coordinating data center needs to project the accrual closure date on the basis of the observed accrual pattern and notify the participating sites several weeks in advance. In the past, projections were simply based on some crude assessment, and conservative measures were incorporated in order to achieve the target accrual size. This approach often resulted in excessive accrual size and subsequently unnecessary financial burden on the study sponsors. Here we proposed a discrete‐time Poisson process‐based method to estimate the accrual rate at time of projection and subsequently the trial closure date. To ensure that target size would be reached with high confidence, we also proposed a conservative method for the closure date projection. The proposed method was illustrated through the analysis of the accrual data of the National Surgical Adjuvant Breast and Bowel Project trial B‐38. The results showed that application of the proposed method could help to save considerable amount of expenditure in patient management without compromising the accrual goal in multi‐center clinical trials. Copyright © 2012 John Wiley & Sons, Ltd.     

Unit root test for short panels with serially correlated errors

This paper proposes a unit root test for short panels with serially correlated errors. The proposed test is based on the instrumental variables (IVs) and the generalized method of moments (GMM) estimators. An advantage of the new test over other tests is that it allows for an ARMA-type serial correlation. A Monte Carlo simulation shows that the new test has good finite sample properties. Several methods to estimate the lag orders of the ARMA structure are briefly discussed.     

A NONPARAMETRIC MIXED‐EFFECTS MODEL FOR CANCER MORTALITY

There are several ways to handle within‐subject correlations with a longitudinal discrete outcome, such as mortality. The most frequently used models are either marginal or random‐effects types. This paper deals with a random‐effects‐based approach. We propose a nonparametric regression model having time‐varying mixed effects for longitudinal cancer mortality data. The time‐varying mixed effects in the proposed model are estimated by combining kernel‐smoothing techniques and a growth‐curve model. As an illustration based on real data, we apply the proposed method to a set of prefecture‐specific data on mortality from large‐bowel cancer in Japan.