Confidence Intervals for Linear Function of Variance Components in Two-Fold Nested Random Effects Model

In the article, we consider the unbalanced case of the two-way nested random effects model under partial balance. Using the method of generalized confidence intervals (GCIs) introduced in Weeranhandi (1993 Weeranhandi , S. ( 1993 ). Generalized confidence intervals . J. Amer. Statist. Assoc. 88 : 899 – 905 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar] 1995 Weeranhandi , S. ( 1995 ). Exact Statistical Methods for Data Analysis . New York : Springer-Verlag . [Google Scholar]), a new method is proposed for constructing confidence intervals on linear function of variance components. To compare the resulted intervals with the Modified Large Sample (MLS) intervals by Hernandez and Burdick (1993 Hernandez , R. P. , Burdick , R. K. ( 1993 ). Confidence intervals on the total variance in unbalanced two-fold nested designs . Biom. J. 35 : 515 – 522 .[Crossref] , [Google Scholar]), a simulation study is conducted. The results indicate that the proposed method performs better than the MLS method, especially for very unbalanced designs.     

Interval Estimation by Frequentist Model Averaging

An important contribution to the literature on frequentist model averaging (FMA) is the work of Hjort and Claeskens (2003 Hjort , N. L. , Claeskens , G. ( 2003 ). Frequestist model average estimators . J. Amer. Statist. Assoc. 98 : 879 – 899 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), who developed an asymptotic theory for frequentist model averaging in parametric models based on a local mis-specification framework. They also proposed a simple method for constructing confidence intervals of the unknown parameters. This article shows that the confidence intervals based on the FMA estimator suggested by Hjort and Claeskens (2003 Hjort , N. L. , Claeskens , G. ( 2003 ). Frequestist model average estimators . J. Amer. Statist. Assoc. 98 : 879 – 899 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are asymptotically equivalent to that obtained from the full model under both parametric and the varying-coefficient partially linear models. Thus, as long as interval estimation rather than point estimation is concerned, the confidence interval based on the full model already fulfills the objective and model averaging provides no additional useful information.     

Semiparametric Efficient Distribution Free Estimation of Panel Models

This article generalizes results from Park et al. (1998 Park , B. U. , Sickles , R. C. , Simar , L. ( 1998 ). Stochastic frontiers: a semiparametric approach . J. Econometrics 84 : 273 – 301 .[Crossref], [Web of Science ®] , [Google Scholar]) and Adams et al. (1999 Adams , R. M. , Berger , A. N. , Sickles , R. C. ( 1999 ). Semiparametric approaches to stochastic panel frontiers with applications in the banking industry . J. Bus. Econ. Statist. 17 : 349 – 358 .[Taylor & Francis Online] , [Google Scholar]) on semiparametric efficient estimation of panel models. The form of semiparametric efficient estimators depends on the statistical assumptions imposed. Normality assumptions on the transitory error are sometimes inappropriate. We relax the normality assumption used in the articles above to derive more general semiparametric efficient estimators. These estimators are illustrated in a Monte Carlo simulation and an analysis of banking productivity.     

A Simple Method for Testing Variance Components in Unbalanced Nested Model

In this article, we consider the unbalanced case of the three fold nested random effects model under partial balance. The distributions of unweighted sums of squares are obtained first. Using the method of generalized p value introduced in Tsui and Weerahandi (1989 Tsui , K. , Weerahandi , S. ( 1989 ). Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters . Journal of the American Statistical Association 84 : 602 – 607 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), a new method is proposed for hypothesis tests involving functions of variance components. To evaluate the sizes of the generalized p value, a simulation study is conducted. The results indicate that the proposed method performs well under all examined conditions.     

On Maximum Likelihood Estimation for the Two Parameter Burr XII Distribution

This article considers three related aspects of maximum likelihood estimation of parameters in the two-parameter Burr XII distribution. Specifically, we first provide further clarification to some limiting results in Wingo (1993 Wingo , D. R. ( 1993 ). Maximum likelihood estimation of Burr XII distribution parameters under Type II censoring . Microelectron. Reliab. 33 : 1251 – 1257 .[Crossref], [Web of Science ®] , [Google Scholar]). We then focus on details in a proof of the uniqueness of the maximum likelihood estimators. Finally, we consider using the likelihood approach for data which does not satisfy Wingo's criterion, and show that this results in fitting either a Pareto distribution or an intuitively sensible degenerate distribution to the data. The discussion here is completely general, and not restricted to data obtained under Type II censoring.     

Goodness-of-Fit Test and Parameter Estimation for a Proportional Odds Model of Random Censorship

Censored data arise naturally in a number of fields, particularly in problems of reliability and survival analysis. There are several types of censoring, in this article, we will confine ourselves to the right randomly censoring type. Recently, Ahmadi et al. (2010 Ahmadi , J. , Doostparast , M. , Parsian , A. ( 2010 ). Bayes estimation based on random censored data for some life time models under symmetric and asymmetric loss functions . Communcations in Statistics-Theory and Methods , 39 : 3058 – 3071 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) considered the problem of estimating unknown parameters in a general framework based on the right randomly censored data. They assumed that the survival function of the censoring time is free of the unknown parameter. This assumption is sometimes inappropriate. In such cases, a proportional odds (PO) model may be more appropriate (Lam and Leung, 2001 Lam , K. F. , Leung , T. L. ( 2001 ). Marginal likelihood estimation for proportional odds models with right censored data . Lifetime Data Analysis 7 : 39 – 54 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Under this model, in this article, point and interval estimations for the unknown parameters are obtained. Since it is important to check the adequacy of models upon which inferences are based (Lawless, 2003 Lawless , J. F. (2003). Statistical Models and Methods for Lifetime Data. , 2nd ed. New York : John Wiley & Sons. [Google Scholar], p. 465), two new goodness-of-fit tests for PO model based on right randomly censored data are proposed. The proposed procedures are applied to two real data sets due to Smith (2002 Smith , P. J. ( 2002 ). Analysis of Failure and Survival Data . London : Chapman & Hall, CRC . [Google Scholar]). A Monte Carlo simulation study is conducted to carry out the behavior of the estimators obtained.     

Comment on Ridge Estimation to the Restricted Linear Model

In this note, we show that the estimator and the following results given by Zhong and Yang (2007 Zhong , Z. , Yang , H. ( 2007 ). Ridge estimation to the restricted linear model . Commun. Statist. Theor. Meth. 36 : 2099 – 2115 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are the same with that of Groß (2003 Groß , J. ( 2003 ). Restricted ridge estimation . Statist. Probab. Lett. 65 : 57 – 64 .[Crossref], [Web of Science ®] , [Google Scholar]).     

General Saddlepoint Approximations: Application to the Anderson-Darling Test Statistic

We consider the relative merits of various saddlepoint approximations for the cumulative distribution function (cdf) of a statistic with a possibly non normal limit distribution. In addition to the usual Lugannani-Rice approximation, we also consider approximations based on higher-order expansions, including the case where the base distribution for the approximation is taken to be non normal. This extends earlier work by Wood et al. (1993 Wood , A. T. A. , Booth , J. G. , Butler , R. W. ( 1993 ). Saddlepoint approximations to the CDF of some statistics with nonnormal limit distributions . Journal of the American Statistical Association 88 : 680 – 686 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). These approximations are applied to the distribution of the Anderson-Darling test statistic. While these generalizations perform well in the middle of the distribution's support, a conventional normal-based Lugannani-Rice approximation (Giles, 2001 Giles , D. E. A. ( 2001 ). A Saddlepoint approximation to the distribution function of the Anderson-Darling test statistic . Communications in Statistics B 30 : 899 – 905 .[Taylor & Francis Online] , [Google Scholar]) is superior for conventional critical regions.     

Bayesian Estimation of Parameters and Comparison of Shared Gamma Frailty Models

In this article, we introduce shared gamma frailty models with three different baseline distributions namely, Weibull, generalized exponential and exponential power distributions. We develop Bayesian estimation procedure using Markov Chain Monte Carlo(MCMC) technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. Also we apply these three models to a real life bivariate survival dataset of McGilchrist and Aisbett (1991 McGilchrist, C. A. and Aisbett, C. W. 1991. Regression with frailty in survival analysis. Biometrics, 47: 461–466. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) related to kidney infection data and a better model is suggested for the data.     

Nonparametric Interval Estimation in One-Way Random-Effects Models

The structural method provided by Hannig et al. (2006 Hannig , J. , Iyer , H. , Patterson , P. (2006). Fiducial generalized confidence intervals. J. Amer. Statist. Assoc. 101:254–269.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) has proved to be a useful tool for constructing confidence intervals. However, it is difficult to apply this method to nonparametric problems since the pivotal quantity required in using it exists only in some special parametric models. Based on an extended structural method, this article discusses nonparametric interval estimation for smooth functions of the variances in one-way random-effects models. We use the bootstrap distribution estimator of a statistic to construct an approximate pivotal equation, and prove that the confidence interval derived by the approximate pivotal equation has asymptotically correct coverage probability. Simulation results are presented and show that the normal fiducial interval is not robust against non normality and that the proposed confidence interval has better finite-sample behaviors than the naive interval based on normal approximation.     

Estimation of Ordered Means of Two Poisson Distributions

In this article, we discuss the estimation of linear functions of two Poisson means, on which an order restriction is given. We give a necessary and sufficient condition on the coefficients of the linear function for the maximum likelihood estimator (MLE) which satisfies the order restriction to dominate the unbiased estimator under squared error loss. Furthermore, simultaneous estimation of two ordered Poisson means is considered and we suggest the Clevenson–Zidek type modification of MLE which dominates the MLE under normalized squared error loss. We also improve the estimator proposed by Clevenson and Zidek (1975 Clevenson , M. , Zidek , J. ( 1975 ). Simultaneous estimation of the means of independent Poisson laws . J. Amer. Statist. Assoc. 70 : 698 – 705 . [CSA] [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) which ignores the order restriction.     

Is Each NPMLE of a Continuous Bivariate Distribution Function with Singly Right-Censored Data Really Inconsistent?

We consider non-parametric estimation of a continuous cdf of a random vector (X ₁, X ₂). With bivariate RC data, it is stated in van der Laan (1996 Van der Laan , M. J. ( 1996 ) Efficient estimation in the bivariate censoring model and repairing NPMLE . Ann. Statist. 24 : 596 – 627 .[Crossref], [Web of Science ®] , [Google Scholar], p. 598¹⁰, Ann. Statist.), Quale et al. (2006 Quale , C. M. , van der Laan , M. J. , Robins , J. R. ( 2006 ). Locally efficient estimation with bivariate right-censored data . JASA. 101 : 1076 – 1084 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], JASA) etc. that “it is well known that the NPMLE for continuous data is inconsistent (Tsai et al. (1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar])).” The claim is based on a result in Tsai et al. (1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar], p.1352, Ann. Statist.) that if X ₁ is right censored but not X ₂, then common ways for defining one NPMLE lead to inconsistency. If X ₁ is right censored and X ₂ is type I right-censored (which includes the case in Tsai et al.), we present a consistent NPMLE. The result corrects a common misinterpretation of Tsai's example (Tsai et al., 1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar], Ann. Statist.).     

Factor Stochastic Volatility in Mean Models: A GMM Approach

This paper provides a semiparametric framework for modeling multivariate conditional heteroskedasticity. We put forward latent stochastic volatility (SV) factors as capturing the commonality in the joint conditional variance matrix of asset returns. This approach is in line with common features as studied by Engle and Kozicki (1993 Engle , R. F. , Kozicki , S. ( 1993 ). Testing for common features . Journal of Business and Economic Statistics 11 ( 4 ): 369 – 395 . [CSA] [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), and it allows us to focus on identication of factors and factor loadings through first- and second-order conditional moments only. We assume that the time-varying part of risk premiums is based on constant prices of factor risks, and we consider a factor SV in mean model. Additional specification of both expectations and volatility of future volatility of factors provides conditional moment restrictions, through which the parameters of the model are all identied. These conditional moment restrictions pave the way for instrumental variables estimation and GMM inference.     

On Testing Sample Selection Bias Under the Multicollinearity Problem

ABSTRACT

This paper reviews and extends the literature on the finite sample behavior of tests for sample selection bias. Monte Carlo results show that, when the “multicollinearity problem” identified by Nawata (1993 Nawata , K. ( 1993 ). A note on the estimation of models with sample-selection biases . Economics Letters 42 : 15 – 24 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) is severe, (i) the t-test based on the Heckman–Greene variance estimator can be unreliable, (ii) the Likelihood Ratio test remains powerful, and (iii) nonnormality can be interpreted as severe sample selection bias by Maximum Likelihood methods, leading to negative Wald statistics. We also confirm previous findings (Leung and Yu, 1996 Leung , S. F. , Yu , S. ( 1996 ). On the choice between sample selection and two-part models . Journal of Econometrics 72 : 197 – 229 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) that the standard regression-based t-test (Heckman, 1979 Heckman , J. J. ( 1979 ). Sample selection bias as a specification error . Econometrica 47 : 153 – 161 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) and the asymptotically efficient Lagrange Multiplier test (Melino, 1982 Melino , A. ( 1982 ). Testing for sample selection bias . Review of Economic Studies 49 : 151 – 153 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), are robust to nonnormality but have very little power.     

Local Linear Estimation of Second-Order Diffusion Models

In this article, we present the local linear estimations for diffusion coefficient and drift coefficient in the second-order diffusion model. We show that under mild conditions, the estimators are weak consistent. We also use a Monte Carlo experiment to compare our estimators with the ones in Nicolau (2007 Nicolau , J. ( 2007 ). Nonparametric estimation of scend-order stochastic differential equations . Econometric Theor. 23 : 880 – 898 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Second-Order Covariance Matrix Formula for Heteroskedastic Generalized Linear Models

For the first time, we provide a matrix formula for second-order covariances of maximum likelihood estimates in heteroskedastic generalized linear models, thus generalizing the results of Cordeiro (2004 Cordeiro , G. M. ( 2004 ). Second-order covariance matrix of maximum likelihood estimates in generalized linear models . Statist. Probab. Lett. 66 : 153 – 160 .[Crossref], [Web of Science ®] , [Google Scholar]) and Cordeiro et al. (2006 Cordeiro , G. M. , Barroso , L. P. , Botter , D. A. (2006). Covariance matrix formula for generalized linear models with unknown dispersion. Commun. Statist. Theor. Meth. 35:113–120.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) related to the generalized linear models with known and unknown dispersion parameter, respectively. The covariance matrix formula does not involve cumulants of log-likelihood derivatives and can be easily obtained using simple matrix operations. We apply our main result to a simple model. Some simulations show that the second-order covariances can be quite pronounced in small to moderate samples. The usual covariances of the maximum likelihood estimates can be corrected by these second-order covariances.     

An Inverse-Probability-Weighted Approach to the Estimation of Successive Duration Times under Left-Truncation and Right-Censoring

In incident cohort studies, survival data often include subjects who have had an initiate event at recruitment and may potentially experience two successive events (first and second) during the follow-up period. Since the second duration process becomes observable only if the first event has occurred, left-truncation and dependent censoring arise if the two duration times are correlated. To confront the two potential sampling biases, Chang and Tzeng (2006 Chang , S.-H. , Tzeng , S.-J. (2006). Noparametric estimation of sojourn time distributions for truncated serial event data- a weight-adjusted approach. Lifetime Data Anal. 53–67.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) provided an inverse-probability-weighted (IPW) approach for estimating the joint probability function of successive duration times. In this note, an alternative IPW approach is proposed. A simulation study is conducted to compare the two IPW approaches.     

Selection of Working Correlation Structure and Best Model in GEE Analyses of Longitudinal Data

The Generalized Estimating Equations (GEE) method is one of the most commonly used statistical methods for the analysis of longitudinal data in epidemiological studies. A working correlation structure for the repeated measures of the outcome variable of a subject needs to be specified by this method. However, statistical criteria for selecting the best correlation structure and the best subset of explanatory variables in GEE are only available recently because the GEE method is developed on the basis of quasi-likelihood theory. Maximum likelihood based model selection methods, such as the widely used Akaike Information Criterion (AIC), are not applicable to GEE directly. Pan (2001 Pan , W. ( 2001 ). Akaike's information criterion in generalized estimating equations . Biometrics 57 : 120 – 125 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) proposed a selection method called QIC which can be used to select the best correlation structure and the best subset of explanatory variables. Based on the QIC method, we developed a computing program to calculate the QIC value for a range of different distributions, link functions and correlation structures. This program was written in Stata software. In this article, we introduce this program and demonstrate how to use it to select the most parsimonious model in GEE analyses of longitudinal data through several representative examples.     

Maximum Likelihood Estimation in Compound Geometric Failure Model with Changing Parameters From Type-I Two-Stage Progressively Censored and Group Censored Samples

Boardman and Kendell (1970 Boardman , T. J. , Kendell , P. J. ( 1970 ). Estimation in compound failure models . Technometrics 12 : 891 – 908 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) considered the problem of estimation with respect to Type-I censoring when an item is subjected to only one of the two causes of failure assuming exponential model. Patel and Gajjar (1992 Patel , M. N. , Gajjar , A. V. ( 1992 ). Maximum likelihood estimation in compound exponential failure model with changing failure rates from Type-I progressively censored and group censored samples . Commun. Statist. Theor. Meth. 21 ( 10 ): 2899 – 2908 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) considered extension of the Boardman and Kendell's results in case of two-stage progressive censoring. Here we have considered geometric competing risk failure model with two independent causes of failures. Maximum likelihood estimation of the parameters is carried out using Type-I two-stage progressively censored and group censored samples. Asymptotic standard errors of the estimators are obtained for both the cases. Two illustrative examples are cited for ungroup and group competing risk models.     

A Broad Class of Partially Specified Autoregressions on Multi-Casting Data

Extending the bifurcating autoregressive (BAR) process (cf. Cowan and Staudte, 1986 Cowan , R. , Staudte , R. G. ( 1986 ). The bifurcating autoregression model in cell lineage studies . Biometrics 42 : 769 – 783 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to multi-casting (multi-splitting) data, Hwang and Choi (2009 Hwang , S. Y. , Choi , M. S. ( 2009 ). Modeling and large sample estimation for multi-casting autoregression . Statist. Prob. Lett. 79 : 1943 – 1950 .[Crossref], [Web of Science ®] , [Google Scholar]) introduced multi-casting autoregression (MCAR, for short) defined on multi-casting tree structured data. This article is concerned with the case when the MCAR model is partially specified only through conditional mean and variance without directly imposing autoregressive (AR) structure. The resulting class of models will be referred to as P-MCAR (partially specified MCAR). The P-MCAR considerably enlarges the class of multi-casting models including (as special cases) MCAR, random coefficient MCAR, conditionally heteroscedastic multi-casting models and binomial-thinning processes. Moment structures for this broad P-MCAR class are investigated. Least squares (LS) estimation method is discussed and asymptotic relative efficiency (ARE) of the generalized-LS over ordinary-LS is obtained in a closed form. A simulation study is conducted to illustrate results.