A Parametric Bootstrap Approach for One-Way ANOVA Under Unequal Variances with Unbalanced Data

This research is to provide a solution of one-way ANOVA without using transformation when variances are heteroscedastic and group sizes are unequal. Parametric bootstrap test (Krishnamoorthy et al., 2007 Krishnamoorthy, K., Lu, F., Mathew, T. (2007). A parametric bootstrap approach for anova with unequal variances: Fixed and random models. Computational Statistics and Data Analysis 51:5731–5742.[Crossref], [Web of Science ®] , [Google Scholar]) has been shown to be competitive with many other methods when testing the equality of group means. We extend the parametric bootstrap algorithm to a multiple comparison procedure. Simulation results show that the parametric bootstrap approach works well for one-way ANOVA.     

A stratified unrelated question randomized response model using Neyman allocation

This paper applies stratified random sampling using Neyman allocation to Mangat et al. (1992 Mangat, N.S., Singh, R., Singh, S. (1992). An improved unrelated question randomized response strategy. Cal. Stat. Assoc. Bull. 42:277–281.[Crossref] , [Google Scholar]) unrelated question randomized response (RR) strategy for both completely truthful reporting and less than completely truthful reporting. It is shown that, for the prior information given, our new model is more efficient in terms of variance (in the case of completely truthful reporting) and mean square error (in terms of less than completely truthful reporting) than Kim and Elam's (2007 Kim, J.M., Elam, M.E. (2007). A stratified unrelated question randomized response model. Stat. Papers 48:215–233.[Crossref], [Web of Science ®] , [Google Scholar]) model. Numerical illustrations and graphs are also given in support of the present study.     

An optional randomized response model for estimating a rare sensitive attribute using Poisson distribution

The crux of this article is to estimate the mean of the number of persons possessing a rare sensitive attribute based on the Mangat (1991 Mangat, N.S. (1991). An optional randomized response sampling technique using non–stigmatized attribute. Statistica. 51(4):595–602. [Google Scholar]) randomization device by utilizing the Poisson distribution in simple random sampling and stratified sampling. Properties of the proposed randomized response (RR) model have been studied along with recommendations. It is also shown that the proposed model is more efficient than that of Land et al. (2011 Land, M., Singh, S., Sedory, S.A. (2011). Estimation of a rare attribute using Poisson distribution. Statistics. 46(3):351–360.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) in simple random sampling and that 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]) in stratified random sampling when the proportion of persons possessing a rare unrelated attribute is known. Numerical illustrations are also given in support of the present study.     

A Bayesian control chart for a common coefficient of variation

By using the medical data analyzed by Kang et al. (2007 Kang, C.W., Lee, M.S., Seong, Y.J., Hawkins, D.M. (2007). A control chart for the coefficient of variation. J. Qual. Technol. 39(2):151–158.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), a Bayesian procedure is applied to obtain control limits for the coefficient of variation. Reference and probability matching priors are derived for a common coefficient of variation across the range of sample values. By simulating the posterior predictive density function of a future coefficient of variation, it is shown that the control limits are effectively identical to those obtained by Kang et al. (2007 Kang, C.W., Lee, M.S., Seong, Y.J., Hawkins, D.M. (2007). A control chart for the coefficient of variation. J. Qual. Technol. 39(2):151–158.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) for the specific dataset they used. This article illustrates the flexibility and unique features of the Bayesian simulation method for obtaining posterior distributions, predictive intervals, and run-lengths in the case of the coefficient of variation. A simulation study shows that the 95% Bayesian confidence intervals for the coefficient of variation have the correct frequentist coverage.     

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.     

Characterization results based on dynamic Tsallis cumulative residual entropy

In this article, the concept of cumulative residual entropy (CRE) given 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. Inf. Theory 50:1220–1228.[Crossref], [Web of Science ®] , [Google Scholar]) is extended to Tsallis entropy function and dynamic version, both residual and past of it. We study some properties and characterization results for these generalized measures. In addition, we provide some characterization results of the first-order statistic based on the Tsallis survival entropy.     

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 dynamic weighted survival entropy of order α

Recently, Abbasnejad et al. (2010 Abbasnejad, M., Arghami, N.R., Morgenthaler, S., Mohtashami Borzadaran, G.R. (2010). On the dynamic survival entropy. Stat. Probab. Lett. 80:1962–1971.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a measure of uncertainty based on survival function, called the survival entropy of order α. A dynamic form of the survival entropy of order α is also proposed by them. In this paper, we derive the weighted form of these measures. The properties of the new measures are also discussed.     

Performance of some ridge regression estimators for the multinomial logit model

This article considers several estimators for estimating the ridge parameter k for multinomial logit model based on the work of Khalaf and Shukur (2005 Khalaf, G., and G. Shukur. 2005. Choosing ridge parameters for regression problems. Commun. Statist. Theor. Meth., 34:1177–1182.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Alkhamisi et al. (2006 Alkhamisi, M., G. Khalaf, and G. Shukur. 2006. Some modifications for choosing ridge parameters. Commun. Statist. Theor. Meth. 35:2005–2020.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and Muniz et al. (2012 Muniz, G., B. M. G. Kibria, K. Månsson, and G. Shukur. 2012. On developing ridge regression parameters: A graphical investigation. in SORT. 36: 115–138.[Web of Science ®] , [Google Scholar]). The mean square error (MSE) is considered as the performance criterion. A simulation study has been conducted to compare the performance of the estimators. Based on the simulation study we found that increasing the correlation between the independent variables and the number of regressors has negative effect on the MSE. However, when the sample size increases the MSE decreases even when the correlation between the independent variables is large. Based on the minimum MSE criterion some useful estimators for estimating the ridge parameter k are recommended for the practitioners.     

Efficient Mixture Design Fitting Quadratic Surface with Quantile Responses Using First-degree Polynomial

This study considers efficient mixture designs for the approximation of the response surface of a quantile regression model, which is a second degree polynomial, by a first degree polynomial in the proportions of q components. Instead of least squares estimation in the traditional regression analysis, the objective function in quantile regression models is a weighted sum of absolute deviations and the least absolute deviations (LAD) estimation technique should be used (Bassett and Koenker, 1982 Bassett, G., Koenker, R. (1982). An empirical quantile function for linear models with i.i.d. errors. Journal of the American Statistical Association 77:407–415.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]; Koenker and Bassett, 1978 Koenker, R., Bassett, G. (1978). Regression quantiles. Econometrica 46(1):33–50.[Crossref], [Web of Science ®] , [Google Scholar]). Therefore, the standard optimal mixture designs like the D-optimal or A-optimal mixture designs for the least squared estimation are not appropriate. This study explores mixture designs that minimize the bias between the approximated 1st-degree polynomial and a 2nd-degree polynomial response surfaces by the LAD estimation. In contrast to the standard optimal mixture designs for the least squared estimation, the efficient designs might contain elementary centroid design points of degrees higher than two. An example of a portfolio with five assets is given to illustrate the proposed efficient mixture designs in determining the marginal contribution of risks by individual assets in the portfolio.     

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.     

Complete convergence for coordinatewise asymptotically negatively associated random vectors in Hilbert spaces

In this article, we study the complete convergence for sequences of coordinatewise asymptotically negatively associated random vectors in Hilbert spaces. We also investigate that some related results for coordinatewise negatively associated random vectors in Huan, Quang, and Thuan (2014 Huan, N. V., N. V. Quang, and N. T. Thuan. 2014. Baum–Katz type theorems for coordinatewise negatively associated random vectors in Hilbert spaces. Acta Mathematica Hungarica 144(1):132–419.[Crossref], [Web of Science ®] , [Google Scholar]) still hold under this concept.     

Optimality of ratio type estimation methods for population mean in the presence of missing data

This article proposes various Searls-type ratio imputation methods (STRIM) on the lines of Ahmed et al. (2006 Ahmed, M. S., O. Al-Titi, Z. Al-Rawi, and W. Abu-Dayyeh. 2006. Estimation of a population mean using different imputation methods. Stat. Trans. 7 (6):1247–1264. [Google Scholar]). It is a well-known fact that the optimal ratio type estimator attains the MSE of regression estimator (or optimal difference estimator) but while using Searls-type transformation (STT) (Searls (1964 Searls, D. T. 1964. The utilization of a known coefficient of variation in the estimation procedure. J. Am. Stat. Assoc. 59:1225–1226.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar])) this may not always happen. These STRIM are shown to perform better than the imputation procedures of Ahmed et al. (2006 Ahmed, M. S., O. Al-Titi, Z. Al-Rawi, and W. Abu-Dayyeh. 2006. Estimation of a population mean using different imputation methods. Stat. Trans. 7 (6):1247–1264. [Google Scholar]). The STRIM may even outperform the Searls type difference imputation methods (STDIM) proposed by us in our earlier work, Bhushan and Pandey (2016 Bhushan, S., and A. P. Pandey. 2016. Optimal imputation of the missing data for estimation of population mean. Journal of Statistics and Management System 19 (6):755–69.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). This study is concluded with the numerical study along with the theoretical comparison.     

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.     

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.     

Optimal allocations of coverage limits for two independent random losses of insurance policy

This article recasts the optimal allocations of coverage limits for two independent random losses. Under some regularity conditions on the two concerned probability density functions, we build the sufficient and necessary condition for the existence of the optimal allocation of coverage limits, and derive the optimal allocation whenever they do exist. The results supplement Lu and Meng (2011 Lu, Z.Y., Meng, L.L. (2011). Stochastic comparisons for allocations of upper limits and deductibles with applications. Insur.: Math. Econ. 48:338–343.[Crossref], [Web of Science ®] , [Google Scholar], Proposition 5.2) and Hu and Wang (2014 Hu, S., Wang, R. (2014). Stochastic comparisons and optimal allocation for policy limits and deductibles. Commun. Stat. – Theory Methods 43:151–164.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 5.1).     

Bayesian Estimation for Linear Functions of Poisson Rates

The probability matching prior for linear functions of Poisson parameters is derived. A comparison is made between the confidence intervals obtained by Stamey and Hamilton (2006 Stamey, J., Hamilton, C. (2006). A note on confidence intervals for a linear function of Poisson rates. Commun. Statist. Simul. &; Computat. 35(4):849–856.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and the intervals derived by us when using the Jeffreys’ and probability matching priors. The intervals obtained from the Jeffreys’ prior are in some cases fiducial intervals (Krishnamoorthy and Lee, 2010 Krishnamoorthy, K., Lee, M. (2010). Inference for functions of parameters in discrete distributions based on fiducial approach: Binomial and Poisson cases. J. Statist. Plann. Infere. 140(5):1182–1192.[Crossref], [Web of Science ®] , [Google Scholar]). A weighted Monte Carlo method is used for the probability matching prior. The power and size of the test, using Bayesian methods, is compared to tests used by Krishnamoorthy and Thomson (2004 Krishnamoorthy, K., Thomson, J. (2004). A more powerful test for comparing two Poisson means. J. Statist. Plann. Infere. 119(1):23–35.[Crossref], [Web of Science ®] , [Google Scholar]). The Jeffreys’, probability matching and two other priors are used.     

Estimating quantiles of several normal populations with a common mean

Suppose we have k( ? 2) normal populations with a common mean and possibly different variances. The problem of estimation of quantile of the first population is considered with respect to a quadratic loss function. In this paper, we have generalized the inadmissibility results obtained by Kumar and Tripathy (2011 Kumar, S., Tripathy, M.R. (2011). Estimating quantiles of normal populations with a common mean. Commun. Stat. - Theory Methods 40:2719–2736.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) for k = 2 to a general k( ? 2). Moreover, a massive simulation study has been done in order to numerically compare the risk values of various proposed estimators for the cases k = 3 and k = 4 and recommendations are made for the use of estimators under certain situations.     

A new estimator of finite population mean based on the dual use of the auxiliary information

When a sufficient correlation between the study variable and the auxiliary variable exists, the ranks of the auxiliary variable are also correlated with the study variable, and thus, these ranks can be used as an effective tool in increasing the precision of an estimator. In this paper, we propose a new improved estimator of the finite population mean that incorporates the supplementary information in forms of: (i) the auxiliary variable and (ii) ranks of the auxiliary variable. Mathematical expressions for the bias and the mean-squared error of the proposed estimator are derived under the first order of approximation. The theoretical and empirical studies reveal that the proposed estimator always performs better than the usual mean, ratio, product, exponential-ratio and -product, classical regression estimators, and Rao (1991 Rao, T.J. (1991). On certail methods of improving ration and regression estimators. Commun. Stat. Theory Methods 20(10):3325–3340.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Singh et al. (2009 Singh, R., Chauhan, P., Sawan, N., Smarandache, F. (2009). Improvement in estimating the population mean using exponential estimator in simple random sampling. Int. J. Stat. Econ. 3(A09):13–18. [Google Scholar]), Shabbir and Gupta (2010 Shabbir, J., Gupta, S. (2010). On estimating finite population mean in simple and stratified random sampling. Commun. Stat. Theory Methods 40(2):199–212.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Grover and Kaur (2011 Grover, L.K., Kaur, P. (2011). An improved estimator of the finite population mean in simple random sampling. Model Assisted Stat. Appl. 6(1):47–55. [Google Scholar], 2014) estimators.     

Bootstrap methods for multivariate hypothesis testing

The nonparametric and parametric bootstrap methods for multivariate hypothesis testing are developed. They are used to approximate the null distribution of the test statistics proposed by Duchesne and Francq (2015 Duchesne, P., Francq, C. (2015). Multivariate hypothesis testing using generalized and {2}-inverses—with applications. Statistics 49:475–496.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), resulting in bootstrap testing procedures. In the problem of testing for the mean vector of a multivariate distribution, the asymptotic validity of the bootstrap methods is proved. The finite sample performance of the new solutions is demonstrated by means of Monte Carlo simulation studies. They indicate that for small-sample size, the bootstrap tests provide a better finite sample properties than the asymptotic tests considered by Duchesne and Francq (2015 Duchesne, P., Francq, C. (2015). Multivariate hypothesis testing using generalized and {2}-inverses—with applications. Statistics 49:475–496.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).