A Modification of a Percentile Estimation Procedure Based on Generalized Polya Urns

Adaptive designs find an important application in the estimation of unknown percentiles for an underlying dose-response curve. A nonparametric adaptive design was suggested by Mugno et al. (2004 Mugno, R.A., Zhus, W., Rosenberger, W.F. (2004). Adaptive urn designs for estimating several percentiles of a dose-response curve. Statist. Med. 23(13):2137–2150.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to simultaneously estimate multiple percentiles of an unknown dose-response curve via generalized Polya urns. In this article, we examine the properties of the design proposed by Mugno et al. (2004 Mugno, R.A., Zhus, W., Rosenberger, W.F. (2004). Adaptive urn designs for estimating several percentiles of a dose-response curve. Statist. Med. 23(13):2137–2150.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) when delays in observing responses are encountered. Using simulations, we evaluate a modification of the design under varying group sizes. Our results demonstrate unbiased estimation with minimal loss in efficiency when compared to the original compound urn design.     

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.     

An efficient use of moment's ratios of scrambling variables in a randomized response technique

This paper aimed at providing an efficient new unbiased estimator for estimating the proportion of a potentially sensitive attribute in survey sampling. The suggested randomization device makes use of the means, variances of scrambling variables, and the two scalars lie between “zero” and “one.” Thus, the same amount of information has been used at the estimation stage. The variance formula of the suggested estimator has been obtained. We have compared the proposed unbiased estimator with that of Kuk (1990 Kuk, A.Y.C. (1990). Asking sensitive questions inderectely. Biometrika 77:436–438.[Crossref], [Web of Science ®] , [Google Scholar]) and Franklin (1989 Franklin, L.A. (1989). A comparision of estimators for randomized response sampling with continuous distribution s from a dichotomous population. Commun. Stat. Theor. Methods 18:489–505.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimators. Relevant conditions are obtained in which the proposed estimator is more efficient than Kuk (1990 Kuk, A.Y.C. (1990). Asking sensitive questions inderectely. Biometrika 77:436–438.[Crossref], [Web of Science ®] , [Google Scholar]) and Franklin (1989 Franklin, L.A. (1989). A comparision of estimators for randomized response sampling with continuous distribution s from a dichotomous population. Commun. Stat. Theor. Methods 18:489–505.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimators. The optimum estimator (OE) in the proposed class of estimators has been identified which finally depends on moments ratios of the scrambling variables. The variance of the optimum estimator has been obtained and compared with that of the Kuk (1990 Kuk, A.Y.C. (1990). Asking sensitive questions inderectely. Biometrika 77:436–438.[Crossref], [Web of Science ®] , [Google Scholar]) and Franklin (1989 Franklin, L.A. (1989). A comparision of estimators for randomized response sampling with continuous distribution s from a dichotomous population. Commun. Stat. Theor. Methods 18:489–505.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) estimator and Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimator. It is interesting to mention that the “optimum estimator” of the class of estimators due to Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) depends on the parameter π under investigation which limits the use of Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) OE in practice while the proposed OE in this paper is free from such a constraint. The proposed OE depends only on the moments ratios of scrambling variables. This is an advantage over the Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimator. Numerical illustrations are given in the support of the present study when the scrambling variables follow normal distribution. Theoretical and empirical results are very sound and quite illuminating in the favor of the present study.     

Using the Inverse Maximum Ratio-Λ as a Technique to Quantify Surface Uniformity

Techniques used in variability assessment are subsequently used to draw conclusions regarding the “spread”/uniformity of data curves. Due to the limitations of these techniques, they are not adequate for circumstances where data manifest with multiple peaks. Examples of these manifestations (in three-dimensional space) include under-foot pressure distributions recorded for different types of footwear (Becerro-de-Bengoa-Vallejo et al., 2014 Biau, D.J. (2011). In brief: Standard deviation and standard error. Clinical Orthopaedics and Related Research 469(9):2661–2664.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Cibulka et al., 1994 Cibulka, M.T., Sinacore, D.R., Mueller, M.J. (1994). Shin splints and forefoot contact running: A case report. Journal of Orthopaedic &; Sports Physical Therapy 20(2):98–102.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Davies et al., 2003 Davies, M.B., Betts, R.P., Scott, I.R. (2003). Optical plantar pressure analysis following internal fixation for displaced intra-articular os calcis fractures. Foot &; Ankle International 24(11):851–856.[PubMed], [Web of Science ®] , [Google Scholar]), surface textures and interfaces designed to impact friction, and and and molecular surface structures such as viral epitopes (Torras and Garcia-Valls, 2004 Torras, C., Garcia-Valls, R. (2004). Quantification of membrane morphology by interpretation of scanning electron microscopy images. Journal of Membrane Science 233(1–2):119–127.[Crossref], [Web of Science ®] , [Google Scholar]; Pacejka, 1997; Fustaffson, 1997). This article proposes a technique for generating a single variable – Λ that will quantify the uniformity of such surfaces. We define and validate this technique using several mathematical and graphical models.     

Buffered Autoregressive Models With Conditional Heteroscedasticity: An Application to Exchange Rates

This article introduces a new model called the buffered autoregressive model with generalized autoregressive conditional heteroscedasticity (BAR-GARCH). The proposed model, as an extension of the BAR model in Li et al. (2015 Li, G.D., Guan, B., Li, W.K., and Yu, P. L.H. (2015), “Hysteretic Autoregressive Time Series Models,” Biometrika, 102, 717–723.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), can capture the buffering phenomena of time series in both the conditional mean and variance. Thus, it provides us a new way to study the nonlinearity of time series. Compared with the existing AR-GARCH and threshold AR-GARCH models, an application to several exchange rates highlights the importance of the BAR-GARCH model.     

On extreme order statistics from heterogeneous beta distributions with applications

This paper treats the problem of stochastic comparisons for the extreme order statistics arising from heterogeneous beta distributions. Some sufficient conditions involved in majorization-type partial orders are provided for comparing the extreme order statistics in the sense of various magnitude orderings including the likelihood ratio order, the reversed hazard rate order, the usual stochastic order, and the usual multivariate stochastic order. The results established here strengthen and extend those including Kochar and Xu (2007 Kochar, S.C., Xu, M. (2007). Stochastic comparisons of parallel systems when components have proportional hazard rates. Probab. Eng. Inf. Sci. 21:597–609.[Crossref], [Web of Science ®] , [Google Scholar]), Mao and Hu (2010 Mao, T., Hu, T. (2010). Equivalent characterizations on orderings of order statistics and sample ranges. Probab. Eng. Inf. Sci. 24:245–262.[Crossref], [Web of Science ®] , [Google Scholar]), Balakrishnan et al. (2014 Balakrishnan, N., Barmalzan, G., Haidari, A. (2014). On usual multivariate stochastic ordering of order statistics from heterogeneous beta variables. J. Multivariate Anal. 127:147–150.[Crossref], [Web of Science ®] , [Google Scholar]), and Torrado (2015 Torrado, N. (2015). On magnitude orderings between smallest order statistics from heterogeneous beta distributions. J. Math. Anal. Appl. 426:824–838.[Crossref], [Web of Science ®] , [Google Scholar]). A real application in system assembly and some numerical examples are also presented to illustrate the theoretical results.     

On generalized exponential chain ratio estimators under stratified two-phase random sampling

Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]) have suggested generalized exponential chain ratio estimators under stratified two-phase sampling scheme for estimating the finite population mean. However, the bias and mean square error (MSE) expressions presented in that work need some corrections, and consequently the study based on efficiency comparison also requires corrections. In this article, we revisit Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]) estimator and provide the correct bias and MSE expressions of their estimator. We also propose an estimator which is more efficient than several competing estimators including the classes of estimators in Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]). Three real datasets are used for efficiency comparisons.     

Multivariate Bayesian U-type asymmetric designs for non parametric response surface prediction under correlated errors

The objective of this paper is to study U-type designs for Bayesian non parametric response surface prediction under correlated errors. The asymptotic Bayes criterion is developed in terms of the asymptotic approach of Mitchell et al. (1994 Mitchell, T., Sacks, J., Ylvisaker, D. (1994). Asymptotic Bayes criteria for nonparametric response surface design. Ann. Stat. 22:634–651.[Crossref], [Web of Science ®] , [Google Scholar]) for a more general covariance kernel proposed by Chatterjee and Qin (2011 Chatterjee, K., Qin, H. (2011). Generalized discrete discrepancy and its applications in experimental designs. J. Stat. Plann. Inference 141:951–960.[Crossref], [Web of Science ®] , [Google Scholar]). A relationship between the asymptotic Bayes criterion and other criteria, such as orthogonality and aberration, is then developed. A lower bound for the criterion is also obtained, and numerical results show that this lower bound is tight. The established results generalize those of Yue et al. (2011 Yue, R.X., Qin, H., Chatterjee, K. (2011). Optimal U-type design for Bayesian nonparametric multiresponse prediction. J. Stat. Plann. Inference 141:2472–2479.[Crossref], [Web of Science ®] , [Google Scholar]) from symmetrical case to asymmetrical U-type designs.     

Improved Exponential Type Estimators of the Mean of a Sensitive Variable in the Presence of Nonsensitive Auxiliary Information

Recently, Koyuncu et al. (2013 Koyuncu, N., Gupta, S., Sousa, R. (2014). Exponential type estimators of the mean of a sensitive variable in the presence of non-sensitive auxiliary information. Communications in Statistics- Simulation and Computation[PubMed], [Web of Science ®] , [Google Scholar]) proposed an exponential type estimator to improve the efficiency of mean estimator based on randomized response technique. In this article, we propose an improved exponential type estimator which is more efficient than the Koyuncu et al. (2013 Koyuncu, N., Gupta, S., Sousa, R. (2014). Exponential type estimators of the mean of a sensitive variable in the presence of non-sensitive auxiliary information. Communications in Statistics- Simulation and Computation[PubMed], [Web of Science ®] , [Google Scholar]) estimator, which in turn was shown to be more efficient than the usual mean estimator, ratio estimator, regression estimator, and the Gupta et al. (2012 Gupta, S., Shabbir, J., Sousa, R., Corte-Real, P. (2012). Regression estimation of the mean of a sensitive variable in the presence of auxiliary information. Communications in Statistics – Theory and Methods 41:2394–2404.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) estimator. Under simple random sampling without replacement (SRSWOR) scheme, bias and mean square error expressions for the proposed estimator are obtained up to first order of approximation and comparisons are made with the Koyuncu et al. (2013 Koyuncu, N., Gupta, S., Sousa, R. (2014). Exponential type estimators of the mean of a sensitive variable in the presence of non-sensitive auxiliary information. Communications in Statistics- Simulation and Computation[PubMed], [Web of Science ®] , [Google Scholar]) estimator. A simulation study is used to observe the performances of these two estimators. Theoretical findings are also supported by a numerical example with real data. We also show how to, extend the proposed estimator to the case when more than one auxiliary variable is available.     

Matrix mean squared error comparisons of some biased estimators with two biasing parameters

To deal with multicollinearity problem, the biased estimators with two biasing parameters have recently attracted much research interest. The aim of this article is to compare one of the last proposals given by Yang and Chang (2010 Yang, H., and X. Chang. 2010. A new two-parameter estimator in linear regression. Communications in Statistics: Theory and Methods 39 (6):923–34.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) with Liu-type estimator (Liu 2003 Liu, K. 2003. Using Liu-type estimator to combat collinearity. Communications in Statistics: Theory and Methods 32 (5):1009–20.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and k ? d class estimator (Sakallioglu and Kaciranlar 2008 Sakallioglu, S., and S. Kaciranlar. 2008. A new biased estimator based on ridge estimation. Statistical Papers 49:669–89.[Crossref], [Web of Science ®] , [Google Scholar]) under the matrix mean squared error criterion. As well as giving these comparisons theoretically, we support the results with the extended simulation studies and real data example, which show the advantages of the proposal given by Yang and Chang (2010 Yang, H., and X. Chang. 2010. A new two-parameter estimator in linear regression. Communications in Statistics: Theory and Methods 39 (6):923–34.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) over the other proposals with increasing multicollinearity level.     

Sample size estimation for comparing rates of change in K-group repeated count outcomes

Sample size estimation for comparing the rates of change in two-arm repeated measurements has been investigated by many investigators. In contrast, the literature has paid relatively less attention to sample size estimation for studies with multi-arm repeated measurements where the design and data analysis can be more complex than two-arm trials. For continuous outcomes, Jung and Ahn (2004 Jung, S., Ahn, C. (2004). K-sample test and sample size calculation for comparing slopes in data with repeated measurements. Biometrical J. 46(5):554–564.[Crossref], [Web of Science ®] , [Google Scholar]) and Zhang and Ahn (2013 Zhang, S., Ahn, C. (2013). Sample size calculation for comparing time-averaged responses in k-group repeated measurement studies. Comput. Stat. Data Anal. 58:283–291.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) have presented sample size formulas to compare the rates of change and time-averaged responses in multi-arm trials, using the generalized estimating equation (GEE) approach. To our knowledge, there has been no corresponding development for multi-arm trials with count outcomes. We present a sample size formula for comparing the rates of change in multi-arm repeated count outcomes using the GEE approach that accommodates various correlation structures, missing data patterns, and unbalanced designs. We conduct simulation studies to assess the performance of the proposed sample size formula under a wide range of designing configurations. Simulation results suggest that empirical type I error and power are maintained close to their nominal levels. The proposed method is illustrated using an epileptic clinical trial example.     

On weighted cumulative residual entropy

In analogy with the weighted Shannon entropy proposed by Belis and Guiasu (1968 Belis, M., Guiasu, S. (1968). A quantitative-qualitative measure of information in cybernetic systems. IEEE Trans. Inf. Th. IT-4:593–594.[Crossref], [Web of Science ®] , [Google Scholar]) and Guiasu (1986 Guiasu, S. (1986). Grouping data by using the weighted entropy. J. Stat. Plann. Inference 15:63–69.[Crossref], [Web of Science ®] , [Google Scholar]), we introduce a new information measure called weighted cumulative residual entropy (WCRE). This is based on the cumulative residual entropy (CRE), which is introduced by Rao et al. (2004 Rao, M., Chen, Y., Vemuri, B.C., Wang, F. (2004). Cumulative residual entropy: a new measure of information. IEEE Trans. Info. Theory 50(6):1220–1228.[Crossref], [Web of Science ®] , [Google Scholar]). This new information measure is “length-biased” shift dependent that assigns larger weights to larger values of random variable. The properties of WCRE and a formula relating WCRE and weighted Shannon entropy are given. Related studies of reliability theory is covered. Our results include inequalities and various bounds to the WCRE. Conditional WCRE and some of its properties are discussed. The empirical WCRE is proposed to estimate this new information measure. Finally, strong consistency and central limit theorem are provided.     

Fubini theorem for non additive measures in the framework of g-expectation

Since the seminal paper of Ghirardato (1997 Ghirardato, P. 1997. On the independence for non-additive measures, with a Fubini theorem. Journal of Economic Theory 73:261–91.[Crossref], [Web of Science ®] , [Google Scholar]), it is known that Fubini theorem for non additive measures can be available only for functions as “slice-comonotonic” in the framework of product algebra. Later, inspired by Ghirardato (1997 Ghirardato, P. 1997. On the independence for non-additive measures, with a Fubini theorem. Journal of Economic Theory 73:261–91.[Crossref], [Web of Science ®] , [Google Scholar]), Chateauneuf and Lefort (2008 Chateauneuf, A., and J. P. Lefort. 2008. Some Fubini theorems on product σ-algebras for non-additive measures. International Journal of Approximate Reasoning 48:686–96.[Crossref], [Web of Science ®] , [Google Scholar]) obtained some Fubini theorems for non additive measures in the framework of product σ-algebra. In this article, we study Fubini theorem for non additive measures in the framework of g-expectation. We give some different assumptions that provide Fubini theorem in the framework of g-expectation.     

On the optimal allocation of active redundancies in series system

This paper studies the allocations of two non identical active redundancies in series systems in terms of the reversed hazard rate order and hazard rate order, which generalizes some results built in Valdés and Zequeira (2003 Valdés, J. E., and R. I. Zequeira 2003. On the optimal allocation of an active redundancy in a two-component series system. Stat. Probab. Lett. 63:325–32.[Crossref], [Web of Science ®] , [Google Scholar], 2006 Valdés, J. E., and R. I. Zequeira 2006. On the optimal allocation of two active redundancies in a two-component series system. Oper. Res. Lett. 34:49–52.[Crossref], [Web of Science ®] , [Google Scholar]).     

A two-stage Land et al.'s randomized response model for estimating a rare sensitive attribute using Poisson distribution

The crux of this paper is to estimate the mean of the number of persons possessing a rare sensitive attribute based on the Mangat (1992 Mangat, N.S. (1992). Two stage reandomized response sampling procedure using unrelated question. J. Ind. Soc. Agric. Stat. 44(1):82–87. [Google Scholar]) randomization device by utilizing the Poisson distribution in survey sampling. It is shown that the proposed model is more efficient than Land et al. (2011 Land, M., Singh, S., Sedory, S.A. (2011). Estimation of a rare attribute using Poisson distribution. Statistics doi:10.1080/02331888.2010.524300[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) when the proportion of persons possessing a rare unrelated attribute is known. Properties of the proposed randomized response model have been studied along with recommendations. We have also extended the proposed model to stratified random sampling on the lines of Lee et al. (2013 Lee, G.S., Uhm, D., Kim, J.M. (2013). Estimation of a rare sensitive attribute in stratified sampling using Poisson distribution. Statistics 47(3):575–589.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). It has been also shown that the proposed estimator is better than Lee et al.'s (2013 Lee, G.S., Uhm, D., Kim, J.M. (2013). Estimation of a rare sensitive attribute in stratified sampling using Poisson distribution. Statistics 47(3):575–589.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) estimator. Numerical illustrations are also given in support of the present study.     

Some extensions of circular balanced and circular strongly balanced repeated measurements designs

Repeated measurement designs are widely used in medicine, pharmacology, animal sciences, and psychology. In this paper the works of Iqbal and Tahir (2009 Iqbal, I., and M. H. Tahir. 2009. Circular strongly balanced repeated measurements designs. Communications in Statistics—Theory and Methods 38:3686–96.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Iqbal, Tahir, and Ghazali (2010 Iqbal, I., M. H. Tahir, and S. S. A. Ghazali. 2010. Circular first- and second-order balanced repeated measurements designs. Communications in Statistics—Theory and Methods 39:228–40.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) are generalized for the construction of circular-balanced and circular strongly balanced repeated measurements designs through the method of cyclic shifts for three periods.     

Successive comparisons between ordered normal means using isotonic regression estimators

In this study we discuss multiple comparison procedures for checking differences among a sequence of normal means with ordered restriction. Lee and Spurrier (1995 Lee, R.E., Spurrier, J.D. (1995). Successive comparisons between ordered treatments. J. Stat. Plann. Inference 43:323–330.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a multiple comparison procedure which tests the difference between two adjacent means using the difference of sample means. In this study we propose a multiple comparison procedure modifying Lee and Spurrier's (1995 Lee, R.E., Spurrier, J.D. (1995). Successive comparisons between ordered treatments. J. Stat. Plann. Inference 43:323–330.[Crossref], [Web of Science ®] , [Google Scholar]) procedure using isotonic regression estimators instead of sample means. We determine the critical value for pairwise comparisons for a specified significance level. Furthermore, we formulate the power of the test. Finally, we give some numerical examples regarding critical values and power of the test intended to compare our procedure with Lee and Spurrier's (1995 Lee, R.E., Spurrier, J.D. (1995). Successive comparisons between ordered treatments. J. Stat. Plann. Inference 43:323–330.[Crossref], [Web of Science ®] , [Google Scholar]) procedure.     

Linear transformation models for survival analysis with tumor growth information in cancer screening study

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 left-truncated 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 the Cox proportional hazards model. Shen (2011a Shen, P.-S. (2011a). Proportional hazards regression for cancer screening data. J. Stat. Comput. Simul. 18:367–377.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) demonstrated 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. In this article, under linear transformation model, we present a general framework to the analysis of data from cancer screening studies. We developed estimation procedures under linear transformation model, which includes Cox's model as a special case. A simulation study is conducted to demonstrate the potential usefulness of the proposed estimators.     

Bimodal symmetric-asymmetric power-normal families

This article proposes new symmetric and asymmetric distributions applying methods analogous as the ones in Kim (2005 Kim, H.J. (2005). On a class of two-piece skew-normal distributions. Statist.: J. Theoret. Appl. Statist. 39:537–553.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Arnold et al. (2009 Arnold, B.C., H.W. Gómez, and H.S. Salinas. (2009). On multiple constraint skewed models. Statist. J. Theoret. Appl. Statist. 43: 279–293.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) to the exponentiated normal distribution studied in Durrans (1992 Durrans, S.R. (1992). Distributions of fractional order statistics in hydrology. Water Resour. Res. 28:1649–1655.[Crossref], [Web of Science ®] , [Google Scholar]), that we call the power-normal (PN) distribution. The proposed bimodal extension, the main focus of the paper, is called the bimodal power-normal model and is denoted by BPN(α) model, where α is the asymmetry parameter. The authors give some properties including moments and maximum likelihood estimation. Two important features of the model proposed is that its normalizing constant has closed and simple form and that the Fisher information matrix is nonsingular, guaranteeing large sample properties of the maximum likelihood estimators. Finally, simulation studies and real applications reveal that the proposed model can perform well in both situations.     

Weighting variables in Kohonen competitive learning algorithms

This paper presents a new variable weight method, called the singular value decomposition (SVD) approach, for Kohonen competitive learning (KCL) algorithms based on the concept of Varshavsky et al. [18 R. Varshavsky, A. Gottlieb, M. Linial, and D. Horn, Novel unsupervised feature filtering of bilogical data, Bioinformatics 22 (2006), pp. 507–513.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]]. Integrating the weighted fuzzy c-means (FCM) algorithm with KCL, in this paper, we propose a weighted fuzzy KCL (WFKCL) algorithm. The goal of the proposed WFKCL algorithm is to reduce the clustering error rate when data contain some noise variables. Compared with the k-means, FCM and KCL with existing variable-weight methods, the proposed WFKCL algorithm with the proposed SVD's weight method provides a better clustering performance based on the error rate criterion. Furthermore, the complexity of the proposed SVD's approach is less than Pal et al. [17 S.K. Pal, R.K. De, and J. Basak, Unsupervised feature evaluation: a neuro-fuzzy approach, IEEE. Trans. Neural Netw. 11 (2000), pp. 366–376.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]], Wang et al. [19 X.Z. Wang, Y.D. Wang, and L.J. Wang, Improving fuzzy c-means clustering based on feature-weight learning, Pattern Recognit. Lett. 25 (2004), pp. 1123–1132.[Crossref], [Web of Science ®] , [Google Scholar]] and Hung et al. [9 W. -L. Hung, M. -S. Yang, and D. -H. Chen, Bootstrapping approach to feature-weight selection in fuzzy c-means algorithms with an application in color image segmentation, Pattern Recognit. Lett. 29 (2008), pp. 1317–1325.[Crossref], [Web of Science ®] , [Google Scholar]].