Quantile-based Chernoff distance for truncated random variables

Several probability distributions such as power-Pareto distribution (see Gilchrist 2000 Gilchrist, W. 2000. Statistical modelling with quantile functions. Boca Raton, FL: Chapman and Hall/CRC.[Crossref] , [Google Scholar] and Hankin and Lee 2006 Hankin, R. K. S., and A. Lee. 2006. A new family of non-negative distributions. Australian and New Zealand Journal of Statistics 48:67–78.[Crossref], [Web of Science ®] , [Google Scholar]), various forms of lambda distributions (see Ramberg and Schmeiser 1974 Ramberg, J. S., and B. W. Schmeiser. 1974. An appropriate method for generating asymmetric random variables. Communications of the ACM 17:78–82.[Crossref], [Web of Science ®] , [Google Scholar] and Freimer et al. 1988 Freimer, M., S. Mudholkar, G. Kollia, and C. T. Lin. 1988. A study of the generalized lambda family. Communications in Statistics - Theory and Methods 17:3547–67.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Govindarajulu distribution (see Nair, Sankaran, and Vineshkumar 2012 Nair, U. N., P. G. Sankaran, and B. Vineshkumar. 2012. The Govindarajulu distribution: some properties and applications. Communications in Statistics—Theory and Methods 41:4391–406.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), etc., do not have manageable distribution functions, though they have tractable quantile functions. Hence, analytical study of the properties of Chernoff distance of two random variables associated with these distributions via traditional distribution function-based tool becomes difficult. To make this simple, in this paper, we introduce quantile-based Chernoff distance for (left or right) truncated random variables and study its various properties. Some useful bounds as well as characterization results are obtained.     

The flexible skew Laplace distribution

ABSTRACT

Skew-symmetric distributions have been discussed by several research-ers. In this article we construct a skew-symmetric Laplace distribution, which is the generalization of distribution given by Ali et al. (2009 Ali, M., Pal, M., Woo, J. (2009). Skewed reflected distributions generated by the Laplace kernel. Aust. J. Statist. 38:45–58. [Google Scholar]) and Nekoukhou and Alamatsaz (2012 Nekoukhou, V., Alamatsaz, M.H. (2012). A family of skew-symmetric-Laplace distributions. Statist. Papers. 53(3):685–696.[Crossref], [Web of Science ®] , [Google Scholar]). This new distribution contains more parameters, and this induces flexibility properties, such as unimodality or bimodality. We study on some properties of this distribution. In the last section we also provide an application with a real data. Concerning example has recently been discussed by Nekoukhou et al. (2013 Nekoukhou, V., Alamatsaz, M.H., Aghajani, A.H. (2013). A flexible skew-generalized normal distribution. Commun. Statist. Theory Methods. 42(13):2324–2334.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to apply to their model. We compare the behavior of our distribution to their distribution on this example.     

Note on posterior inference for the Bingham distribution

The properties of high-dimensional Bingham distributions have been studied by Kume and Walker (2014 Kume, A., and S. G. Walker. 2014. On the Bingham distribution with large dimension. Journal of Multivariate Analysis 124:345–52.[Crossref], [Web of Science ®] , [Google Scholar]). Fallaize and Kypraios (2016 Fallaize, C. J., and T. Kypraios. 2016. Exact Bayesian inference for the Bingham distribution. Statistics and Computing 26:349–60.[Crossref], [Web of Science ®] , [Google Scholar]) propose the Bayesian inference for the Bingham distribution and they use developments in Bayesian computation for distributions with doubly intractable normalizing constants (Møller et al. 2006 Møller, J., A. N. Pettitt, R. Reeves, and K. K. Berthelsen. 2006. An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Biometrika 93 (2):451–458.[Crossref], [Web of Science ®] , [Google Scholar]; Murray, Ghahramani, and MacKay 2006 Murray, I., Z. Ghahramani, and D. J. C. MacKay. 2006. MCMC for doubly intractable distributions. In Proceedings of the 22nd annual conference on uncertainty in artificial intelligence (UAI-06), 359–66. AUAI Press. [Google Scholar]). However, they rely heavily on two Metropolis updates that they need to tune. In this article, we propose instead a model selection with the marginal likelihood.     

A Comparison of Procedures for Controlling the False Discovery Rate in the Presence of Small Variance Genes: A Simulation Study

The Significance Analysis of Microarrays (SAM; Tusher et al., 2001 Tusher , V. G. , Tibshirani , R. , Chu , G. ( 2001 ). Significance analysis of microarrys applied to the ionizing radiation response . Proceedings of the National Academy of Sciences 98 : 5116 – 5121 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) method is widely used in analyzing gene expression data while controlling the FDR by using resampling-based procedure in the microarray setting. One of the main components of the SAM procedure is the adjustment of the test statistic. The introduction of the fudge factor to the test statistic aims at deflating the large value of test statistics due to the small standard error of gene-expression. Lin et al. (2008 Lin , D. , Shkedy , Z. , Burzykowski , T. , Göhlmann , H. W. H. , De Bondt , A. , Perera , T. , Geerts , T. , Bijnens , L. ( 2008 ). Significance analysis of microarray (SAM) for comparisons of several treatments with one control . Biometric Journal, MCP 50 ( 5 ): 801 – 823 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) pointed out that the fudge factor does not effectively improve the power and the control of the FDR as compared to the SAM procedure without the fudge factor in the presence of small variance genes. Motivated by the simulation results presented in Lin et al. (2008 Lin , D. , Shkedy , Z. , Burzykowski , T. , Göhlmann , H. W. H. , De Bondt , A. , Perera , T. , Geerts , T. , Bijnens , L. ( 2008 ). Significance analysis of microarray (SAM) for comparisons of several treatments with one control . Biometric Journal, MCP 50 ( 5 ): 801 – 823 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), in this article, we extend our study to compare several methods for choosing the fudge factor in the modified t-type test statistics and use simulation studies to investigate the power and the control of the FDR of the considered methods.     

A new lifetime model with different types of failure rate

A new class of lifetime distributions, which can exhibit with upside-down bathtub-shaped, bathtub-shaped, decreasing, and increasing failure rates, is introduced. The new distribution is constructed by compounding generalized Weibull and logarithmic distributions, leading to improvement on the lifetime distribution considered in Dimitrakopoulou et al. (2007 Dimitrakopoulou, T., K. Adamidis, and S. Loukas. 2007. A lifetime distribution with an upside-down bathtub-shaped hazard function. IEEE Transactions on Reliability 56:308–11.[Crossref], [Web of Science ®] , [Google Scholar]) by having no restriction on the shape parameter and extending the result studied by Tahmasbi and Rezaei (2008 Tahmasbi, R., and S. Rezaei. 2008. A two-parameter lifetime distribution with decreasing failure rate. Computational Statistics and Data Analysis 52:3889–901.[Crossref], [Web of Science ®] , [Google Scholar]) in the general form. The proposed model includes the exponential–logarithmic and Weibull–logarithmic distributions as special cases. Various statistical properties of the proposed class are discussed. Furthermore, estimation via the maximum likelihood method and the Fisher information matrix are discussed. Applications to real data demonstrate that the new class of distributions is more flexible than other recently proposed classes.     

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

ABSTRACT

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

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.     

MULTI-LEVEL FACTORIAL DESIGNS WITH MINIMUM NUMBERS OF LEVEL CHANGES

The order of experimental runs in a fractional factorial experiment is essential when the cost of level changes in factors is considered. The generalized foldover scheme given by [1] Coster, D. C. and Cheng, C. S. 1988. Minimum cost trend free run orders of fractional factorial designs. The Annals of Statistics, 16: 1188–1205. [Crossref], [Web of Science ®] , [Google Scholar]gives an optimal order to experimental runs in an experiment with specified defining contrasts. An experiment can be specified by a design requirement such as resolution or estimation of some interactions. To meet such a requirement, we can find several sets of defining contrasts. Applying the generalized foldover scheme to these sets of defining contrasts, we obtain designs with different numbers of level changes and then the design with minimum number of level changes. The difficulty is to find all the sets of defining contrasts. An alternative approach is investigated by [2] Cheng, C. S., Martin, R. J. and Tang, B. 1998. Two-level factorial designs with extreme numbers of level changes. The Annals of Statistics, 26: 1522–1539. [Crossref], [Web of Science ®] , [Google Scholar]for two-level fractional factorial experiments. In this paper, we investigate experiments with all factors in slevels.     

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

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

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

ABSTRACT

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

Convergence of weighted sums for sequences of pairwise NQD random variables

ABSTRACT

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

Combined estimating function for random coefficient models with correlated errors

ABSTRACT

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

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.     

Moments for Some Kumaraswamy Generalized Distributions

Explicit expansions for the moments of some Kumaraswamy generalized (Kw-G) distributions (Cordeiro and de Castro, 2011 Cordeiro, G.M., de Castro, M. (2011). A new family of generalized distributions. J. Statist. Computat. Simul. 81:883–898.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are derived using special functions. We explore the Kw-normal, Kw-gamma, Kw-beta, Kw-t, and Kw-F distributions. These expressions are given as infinite weighted linear combinations of well-known special functions for which numerical routines are readily available.     

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

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

On Augmented Franses Tests for Seasonal Unit Roots

This article extends the results reported in del Barrio Castro, Osborn and Taylor (2012 del Barrio Castro, T., Osborn, D.R., Taylor, A. M.R. (2012). On augmented HEGY tests for seasonal unit roots. Econometric Theor. 18:1121–1143.[Crossref], [Web of Science ®] , [Google Scholar]) to the approach followed by Franses (1991a Franses, P. H. (1991a). Model selection and seasonality in time series. Tibergen Institute Series, 18. [Google Scholar],b Franses, P.H. (1991b). Seasonality, non-stationarity and the forecasting of monthly time series. Int. J. Forecast. 7:199–208.[Crossref], [Web of Science ®] , [Google Scholar]) to test for seasonal unit roots, providing the asymptotic representation to the seasonal unit roots tests proposed by Franses for a general number of seasons S.     

ESTIMATION OF PARAMETERS IN ARFIMA PROCESSES: A SIMULATION STUDY

It is known that, in the presence of short memory components, the estimation of the fractional parameter d in an Autoregressive Fractionally Integrated Moving Average, ARFIMA(p, d, q), process has some difficulties (see [1] Smith, J., Taylor, N. and Yadav, S. 1997. Comparing the bias and misspecification in ARFIMA models. Journal of Time Series Analysis, 18(5): 507–527. [Crossref] , [Google Scholar]). In this paper, we continue the efforts made by Smith et al. [1] Smith, J., Taylor, N. and Yadav, S. 1997. Comparing the bias and misspecification in ARFIMA models. Journal of Time Series Analysis, 18(5): 507–527. [Crossref] , [Google Scholar] and Beveridge and Oickle [2] Beveridge, S. and Oickle, C. 1993. Estimating fractionally integrated time series models. Economics Letters, 43: 137–142.  [Google Scholar] by conducting a simulation study to evaluate the convergence properties of the iterative estimation procedure suggested by Hosking [3] Hosking, J. 1981. Fractional differencing. Biometrika, 68(1): 165–176. [Crossref], [Web of Science ®] , [Google Scholar]. In this context we consider some semiparametric approaches and a parametric method proposed by Fox-Taqqu[4] Fox, R. and Taqqu, M. S. 1986. Large-sample properties of parameter estimates for strongly dependent stationary gaussian time series. The Annals of Statistics, 14(2): 517–532. [Crossref], [Web of Science ®] , [Google Scholar]. We also investigate the method proposed by Robinson [5] Robinson, P. M. 1995a. Log-periodogram regression of time series with long range dependence. The Annals of Statistics, 23(3): 1048–1072. [Crossref], [Web of Science ®] , [Google Scholar] and a modification using the smoothed periodogram function.     

Bayes minimax ridge regression estimators

The problem of estimating of the vector β of the linear regression model y = Aβ + ? with ? ～ N_p(0, σ²I_p) under quadratic loss function is considered when common variance σ² is unknown. We first find a class of minimax estimators for this problem which extends a class given by Maruyama and Strawderman (2005 Maruyama, Y., and W. E. Strawderman. 2005. A new class of generalized Bayes minimax ridge regression estimators. Annals of Statistics 33:1753–70.[Crossref], [Web of Science ®] , [Google Scholar]) and using these estimators, we obtain a large class of (proper and generalized) Bayes minimax estimators and show that the result of Maruyama and Strawderman (2005 Maruyama, Y., and W. E. Strawderman. 2005. A new class of generalized Bayes minimax ridge regression estimators. Annals of Statistics 33:1753–70.[Crossref], [Web of Science ®] , [Google Scholar]) is a special case of our result. We also show that under certain conditions, these generalized Bayes minimax estimators have greater numerical stability (i.e., smaller condition number) than the least-squares estimator.     

On simulation of normal records

ABSTRACT

In the present paper, we discuss algorithms of record generation when records are taken from a normal population. We propose three new generation algorithms, compare their efficiency and find the most efficient algorithm (Algorithm 2.1). We then compare these algorithms with known generation algorithms presented in the work of Balakrishnan, So, and Zhu (2016 Balakrishnan, N., H. Y. So, and X. J. Zhu. 2016. On Box-Muller transformation and simulation of normal record data. Communication in Statistics – Simulation and Computations 45(10):3670–82.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

An improved estimation in stratified random sampling

ABSTRACT

The article suggests a class of estimators of population mean in stratified random sampling using auxiliary information with its properties. In addition, various known estimators/classes of estimators are identified as members of the suggested class. It has been shown that the suggested class of estimators under optimum condition performs better than the usual unbiased, usual combined ratio, usual combined regression, Kadilar and Cingi (2005 Kadilar, C., Cingi, H. (2005). A new ratio estimator in stratified sampling. Commun. Stat. Theory Methods 34:597–602.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Singh and Vishwakarma (2006 Singh, H.P., Vishwakarma, G.K. (2006). Combined ratio-product estimator of finite population mean in stratified sampling. Metodologia de Encuestas Monografico: Incidencias en el trabjo de Campo 7(1):32–40. [Google Scholar]) estimators and the members belonging to the classes of estimators envisaged by Kadilar and Cingi (2003 Kadilar, C., Cingi, H. (2003). Ratio estimator in stratified sampling. Biomet. J. 45:218–225.[Crossref], [Web of Science ®] , [Google Scholar]), Singh, Tailor et al. (2008 Singh, H.P., Agnihotri, N. (2008). A general procedure of estimating population mean using auxiliary information in sample surveys. Stat. Trans. 9(1):71–87. [Google Scholar]), Singh et al. (2009 Singh, R., Kumar, M., Chaudhary, M.K., Kadilar, C. (2009). Improved exponential estimator in stratified random sampling. Pak. J. Stat. Oper. Res. 5(2):67–82.[Crossref] , [Google Scholar]), Singh and Vishwakarma (2010 Singh, H.P., Vishwakarma, G.K. (2010). A general procedure for estimating the population mean in stratified sampling using auxiliary information. METRON 67(1):47–65.[Crossref] , [Google Scholar]) and Koyuncu and Kadilar (2010) Koyuncu, N., Kadilar, C. (2010). On improvement in estimating population mean in stratified random sampling. J. Appl. Stat. 37(6):999–1013.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar].