Skew-Normal ARMA Models with Nonlinear Heteroscedastic Predictors

Multivariate skew-normal (SN) distributions (Azzalini and Dalla Valle, 1996 Azzalini , A. , Dalla Valle , A. ( 1996 ). The multivariate skew-normal distribution . Biometrika 83 : 715 – 726 .[Crossref], [Web of Science ®] , [Google Scholar]) enjoy some of the useful properties of normal distributions, have nonlinear heteroscedastic predictors but lack the closure property of normal distributions (the sum of independent SN random variables is not SN). Recently, there has been a proliferation of classes of SN distributions with certain closure properties, one of the most promising being the closed skew-normal (CSN) distributions of González-Farías et al. (2004 González-Farías , G. , Dominguez-Molina , J. A. , Gupta , A. K. ( 2004 ). Additive properties of skew-normal random vectors . J. Statist. Plann. Infer. 126 : 521 – 534 .[Crossref], [Web of Science ®] , [Google Scholar]). We study the construction of stationary SN ARMA models for colored SN noise and show that their finite-dimensional distributions are skew-normal, seldom strictly stationary and their covariance functions differ from their normal ARMA counterparts in that they do not converge to zero for large lags. The situation is better for ARMA models driven by CSN noise, but at the additional cost of considerable computational complexity and a less explicit skewness parameter. In view of these results, the widespread use of such classes of SN distributions in the framework of ARMA models seem doubtful.     

Extensions of Stein's Lemma for the Skew-Normal Distribution

When two random variables have a bivariate normal distribution, Stein's lemma (Stein, 1973 Stein , C. M. ( 1973 ). Estimation of the mean of a multivariate normal distribution . Proc. Prague Symp. Asymptotic Statist. 345 – 381 . [Google Scholar] 1981 Stein , C. M. ( 1981 ). Estimation of the mean of a multivariate normal distribution . Ann. Statist. 9 : 1135 – 1151 .[Crossref], [Web of Science ®] , [Google Scholar]), provides, under certain regularity conditions, an expression for the covariance of the first variable with a function of the second. An extension of the lemma due to Liu (1994 Liu , J. S. ( 1994 ). Siegel's formula via Stein's identities . Statist. Probab. Lett. 21 : 247 – 251 .[Crossref], [Web of Science ®] , [Google Scholar]) as well as to Stein himself establishes an analogous result for a vector of variables which has a multivariate normal distribution. The extension leads in turn to a generalization of Siegel's (1993 Siegel , A. F. ( 1993 ). A surprising covariance involving the minimum of multivariate normal variables . J. Amer. Statist. Assoc. 88 : 77 – 80 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) formula for the covariance of an arbitrary element of a multivariate normal vector with its minimum element. This article describes extensions to Stein's lemma for the case when the vector of random variables has a multivariate skew-normal distribution. The corollaries to the main result include an extension to Siegel's formula. This article was motivated originally by the issue of portfolio selection in finance. Under multivariate normality, the implication of Stein's lemma is that all rational investors will select a portfolio which lies on Markowitz's mean-variance efficient frontier. A consequence of the extension to Stein's lemma is that under multivariate skew-normality, rational investors will select a portfolio which lies on a single mean-variance-skewness efficient hyper-surface.     

On a Generalization of Bivariate Cauchy Distribution

This paper addresses a generalization of the bivariate Cauchy distribution discussed by Fang et al. (1990 Fang , K. T. , Kotz , S. , Ng , K. W. ( 1990 ). Symmetric Multivariate and Related Distributions . London : Chapman and Hall .[Crossref] , [Google Scholar]), derived from a trivariate normal distribution with a general correlation matrix. We obtain explicit expressions for the joint distribution function and joint density function, and show that they reduce in a special case to the corresponding expressions of Fang et al. (1990 Fang , K. T. , Kotz , S. , Ng , K. W. ( 1990 ). Symmetric Multivariate and Related Distributions . London : Chapman and Hall .[Crossref] , [Google Scholar]). Finally, we show that this generalized distribution is useful in determining the orthant probability of a bivariate skew-normal distribution of Azzalini and Dalla Valle (1996 Azzalini , A. , Dalla Valle , A. ( 1996 ). The multivariate skew-normal distribution . Biometrika 83 : 715 – 726 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Local Influence Analysis for Skew-Normal Linear Mixed Models

In this article, we consider local influence analysis for the skew-normal linear mixed model (SN-LMM). As the observed data log-likelihood associated with the SN-LMM is intractable, Cook's well-known approach cannot be applied to obtain measures of local influence. Instead, we develop local influence measures following the approach of Zhu and Lee (2001 Zhu , H. , Lee , S. ( 2001 ). Local influence for incomplete-data models . J. Roy. Statist. Soc. Ser. B 63 : 111 – 126 .[Crossref] , [Google Scholar]). This approach is based on the use of an EM-type algorithm and is measurement invariant under reparametrizations. Four specific perturbation schemes are discussed. Results obtained for a simulated data set and a real data set are reported, illustrating the usefulness of the proposed methodology.     

A Flexible Skew-Generalized Normal Distribution

Skew-symmetric distributions of various types have been the center of attraction by many researchers in the literature. In this article, we will introduce a uni/bimodal generalization of the Azzalini's skew-normal distribution which is indeed an extension of the skew-generalized normal distribution obtained by Arellano-Valle et al. (2004 Arellano-Valle , R. B. , Gómez , H. W. , Quintana , F. A. ( 2004 ). A new class of Skew-normal distributions . Commun. Statist. 33 : 1465 – 1480 .[Taylor & Francis Online] , [Google Scholar]). Our new distribution contains more parameters and thus it is more flexible in data modeling. Indeed, certain univariate case of the so called flexible skew-symmetric distribution of Ma and Genton (2004 Ma , Y. , Genton , M. G. ( 2004 ). Flexible class of skew-symmetric distributions . Scan. J. Statist. 31 : 459 – 468 .[Crossref], [Web of Science ®] , [Google Scholar]) is also a particular case of our proposed model. We will first study some basic distributional properties of the new extension, such as its distribution function, limiting behavior and moments. Then, we will investigate some useful results regarding its relation with other known distributions, such as student's t and skew-Cauchy distributions. In addition, we will present certain methods to generate the new distribution and, finally, we shall apply the model to a real data set to illustrate its behavior comparing to some rival models.     

Parameter recovery for a skew-normal IRT model under a Bayesian approach: hierarchical framework,prior and kernel sensitivity and sample size

Item response theory (IRT) comprises a set of statistical models which are useful in many fields, especially when there is an interest in studying latent variables (or latent traits). Usually such latent traits are assumed to be random variables and a convenient distribution is assigned to them. A very common choice for such a distribution has been the standard normal. Recently, Azevedo et al. [Bayesian inference for a skew-normal IRT model under the centred parameterization, Comput. Stat. Data Anal. 55 (2011), pp. 353–365] proposed a skew-normal distribution under the centred parameterization (SNCP) as had been studied in [R.B. Arellano-Valle and A. Azzalini, The centred parametrization for the multivariate skew-normal distribution, J. Multivariate Anal. 99(7) (2008), pp. 1362–1382], to model the latent trait distribution. This approach allows one to represent any asymmetric behaviour concerning the latent trait distribution. Also, they developed a Metropolis–Hastings within the Gibbs sampling (MHWGS) algorithm based on the density of the SNCP. They showed that the algorithm recovers all parameters properly. Their results indicated that, in the presence of asymmetry, the proposed model and the estimation algorithm perform better than the usual model and estimation methods. Our main goal in this paper is to propose another type of MHWGS algorithm based on a stochastic representation (hierarchical structure) of the SNCP studied in [N. Henze, A probabilistic representation of the skew-normal distribution, Scand. J. Statist. 13 (1986), pp. 271–275]. Our algorithm has only one Metropolis–Hastings step, in opposition to the algorithm developed by Azevedo et al., which has two such steps. This not only makes the implementation easier but also reduces the number of proposal densities to be used, which can be a problem in the implementation of MHWGS algorithms, as can be seen in [R.J. Patz and B.W. Junker, A straightforward approach to Markov Chain Monte Carlo methods for item response models, J. Educ. Behav. Stat. 24(2) (1999), pp. 146–178; R.J. Patz and B.W. Junker, The applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses, J. Educ. Behav. Stat. 24(4) (1999), pp. 342–366; A. Gelman, G.O. Roberts, and W.R. Gilks, Efficient Metropolis jumping rules, Bayesian Stat. 5 (1996), pp. 599–607]. Moreover, we consider a modified beta prior (which generalizes the one considered in [3 Azevedo, C. L.N., Bolfarine, H. and Andrade, D. F. 2011. Bayesian inference for a skew-normal IRT model under the centred parameterization. Comput. Stat. Data Anal., 55: 353–365. [Crossref], [Web of Science ®] , [Google Scholar]]) and a Jeffreys prior for the asymmetry parameter. Furthermore, we study the sensitivity of such priors as well as the use of different kernel densities for this parameter. Finally, we assess the impact of the number of examinees, number of items and the asymmetry level on the parameter recovery. Results of the simulation study indicated that our approach performed equally as well as that in [3 Azevedo, C. L.N., Bolfarine, H. and Andrade, D. F. 2011. Bayesian inference for a skew-normal IRT model under the centred parameterization. Comput. Stat. Data Anal., 55: 353–365. [Crossref], [Web of Science ®] , [Google Scholar]], in terms of parameter recovery, mainly using the Jeffreys prior. Also, they indicated that the asymmetry level has the highest impact on parameter recovery, even though it is relatively small. A real data analysis is considered jointly with the development of model fitting assessment tools. The results are compared with the ones obtained by Azevedo et al. The results indicate that using the hierarchical approach allows us to implement MCMC algorithms more easily, it facilitates diagnosis of the convergence and also it can be very useful to fit more complex skew IRT models.     

Relations for Moments of Progressively Type-II Censored Order Statistics from Log-Logistic Distribution with Applications to Inference

In this article, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a log-logistic distribution. The use of these relations in a systematic recursive manner would enable the computation of all the means, variances and covariances of progressively Type-II right censored order statistics from the log-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R ₁,…, R _m ). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan and Malik (1987 Balakrishnan , N. , Malik , H. J. ( 1987 ). Moments of order statistics from truncated log-logistic distribution . J. Statist. Plann. Infer. 17 : 251 – 267 .[Crossref], [Web of Science ®] , [Google Scholar]) and Balakrishnan et al. (1987 Balakrishnan , N. , Malik , H. J. , Puthenpura , S. ( 1987 ). Best linear unbiased estimation of location and scale parameters of the log-logistic distribution . Commun. Statist. Theor. Meth. 16 : 3477 – 3495 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The moments so determined are then utilized to derive best linear unbiased estimators for the scale- and location-scale log-logistic distributions. A comparison of these estimates with the maximum likelihood estimates is made through Monte Carlo simulation. The best linear unbiased predictors of progressively censored failure times is then discussed briefly. Finally, a numerical example is presented to illustrate all the methods of inference developed here.     

Limit Theorems for the Counting Process of Near-Records

Near-records of a sequence, as defined in Balakrishnan et al. (2005 Balakrishnan , N. , Pakes , A. G. , Stepanov , A. ( 2005 ). On the number and sum of near-record observations . Advances in Applied Probability 37 : 765 – 780 .[Crossref], [Web of Science ®] , [Google Scholar]), are observations lying within a fixed distance of the current record. In this article we study the asymptotic behavior of the number of near-records, among the first n observations in a sequence of independent, identically distributed and absolutely continuous random variables. We give conditions for the finiteness of the total number of near-records as well as laws of large numbers for their counting process. For distributions with a finite number of near-records, we carry out a simulation study suggesting that the total number of near-records has a geometric distribution.     

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.).     

Using Frailty Models to Identify the Longitudinal Biomarkers in Survival Analysis

We introduce a score test to identify longitudinal biomarkers or surrogates for a time to event outcome. This method is an extension of Henderson et al. (2000 Henderson , R. , Diggle , P. , Dobson , A. ( 2000 ). Joint modelling of longitudinal measurements and event time data . Biostatistics 1 ( 4 ): 465 – 480 .[Crossref], [PubMed] , [Google Scholar], 2002 Henderson , R. , Diggle , P. , Dobson , A. ( 2002 ). Identification and efficacy of longitudinal markers for survival . Biostatistics 3 ( 1 ): 33 – 50 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). In this article, a score test is based on a joint likelihood function which combines the likelihood functions of the longitudinal biomarkers and the survival times. Henderson et al. (2000 Henderson , R. , Diggle , P. , Dobson , A. ( 2000 ). Joint modelling of longitudinal measurements and event time data . Biostatistics 1 ( 4 ): 465 – 480 .[Crossref], [PubMed] , [Google Scholar], 2002 Henderson , R. , Diggle , P. , Dobson , A. ( 2002 ). Identification and efficacy of longitudinal markers for survival . Biostatistics 3 ( 1 ): 33 – 50 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) assumed that the same random effect exists in the longitudinal component and in the Cox model and then they can derive a score test to determine if a longitudinal biomarker is associated with time to an event. We extend this work and our score test is based on a joint likelihood function which allows other random effects to be present in the survival function.

Considering heterogeneous baseline hazards in individuals, we use simulations to explore how the factors can influence the power of a score test to detect the association of a longitudinal biomarker and the survival time. These factors include the functional form of the random effects from the longitudinal biomarkers, in the different number of individuals, and time points per individual. We illustrate our method using a prothrombin index as a predictor of survival in liver cirrhosis patients.     

Improved Bar-Lev,Bobovitch, and Boukai Randomized Response Models

In this article, two new improved randomized response models have been proposed. The proposed models are found to be more efficient than the recent randomized response model studied by Bar-Lev et al. (2004 Bar-Lev , S. K. , Bobovitch , E. , Boukai , B. ( 2004 ). A note on randomized response models for quantitative data . Metrika 60 : 255 – 260 .[Crossref], [Web of Science ®] , [Google Scholar]). The relative efficiency of the proposed models has been studied with respect to the Bar-Lev et al. (2004 Bar-Lev , S. K. , Bobovitch , E. , Boukai , B. ( 2004 ). A note on randomized response models for quantitative data . Metrika 60 : 255 – 260 .[Crossref], [Web of Science ®] , [Google Scholar]) model under different situations.     

Aspects of Dependence in Generalized Farlie-Gumbel-Morgenstern Distributions

In this article, we obtained a dependence measure for generalized Farlie-Gumbel-Morgenstern (FGM) family in view of Kochar and Gupta (1987 Kochar , S. G. , Gupta , R. P. ( 1987 ). Competitors of Kendall-tau test for testing independence against PQD . Biometrika 74 ( 3 ): 664 – 669 .[Crossref], [Web of Science ®] , [Google Scholar]) and then compared this measure with Spearman's rho and Kendall's tau in FGM family. Moreover, we evaluated the empirical power of the class of distribution-free tests proposed by Kochar and Gupta (1987 Kochar , S. G. , Gupta , R. P. ( 1987 ). Competitors of Kendall-tau test for testing independence against PQD . Biometrika 74 ( 3 ): 664 – 669 .[Crossref], [Web of Science ®] , [Google Scholar], 1990 Kochar , S. G. , Gupta , R. P. ( 1990 ). Distribution-free tests based on sub-sample extrema for testing against positive dependence . Australian Journal of Statistics 32 : 45 – 51 .[Crossref] , [Google Scholar]) based on exact distribution of a U-statistics. This is derived via a simulation study for sample of sizes n = 6, 8, 10, 12, 16, and 20. Also, we compared our simulation results with those achieved by Amini et al. (2010 Amini , M. , Jabbari , H. , Mohtashami Borzadaran , G. R. , Azadbakhsh , M. ( 2010 ). Power comparison of independence test for the Farlie-Gumbel-Moregenstern family . Communications of the Korean Statistical Society 17 ( 4 ): 493 – 505 .[Crossref] , [Google Scholar]) and Güven and Kotz (2008 Güven , B. , Kotz , S. ( 2008 ). Test of independence for generalized Farlie-Gumbel-Morgenstern distributions . Journal of Computational and Applied Mathematics 212 : 102 – 111 .[Crossref], [Web of Science ®] , [Google Scholar]).     

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.     

A Comparison of Procedures for Controlling the False Discovery Rate in the Presence of Small Variance Genes: A Simulation Study

The Significance Analysis of Microarrays (SAM; Tusher et al., 2001 Tusher , V. G. , Tibshirani , R. , Chu , G. ( 2001 ). Significance analysis of microarrys applied to the ionizing radiation response . Proceedings of the National Academy of Sciences 98 : 5116 – 5121 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) method is widely used in analyzing gene expression data while controlling the FDR by using resampling-based procedure in the microarray setting. One of the main components of the SAM procedure is the adjustment of the test statistic. The introduction of the fudge factor to the test statistic aims at deflating the large value of test statistics due to the small standard error of gene-expression. Lin et al. (2008 Lin , D. , Shkedy , Z. , Burzykowski , T. , Göhlmann , H. W. H. , De Bondt , A. , Perera , T. , Geerts , T. , Bijnens , L. ( 2008 ). Significance analysis of microarray (SAM) for comparisons of several treatments with one control . Biometric Journal, MCP 50 ( 5 ): 801 – 823 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) pointed out that the fudge factor does not effectively improve the power and the control of the FDR as compared to the SAM procedure without the fudge factor in the presence of small variance genes. Motivated by the simulation results presented in Lin et al. (2008 Lin , D. , Shkedy , Z. , Burzykowski , T. , Göhlmann , H. W. H. , De Bondt , A. , Perera , T. , Geerts , T. , Bijnens , L. ( 2008 ). Significance analysis of microarray (SAM) for comparisons of several treatments with one control . Biometric Journal, MCP 50 ( 5 ): 801 – 823 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), in this article, we extend our study to compare several methods for choosing the fudge factor in the modified t-type test statistics and use simulation studies to investigate the power and the control of the FDR of the considered methods.     

Improved strategy to collect sensitive data by using geometric distribution as a randomization device

Abstract

In this article, we improvise Singh and Grewal (2013 Singh, S., and I. S. Grewal. 2013. Geometric distribution as a randomization device implemented in the Kuk’s model. International Journal of Contemporary Mathematical Sciences 8:243–8.[Crossref] , [Google Scholar]) and Hussain et al. (2016 Hussain, Z., J. Shabbir, Z. Pervez, S. F. Shah, and M. Khan. 2016. Generalized geometric distribution of order k: A flexible choice to randomize the response. Communications in Statistics: Simulation and Computation 46:4708–21.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) techniques by introducing a new two-stage randomization response process. Using the proposed new technique, we achieve better efficiency and increasing protection of privacy of respondents than the Kuk (1990 Kuk, A. Y. C. 1990. Asking sensitive questions indirectly. Biometrika 77 (2):436–8.[Crossref], [Web of Science ®] , [Google Scholar]), Singh and Grewal (2013 Singh, S., and I. S. Grewal. 2013. Geometric distribution as a randomization device implemented in the Kuk’s model. International Journal of Contemporary Mathematical Sciences 8:243–8.[Crossref] , [Google Scholar]) and Hussain et al. (2016 Hussain, Z., J. Shabbir, Z. Pervez, S. F. Shah, and M. Khan. 2016. Generalized geometric distribution of order k: A flexible choice to randomize the response. Communications in Statistics: Simulation and Computation 46:4708–21.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) models. The relative efficiency and protection of the respondents of the proposed two-stage randomization device have been investigated through simulation study, and the situations are reported where the proposed estimator performs better than its competitors. The SAS code used to investigate the performance of the proposed strategy are also provided.     

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.     

Generalized Scrambling in Quantitative Optional Randomized Response Models

Huang (2010 Huang , K. C. ( 2010 ). Unbiased estimators of mean, variance and sensitivity level for quantitative characteristics in finite population sampling . Metrika 71 : 341 – 352 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed an optional randomized response model using a linear combination scrambling which is a generalization of the multiplicative scrambling of Eichhorn and Hayre (1983 Eichhorn , B. H. , Hayre , L. S. ( 1983 ). Scrambled randomized response methods for obtaining sensitive quantitative data . J. Statist. Plann. Infer. 7 : 307 – 316 .[Crossref], [Web of Science ®] , [Google Scholar]) and the additive scrambling of Gupta et al. (2006, 2010). In this article, we discuss two main issues. (1) Can the Huang (2010 Huang , K. C. ( 2010 ). Unbiased estimators of mean, variance and sensitivity level for quantitative characteristics in finite population sampling . Metrika 71 : 341 – 352 .[Crossref], [Web of Science ®] , [Google Scholar]) model be improved further by using a two-stage approach?; (2) Does the linear combination scrambling provide any benefit over the additive scrambling of Gupta et al. (2010 Gupta , S. N. , Shabbir , J. , Sehra , S. ( 2010 ). Mean and sensitivity estimation in optional randomized response models . J. Statist. Plann. Infer. 140 : 2870 – 2874 .[Crossref], [Web of Science ®] , [Google Scholar])? We will note that the answer to the first question is “yes” but the answer to the second question is “no.”     

A Note on Importance Resampling for Multi-Dimensional Statistics

Johns (1988 Johns , M. V. (1988). Importance sampling for bootstrap confidence intervals. Journal of the American Statistical Association 83:709–714.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Davison (1988 Davison , A. C. ( 1988 ). Discussion of paper by D. V. Hinkley . Journal of the Royal Statistical Society Series B 50 : 356 – 357 . [Google Scholar]), and Do and Hall (1991 Do , K. A. , Hall , P. ( 1991 ). On importance sampling for the bootstrap . Biometrika 78 : 161 – 167 .[Crossref], [Web of Science ®] , [Google Scholar]) used importance sampling for calculating bootstrap distributions of one-dimensional statistics. Realizing that their methods can not be extended easily to multi-dimensional statistics, Fuh and Hu (2004 Fuh , C. D. , Hu , I. ( 2004 ). Efficient importance sampling for events of moderate deviations with applications . Biometrika 91 : 471 – 490 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed an exponential tilting formula for statistics of multi-dimension, which is optimal in the sense that the asymptotic variance is minimized for estimating tail probabilities of asymptotically normal statistics. For one-dimensional statistics, Hu and Su (2008 Hu , J. , Su , Z. ( 2008 ). Adaptive resampling algorithms for estimating bootstrap distributions . Journal of Statistical Planning and Inference 138 ( 6 ): 1763 – 1777 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed a multi-step variance minimization approach that can be viewed as a generalization of the two-step variance minimization approach proposed by Do and Hall (1991 Do , K. A. , Hall , P. ( 1991 ). On importance sampling for the bootstrap . Biometrika 78 : 161 – 167 .[Crossref], [Web of Science ®] , [Google Scholar]). In this article, we generalize the approach of Hu and Su (2008 Hu , J. , Su , Z. ( 2008 ). Adaptive resampling algorithms for estimating bootstrap distributions . Journal of Statistical Planning and Inference 138 ( 6 ): 1763 – 1777 .[Crossref], [Web of Science ®] , [Google Scholar]) to multi-dimensional statistics, which applies to general statistics and does not resort to asymptotics. Empirical results on a real survival data set show that the proposed algorithm provides significant computational efficiency gains.     

Empirical Likelihood Ratio for Linear Transformation Models with Doubly Censored Data

Double censoring arises when T represents an outcome variable that can only be accurately measured within a certain range, [L, U], where L and U are the left- and right-censoring variables, respectively. When L is always observed, we consider the empirical likelihood inference for linear transformation models, based on the martingale-type estimating equation proposed by Chen et al. (2002 Chen , K. , Jin , Z. , Ying , Z. ( 2002 ). Semiparametric analysis of transformation models with censored data . Biometrika 89 : 659 – 668 .[Crossref], [Web of Science ®] , [Google Scholar]). It is demonstrated that both the approach of Lu and Liang (2006 Lu , W. , Liang , Y. ( 2006 ). Empirical likelihood inference for linear transformation models . Journal of Multivariate Analysis 97 : 1586 – 1599 .[Crossref], [Web of Science ®] , [Google Scholar]) and that of Yu et al. (2011 Yu , W. , Sun , Y. , Zheng , M. ( 2011 ). Empirical likelihood method for linear transformation models . Annals of the Institute of Statistical Mathematics 63 : 331 – 346 .[Crossref], [Web of Science ®] , [Google Scholar]) can be extended to doubly censored data. Simulation studies are conducted to investigate the performance of the empirical likelihood ratio methods.