Group Variable Selection with Oracle Property by Weight-Fused Adaptive Elastic Net Model for Strongly Correlated Data

This paper is the generalization of weight-fused elastic net (Fu and Xu, 2012 Fu, G., Xu, Q. (2012). Grouping variable selection by weight fused elastic net for multi-collinear data. Communications in Statistics-Simulation and Computation 41(2):205–221.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), which performs group variable selection by combining weight-fused LASSO(wfLasso) and elastic net (Zou and Hastie, 2005 Zou, H., Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67(2):301–320.[Crossref], [Web of Science ®] , [Google Scholar]) penalties. In this study, the elastic net penalty is replaced by adaptive elastic net penalty (AdaEnet) (Zou and Zhang, 2009 Zou, H., Zhang, H. (2009). On the adaptive elastic-net with a diverging number of parameters. Annals of Statistics 37(4):1733–1751.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), and a new group variable selection algorithm with oracle property (Fan and Li, 2001 Fan, J., Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96(456):1348–1360.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]; Zou, 2006 Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association 101(476):1418–1429.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is obtained.     

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.     

Asymptotics of maxima and sums for a type of strongly dependent isotropic Gaussian random fields

For a type of strongly dependent isotropic Gaussian random fields introduced by Mittal (1976 Mittal, Y. 1976. A class of isotropic covariances functions. Pacific Journal of Mathematics 64:517–38.[Crossref], [Web of Science ®] , [Google Scholar]), the joint limiting distribution of the maximum and the sum for the Gaussian random fields is derived. The asymptotic relation between the maximum and sum of the continuous time strongly dependent isotropic Gaussian random fields and the maximum and sum of this fields sampled at discrete time points is also obtained.     

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.     

Inference in log-alpha-power and log-skew-normal multivariate models

ABSTRACT

Random vectors with positive components are common in many applied fields, for example, in meteorology, when daily precipitation is measured through a region Marchenko and Genton (2010 Marchenko, Y., Genton, M. (2010). Multivariate log-skew-elliptical distributions with applications to precipitation data. Environmetrics 21:318–340.[Crossref], [Web of Science ®] , [Google Scholar]). Frequently, the log-normal multivariate distribution is used for modeling this type of data. This modeling approach is not appropriate for data with high asymmetry or kurtosis. Consequently, more flexible multivariate distributions than the log-normal multivariate are required. As an alternative to this distribution, we propose the log-alpha-power multivariate and log-skew-normal multivariate models. The first model is an extension for positive data of the fractional order statistics model Durrans (1992 Durrans, S. (1992). Distributions of fractional order statistics in hydrology. Water Resour. Res. 28:1649–1655.[Crossref], [Web of Science ®] , [Google Scholar]). The second one is an extension of the log-skew-normal model studied by Mateu-Figueras and Pawlowsky-Glahn (2007 Mateu-Figueras, G., Pawlowsky-Glahn, V. (2007). The skew-normal distribution on the simplex. Commun. Stat.-Theory Methods 36:1787–1802.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We study parameter estimation for these models by means of pseudo-likelihood and maximum likelihood methods. We illustrate the proposal analyzing a real dataset.     

Combined estimating function for random coefficient models with correlated errors

ABSTRACT

Estimating functionshave been shown to be convenient to study inference for non linear time series models. Recently, Thavaneswaran et al. (2012 Thavaneswaran, A., Liang, Y., Frank, J. (2012). Inference for random coefficient volatility models. Stat. Probab. Lett. 82(12):2086–2090.[Crossref], [Web of Science ®] , [Google Scholar]) used combined estimating functions to study inference for random coefficient autoregressive (RCA) models with generalized autoregressive heteroscedasticity errors. While most RCA modeling assumes that the random term and the error are independent, Chandra and Taniguchi (2001 Chandra, S.A., Taniguchi, M. (2001). Estimating functions for nonlinear time series models. Ann. Inst. Stat. Math 53(1):125–141.[Crossref], [Web of Science ®] , [Google Scholar]) studied inference for RCA models with correlated errors using linear estimating functions. In this paper, we derive the quadratic estimating functions for the joint estimation of the conditional mean, variance, and correlation parameters of the RCA models with correlated errors.     

Quantile Regression with Left-Truncated and Right-Censored Data in a Reproducing Kernel Hilbert Space

Li et al. (2007 Li, Y., Liu, Y., Zhu, J. (2007). Quantile regression in reproducing kernel Hilbert spaces. J. Amer. Statist. Assoc. 102:255–268.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) developed an estimation method for quantile functions in a reproducing kernel Hilbert space for complete data, and Park and Kim (2011 Park, J., Kim, J. (2011). Quantile regression with an epsilon-insensitive loss in a reproducing kernel Hilbert space. Statist. Probab. Lett. 81:62–70.[Crossref], [Web of Science ®] , [Google Scholar]) proposed an estimation method using the ε-insensitive loss. This article extends these estimation methods to left-truncated and right-censored data. As a measure of goodness of fit, the check loss and the ε-insensitive loss were used to estimate the quantile function. The ε-insensitive loss can shrink the estimated coefficients toward zero; hence, it can reduce the variability of the estimates. Simulation studies show that the estimated quantile functions based on the ε-insensitive loss perform slightly better when ε is adequately chosen.     

Inferences on partial linear models with right censored data by empirical likelihood

ABSTRACT

This article considers inference for partial linear models with right censored data. We use empirical likelihood based on the Buckley and James (1979 Buckley, J., James, I. (1979). Linear regression with censored data. Biometrika 66:429–436.[Crossref], [Web of Science ®] , [Google Scholar]) estimating equation to derive the confidence region for the regression parameter. We introduce an adjusted empirical likelihood ratio statistic for the parameter of interest and show that its limiting distribution is standard chi-square. A simulation is carried out to compare our method with the synthetic data approach in Wang and Li (2002 Wang, Q.-H., Li, G. (2002). Empirical Likelihood Semiparametric Regression Analysis under Random Censorship. J. Multivariate Anal. 83:469–486.[Crossref], [Web of Science ®] , [Google Scholar]).     

Convergence of weighted sums for sequences of pairwise NQD random variables

ABSTRACT

The authors discuss the convergence for weighted sums of pairwise negatively quadrant dependent (NQD) random variables and obtain some new results which extend and improve the result of Bai and Cheng (2000) Bai, Z.D., Cheng, P.E. (2000). Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett. 46:105–112.[Crossref], [Web of Science ®] , [Google Scholar]. In addition, we relax some restrictions of the conditions in their result. Some new methods are used in this article which differ from that of Bai and Cheng (2000) Bai, Z.D., Cheng, P.E. (2000). Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett. 46:105–112.[Crossref], [Web of Science ®] , [Google Scholar].     

Baker- Lin-Huang Type Bivariate Distributions Based on Order Statistics

Baker (2008 Baker, R. (2008). An order-statistics-based method for constructing multivariate distributions with fixed marginals. J. Multivariate Anal. 99: 23122327.[Crossref], [Web of Science ®] , [Google Scholar]) introduced a new class of bivariate distributions based on distributions of order statistics from two independent samples of size n. Lin and Huang (2010 Lin, G.D., Huang, J.S. (2010). A note on the maximum correlation for Baker’s bivariate distributions with fixed marginals. J. Multivariate Anal. 101: 22272233.[Crossref], [Web of Science ®] , [Google Scholar]) discovered an important property of Baker’s distribution and showed that the Pearson’s correlation coefficient for this distribution converges to maximum attainable value, i.e., the correlation coefficient of the Fréchet upper bound, as n increases to infinity. Bairamov and Bayramoglu (2013 Bairamov, I., Bayramoglu, K. (2013). From Huang-Kotz distribution to Baker’s distribution. J. Multivariate Anal. 113: 106115.[Crossref], [Web of Science ®] , [Google Scholar]) investigated a new class of bivariate distributions constructed by using Baker’s model and distributions of order statistics from dependent random variables, allowing higher correlation than that of Baker’s distribution. In this article, a new class of Baker’s type bivariate distributions with high correlation are constructed based on distributions of order statistics by using an arbitrary continuous copula instead of the product copula.     

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.     

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.     

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.     

Bootstrap test of hypothesis for the multi-state models in survival analysis

Abstract

In the present paper we develop bootstrap tests of hypothesis, based on simulation, for the transition probability matrix arising in the context of a multi-state model. The bootstrap test statistic is based on the paper of Tattar and Vaman (2008 Tattar, P. N., Vaman, H. J. (2008). Testing transition probability matrix of a multi-state model with censored data. Lifetime Data Anal. 14(2):216–230.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), which develops a statistic for the testing problems concerning the transition probability matrix of the non homogeneous Markov process.     

Estimation of the covariance matrix with two-step monotone missing data

Abstract

We suggest shrinkage based technique for estimating covariance matrix in the high-dimensional normal model with missing data. Our approach is based on the monotone missing scheme assumption, meaning that missing values patterns occur completely at random. Our asymptotic framework allows the dimensionality p grow to infinity together with the sample size, N, and extends the methodology of Ledoit and Wolf (2004) Ledoit, O., Wolf, M. (2004). A well-conditioned estimator for large dimensional covariance matrices. J. Multivariate Anal. 88:365–411.[Crossref], [Web of Science ®] , [Google Scholar] to the case of two-step monotone missing data. Two new shrinkage-type estimators are derived and their dominance properties over the Ledoit and Wolf (2004) Ledoit, O., Wolf, M. (2004). A well-conditioned estimator for large dimensional covariance matrices. J. Multivariate Anal. 88:365–411.[Crossref], [Web of Science ®] , [Google Scholar] estimator are shown under the expected quadratic loss. We perform a simulation study and conclude that the proposed estimators are successful for a range of missing data scenarios.     

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.     

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.     

A stratified Tracy and Osahan's two-stage randomized response model

ABSTRACT

This paper proposes an alternative two-stage stratified randomized response model based on Tracy and Osahan (1999 Tracy, D.S., Osahan, S.S. (1999). An improved randomized response technique. Pak. J. Stat. 15(1):1–6. [Google Scholar]) model that has an optimal allocation and large gain in precision. It is also shown that the proposed model is more efficient than Kim and Warde (2004 Kim, J., Warde, W. (2004). A stratified Warner randomized response model. J. Stat. Plan. Infer. 120:155–165.[Crossref], [Web of Science ®] , [Google Scholar]) and Kim and Elam (2005 Kim, J.M., Elam, M.E. (2005). A two-stage stratified Warner's randomized response model using optimal allocation. Metrika 61:1–7.[Crossref], [Web of Science ®] , [Google Scholar]) stratified randomized response models under the conditions presented in both the cases of completely truthful reporting and that of not completely truthful reporting by the respondents. Numerical illustrations and graphs are also given in support of the present study.     

Designs Robust against Violation of Normality Assumption in the Standard F-test

In this article, designs are found for which the F-test of analysis of variance is insensitive to violation of normality assumption. Atiqullah (1962 Atiquallah, M. (1962). The estimation of residual variance in quadratically balanced least-squares problems and the robustness of the F-test. Biometrika 49: 83–91.[Crossref], [Web of Science ®] , [Google Scholar]) proved that the F-test for treatments adjusting for blocks in the intra-block analysis of a balanced incomplete block design is robust against non-normality in the observations. Here an attempt has been made to identify other designs robust in this sense. In particular, it is observed that for testing relevant hypothesis, a partially balanced incomplete block design in block design setup, under certain conditions, is robust. Robustness of a balanced treatment incomplete block design and a partially balanced treatment incomplete block design (Biswas, 2012 Biswas, A. (2012). Block designs robust against the presence of an aberration in a treatment-control setup. Commun Statist.Theor Methods 41: 920–933.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), in treatment-control design setup, is also studied. Moreover, a new measure of robustness is introduced for further study. The performance of the F-test in presence of non-normality in the observations for a quadratically balanced design is also examined.