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.     

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.     

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.     

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]).     

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.     

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.     

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.     

Multiple dependent state repetitive sampling plans based on one-sided process capability indices

In this research, multiple dependent state and repetitive group sampling are used to design a variable sampling plan based on one-sided process capability indices, which consider the quality of the current lot as well as the quality of the preceding lots. The sample size and critical values of the proposed plan are determined by minimizing the average sample number while satisfying the producer's risk and consumer's risk at corresponding quality levels. In addition, comparisons are made with the existing sampling plans [Pearn and Wu (2006a Pearn, W. L., and C. W. Wu. 2006a. Critical acceptance values and sample sizes of a variables sampling plan for very low fraction of defectives. Omega: International Journal of Management Science 34 (1):90–101.[Crossref], [Web of Science ®] , [Google Scholar]), Yen et al. (2015 Yen, C. H., C. H. Chang, and M. Aslam. 2015. Repetitive variable acceptance sampling plan for one-sided specification. Journal of Statistical Computation and Simulation 85 (6):1102–16.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar])] in terms of average sample number and operating characteristic curve. Finally, an example is provided to illustrate the proposed plan.     

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 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.     

An Approximation Model of the Collective Risk Model with INAR(1) Claim Process

Cossette et al. (2010 Cossette, H., Marceau, E., Maume-Deschamps, V. (2010). Discerte-time risk models based on time series for count random variables. ASTIN Bull. 40:123–150.[Crossref], [Web of Science ®] , [Google Scholar], 2011 Cossette, H., Marceau, E., Toureille, F. (2011). risk models based on time series for count random variables. Insur. Math. Econ. 48:19–28.[Crossref], [Web of Science ®] , [Google Scholar]) gave a novel collective risk model where the total numbers of claims satisfy the first-order integer-valued autoregressive process. For a risk model, it is interesting to investigate the upper bound of ruin probability. However, the loss increments of the above model are dependent; it is difficult to derive the upper bound of ruin probability. In this article, we propose an approximation model with stationary independent increments. The upper bound of ruin probability and the adjustment coefficient are derived. The approximation model is illustrated via four simulated examples. Results show that the gap of the approximation model and dependent model can be ignored by adjusting values of parameters.     

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.     

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.     

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.     

A Comparative Note about Estimation of the Fractional Parameter under Additive Outliers

In recent articles, Fajardo et al. (2009 Fajardo Molinares, F., Reisen, V.A., Cribari-Neto, F. (2009). Robust estimation in long-memory processes under additive outliers. Journal of Statistical Planning and Inference 139:2511–2525.[Crossref], [Web of Science ®] , [Google Scholar]) and Reisen and Fajardo (2012) propose an alternative semiparametric estimator of the fractional parameter in ARFIMA models which is robust to the presence of additive outliers. The results are very interesting, however, they use samples of 300 or 800 observations which are rarely found in macroeconomics. In order to perform a comparison, I estimate the fractional parameter using the procedure of Geweke and Porter-Hudak (1983 Geweke, J., Porter-Hudak, S. (1983). The estimation and application of long memory time series model. Journal of Time Series Analysis 4:221–238.[Crossref] , [Google Scholar]) augmented with dummy variables associated with the (previously) detected outliers using the statistic τ_d suggested by Perron and Rodríguez (2003 Perron, P., Rodríguez, G. (2003). Searching for additive outliers in nonstationary time series. Journal of Time Series Analysis 24(2):193–220.[Crossref], [Web of Science ®] , [Google Scholar]). Comparing with Fajardo et al. (2009 Fajardo Molinares, F., Reisen, V.A., Cribari-Neto, F. (2009). Robust estimation in long-memory processes under additive outliers. Journal of Statistical Planning and Inference 139:2511–2525.[Crossref], [Web of Science ®] , [Google Scholar]) and Reisen and Fajardo (2012), I found better results for the mean and bias of the fractional parameter when T = 100 and the results in terms of the standard deviation and the MSE are very similar. However, for higher sample sizes such as 300 or 800, the robust procedure performs better. Empirical applications for seven monthly Latin-American inflation series with very small sample sizes contaminated by additive outliers are 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.     

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.     

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.     

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.     

On the Optimality of Circular Block Designs Under a Mixed Interference Model

Filipiak and Markiewicz (2012 Filipiak, K., Markiewicz, A. (2012). On universal optimality of circular weakly neighbor balanced designs under an interference model. Comm. Stat. Theor Methods 41: 2356–2366.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) proved the universal optimality of circular weakly neighbor balanced designs (CWNBDs) under the interference model with fixed neighbor effects among the class of complete block designs. In two special cases where a CWNBD cannot exist, Filipiak et al. (2012 Filipiak, K., Markiewicz, A., Ró?ański, R. (2012). Maximal determinant over a certain class of matrices and its application to D-optimality of designs. Linear Algebra Appl. 436(4): 874–887.[Crossref], [Web of Science ®] , [Google Scholar]) characterized D-optimal designs. The aim of this paper is to show the universal optimality of CWNBDs and to characterize D-optimal designs under the interference model with random neighbor effects.