An asymptotic theory for semiparametric generalized least squares estimation in partially linear regression models

Consider a partially linear regression model with an unknown vector parameter β, an unknown functiong(·), and unknown heteroscedastic error variances. In this paper we develop an asymptotic semiparametric generalized least squares estimation theory under some weak moment conditions. These moment conditions are satisfied by many of the error distributions encountered in practice, and our theory does not require the number of replications to go to infinity.     

On the Weibull-Gamma frailty model, its infinite moments, and its connection to generalized log-logistic, logistic, Cauchy, and extreme-value distributions

It is shown that the commonly used Weibull-Gamma frailty model has a finite number of finite moments only and that its marginal distribution generalizes the log-logistic distribution. In some cases there is not even a finite variance, and there are cases without a single finite moment. Upon transformation to the entire real line, generalized logistic and generalized Cauchy distributions are introduced and their connection with the previous ones established, as well as with the extreme-value distribution. Apart from intrinsic and classroom value, the family can be of use when formulating non-informative priors in Bayesian data analysis. Also, gauging the amount of finite moments is important when checking regularity conditions in the Weibull-Gamma model. Our findings are illustrated using data from survival in cancer patients.     

Transformed generalized linear models

The estimation of data transformation is very useful to yield response variables satisfying closely a normal linear model. Generalized linear models enable the fitting of models to a wide range of data types. These models are based on exponential dispersion models. We propose a new class of transformed generalized linear models to extend the Box and Cox models and the generalized linear models. We use the generalized linear model framework to fit these models and discuss maximum likelihood estimation and inference. We give a simple formula to estimate the parameter that index the transformation of the response variable for a subclass of models. We also give a simple formula to estimate the r$r$th moment of the original dependent variable. We explore the possibility of using these models to time series data to extend the generalized autoregressive moving average models discussed by Benjamin et al. [Generalized autoregressive moving average models. J. Amer. Statist. Assoc. 98, 214–223]. The usefulness of these models is illustrated in a simulation study and in applications to three real data sets.     

The generalized inverse Weibull distribution

The inverse Weibull distribution has the ability to model failure rates which are quite common in reliability and biological studies. A three-parameter generalized inverse Weibull distribution with decreasing and unimodal failure rate is introduced and studied. We provide a comprehensive treatment of the mathematical properties of the new distribution including expressions for the moment generating function and the rth generalized moment. The mixture model of two generalized inverse Weibull distributions is investigated. The identifiability property of the mixture model is demonstrated. For the first time, we propose a location-scale regression model based on the log-generalized inverse Weibull distribution for modeling lifetime data. In addition, we develop some diagnostic tools for sensitivity analysis. Two applications of real data are given to illustrate the potentiality of the proposed regression model.     

Book reviews

We propose two moment ratios based on the first four moments. These moment ratios are useful in identifying different members from a class of discrete or continuous distributions. These ratios are also useful in approximating the Neyman type A and the generalized Poisson distribution by the negative binomial distribution.     

Generalized discrete discrepancy and its applications in experimental designs

In recent years, there has been increasing interest in the study of discrete discrepancy. In this paper, the popular discrete discrepancy is extended to the so-called generalized discrete discrepancy. Connections among generalized discrete discrepancy and other optimality criteria, such as orthogonality, generalized minimum aberration and minimum moment aberration, are investigated. These connections provide strong statistical justification of generalized discrete discrepancy. A lower bound of generalized discrete discrepancy is also obtained, which serves as an important benchmark of design uniformity.     

GMM estimation of a realized stochastic volatility model: A Monte Carlo study

This article investigates alternative generalized method of moments (GMM) estimation procedures of a stochastic volatility model with realized volatility measures. The extended model can accommodate a more general correlation structure. General closed form moment conditions are derived to examine the model properties and to evaluate the performance of various GMM estimation procedures under Monte Carlo environment, including standard GMM, principal component GMM, robust GMM and regularized GMM. An application to five company stocks and one stock index is also provided for an empirical demonstration.     

Exact Inference for a Simple Step-stress Model with Generalized Type-I Hybrid Censored Data from the Exponential Distribution

In this article, the simple step-stress model is considered based on generalized Type-I hybrid censored data from the exponential distribution. The maximum likelihood estimators (MLEs) of the unknown parameters are derived assuming a cumulative exposure model. We then derive the exact distributions of the MLEs of the parameters using conditional moment generating functions. The Bayesian estimators of the parameters are derived and then compared with the MLEs. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs, Bayesian, and the parametric bootstrap methods. The problem of determining the optimal stress-changing point is discussed and the MLEs of the pth quantile and reliability functions at the use condition are obtained. Finally, Monte Carlo simulation and some numerical results are presented for illustrating all the inferential methods developed here.     

Testing for normally in censored regressions

This paper derives a Lagrange Multiplier test for normality in censored regressions. The test is derived against the generalized log-gamma distribution, in which normal is a special case. The resulting test statistic coincides to some extent with previously suggested score and conditional moment tests. Estimation of the variance is performed by using the matrix of second order derivatives in order to get an easy to use test statistic. Small sample performance of the test is studied and compared to other tests by Monte Carlo experiments.     

On diagnostic devices for proposing half-logistic and inverse half-logistic models using generalized (k) record values

In this work we consider the generalized upper (k) record values (GURV’s) and generalized lower (k) record values (GLRV’s) arising from half-logistic distribution (HLD) and inverse half-logistic distribution (IHLD). We derive some characterization results of HLD based on some moment relations of generalized upper (k) record values and those of generalized lower (k) record values and accordingly devised some diagnostic tools to identify HLD as a model to the distribution of a population. Similar characterization theorems and diagnostic tools are developed for IHLD as well. Simulation studies are conducted to validate the diagnostic tools devised for both HLD and IHLD.     

Optimal fractional factorial designs and their construction

Optimal symmetrical fractional factorial designs with n$n$ runs and m$m$ factors of s$s$ levels each are constructed. We consider only designs such that no two factors are aliases. The minimum moment aberration criterion proposed by Xu (2003) is used to judge the optimality of the designs. The minimum moment aberration criterion is equivalent to the popular generalized minimum aberration criterion proposed by Xu and Wu (2001), but the minimum moment criterion is simpler to formulate and employ computationally. Some optimal designs are constructed by using generalized Hadamard matrices.     

Performance of a bartlett-type modification for the deviance

Cordeiro (1983) has derived the expected value of the deviance for generalized linear models correct to terms of order n ^-1 being the sample size. Then a Bartlett-type factor is available for correcting the first moment of the deviance and for fitting its distribution. If the model is correct, the deviance is not, in general, distributed as chi-squared even asymptotically and very little is known about the adequacy of the X ² approximation. This paper through simulation studies examines the behaviour of the deviance and a Bartlett adjusted deviance for testing the goodness-of-fit of a generalized linear model. The practical use of such adjustment is illustrated for some gamma and Poisson models. It is suggested that the null distribution of the adjusted deviance is better approximated by chi-square than the distribution of the deviance.     

Baysian estimation of the parameters in two modified poission distributions

Posterior distributions and moment are derived for the generalized Poisson and the excess zeroes Poisson distributions.Three examples are presented where both maximum likelihood and posterior estimates are given.     

Higher order expansion for moments of extreme for generalized Maxwell distribution

In this paper, the higher order asymptotic expansion of the moments of extreme from generalized Maxwell distribution is gained, by which one establishes the rate of convergence of the moment of the normalized partial maximum to the moment of the associated Gumbel extreme value distribution.     

Ultrastructural elliptical models

Dolby's (1976) ultrastructural model with no replications is investigated within the class of the elliptical distributions. General asymptotic results are given for the sample covariance matrix S in the presence of incidental parameters. These results are used to study the asymptotic behaviour of some estimators of the slope parameter, unifying and extending existing results in the literature. In particular, under some regularity conditions they are shown to be consistent and asymptotically normal. For the special case of the structural model, some asymptotic relative efficiencies are also reported which show that generalized least squares and the method of moment estimators can be highly inefficient under nonnormality.     

LIKELIHOOD MOMENT ESTIMATION FOR THE GENERALIZED PARETO DISTRIBUTION

Traditional methods for estimating parameters in the generalized Pareto distribution have theoretical and computational defects. The moment estimator and the probability‐weighted moment estimator have low asymptotic efficiencies. They may not exist or may give nonsensical estimates. The maximum likelihood estimator, which sometimes does not exist, is asymptotically efficient, but its computation is complex and has convergence problems. The likelihood moment estimator is proposed, which is computationally easy and has high asymptotic efficiency.     

A New Pearson-Type QMLE for Conditionally Heteroscedastic Models

This article proposes a novel Pearson-type quasi-maximum likelihood estimator (QMLE) of GARCH(p, q) models. Unlike the existing Gaussian QMLE, Laplacian QMLE, generalized non-Gaussian QMLE, or LAD estimator, our Pearsonian QMLE (PQMLE) captures not just the heavy-tailed but also the skewed innovations. Under strict stationarity and some weak moment conditions, the strong consistency and asymptotic normality of the PQMLE are obtained. With no further efforts, the PQMLE can be applied to other conditionally heteroscedastic models. A simulation study is carried out to assess the performance of the PQMLE. Two applications to four major stock indexes and two exchange rates further highlight the importance of our new method. Heavy-tailed and skewed innovations are often observed together in practice, and the PQMLE now gives us a systematic way to capture these two coexisting features.     

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.     

Generalized spectral tests for serial dependence

Two tests for serial dependence are proposed using a generalized spectral theory in combination with the empirical distribution function. The tests are generalizations of the Cramér-von Mises and Kolmogorov-Smirnov tests based on the standardized spectral distribution function. They do not involve the choice of a lag order, and they are consistent against all types of pairwise serial dependence, including those with zero autocorrelation. They also require no moment condition and are distribution free under serial independence. A simulation study compares the finite sample performances of the new tests and some closely related tests. The asymptotic distribution theory works well in finite samples. The generalized Cramér-von Mises test has good power against a variety of dependent alternatives and dominates the generalized Kolmogorov-Smirnov test. A local power analysis explains some important stylized facts on the power of the tests based on the empirical distribution function.