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 Identification of Potential Longitudinal Biomarkers and Measurements of Effectiveness for Biomarkers as Surrogates in Multivariate Survival Data

A proposed method based on frailty models is used to identify longitudinal biomarkers or surrogates for a multivariate survival. This method is an extention of earlier models by Wulfsohn and Tsiatis (1997 Wulfsohn , M. S. , Tsiatis , A. A. ( 1997 ). A joint model for survival and longitudinal data measured with error . Biometrics 53 ( 1 ): 330 – 339 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Song et al. (2002 Song , X. , Davidian , M. , Tsiatis , A. A. ( 2002 ). A Semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data . Biometrics 58 ( 4 ): 742 – 753 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). In this article, similar to Henderson et al. (2002 Henderson , R. , Diggle , P. J. , Dobson , A. ( 2002 ). Identification and efficacy of longitudinal markers for survival . Biostatistics 3 ( 1 ): 33 – 50 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), a joint likelihood function combines the likelihood functions of the longitudinal biomarkers and the multivariate survival times. We use simulations to explore how the number of individuals, the number of time points per individual and the functional form of the random effects from the longitudianl biomarkers influence the power to detect the association of a longitudinal biomarker and the multivariate survival time. The proposed method is illustrate by using the gastric cancer data.     

A side sensitive modified group runs control chart to detect shifts in the process mean

Gadre and Rattihalli [5 Gadre, M. P. and Rattihalli, R. N. 2006. Modified group runs control charts to detect increases in fraction non-conforming and shifts in the process mean. Commun. Stat. Simul. Comput., 35: 225–240. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]] have introduced the Modified Group Runs (MGR) control chart to identify the increases in fraction non-conforming and to detect shifts in the process mean. The MGR chart reduces the out-of-control average time-to-signal (ATS), as compared with most of the well-known control charts. In this article, we develop the Side Sensitive Modified Group Runs (SSMGR) chart to detect shifts in the process mean. With the help of numerical examples, it is illustrated that the SSMGR chart performs better than the Shewhart's X¯ chart, the synthetic chart [12 Wu, Z. and Spedding, T. A. 2000. A synthetic control chart for detecting small shifts in the process mean. J. Qual. Technol., 32: 32–38. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]], the Group Runs chart [4 Gadre, M. P. and Rattihalli, R. N. 2004. A group runs control chart for detecting shifts in the process mean. Econ. Qual. Control, 19: 29–43. [Crossref] [Google Scholar]], the Side Sensitive Group Runs chart [6 Gadre, M. P. and Rattihalli, R. N. 2007. A side sensitive group runs control chart for detecting shifts in the process mean. Stat. Methods Appl., 16: 27–37. [Crossref], [Web of Science ®] [Google Scholar]], as well as the MGR chart [5 Gadre, M. P. and Rattihalli, R. N. 2006. Modified group runs control charts to detect increases in fraction non-conforming and shifts in the process mean. Commun. Stat. Simul. Comput., 35: 225–240. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]]. In some situations it is also superior to the Cumulative Sum chart p9 Page, E. S. 1954. Continuous inspection schemes. Biometrika, 41: 100–114. [Crossref], [Web of Science ®] [Google Scholar]] and the exponentially weighed moving average chart [10 Roberts, S. W. 1959. Control chart tests based on geometric moving averages. Technometrics, 1: 239–250. [Taylor & Francis Online] [Google Scholar]]. In the steady state also, its performance is better than the above charts.     

Quantile-based Chernoff distance for truncated random variables

Several probability distributions such as power-Pareto distribution (see Gilchrist 2000 Gilchrist, W. 2000. Statistical modelling with quantile functions. Boca Raton, FL: Chapman and Hall/CRC.[Crossref] , [Google Scholar] and Hankin and Lee 2006 Hankin, R. K. S., and A. Lee. 2006. A new family of non-negative distributions. Australian and New Zealand Journal of Statistics 48:67–78.[Crossref], [Web of Science ®] , [Google Scholar]), various forms of lambda distributions (see Ramberg and Schmeiser 1974 Ramberg, J. S., and B. W. Schmeiser. 1974. An appropriate method for generating asymmetric random variables. Communications of the ACM 17:78–82.[Crossref], [Web of Science ®] , [Google Scholar] and Freimer et al. 1988 Freimer, M., S. Mudholkar, G. Kollia, and C. T. Lin. 1988. A study of the generalized lambda family. Communications in Statistics - Theory and Methods 17:3547–67.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Govindarajulu distribution (see Nair, Sankaran, and Vineshkumar 2012 Nair, U. N., P. G. Sankaran, and B. Vineshkumar. 2012. The Govindarajulu distribution: some properties and applications. Communications in Statistics—Theory and Methods 41:4391–406.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), etc., do not have manageable distribution functions, though they have tractable quantile functions. Hence, analytical study of the properties of Chernoff distance of two random variables associated with these distributions via traditional distribution function-based tool becomes difficult. To make this simple, in this paper, we introduce quantile-based Chernoff distance for (left or right) truncated random variables and study its various properties. Some useful bounds as well as characterization results are obtained.     

Regression M-estimators with One Modified Form of Doubly Censored Data

In this article, we consider the M-estimators for the linear regression model when both response and covariate variables are subject to double censoring. The proposed estimators are constructed as some functional of three types of estimators for a bivariate survival distribution. The first two estimators are the generalizations of the Campbell and Földes (1982 Campbell, G. and Földes, A. 1982. “Large sample properties of nonparametric statistical inference”. In Nonparametric Statistical Inference., Edited by: Gnredenko, B. V., Puri, M. L. and Vineze, I. 103–122. Amsterdam: North-Holland.  [Google Scholar]) and Dabrowska (1988 Dabrowska, D. M. 1988. Kaplan-Meier estimate on the plane. Annals of Statistics, 18: 1475–1489. [Crossref], [Web of Science ®] , [Google Scholar]) estimators proposed by Shen (2009 Shen, P. S. 2009. Nonparametric estimation of the bivariate survival function one modified form of doubly censored data. Computational Statistics, 25: 203–313. [Crossref], [Web of Science ®] , [Google Scholar]). The third estimator is the generalization of the Prentice and Cai (1992 Prentice, R. L. and Cai, J. 1992. Covariance and survivor function estimation using censored multivariate failure time data. Biometrika, 79: 495–512. [Crossref], [Web of Science ®] , [Google Scholar]) estimator. The consistency of the proposed M-estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators. Furthermore, the simple bootstrap methods are used to estimate standard deviations and construct interval estimators.     

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.).     

Estimating Incidence Rate on Current Status Data with Application to a Phase IV Cancer Trial

New drug discovery in the pediatrics has dramatically improved survival, but with long- term adverse events. This motivates the examination of adverse outcomes such as long-term toxicity in a phase IV trial. An ideal approach to monitor long-term toxicity is to systematically follow the survivors, which is generally not feasible. Instead, cross-sectional surveys are conducted in Hudson et al. (2007 Hudson , M. M. , Rai , S. N. , Nunez , C. , Merchant , T. E. , Marina , N. M. , Zalamea , N. , Cox , C. , Phipps , S. , Pompeu , R. , Rosenthal , D. ( 2007 ). Noninvasive evaluation of late anthracycline cardiac toxicity in childhood cancer survivors . J. Clin. Oncol. 25 : 3635 – 3643 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), with one of the objectives to estimate the cumulative incidence rates along with specific interest in fixed-term (5 or 10 year) rates. We present inference procedures based on current status data to our motivating example with very interesting findings.     

Robustness of the EWMA control chart for individual observations

The traditional exponentially weighted moving average (EWMA) chart is one of the most popular control charts used in practice today. The in-control robustness is the key to the proper design and implementation of any control chart, lack of which can render its out-of-control shift detection capability almost meaningless. To this end, Borror et al. [5 Borror, C. M., Montgomery, D. C. and Runger, G. C. 1999. Robustness of the EWMA control chart to non-normality. J. Qual. Technol., 31(3): 309–316. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]] studied the performance of the traditional EWMA chart for the mean for i.i.d. data. We use a more extensive simulation study to further investigate the in-control robustness (to non-normality) of the three different EWMA designs studied by Borror et al. [5 Borror, C. M., Montgomery, D. C. and Runger, G. C. 1999. Robustness of the EWMA control chart to non-normality. J. Qual. Technol., 31(3): 309–316. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]]. Our study includes a much wider collection of non-normal distributions including light- and heavy-tailed and symmetric and asymmetric bi-modal as well as the contaminated normal, which is particularly useful to study the effects of outliers. Also, we consider two separate cases: (i) when the process mean and standard deviation are both known and (ii) when they are both unknown and estimated from an in-control Phase I sample. In addition, unlike in the study done by Borror et al. [5 Borror, C. M., Montgomery, D. C. and Runger, G. C. 1999. Robustness of the EWMA control chart to non-normality. J. Qual. Technol., 31(3): 309–316. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]], the average run-length (ARL) is not used as the sole performance measure in our study, we consider the standard deviation of the run-length (SDRL), the median run-length (MDRL), and the first and the third quartiles as well as the first and the 99th percentiles of the in-control run-length distribution for a better overall assessment of the traditional EWMA chart's in-control performance. Our findings sound a cautionary note to the (over) use of the EWMA chart in practice, at least with some types of non-normal data. A summary and recommendations are provided.     

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.     

The performance of the truncated mixed control chart

Abstract

When the mixed chart proposed by Aslam et al. (2015 Aslam, M., M. Azam, N. Khan, and C.-H. Jun. 2015. A mixed control chart to monitor the process. International Journal of Production Research 53 (15):4684–93. doi:10.1080/00207543.2015.1031354.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is in use, the sample items are classified as defective or not defective and, depending on the number of defectives, the quality characteristic X of the sample items are also measured. In this case, an Xbar chart decides the state of the process. The previous conforming/non-conforming classification truncates the X distribution and, because of that, the mathematical development to obtain the ARLs is complex. Aslam et al. (2015 Aslam, M., M. Azam, N. Khan, and C.-H. Jun. 2015. A mixed control chart to monitor the process. International Journal of Production Research 53 (15):4684–93. doi:10.1080/00207543.2015.1031354.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) didn’t pay attention to the fact that the X distribution is truncated and, due to that, they obtained incorrect ARLs.     

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.     

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.     

A quantile-based study on ageing intensity function

Jiang, Ji, and Xiao (2003 Jiang, R., P. Ji, and X. Xiao. 2003. Aging property of unimodal failure rate models. Reliability Engineering and System Safety 79(1):113–6.[Crossref], [Web of Science ®] , [Google Scholar]) has introduced a quantitative measure known as the ageing intensity function for evaluating the ageing properties of a component/system. In recent years, there has been a great interest on the study of quantile function, an equivalent alternative to the distribution function approach. Unlike the distribution function approach, the quantile method possess some unique properties (see Gilchrist 2000 Gilchrist, W. 2000. Statistical modelling with quantile functions. Boca Raton, Florida: Chapman and Hall/CRC.[Crossref] , [Google Scholar], Nair, Sankaran, and Balakrishnan 2013 Nair N. U., P. G. Sankaran, N. Balakrishnan. 2013. Quantile-based reliability concepts. In: Quantile-Based Reliability Analysis. Statistics for Industry and Technology. New York, NY: Birkhäuser.[Crossref] , [Google Scholar]). Motivated with this, in the present paper we introduce a quantile-based ageing intensity function and study its various ageing properties. We also study some stochastic comparison of random variables based on the proposed measure.     

On Complete Convergence for Arrays of Dependent Random Variables

A complete convergence theorem for an array of rowwise independent random variables was established by Sung et al. (2005 Sung , S. H. , Volodin , A. I. , Hu , T.-C. ( 2005 ). More on complete convergence for arrays . Statist. Probab. Lett. 71 : 303 – 311 .[Crossref], [Web of Science ®] , [Google Scholar]). This result has been generalized and extended by Kruglov et al. (2006 Kruglov , V. M. , Volodin , A. I. , Hu , T.-C. ( 2006 ). On complete convergence for arrays . Statist. Probab. Lett. 76 : 1631 – 1640 .[Crossref], [Web of Science ®] , [Google Scholar]) and Chen et al. (2007 Chen , P. , Hu , T.-C. , Liu , X. , Volodin , A. ( 2007 ). On complete convergence for arrays of rowwise negatively associated random variables . Theor. Probab. Appl. 52 : 393 – 397 . [Google Scholar]). In this article, we extend the results of Sung et al. (2005 Sung , S. H. , Volodin , A. I. , Hu , T.-C. ( 2005 ). More on complete convergence for arrays . Statist. Probab. Lett. 71 : 303 – 311 .[Crossref], [Web of Science ®] , [Google Scholar]), Kruglov et al. (2006 Kruglov , V. M. , Volodin , A. I. , Hu , T.-C. ( 2006 ). On complete convergence for arrays . Statist. Probab. Lett. 76 : 1631 – 1640 .[Crossref], [Web of Science ®] , [Google Scholar]), and Chen et al. (2007 Chen , P. , Hu , T.-C. , Liu , X. , Volodin , A. ( 2007 ). On complete convergence for arrays of rowwise negatively associated random variables . Theor. Probab. Appl. 52 : 393 – 397 . [Google Scholar]) to an array of dependent random variables satisfying Hoffmann-Jørgensen type inequalities.     

On Complete Convergence in Marcinkiewicz-Zygmund Type SLLN for END Random Variables and Its Applications

Abstract

In this paper, the complete convergence for maximal weighted sums of extended negatively dependent (END, for short) random variables is investigated. Some sufficient conditions for the complete convergence and some applications to a nonparametric model are provided. The results obtained in the paper generalize and improve the corresponding ones of Wang et al. (2014 Wang, X. J., X. Deng, L. L. Zheng, and S. H. Hu. 2014. Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications. A Journal of Theoretical and Applied Statistics 48(4):834–50. [Google Scholar]b) and Shen, Xue, and Wang (2017 Shen, A., M. Xue, and W. Wang. 2017. Complete convergence for weighted sums of extended negatively dependent random variables. Communications in Statistics – Theory and Methods 46(3):1433–44.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

Quantile-Based Reliability Analysis

Quantile functions are equivalent alternatives to distribution functions in modeling and analysis of statistical data. The present article discusses the role of quantile functions in reliability studies. We present the hazard, mean residual, variance residual, and percentile residual quantile functions, their mutual relationships and expressions for the quantile functions in terms of these functions. Further, some theoretical results relating to the Hankin and Lee (2006 Hankin , R. K. S. , Lee , A. ( 2006 ). A new family of non-negative distributions . Austral. N.Z. J. Statist. 48 : 67 – 78 .[Crossref], [Web of Science ®] , [Google Scholar]) lambda distribution are discussed.     

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.     

Lag Length Selection in DF-GLS Unit Root Tests

We reinvestigate the empirical problem of lag length selection in unit root tests when using the augmented Dickey–Fuller test based on GLS-detrending. We extend the Ng and Perron (1995 Ng , S. , Perron , P. ( 1995 ). Unit root tests in ARMA models with data-dependent methods for the selection of the truncation lag . Journal of American Statistical Association 90 : 268 – 281 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) work on this issue by applying the finite sample critical values calculated using the formulae proposed by Cheung and Lai (1995 Cheung , Y. W. , Lai , K. S. ( 1995 ). Lag order and critical values of a modified Dickey–Fuller test . Oxford Bulletin of Business and Economics 57 : 411 – 418 .[Crossref] , [Google Scholar]). Unlike Ng and Perron (2001 Ng , S. , Perron , P. (2001). Lag length selection and the construction of unit root tests with good size and power. Econometrica 69:1519–1554.[Crossref], [Web of Science ®] , [Google Scholar]) we find through simulation studies that the method of selecting lag length using the sequential t-test in the ADF regression of GLS-detrended series performs the best in most cases.     

Estimators for the Population Mean in the Case of Missing Data

We propose a class of estimators for the population mean when there are missing data in the data set. Obtaining the mean square error equations of the proposed estimators, we show the conditions where the proposed estimators are more efficient than the sample mean, ratio-type estimators, and the estimators in Singh and Horn (2000 Singh , S. , Horn , S. ( 2000 ). Compromised imputation in survey sampling . Metrika 51 : 267 – 276 .[Crossref], [Web of Science ®] , [Google Scholar]) and Singh and Deo (2003 Singh , S. , Deo , B. (2003). Imputation by power transformation. Statist. Pap. 44:555–579.[Crossref], [Web of Science ®] , [Google Scholar]) in the case of missing data. These conditions are also supported by a numerical example.