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.     

A Multivariate Threshold Varying Conditional Correlations Model

In this article, a multivariate threshold varying conditional correlation (TVCC) model is proposed. The model extends the idea of Engle (2002 Engle , R. F. ( 2002 ). Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models . Journal of Business and Economic Statistics 20 ( 3 ): 339 – 350 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Tse and Tsui (2002 Tse , Y. K. , Tsui , A. K. C. ( 2002 ). A multivariate GARCH model with time-varying correlations . Journal of Business and Economic Statistics , July 2002 , 20 ( 3 ): 351 – 362 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to a threshold framework. This model retains the interpretation of the univariate threshold GARCH model and allows for dynamic conditional correlations. Techniques of model identification, estimation, and model checking are developed. Some simulation results are reported on the finite sample distribution of the maximum likelihood estimate of the TVCC model. Real examples demonstrate the asymmetric behavior of the mean and the variance in financial time series and the ability of the TVCC model to capture these phenomena.     

From Ordinary to Shrinkage Square-Root Estimators

Consider a skewed population. Suppose an intelligent guess could be made about an interval that contains the population mean. There may exist biased estimators with smaller mean squared error than the arithmetic mean within such an interval. This article indicates when it is advisable to shrink the arithmetic mean towards a guessed interval using root estimators. The goal is to obtain an estimator that is better near the average of natural origins. An estimator proposed. This estimator contains the Thompson (1968 Thompson , J. R. ( 1968 ). Accuracy borrowing in the estimation of the mean by shrinkage towards an interval . J. Amer. Statist. Assoc. 63 : 953 – 963 . [CSA] [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) ordinary shrinkage estimator, the Jenkins et al. (1973 Jenkins , O. C. , Ringer , L. J. , Hartley , H. O. ( 1973 ). Root estimators . J Amer. Statist. Assoc. 68 : 414 – 419 . [CSA] [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) square-root estimator, and the arithmetic sample mean as special cases. The bias and the mean squared error of the proposed more general estimator is compared with the three special cases. Shrinkage coefficients that yield minimum mean squared error estimators are obtained. The proposed estimator is considerably more efficient than the three special cases. This remains true for highly skewed populations. The merits of the proposed shrinkage square-root estimator are supported by the results of numerical and simulation studies.     

On Estimating the Finite Population Mean Using Two Auxiliary Variables in Two Phase Sampling in the Presence of Non Response

This article considers the problem of estimating the population mean using information on two auxiliary variables in the presence of non response under two-phase sampling. Some improved ratio-in-regression type estimators have been proposed in four different situations of non response along with their properties under large sample approximation. Efficiency comparisons of the proposed estimators with the usual unbiased estimator by Hansen and Hurwitz (1946 Hansen , M. H. , Hurwitz , W. N. ( 1946 ). The problem of non response in sample surveys . J. Amer. Statist. Assoc. 41 : 517 – 529 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), conventional ratio and regression estimators using single auxiliary variable and Singh and Kumar (2010b Singh , H. P. , Kumar , S. ( 2010b ). Improved estimation of population mean under double sampling with sub-sampling the non-respondents . J. Statist. Plann. Infer. 140 ( 9 ): 2536 – 2550 .[Crossref], [Web of Science ®] , [Google Scholar]) estimators using two auxiliary variables have been made. Finally, these theoretical findings are illustrated by a numerical example.     

Improved Estimation of Finite Population Variance Using Auxiliary Information

This article addresses the problem of estimating of finite population variance using auxiliary information in simple random sampling. A ratio-cum-difference type class of estimators for population variance has been suggested with its properties under large sample approximation. It has been shown that the suggested class of estimators is more efficient than usual unbiased, difference, Das and Tripathi (1978 Das , A. K. , Tripathi , T. P. ( 1978 ). Use of auxiliary information in estimating the finite population variance . Sankhya C 40 : 139 – 148 . [Google Scholar]), Isaki (1983 Isaki , C. T. ( 1983 ). Variance estimation using auxiliary information . J. Amer. Statist. Assoc. 78 : 117 – 123 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Singh et al. (1988 Singh , H. P. , Upadhyaya , L. N. , Namjoshi , U. D. ( 1988 ). Estimation of finite population variance . Curr. Sci. 57 : 1331 – 1334 .[Web of Science ®] , [Google Scholar]), Kadilar and Cingi (2006 Kadilar , C. , Cingi , H. ( 2006 ). Ratio estimators for the population variance in simple and stratified random sampling . Appl. Math. Comp. 173 ( 2 ): 1047 – 1059 .[Crossref], [Web of Science ®] , [Google Scholar]), and other estimators/classes of estimators. In addition, we support this theoretical result with the aid of a empirical study.     

Regression Analysis with Covariates Missing at Random: A Piece-wise Nonparametric Model for Missing Covariates

Statistical analysis for the regression model f _β(y | x, z) with missing values in the covariate vector X requires modeling of the covariate distribution g(x | z). Likelihood methods, including Ibrahim (1990 Ibrahim , J. G. ( 1990 ). Incomplete data in generalized linear models . J. Amer. Statist. Assoc. 85 : 765 – 769 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Chen (2004 Chen , H. Y. (2004). Nonparametric and semiparametric models for missing covariates in parametric regression. J. Amer. Statist. Assoc. 99:1176–1189.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), and Zhao (2005 Zhao , Y. ( 2005 ). Design and Efficient Estimation in Regression Analysis with Missing Data in Two-Phase Studies. Ph.D. thesis , University of Waterloo . [Google Scholar]), need either X or Z to be discrete. This article considers extending the likelihood methods to deal with cases where both X and Z may be continuous. We propose modeling the covariate distribution g(x | z) using a piece-wise nonparametric model, then a maximum likelihood estimate (MLE) of β can be computed following the maximum likelihood estimating procedure of Chen (2004 Chen , H. Y. (2004). Nonparametric and semiparametric models for missing covariates in parametric regression. J. Amer. Statist. Assoc. 99:1176–1189.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) or Zhao (2005 Zhao , Y. ( 2005 ). Design and Efficient Estimation in Regression Analysis with Missing Data in Two-Phase Studies. Ph.D. thesis , University of Waterloo . [Google Scholar]). The resulting estimation method is easy to implement and the asymptotic properties of the MLE follow under certain conditions. Extensive simulation studies for different models indicate that the proposed method is acceptable for practical implementation. A real data example is used to illustrate the method.     

Relative Errors of Difference-Based Variance Estimators in Nonparametric Regression

Difference-based estimators for the error variance are popular since they do not require the estimation of the mean function. Unlike most existing difference-based estimators, new estimators proposed by Müller et al. (2003 Müller , U. , Schick , A. , Wefelmeyer , W. ( 2003 ). Estimating the error variance in nonparametric regression by a covariate-matched U-statistic . Statistics 37 : 179 – 188 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Tong and Wang (2005 Tong , T. , Wang , Y. ( 2005 ). Estimating residual variance in nonparametric regression using least squares . Biometrika 92 : 821 – 830 .[Crossref], [Web of Science ®] , [Google Scholar]) achieved the asymptotic optimal rate as residual-based estimators. In this article, we study the relative errors of these difference-based estimators which lead to better understanding of the differences between them and residual-based estimators. To compute the relative error of the covariate-matched U-statistic estimator proposed by Müller et al. (2003 Müller , U. , Schick , A. , Wefelmeyer , W. ( 2003 ). Estimating the error variance in nonparametric regression by a covariate-matched U-statistic . Statistics 37 : 179 – 188 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), we develop a modified version by using simpler weights. We further investigate its asymptotic property for both equidistant and random designs and show that our modified estimator is asymptotically efficient.     

Additive Mixed Effect Model for Clustered Doubly Censored Data

Cai and Zeng (2011 Cai, J. and Zeng, D. 2011. Additive mixed effect model for clustered failure time data. Biometrics, 67(4): 1340–1351. [Crossref], [PubMed] , [Google Scholar]) proposed an additive mixed effect model to analyze clustered right-censored data. In this article, we demonstrate that the approach of Cai and Zeng (2011 Cai, J. and Zeng, D. 2011. Additive mixed effect model for clustered failure time data. Biometrics, 67(4): 1340–1351. [Crossref], [PubMed] , [Google Scholar]) can be extended to clustered doubly censored data. Furthermore, when both left- and right-censoring variables are always observed, we propose alternative estimators using the approach of Cai and Cheng (2004 Cai, T. and Cheng, S. C. 2004. Semiparametric regression analysis for doubly censored data. Biometrika, 91: 277–290. [Crossref], [Web of Science ®] , [Google Scholar]). A simulation study is conducted to investigate the performance of the proposed estimators.     

Efficient Ratio and Product Estimators in Stratified Random Sampling

This paper suggests an efficient class of ratio and product estimators for estimating the population mean in stratified random sampling using auxiliary information. It is interesting to mention that, in addition to many, Koyuncu and Kadilar (2009 Koyuncu , N. , Kadilar , C. ( 2009 ). Ratio and product estimators in stratified random sampling . J. Statist. Plann. Infer. 139 : 2552 – 2558 .[Crossref], [Web of Science ®] , [Google Scholar]), Kadilar and Cingi (2003 Kadilar , C. , Cingi , H. ( 2003 ). Ratio estimator in stratified sampling . Biometr. J. 45 : 218 – 225 .[Crossref], [Web of Science ®] , [Google Scholar], 2005 Kadilar , C. , Cingi , H. ( 2005 ). A new estimator in stratified random sampling . Commun. Statist. Theor. Meth. 34 : 597 – 602 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), and Singh and Vishwakarma (2007 Singh , H. P. , Vishwakarma , G. K. ( 2007 ). Modified exponential ratio and product estimators for finite population mean in double sampling . Austr. J. Statist. 36 ( 3 ): 217 – 225 . [Google Scholar]) estimators are identified as members of the proposed class of estimators. The expressions of bias and mean square error (MSE) of the proposed estimators are derived under large sample approximation in general form. Asymptotically optimum estimator (AOE) in the class is identified alongwith its MSE formula. It has been shown that the proposed class of estimators is more efficient than combined regression estimator and Koyuncu and Kadilar (2009 Koyuncu , N. , Kadilar , C. ( 2009 ). Ratio and product estimators in stratified random sampling . J. Statist. Plann. Infer. 139 : 2552 – 2558 .[Crossref], [Web of Science ®] , [Google Scholar]) estimator. Moreover, theoretical findings are supported through a numerical example.     

The Exact General Formulas for the Moments of a Ridge Regression Estimator when the Regression Error Terms Follow a Multivariate t Distribution

Huang (1999 Huang , J. C. ( 1999 ). Improving the estimation precision for a selected parameter in multiple regression analysis: an algebraic approach . Econ. Lett. 62 : 261 – 264 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed a feasible ridge regression (FRR) estimator to estimate a specific regression coefficient. Assuming that the error terms follow a normal distribution, Huang (1999 Huang , J. C. ( 1999 ). Improving the estimation precision for a selected parameter in multiple regression analysis: an algebraic approach . Econ. Lett. 62 : 261 – 264 .[Crossref], [Web of Science ®] , [Google Scholar]) examined the small sample properties of the FRR estimator. In this article, assuming that the error terms follow a multivariate t distribution, we derive an exact general formula for the moments of the FRR estimator to estimate a specific regression coefficient. Using the exact general formula, we obtain exact formulas for the bias, mean squared error (MSE), skewness, and kurtosis of the FRR estimator. Since these formulas are very complex, we compare the bias, MSE, skewness, and kurtosis of the FRR estimator with those of ordinary least square (OLS) estimator by numerical evaluations. Our numerical results show that the range of MSE dominance of the FRR estimator over the OLS estimator is widen under a fat tail distributional assumption.     

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.     

The Modified Pulak and Al-Sultan's Model for Determining the Optimum Process Parameters

Economic selection of process parameters has been an important topic in modern statistical process control. The optimum process parameters setting have a major effect on the expected profit/cost per item. There are some concerns on the problem of setting process parameters. Boucher and Jafari (1991 Boucher , T. O. , Jafari , M. A. ( 1991 ). The optimum target value for single filling operations with quality sampling plans . J. Qual. Technol. 23 : 44 – 47 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) first considered the attribute single sampling plan applied in the selection of process target. Pulak and Al-Sultan (1996 Pulak , M. F. S. , Al-Sultan , K. S. ( 1996 ). The optimum targeting for a single filling operation with rectifying inspection . Omega 24 : 727 – 733 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) extended Boucher and Jafari's model and presented the rectifying inspection plan for determining the optimum process mean. In this article, we further propose a modified Pulak and Al-Sultan model for determining the optimum process mean and standard deviation under the rectifying inspection plan with the average outgoing quality limit (AOQL) protection. Taguchi's (1986 Taguchi , G. ( 1986 ). Introduction to Quality Engineering . Asian Productivity Organization . [Google Scholar]) symmetric quadratic quality loss function is adopted for evaluating the product quality. By solving the modified model, we can obtain the optimum process parameters with the maximum expected profit per item and the specified quality level can be reached.     

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]).     

Improved Predictive Inference for Responses Based on Location Parameter Median from the Rayleigh Model Given a Doubly Censored Sample

The purpose of this article is to investigate the predictive inference for responses from the location parameter mean as well as from the median given a doubly censored sample from the two-parameter Rayleigh model. The predictive results by Khan et al. (2010 Khan , H. M. R. , Provost , S. B. , Ashima , S. ( 2010 ). Predictive Inference from a Two-Parameter Rayleigh Life Model Given a Doubly Censored Sample . Commun. Statist. Theor. Meth. 39 ( 7 ): 1237 – 1246 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are used to obtain the predictive inference for responses from the median, where Khan et al. (2010 Khan , H. M. R. , Provost , S. B. , Ashima , S. ( 2010 ). Predictive Inference from a Two-Parameter Rayleigh Life Model Given a Doubly Censored Sample . Commun. Statist. Theor. Meth. 39 ( 7 ): 1237 – 1246 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) obtained the future estimates from the mean. A numerical example representing 66 liver cancer patients is used for predictive analysis. It is concluded that the predictive inference from the median gives precise results as compared with the location parameter mean.     

Bayesian Capture-Recapture Analysis: An Application in Modeling Semelparity of a Neotropical Didelphid Marsupial

We develop a series of Bayesian statistical models for estimating survival of a neotropic didelphid marsupial, the Brazilian gracile mouse opossum (Gracilinanus microtarsus). These models are based on the Cormack–Jolly–Seber model (Cormack, 1964 Cormack , R. M. ( 1964 ). Estimates of survival from the sighting of marked animals . Biometrika 51 : 429 – 438 .[Crossref], [Web of Science ®] , [Google Scholar]; Jolly 1965 Jolly , G. M. ( 1965 ). Explicit estimates from capture-recapture data with both death and immigration stochastic model . Biometrika 52 : 225 – 247 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Seber 1965 Seber , G. A. F. ( 1965 ). A note on the multiple recapture census . Biometrika 52 : 249 – 259 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) with both survival and recapture rates expressed as a function of covariates using a logit link. The proposed models allow taking into account heterogeneity in capture probability caused by the existence of different groups of individuals in the population. The models were applied to two cohorts (Cohort, 2000, 2001) with the first one including 14 and the second one 15 sampling occasions. The best models for each of the cohorts indicate that G. microtarsus is best described as partially semelparous, a condition in which mortality after the first mating is high but graded over time, with a fraction of males surviving for a second breeding season (Boonstra, 2005 Boonstra , R. ( 2005 ). Equipped for life: the adaptive role of the stress axis in male mammals . Journal of Mammalogy 86 : 236 – 247 .[Crossref], [Web of Science ®] , [Google Scholar]).     

The Alpha-power Tobit Model

The main object of this article is to propose an extension of the tobit model for which the error distribution follows the power-normal distribution (Gupta and Gupta, 2008 Gupta , D. , Gupta , R. C. ( 2008 ). Analyzing skewed data by power normal model . Test 17 : 197 – 210 .[Crossref], [Web of Science ®] , [Google Scholar]). Inference is dealt with by using the likelihood approach. Simulation studies and application to a real data set are used to demonstrate the usefulness of the extension.     

The Performance of Panel Cointegration Methods: Results from a Large Scale Simulation Study

This article presents results concerning the performance of both single equation and system panel cointegration tests and estimators. The study considers the tests developed in Pedroni (1999 Pedroni , P. ( 1999 ). Critical values for cointegration tests in heterogeneous panels with multiple regressors . Oxford Bulletin of Economics and Statistics 61 : 653 – 670 .[Crossref], [Web of Science ®] , [Google Scholar], 2004 Pedroni , P. ( 2004 ). Panel cointegration. Asymptotic and finite sample properties of pooled time series tests with an application to the PPP hypothesis . Econometric Theory 20 : 597 – 625 .[Crossref], [Web of Science ®] , [Google Scholar]), Westerlund (2005 Westerlund , J. ( 2005 ). New simple tests for panel cointegration . Econometric Reviews 24 : 297 – 316 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Larsson et al. (2001 Larsson , R. , Lyhagen , J. , Löthgren , M. ( 2001 ). Likelihood-based cointegration tests in heterogeneous panels . Econometrics Journal 4 : 109 – 142 .[Crossref] , [Google Scholar]), and Breitung (2005 Breitung , J. ( 2005 ). A parametric approach to the estimation of cointegration vectors in panel data . Econometric Reviews 24 : 151 – 173 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and the estimators developed in Phillips and Moon (1999 Phillips , P. C. B. , Moon , H. R. ( 1999 ). Linear regression limit theory for nonstationary panel data . Econometrica 67 : 1057 – 1111 .[Crossref], [Web of Science ®] , [Google Scholar]), Pedroni (2000 Pedroni , P. ( 2000 ). Fully modified OLS for heterogeneous cointegrated panels . In: Baltagi , B. H. , ed. Nonstationary Panels, Panel Cointegration, and Dynamic Panels . Amsterdam : Elsevier , pp. 93 – 130 .[Crossref] , [Google Scholar]), Kao and Chiang (2000 Kao , C. , Chiang , M.-H. ( 2000 ). On the estimation and inference of a cointegrated regression in panel data . In: Baltagi , B. H. , ed. Nonstationary Panels, Panel Cointegration, and Dynamic Panels . Amsterdam : Elsevier , pp. 179 – 222 .[Crossref] , [Google Scholar]), Mark and Sul (2003 Mark , N. C. , Sul , D. ( 2003 ). Cointegration vector estimation by panel dynamic OLS and long-run money demand . Oxford Bulletin of Economics and Statistics 65 : 655 – 680 .[Crossref], [Web of Science ®] , [Google Scholar]), Pedroni (2001 Pedroni , P. ( 2001 ). Purchasing power parity tests in cointegrated panels . Review of Economics and Statistics 83 : 1371 – 1375 . [Google Scholar]), and Breitung (2005 Breitung , J. ( 2005 ). A parametric approach to the estimation of cointegration vectors in panel data . Econometric Reviews 24 : 151 – 173 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We study the impact of stable autoregressive roots approaching the unit circle, of I(2) components, of short-run cross-sectional correlation and of cross-unit cointegration on the performance of the tests and estimators. The data are simulated from three-dimensional individual specific VAR systems with cointegrating ranks varying from zero to two for fourteen different panel dimensions. The usual specifications of deterministic components are considered.     

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.).     

A Class of Improved Estimators for the Scale Parameter of a Mixture Model of Exponential Distribution with Unknown Location

Under Stein's loss, a class of improved estimators for the scale parameter of a mixture of exponential distribution with unknown location is constructed. The method is analogous to Maruyama's (1998 Maruyama , Y. ( 1998 ). Minimax estimators of a normal variance . Metrika 48 : 209 – 214 .[Crossref], [Web of Science ®] , [Google Scholar]) construction for the variance of a normal distribution and also an extension of the result produced in Petropoulos and Kourouklis (2002 Petropoulos , C. , Kourouklis , S. ( 2002 ). A class of improved estimators for the scale parameter of an exponential distribution with unknown location . Commun. Statist. Theor. Meth. 31 : 325 – 335 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Also, robustness properties are considered.     

A Regression Model for the AUC of Clustered Ordinal Test Results and Working Independent Optimal Weights

We study a regression model on the area under the receiver operating characteristic curves (AUC) for clustered (or repeatedly measured) test results. To account for cluster information, we consider a weighted estimating equation for Dodd and Pepe (2003 Dodd , L. , Pepe , M. ( 2003 ). Semiparametric regression for the area under the receiver operating charateristic curve . Journal of the American Statistical Association 98 : 409 – 417 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar])'s regression model with working independence weights. We find the optimal weight in the given class of working independence weights to minimize the variance (or MSE) of regression estimators. We apply the proposed procedure to analyzing our recent experiment on diagnosing a liver disorder. In this experiment, we investigated MRI images of patients having symptoms of potential liver disorder to compare the performance of different MRI picturing methods in testing for liver disorders.