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.     

Simultaneous Confidence Intervals for the Parameters of Pareto Distribution under Progressive Censoring

In this article, we consider the progressive Type II right censored sample from Pareto distribution. We introduce a new approach for constructing the simultaneous confidence interval of the unknown parameters of this distribution under progressive censoring. A Monte Carlo study is also presented for illustration. It is shown that this confidence region has a smaller area than that introduced by Ku? and Kaya (2007 Ku? , C. , Kaya , M. F. ( 2007 ). Estimation for the parameters of the Pareto distribution under progressive censoring . Commun. Statist. Theor. Meth. 36 : 1359 – 1365 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

Influence diagnostics in Birnbaum–Saunders nonlinear regression models

We consider the issue of assessing influence of observations in the class of Birnbaum–Saunders nonlinear regression models, which is useful in lifetime data analysis. Our results generalize those in Galea et al. [8 Galea, M., Leiva, V. and Paula, G. A. 2004. Influence diagnostics in log-Birnbaum–Saunders regression models. J. Appl. Stat., 31: 1049–1064. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]] which are confined to Birnbaum–Saunders linear regression models. Some influence methods, such as the local influence, total local influence of an individual and generalized leverage are discussed. Additionally, the normal curvatures for studying local influence are derived under some perturbation schemes. We also give an application to a real fatigue data set.     

On Rates of Convergence for Posterior Distributions Under Misspecification

We extend the approach of Walker (2003 Walker , S. ( 2003 ). On sufficient conditions for Bayesian consistency . Biometrika 90 ( 2 ): 482 – 488 .[Crossref], [Web of Science ®] , [Google Scholar]); (2004 Walker , S. ( 2004 ). New approaches to Bayesian consistency . Ann. Statist. 32 ( 5 ): 2028 – 2043 .[Crossref], [Web of Science ®] , [Google Scholar]) to the case of misspecified models. A sufficient condition for establishing rates of convergence is given based on a key identity involving martingales, which does not require construction of tests. We also show roughly that the result obtained by using tests can also be obtained by our approach, which demonstrates the potential wider applicability of this method.     

A Simple Method for Constructing Multidimensional Distributions of Correlated Categorical Data

We suggest a method for constructing a multidimensional distribution of correlated categorical data with fixed marginal distributions and specified degrees of association based on the log-linear models. A convex combination approach by Lee (1997 Lee , A. J. ( 1997 ). Some simple methods for generating correlated categorical variates . Computational Statistics & Data Analysis 26 : 133 – 148 .[Crossref], [Web of Science ®] , [Google Scholar]) is applied to get a joint distribution with fixed Pearson chi-square coefficient. By using the suggested method, we can generate three-dimensional distributions which have a fixed association among three variables. Therefore, the suggested method could be extended to higher dimensions.     

The Finite Time Ruin Probability of a New Risk Model Based on Entrance Process

In this article, we consider a new insurance risk model based on the entrance process proposed in Li et al. (2005 Li , Z. , Zhu , J. , Chen , F. ( 2005 ). Study of a risk model based on the entrance process . Statist. Probab. Lett. 72 : 1 – 10 .[Crossref], [Web of Science ®] , [Google Scholar]), and investigate the finite time ruin probabilities of this model. It is showed that an exponential upper bound for the finite time ruin probability exists, when the distributions of the claim size are light tailed. Furthermore, when the distributions of the claim size are heavy tailed, an asymptotic formula for the finite time ruin probability is obtained.     

Upper Bounds for the Ruin Probabilities of the Entrance-Based Risk Model

Li et al. (2005 Li , Z. , Zhu , J. , Chen , F. ( 2005 ). Study of a risk model based on the entrance process . Statist. Probab. Lett. 72 ( 1 ): 1 – 10 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed a risk model based on the entrance process and studied the asymptotic behavior of the surplus when time goes to infinity. This article considers the ruin problem in that model. Some simple characteristics (stochastic intensity, compensator, mean process, etc.) of the risk process and other related processes are also considered. Under small claim condition, exponential upper bounds for the ruin probability are obtained.     

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.     

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.     

Strong Limit Theorems for Pairwise NQD Random Variables

A number of strong laws of large numbers for sequences of pairwise negative quadrant dependent (NQD) random variables have been established by using the generalized three series theorem. In this article, we obtain a strong law of large numbers by using the truncation technique and method of subsequences instead of the generalized three series theorem. Our result generalizes and improves on the corresponding one in Li and Yang (2008 Li , R. , Yang , W. ( 2008 ). Strong convergence of pairwise NQD random sequences . J. Math. Anal. Appl. 344 : 741 – 747 .[Crossref], [Web of Science ®] , [Google Scholar]). We also obtain a complete convergence result for an array of rowwise pairwise NQD random variables.     

Tests for Multiple Outliers in an Exponential Sample

By applying the recursion of Huffer (1988 Huffer, F. 1988. Divided differences and the joint distribution of linear combinations of spacings. Journal of Applied Probability, 25: 346–354. [Crossref], [Web of Science ®] , [Google Scholar]) repeatedly, we propose an algorithm for evaluating the null joint distribution of Dixon-type test statistics for testing discordancy of k upper outliers in exponential samples. By using the critical values of Dixon-type test statistics determined from the proposed algorithm and those of Cochran-type test statistics presented earlier by Lin and Balakrishnan (2009 Lin, C. T. and Balakrishnan, N. 2009. Exact computation of the null distribution of a test for multiple outliers in an exponential sample. Computational Statistics & Data Analysis, 53: 3281–3290. [Crossref], [Web of Science ®] , [Google Scholar]), we carry out an extensive Monte Carlo study to investigate the powers and the error probabilities for the effects of masking and swamping when the number of outliers k = 2 and 3. Based on our empirical findings, we recommend Rosner’s (1975 Rosner, B. 1975. On the detection of many outliers. Technometrics, 17: 221–227. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) sequential test procedure based on Dixon-type test statistics for testing multiple outliers from an exponential distribution.     

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.     

A Similarity Measure Based on Species Proportions

ABSTRACT

There are several indices for measuring the similarity of two populations, including the ratio of the number of shared species to the number of distinct species (Jaccard's index) and the conditional probability of observing a shared species (Smith et al., 1996 Smith , W. , Solow , A. R. , Preston , P. E. ( 1996 ). An estimator of species overlap using a modified beta-binomial model. Biometrics 52 : 1472 – 1477 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). However, these indices only take into account the number of species and species proportions of shared species. In this article, we propose a new similarity index which includes the species proportions of both the shared and non shared species in each population, and also propose a Nonparametric Maximum Likelihood Estimator (NPMLE) for this index. Bootstrap and delta methods are used to evaluate the standard errors of the NPMLE. Based on a loss function, we also compare a class of nonparametric estimators for the proposed index in various situations.     

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.     

Generalized Symmetric Divergence Measures and the Probability of Error

There are three classical divergence measures exist in the literature on information theory and statistics. These are namely, Jeffryes-Kullback-Leiber (Jeffreys, 1946 Jeffreys , H. ( 1946 ). An invariant form for the prior probability in estimation problems . Proc. Roy. Soc. Lon. Ser. A 186 : 453 – 461 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Kullback and Leibler, 1951 Kullback , S. , Leibler , R. A. ( 1951 ). On information and sufficiency . Ann. Math. Statist. 22 : 79 – 86 .[Crossref] , [Google Scholar]) J-divergence. Sibson-Burbea-Rao (Sibson, 1969 Sibson , R. ( 1969 ). Information radius . Z. Wahrs. und verw Geb. 14 : 149 – 160 .[Crossref], [Web of Science ®] , [Google Scholar]), Jensen-Shannon divegernce, (Burbea and Rao, 1982 Burbea , J. , Rao , C. R. ( 1982 ). On the convexity of some divergence measures based on entropy functions . IEEE Trans. Inform. Theor. IT-28 : 489 – 495 .[Crossref], [Web of Science ®] , [Google Scholar]), and Taneja (1995 Taneja , I.J. ( 1995 ). New developments in generalized information measures , In: Hawkes , P. W. , Ed., Advances in Imaging and Electron Physics . 91:37–136 . [Google Scholar]). Arithmetic-Geometric divergence. These three measures bear an interesting relationship among each other. The divergence measures like Hellinger (1909 Hellinger , H. ( 1909 ). Neue Begründung der theorie der quadratischen formen von unendlichen vielen Veränderlichen . J. Reine Aug. Math. 136 : 210 – 271 .[Crossref] , [Google Scholar]) discrimination, symmetric χ²-divergence, and triangular discrimination are also known in the literature. In this article, we have considered generalized symmetric divergence measures having the measures given above as particular cases. Bounds on the probability of error are obtained in terms of generalized symmetric divergence measures. Study of bounds on probability of error is extended for the difference of divergence measures.     

Hazards Regression for Cancer Studies

The complication in analyzing tumor data is that the tumors detected in a screening program tend to be slowly progressive tumors, which is the so-called length-biased sampling that is inherent in screening studies. Under the assumption that all subjects have the same tumor growth function, Ghosh (2008 Ghosh , D. ( 2008 ). Proportional hazards regression for cancer studies . Biometrics 64 : 141 – 148 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) developed estimation procedures for proportional hazards model. In this article, by modeling growth function as a function of covariates, we demonstrate that Ghosh (2008 Ghosh , D. ( 2008 ). Proportional hazards regression for cancer studies . Biometrics 64 : 141 – 148 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar])'s approach can be extended to the case when each subject has a specific growth function. A simulation study is conducted to demonstrate the potential usefulness of the proposed estimators for the regression parameters in the proportional and additive hazards model.     

A Nonparametric Shewhart-Type Synthetic Control Chart

In this article, we provide a nonparametric Shewhart-type synthetic control chart based on the signed-rank statistic to monitor shifts in the known in-control process median. The synthetic control chart is a combination of a signed-rank chart due to Bakir (2004 Bakir , S. T. ( 2004 ). A distribution-free Shewhart quality control chart based on signed-ranks . Quality Engineering 16 : 613 – 623 .[Taylor & Francis Online] , [Google Scholar]) and a conforming run length chart due to Bourke (1991 Bourke , P. D. ( 1991 ). Detecting a shift in fraction nonconforming using run-length control charts with 100% inspection . Journal of Quality Technology 23 : 225 – 238 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The operation and design of the chart are discussed and the performance of the chart has been studied. The chart has an attractive average run length behavior as compared to the parametric control chart for a class of symmetric continuous process distributions. The proposed chart performs better than the nonparametric signed-rank chart given by Bakir (2004 Bakir , S. T. ( 2004 ). A distribution-free Shewhart quality control chart based on signed-ranks . Quality Engineering 16 : 613 – 623 .[Taylor & Francis Online] , [Google Scholar]) and Chakraborti and Eryilmaz (2007 Chakraborti , S. , Eryilmaz , S. (2007). A nonparametric Shewhart-type signed-rank control chart based on runs. Communications in Statistics—Simulation and Computation 36:335–356.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

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]).     

An Improved Solution for Multiple Comparisons with a Control in Response Surface Methodology with Two Predictor Variables

Sa and Edwards (1993 Sa , P. , Edwards , D. ( 1993 ). Multiple comparisons with a control in response surface methodology . Technometrics 35 ( 4 ): 436 – 445 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) first proposed the Multiple Comparisons with a Control problem in Response Surface Methodology. They provided an exact solution for one predictor variable and a conservative solution when number of predictor variables is more than one. Merchant et al. (1998 Merchant , A. , McCann , M. , Edwards , D. ( 1998 ). Improved multiple comparisons with a control in response surface analysis . Technometrics 40 ( 4 ): 291 – 303 .[Crossref], [Web of Science ®] , [Google Scholar]) improved the solution for the latter case. This article improves Merchant et al.'s solution for the case of rotatable designs in two predictor variables.     

Nonparametric Knn estimation with monotone constraints

ABSTRACT

The K-nearest-neighbor (Knn) method is known to be more suitable in fitting nonparametrically specified curves than the kernel method (with a globally fixed smoothing parameter) when data sets are highly unevenly distributed. In this paper, we propose to estimate a nonparametric regression function subject to a monotonicity restriction using the Knn method. We also propose using a new convergence criterion to measure the closeness between an unconstrained and the (monotone) constrained Knn-estimated curves. This method is an alternative to the monotone kernel methods proposed by Hall and Huang (2001 Hall, P., Huang, L.-S. (2001). Nonparametric kernel regression subject to monotonicity constraints. The Annals of Statistics 29:624–647.[Crossref], [Web of Science ®] , [Google Scholar]), and Du et al. (2013 Du, P., Parmeter, C. F., Racine, J. S. (2013). Nonparametric kernel regression with multiple predictors and multiple shape constraints. Statistic Sinica 23:1347–1371.[Web of Science ®] , [Google Scholar]). We use a bootstrap procedure for testing the validity of the monotone restriction. We apply our method to the “Job Market Matching” data taken from Gan and Li (2016 Gan, L., Li, Q. (2016). Efficiency of thin and thick market. Journal of Econometrics 192(1):40–54.[Crossref], [Web of Science ®] , [Google Scholar]) and find that the unconstrained/constrained Knn estimators work better than kernel estimators for this type of highly unevenly distributed data.