Equivalent Conditions of Complete Moment Convergence of Weighted Sums for ϕ-Mixing Sequence of Random Variables

In this article, the complete moment convergence of weighted sums for ?-mixing sequence of random variables is investigated. By applying moment inequality and truncation methods, the equivalent conditions of complete moment convergence of weighted sums for ?-mixing sequence of random variables are established. These results promote and improve the corresponding results obtained by Li et al. (1995 Li, D.L., Rao, M.B., Jiang, T.F., Wang, X.C. (1995). Complete convergence and almost sure convergence of weighted sums of random variables. J. Theoret. Probab. 8:49–76.[Crossref], [Web of Science ®] , [Google Scholar]) and Gut (1993 Gut, A. (1993). Complete convergence and Cesàro summation for i.i.d. random variables. Probab. Theory Related Fields 97:169–178.[Crossref], [Web of Science ®] , [Google Scholar]) from i.i.d. to ?-mixing setting. Moreover, we obtain the complete moment convergence of moving average processes based on ?-mixing random variables, which extends the result of Kim et al. (2008 Kim, T.S., Ko, M.H. (2008). Complete moment convergence of moving average processes under dependence assumptions. Statist. Probab. Lett. 78:839–846.[Crossref], [Web of Science ®] , [Google Scholar]) in the sense that it does not require a specific mixing rate.     

A Strong Limit Theorem for Weighted Sums of Negatively Dependent Random Variables

In this article, we establish a new complete convergence theorem for weighted sums of negatively dependent random variables. As corollaries, many results on the almost sure convergence and complete convergence for weighted sums of negatively dependent random variables are obtained. In particular, the results of Jing and Liang (2008 Jing, B.Y., Liang, H.Y. (2008). Strong limit theorems for weighted sums of negatively associated random variables. J. Theor. Probab. 21:890–909.[Crossref], [Web of Science ®] , [Google Scholar]), Sung (2012 Sung, S.H. (2012). Complete convergence for weighted sums of negatively dependent random variables. Stat. Pap. 53:73–82.[Crossref], [Web of Science ®] , [Google Scholar]), and Wu (2010) can be obtained.     

Generalized linear regression models incorporating original outcome distributions

ABSTRACT

In this article, we propose an approach for incorporating continuous and discrete original outcome distributions into the usual exponential family regression models. The new approach is an extension of the works of Suissa (1991 Suissa, S. (1991). Binary methods for continuous outcomes: A parametric alternative. J. Clin. Epidemiol. 44:241–248.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Suissa and Blais (1995 Suissa, S., Blais, L. (1995). Binary regression with continuous outcomes. Stat. Med. 14:247–255.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), which present methods to estimate the risk of an event defined in a sample subspace of an original continuous outcome variable. Simulation studies are presented in order to illustrate the performance of the developed methodology. Real data sets are analyzed by using the proposed models.     

EVALUATION OF POWER OF TESTS INVOLVING COVARIATES UNDER NONCONSTANT VARIANCE

In many experiments where pre-treatment and post-treatment measurements are taken, investigators wish to determine if there is a difference between two treatment groups. For this type of data, the post-treatment variable is used as the primary comparison variable and the pre-treatment variable is used as a covariate. Although most of the discussion in this paper is written with the pre-treatment variable as the covariate the results are applicable to other choices of a covariate. Tests based on residuals have been proposed as alternatives to the usual covariance methods. Our objective is to investigate how the powers of these tests are affected when the conditional variance of the post-treatment variable depends on the magnitude of the pre-treatment variable. In particular, we investigate two cases. [1] Crager, Michael R. 1987. Analysis of Covariance in Parallel-Group Clinical Trials With Pretreatment Baselines. Biometrics, 43: 895–901. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar] The conditional variance of the post-treatment variable gradually increases as the magnitude of the pre-treatment variable increases. (In many biological models this is the case.) [2] Knoke, James D. 1991. Nonparametric Analysis of Covariance for Comparing Change in Randomized Studies with Baseline Values Subject to Error. Biometrics, 47: 523–533. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar] The conditional variance of the post-treatment variable is dependent upon natural or imposed subgroups contained within the pre-treatment variable. Power comparisons are made using Monte Carlo techniques.     

Complete convergence and complete moment convergence for arrays of rowwise negatively superadditive dependent random variables

In this paper, some complete convergence and complete moment convergence results for arrays of rowwise negatively superadditive dependent (NSD, in short) random variables are studied. The obtained theorems not only extend the result of Gan and Chen (2007 Gan, S. X., and P. Y. Chen. 2007. On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables. Acta Mathematica Scientia. Series B 27 (2):283–90.[Crossref], [Web of Science ®] , [Google Scholar]) to the case of NSD random variables, but also improve them.     

Complete moment convergence of moving-average process generated by a class of random variables

In this article, we establish the complete moment convergence of a moving-average process generated by a class of random variables satisfying the Rosenthal-type maximal inequality and the week mean dominating condition. On the one hand, we give the correct proof for the case p = 1 in Ko (2015 Ko, M.H. (2015). Complete moment convergence of moving average process generated by a class of random variables. J. Inequalities Appl. 2015(1):1–9. Article ID 225.[Crossref], [Web of Science ®] , [Google Scholar]); on the other hand, we also consider the case αp = 1 which was not considered in Ko (2015 Ko, M.H. (2015). Complete moment convergence of moving average process generated by a class of random variables. J. Inequalities Appl. 2015(1):1–9. Article ID 225.[Crossref], [Web of Science ®] , [Google Scholar]). The results obtained in this article generalize some corresponding ones for some dependent sequences.     

On Complete Convergence for Weighted Sums of NA Random Fields

In this paper, we prove the complete convergence for the weighted sums of negatively associated random variables with multidimensional indices. The main result generalizes Theorem 2.1 in Kuczmaszewska and Lagodowski (2011 Kuczmaszewska, A., Lagodowski, Z.A. (2011). Convergence rates in the SLLN for some classes of dependent random field. J. Math. Anal. Appl. 380:571–584.[Crossref], [Web of Science ®] , [Google Scholar]) to the case of weighted sums.     

Complete convergence for weighted sums of arrays of APND random variables

In this note, we introduce a new class of dependent random variables (henceforth rvs), together with some its basic properties. This class includes independent rvs and pairwise negatively dependent rvs. Some extensions of Ranjbar et al. (2008) are discussed. The complete convergence for the new class of rvs is also proved, and some results of Beak and Park (2010 Beak, J.-II., and S. T. Park. 2010. Convergence of weighted sums for arrays of negatively dependent random variables and its applications. J. Stat. Plann. Inference 140:2461–2469.[Crossref], [Web of Science ®] , [Google Scholar]) are extended to this class conveniently.     

Asymptotics of improved generalized moment estimators for spatial autoregressive error models

Abstract

This article considers linear models with a spatial autoregressive error structure. Extending Arnold and Wied (2010) Arnold, M., Wied, D. (2010). Improved GMM estimation of the spatial autoregressive error model. Econ. Lett. 108:65–68.[Crossref], [Web of Science ®] , [Google Scholar], who develop an improved generalized method of moment (GMM) estimator for the parameters of the disturbance process to reduce the bias of existing estimation approaches, we establish the asymptotic normality of a new weighted version of this improved estimator and derive the efficient weighting matrix. We also show that this efficiently weighted GMM estimator is feasible as long as the regression matrix of the underlying linear model is non stochastic and illustrate the performance of the new estimator by a Monte Carlo simulation and an application to real data.     

On Fitting Transformation Model to Survey Data

In this article, we consider fitting a semiparametric linear model to survey data with censored observations. The specific goal of the paper is to extend the methods of Cheng et al. (1995 Cheng, S.C., Wei, L.J., Ying, Z. (1995). Analysis of transformation models with censored data. Biometrika 82(4):835–845.[Crossref], [Web of Science ®] , [Google Scholar]) and 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]) to the case when the sample has been drawn from a population using a complex sampling design. Similar to the approach of Lin (2000 Lin, D.Y. (2000). On fitting Cox’s proportional hazards models to survey data. Biometrika 87:37–47.[Crossref], [Web of Science ®] , [Google Scholar]), we regard the survey population as a random sample from an infinite universe and accounts for this randomness in the statistical inference. A simulation study is conducted to investigate the performance of the proposed estimators.     

Alternative Approximations to Value-At-Risk: A Comparison

This article compares three value-at-risk (VaR) approximation methods suggested in the literature: Cornish and Fisher (1937 Cornish, E.A., Fisher, R.A. (1937). Moments and cumulants in the specification of distributions. Revue de l’Institut International de Statistique 5:307–320.[Crossref] , [Google Scholar]), Sillitto (1969 Sillitto, G.P. (1969). Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample. Biometrika 56:641–650.[Crossref], [Web of Science ®] , [Google Scholar]), and Liu (2010 Liu, W.-H. (2010). Estimation and testing of portfolio value-at-risk based on L-comoment matrices. Journal of Futures Markets 30:897–908.[Crossref], [Web of Science ®] , [Google Scholar]). Simulation results are obtained for three families of distributions: student-t, skewed-normal, and skewed-t. We recommend the Sillitto approximation as the best method to evaluate the VaR when the financial return has an unknown, skewed, and heavy-tailed distribution.     

An adaptive multivariate CUSUM control chart for signaling a range of location shifts

ABSTRACT

In this work, we proposed an adaptive multivariate cumulative sum (CUSUM) statistical process control chart for signaling a range of location shifts. This method was based on the multivariate CUSUM control chart proposed by Pignatiello and Runger (1990 Pignatiello, J.J., Runger, G.C. (1990). Comparisons of multivariate CUSUM charts. J. Qual. Technol. 22(3):173–186.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), but we adopted the adaptive approach similar to that discussed by Dai et al. (2011 Dai, Y., Luo, Y., Li, Z., Wang, Z. (2011). A new adaptive CUSUM control chart for detecting the multivariate process mean. Qual. Reliab. Eng. Int. 27(7):877–884.[Crossref], [Web of Science ®] , [Google Scholar]), which was based on a different CUSUM method introduced by Crosier (1988 Crosier, R.B. (1988). Multivariate generalizations of cumulative sum quality-control schemes. Technometrics 30(3):291–303.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The reference value in this proposed procedure was changed adaptively in each run, with the current mean shift estimated by exponentially weighted moving average (EWMA) statistic. By specifying the minimal magnitude of the mean shift, our proposed control chart achieved a good overall performance for detecting a range of shifts rather than a single value. We compared our adaptive multivariate CUSUM method with that of Dai et al. (2001 Dai, Y., Luo, Y., Li, Z., Wang, Z. (2011). A new adaptive CUSUM control chart for detecting the multivariate process mean. Qual. Reliab. Eng. Int. 27(7):877–884.[Crossref], [Web of Science ®] , [Google Scholar]) and the non adaptive versions of these two methods, by evaluating both the steady state and zero state average run length (ARL) values. The detection efficiency of our method showed improvements over the comparative methods when the location shift is unknown but falls within an expected range.     

Complete and complete moment convergence for i.i.d. random variables under exponential moment conditions

In this paper, we establish a complete convergence result and a complete moment convergence result for i.i.d. random variables under moment condition which is slightly weaker than the existence of the moment generating function. The main results extend and improve the related known results of Lanzinger (1998 Lanzinger, H. (1998). A Baum-Katz theorem for random variables under exponential moment conditions. Stat. Probab. Lett. 39(2):89–95.[Crossref], [Web of Science ®] , [Google Scholar]) and Gut and Stadtmüller (2011 Gut, A., Stadtmüller, U. (2011). An intermediate Baum-Katz theorem. Stat. Probab. Lett. 81(10):1486–1492.[Crossref], [Web of Science ®] , [Google Scholar]).     

On goodness-of-fit tests for Aalen's additive model based on left-truncated right-censored or doubly censored data

ABSTRACT

Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) proposed goodness-of-fit tests for Aalen's additive risk model. In this article, we demonstrate that the approach of Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) can be applied to left-truncated right-censored (LTRC) data and doubly censored data. A simulation study is conducted to investigate the performance of the proposed tests. The proposed tests are illustrated using heart transplant data.     

A Simple Corrected Score for Logistic Regression with Errors-in-Covariates

We develop a simple corrected score for logistic regression with errors-in-covariates. The new method is an extension of the consistent functional methods proposed by Huang and Wang (2001) and is closely related to the corrected score method by Nakamura (1990 Nakamura, T. (1990). Corrected score function for errors-in-variables models: Methodology and application to generalized linear models. Biometrika. 77:127–137.[Crossref], [Web of Science ®] , [Google Scholar]) and Stefanski (1989 Stefanski, L.A. (1989). Unbiased estimation of a nonlinear function a normal mean with application to measurement error models. Commun. Stat. Ser. A - Theory Methods. 18:4335–4358.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The new method requires that the measurement error distribution is known, but does not require normality. The new method yields a consistent and asymptotically normal estimator under regularity conditions. We examine the finite-sample performance of the new estimator through simulation studies. Finally, we illustrate the new method by applying it to an AIDS study.     

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.     

Designs Robust Against the Presence of an Aberration in a Diallel Cross Design Setup

In this article, we find designs insensitive to the presence of an outlier in a diallel cross design setup for estimating a complete set of orthonormal contrasts among the effects of the general combining abilities of a set of parental lines. The criterion of robustness, suggested by Mandal (1989 Mandal , N. K. ( 1989 ). On robust designs . Cal. Stat. Assoc. Bull. 38 : 115 – 119 . [Google Scholar]) in block design setup and used by Biswas (2012 Biswas , A. ( 2012 ). Block designs robust against the presence of an aberration in a treatment-control setup . Commun. Statist. Theor. Meth. 41 : 920 – 933 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in treatment-control setup, is adapted here. Complete diallel cross designs, suggested by Gupta and Kageyama (1994 Gupta , S. , Kageyama , S. ( 1994 ). Optimal complete diallel crosses . Biometrika 81 : 420 – 424 .[Crossref], [Web of Science ®] , [Google Scholar]), and partial diallel cross designs, suggested by Gupta et al. (1995 Gupta , S. , Das , A. , Kageyama , S. ( 1995 ). Single replicate orthogonal block designs for circulant partial diallel crosses . Commun. Statist. A 24 : 2601 – 2607 .[Taylor & Francis Online] , [Google Scholar]) and Mukerjee (1997 Mukerjee , R. ( 1997 ). Optimal partial diallel crosses . Biometrika 84 : 939 – 948 .[Crossref], [Web of Science ®] , [Google Scholar]), are found to be robust under certain conditions.     

MULTI-LEVEL FACTORIAL DESIGNS WITH MINIMUM NUMBERS OF LEVEL CHANGES

The order of experimental runs in a fractional factorial experiment is essential when the cost of level changes in factors is considered. The generalized foldover scheme given by [1] Coster, D. C. and Cheng, C. S. 1988. Minimum cost trend free run orders of fractional factorial designs. The Annals of Statistics, 16: 1188–1205. [Crossref], [Web of Science ®] , [Google Scholar]gives an optimal order to experimental runs in an experiment with specified defining contrasts. An experiment can be specified by a design requirement such as resolution or estimation of some interactions. To meet such a requirement, we can find several sets of defining contrasts. Applying the generalized foldover scheme to these sets of defining contrasts, we obtain designs with different numbers of level changes and then the design with minimum number of level changes. The difficulty is to find all the sets of defining contrasts. An alternative approach is investigated by [2] Cheng, C. S., Martin, R. J. and Tang, B. 1998. Two-level factorial designs with extreme numbers of level changes. The Annals of Statistics, 26: 1522–1539. [Crossref], [Web of Science ®] , [Google Scholar]for two-level fractional factorial experiments. In this paper, we investigate experiments with all factors in slevels.     

An objective Bayesian estimation in a two-period crossover design

Two-period crossover design is one of the commonly used designs in clinical trials. But, the estimation of treatment effect is complicated by the possible presence of carryover effect. It is known that ignoring the carryover effect when it exists can lead to poor estimates of the treatment effect. The classical approach by Grizzle (1965 Grizzle, J.E. (1965). The two-period change-over design and its use in clinical trials. Biometrics 21:467–480. See Grizzle (1974) for corrections.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) consists of two stages. First, a preliminary test is conducted on carryover effect. If the carryover effect is significant, analysis is based only on data from period one; otherwise, analysis is based on data from both periods. A Bayesian approach with improper priors was proposed by Grieve (1985 Grieve, A.P. (1985). A Bayesian analysis of the two-period crossover design for clinical trials. Biometrics 41:979–990.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) which uses a mixture of two models: a model with carryover effect and another without. The indeterminacy of the Bayes factor due to the arbitrary constant in the improper prior was addressed by assigning a minimally discriminatory value to the constant. In this article, we present an objective Bayesian estimation approach to the two-period crossover design which is also based on a mixture model, but using the commonly recommended Zellner–Siow g-prior. We provide simulation studies and a real data example and compare the numerical results with Grizzle (1965 Grizzle, J.E. (1965). The two-period change-over design and its use in clinical trials. Biometrics 21:467–480. See Grizzle (1974) for corrections.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar])’s and Grieve (1985 Grieve, A.P. (1985). A Bayesian analysis of the two-period crossover design for clinical trials. Biometrics 41:979–990.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar])’s approaches.     

Further Results for a General Family of Bivariate Copulas

A bivariate family of copulas has been initiated by Cuadras-Augé (1981 Cuadras, C.M., Augé, J. (1981). A continuous general multivariate distribution and its properties. Commun. Statist. (A) Theor. Meth. 10:339–353.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Marshall (1996 Marshall, A.W. (1996). Copulas, marginals, and joint distributions. In: Distributions with fixed marginals and related topics. IMS Lecture Notes Monogr. Ser. 28:213–222.[Crossref] , [Google Scholar]). Recently, Durante (2007 Durante, F. (2007). A new family of symmetric bivariate copulas. C. R. Math. Acad. Sci. Paris 344:195–198.[Crossref], [Web of Science ®] , [Google Scholar]) considered this family as a general family of symmetric bivariate copulas indexed by a generator function and studied some of its dependence properties. In this article, we obtain and describe further aspects of dependence for this family. For example, we have proved that the family has positive likelihood ratio dependence structure if and only if the family reduces to some well-known copulas. We also derive several proper forms for the generator function of this family. Considering a multivariate extension of the bivariate family of copulas provided by Durante et al. (2007 Durante, F., Quesada-Molina, J.J., Flores, M. (2007). On a family of multivariate copulas for aggregation processes. Inform. Sci. 177(24):5715–5724.[Crossref], [Web of Science ®] , [Google Scholar]), some dependence properties are studied. Finally, some positive dependence stochastic orderings for two random vectors having a copula from the proposed families, are discussed.