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

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

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.     

Heavy traffic analysis of roving server networks

This article studies the heavy-traffic (HT) behavior of queueing networks with a single roving server. External customers arrive at the queues according to independent renewal processes and after completing service, a customer either leaves the system or is routed to another queue. This type of customer routing in queueing networks arises very naturally in many application areas (in production systems, computer- and communication networks, maintenance, etc.). In these networks, the single most important characteristic of the system performance is oftentimes the path time, i.e., the total time spent in the system by an arbitrary customer traversing a specific path. The current article presents the first HT asymptotic for the path-time distribution in queueing networks with a roving server under general renewal arrivals. In particular, we provide a strong conjecture for the system’s behavior under HT extending the conjecture of Coffman et al.^{[8 Coffman, Jr, E. G.; Puhalskii, A. A.; Reiman, M. I,. Polling systems with zero switchover times: A heavy-traffic averaging principle. Ann. App. Prob. 1995, 5(3), 681–719.[Crossref], [Web of Science ®] [Google Scholar],9 Coffman, Jr., E. G.; Puhalskii, A. A.; Reiman, M. I,. Polling systems in heavy-traffic: A Bessel process limit. Math. Oper. Res. 1998, 23, 257–304.[Crossref], [Web of Science ®] [Google Scholar]]} to the roving server setting of the current article. By combining this result with novel light-traffic asymptotics, we derive an approximation of the mean path time for arbitrary values of the load and renewal arrivals. This approximation is not only highly accurate for a wide range of parameter settings, but is also exact in various limiting cases.     

Aspects of Dependence in Generalized Farlie-Gumbel-Morgenstern Distributions

In this article, we obtained a dependence measure for generalized Farlie-Gumbel-Morgenstern (FGM) family in view of Kochar and Gupta (1987 Kochar , S. G. , Gupta , R. P. ( 1987 ). Competitors of Kendall-tau test for testing independence against PQD . Biometrika 74 ( 3 ): 664 – 669 .[Crossref], [Web of Science ®] , [Google Scholar]) and then compared this measure with Spearman's rho and Kendall's tau in FGM family. Moreover, we evaluated the empirical power of the class of distribution-free tests proposed by Kochar and Gupta (1987 Kochar , S. G. , Gupta , R. P. ( 1987 ). Competitors of Kendall-tau test for testing independence against PQD . Biometrika 74 ( 3 ): 664 – 669 .[Crossref], [Web of Science ®] , [Google Scholar], 1990 Kochar , S. G. , Gupta , R. P. ( 1990 ). Distribution-free tests based on sub-sample extrema for testing against positive dependence . Australian Journal of Statistics 32 : 45 – 51 .[Crossref] , [Google Scholar]) based on exact distribution of a U-statistics. This is derived via a simulation study for sample of sizes n = 6, 8, 10, 12, 16, and 20. Also, we compared our simulation results with those achieved by Amini et al. (2010 Amini , M. , Jabbari , H. , Mohtashami Borzadaran , G. R. , Azadbakhsh , M. ( 2010 ). Power comparison of independence test for the Farlie-Gumbel-Moregenstern family . Communications of the Korean Statistical Society 17 ( 4 ): 493 – 505 .[Crossref] , [Google Scholar]) and Güven and Kotz (2008 Güven , B. , Kotz , S. ( 2008 ). Test of independence for generalized Farlie-Gumbel-Morgenstern distributions . Journal of Computational and Applied Mathematics 212 : 102 – 111 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Empirical Likelihood Ratio for Linear Transformation Models with Doubly Censored Data

Double censoring arises when T represents an outcome variable that can only be accurately measured within a certain range, [L, U], where L and U are the left- and right-censoring variables, respectively. When L is always observed, we consider the empirical likelihood inference for linear transformation models, based on the martingale-type estimating equation proposed by Chen et al. (2002 Chen , K. , Jin , Z. , Ying , Z. ( 2002 ). Semiparametric analysis of transformation models with censored data . Biometrika 89 : 659 – 668 .[Crossref], [Web of Science ®] , [Google Scholar]). It is demonstrated that both the approach of Lu and Liang (2006 Lu , W. , Liang , Y. ( 2006 ). Empirical likelihood inference for linear transformation models . Journal of Multivariate Analysis 97 : 1586 – 1599 .[Crossref], [Web of Science ®] , [Google Scholar]) and that of Yu et al. (2011 Yu , W. , Sun , Y. , Zheng , M. ( 2011 ). Empirical likelihood method for linear transformation models . Annals of the Institute of Statistical Mathematics 63 : 331 – 346 .[Crossref], [Web of Science ®] , [Google Scholar]) can be extended to doubly censored data. Simulation studies are conducted to investigate the performance of the empirical likelihood ratio methods.     

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.     

Generalized linear regression models incorporating original outcome distributions

ABSTRACT

In this article, we propose an approach for incorporating continuous and discrete original outcome distributions into the usual exponential family regression models. The new approach is an extension of the works of Suissa (1991 Suissa, S. (1991). Binary methods for continuous outcomes: A parametric alternative. J. Clin. Epidemiol. 44:241–248.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Suissa and Blais (1995 Suissa, S., Blais, L. (1995). Binary regression with continuous outcomes. Stat. Med. 14:247–255.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), which present methods to estimate the risk of an event defined in a sample subspace of an original continuous outcome variable. Simulation studies are presented in order to illustrate the performance of the developed methodology. Real data sets are analyzed by using the proposed models.     

Is Each NPMLE of a Continuous Bivariate Distribution Function with Singly Right-Censored Data Really Inconsistent?

We consider non-parametric estimation of a continuous cdf of a random vector (X ₁, X ₂). With bivariate RC data, it is stated in van der Laan (1996 Van der Laan , M. J. ( 1996 ) Efficient estimation in the bivariate censoring model and repairing NPMLE . Ann. Statist. 24 : 596 – 627 .[Crossref], [Web of Science ®] , [Google Scholar], p. 598¹⁰, Ann. Statist.), Quale et al. (2006 Quale , C. M. , van der Laan , M. J. , Robins , J. R. ( 2006 ). Locally efficient estimation with bivariate right-censored data . JASA. 101 : 1076 – 1084 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], JASA) etc. that “it is well known that the NPMLE for continuous data is inconsistent (Tsai et al. (1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar])).” The claim is based on a result in Tsai et al. (1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar], p.1352, Ann. Statist.) that if X ₁ is right censored but not X ₂, then common ways for defining one NPMLE lead to inconsistency. If X ₁ is right censored and X ₂ is type I right-censored (which includes the case in Tsai et al.), we present a consistent NPMLE. The result corrects a common misinterpretation of Tsai's example (Tsai et al., 1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar], Ann. Statist.).     

Peaks-Over-Threshold Modeling Under Random Censoring

Recently, the topic of extreme value under random censoring has been considered. Different estimators for the index have been proposed (see Beirlant et al., 2007 Beirlant , J. , Guillou , A. , Dierckx , G. , Fils-Villetard , A. ( 2007 ). Estimation of the extreme value index and extreme quantiles under random censoring . Extremes 10 : 151 – 174 .[Crossref] , [Google Scholar]). All of them are constructed as the classical estimators (without censoring) divided by the proportion of non censored observations above a certain threshold. Their asymptotic normality was established by Einmahl et al. (2008 Einmahl , J. H. J. , Fils-Villetard , A. , Guillou , A. ( 2008 ). Statistics of extremes under random censoring . Bernoulli 14 ( 1 ): 207 – 227 . [Google Scholar]). An alternative approach consists of using the Peaks-Over-Threshold method (Balkema and de Haan, 1974 Balkema , A. , de Haan , L. ( 1974 ). Residual life at great age . Ann. Probab. 2 : 792 – 804 .[Crossref], [Web of Science ®] , [Google Scholar]; Smith, 1987 Smith , R. L. ( 1987 ). Estimating tails of probability distributions . Ann. Statist. 15 : 1174 – 1207 .[Crossref], [Web of Science ®] , [Google Scholar]) and to adapt the likelihood to the context of censoring. This leads to ML-estimators whose asymptotic properties are still unknown. The aim of this article is to propose one-step approximations, based on the Newton-Raphson algorithm. Based on a small simulation study, the one-step estimators are shown to be close approximations to the ML-estimators. Also, the asymptotic normality of the one-step estimators has been established, whereas in case of the ML-estimators it is still an open problem. The proof of our result, whose approach is new in the Peaks-Over-Threshold context, is in the spirit of Lehmann's theory (1991 Lehmann , E. L. ( 1991 ). Theory of Point Estimation . Pacific Grove , CA : Wadsworth & Brooks/Cole Advanced Books & Software .[Crossref] , [Google Scholar]).     

Improved Bar-Lev,Bobovitch, and Boukai Randomized Response Models

In this article, two new improved randomized response models have been proposed. The proposed models are found to be more efficient than the recent randomized response model studied by Bar-Lev et al. (2004 Bar-Lev , S. K. , Bobovitch , E. , Boukai , B. ( 2004 ). A note on randomized response models for quantitative data . Metrika 60 : 255 – 260 .[Crossref], [Web of Science ®] , [Google Scholar]). The relative efficiency of the proposed models has been studied with respect to the Bar-Lev et al. (2004 Bar-Lev , S. K. , Bobovitch , E. , Boukai , B. ( 2004 ). A note on randomized response models for quantitative data . Metrika 60 : 255 – 260 .[Crossref], [Web of Science ®] , [Google Scholar]) model under different situations.     

On Simultaneous Estimation of the Number of Signals and Frequencies of Multiple Sinusoids When Some Observations Are Missing

In an earlier article (Bai et al., 1999 Bai , Z. D. , Rao , C. R. , Wu , Y. H. , Zen , M. M. , Zhao , L. C. ( 1999 ). The simultaneous estimation of the number of signals and frequencies of multiple sinusods when some observations are missing: I. Asymptotics . Proc. Natl. Acad. Sci. 96 : 11,106 – 11,110 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), the problem of simultaneous estimation of the number of signals and frequencies of multiple sinusoids is considered in the case that some observations are missing. The number of signals is estimated with an information theoretic criterion and the frequencies are estimated with eigenvariation linear prediction. Asymptotic properties of the procedure are investigated but the Monte Carlo simulation is not performed. In this article, a slightly different but scale invariant criterion for detection is proposed and the estimation of frequencies remains the same. Asymptotic properties of this new procedure are provided. Monte Carlo Simulation for both procedures is carried out. Furthermore, comparison on the real signals is also given.     

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.     

Methods for Bivariate Survival Data with Mismeasured Covariates Under an Accelerated Failure Time Model

Accelerated failure time models are useful in survival data analysis, but such models have received little attention in the context of measurement error. In this paper we discuss an accelerated failure time model for bivariate survival data with covariates subject to measurement error. In particular, methods based on the marginal and joint models are considered. Consistency and efficiency of the resultant estimators are investigated. Simulation studies are carried out to evaluate the performance of the estimators as well as the impact of ignoring the measurement error of covariates. As an illustration we apply the proposed methods to analyze a data set arising from the Busselton Health Study (Knuiman et al., 1994 Knuiman , M. W. , Cullent , K. J. , Bulsara , M. K. , Welborn , T. A. , Hobbs , M. S. T. ( 1994 ). Mortality trends, 1965 to 1989, in Busselton, the site of repeated health surveys and interventions . Austral. J. Public Health 18 : 129 – 135 . [CSA] [Crossref], [PubMed] , [Google Scholar]).     

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.     

On a Generalization of Bivariate Cauchy Distribution

This paper addresses a generalization of the bivariate Cauchy distribution discussed by Fang et al. (1990 Fang , K. T. , Kotz , S. , Ng , K. W. ( 1990 ). Symmetric Multivariate and Related Distributions . London : Chapman and Hall .[Crossref] , [Google Scholar]), derived from a trivariate normal distribution with a general correlation matrix. We obtain explicit expressions for the joint distribution function and joint density function, and show that they reduce in a special case to the corresponding expressions of Fang et al. (1990 Fang , K. T. , Kotz , S. , Ng , K. W. ( 1990 ). Symmetric Multivariate and Related Distributions . London : Chapman and Hall .[Crossref] , [Google Scholar]). Finally, we show that this generalized distribution is useful in determining the orthant probability of a bivariate skew-normal distribution of Azzalini and Dalla Valle (1996 Azzalini , A. , Dalla Valle , A. ( 1996 ). The multivariate skew-normal distribution . Biometrika 83 : 715 – 726 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Testing for Random Effects in Growth Curve Models

Growth curve models (GCMs) are useful and Demidenko (2004 Demidenko, E. (2004). Mixed Models: Theory and Applications. New York: Wiley.[Crossref] , [Google Scholar]) considered the presence of random effects under the normal assumptions about random effects and random errors. It is also of interest to remove distribution assumptions to investigate the same problem. A difference-based test is constructed for GCMs, which can be regarded as an extension of Li and Zhu (2010 Li, Z.X., Zhu, L.X. (2010). On variance components in semiparametric mixed models for longitudinal data. Scand. J. Statist. 37:442–457.[Crossref], [Web of Science ®] , [Google Scholar])’s method and a complement to Demidenko (2004 Demidenko, E. (2004). Mixed Models: Theory and Applications. New York: Wiley.[Crossref] , [Google Scholar]) where his test is exact in small samples. Without any distribution assumptions, our test derived for GCMs is asymptotically a standard normal. The power properties are also investigated. Besides, simulations are carried out to examine its performance.     

Bayesian Inference in Dynamic Disequilibrium Models: An Application to the Polish Credit Market

We propose a Bayesian approach for inference in a dynamic disequilibrium model. To circumvent the difficulties raised by the Maddala and Nelson (1974 Maddala , G. , Nelson , F. ( 1974 ). Maximum likelihood methods for models of markets in disequilibrium . Econometrica 42 ( 6 ): 1013 – 1030 .[Crossref], [Web of Science ®] , [Google Scholar]) specification in the dynamic case, we analyze a dynamic extended version of the disequilibrium model of Ginsburgh et al. (1980 Ginsburgh , V. , Tishler , A. , Zang , I. ( 1980 ). Alternative estimation methods for two regime models . European Economic Review 13 : 207 – 228 .[Crossref], [Web of Science ®] , [Google Scholar]). We develop a Gibbs sampler based on the simulation of the missing observations. The feasibility of the approach is illustrated by an empirical analysis of the Polish credit market, for which we conduct a specification search using the posterior deviance criterion of Spiegelhalter et al. (2002 Spiegelhalter , D. J. , Best , N. G. , Carlin , B. P. , Van Der Linde , A. ( 2002 ). Bayesian measures of complexity and fit . Journal of the Royal Statistical Society, Ser. B 64 : 583 – 639 .[Crossref] , [Google Scholar]).     

A Nonparametric Test for the Ordered Alternative Based on Kendall's Correlation Coefficient

Several methods have been developed for testing the ordered alternative. These include the Jonckheere–Terpstra (JT) test (Jonckheere, 1954 Jonckheere , A. R. ( 1954 ). A distribution free k-sample test against ordered alternatives . Biometrika 41 : 133 – 145 .[Crossref], [Web of Science ®] , [Google Scholar]; Terpstra, 1952 Terpstra , T. ( 1952 ). The asymptotic normality and consistency of Kendall's test against trend when ties are present in one ranking . Indigationes Mathematicae 14 : 327 – 333 . [Google Scholar]), a modified JT test (MJT) (Tryon and Hettmansperger, 1987 Tryon , V. P. , Hettmansperger , T. P. ( 1987 ). A class of nonparametric tests for homogeneity against ordered alternatives . Annals of Statistics 1 : 1061 – 1070 . [Google Scholar]), and a test proposed by Terpstra and Magel (TM) (Terpstra and Magel, 2003 Terpstra , J. T. , Magel , R. C. ( 2003 ). A new nonparametric test for the ordered alternative problem . Journal of Nonparametric Statistics 15 : 289 – 301 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), among others. This article proposes a new method for testing the ordered alternative. The proposed test is based on Kendall's tau statistic. The asymptotic distribution of the test statistic is given. A Monte Carlo simulation study is conducted comparing the estimated powers of the proposed test with existing tests under a variety of sample sizes and distributions.     

Computing the Exact Fisher Information Matrix of Periodic State-Space Models

The purpose of this article is to develop algorithms for computing the exact Fisher information matrix of periodic time-varying state-space models. We first present a relatively simple recursive algorithm which computes the elements of the exact information matrix without involving numerical differentiation, since all required derivatives are analytically evaluated. The proposed algorithm extends the procedure due to Cavanaugh and Shumway (1996 Cavanaugh , J. E. , Shumway , R. H. ( 1996 ). On computing the expected Fisher information matrix for state-space model parameters . Statist. Probab. Lett. 26 : 347 – 355 .[Crossref], [Web of Science ®] , [Google Scholar]) to the periodic state-space framework. Exploiting the approach used in Klein et al. (2000 Klein , A. , Mélard , G. , Zahaf , T. ( 2000 ). Construction of the exact Fisher information matrix of Gaussian time series models by means of matrix differential rules . Linear Alg. Applic. 321 : 209 – 232 .[Crossref], [Web of Science ®] , [Google Scholar]), a second algorithm is proposed in order to obtain the exact information matrix as a whole instead of element by element. The algorithms are first developed in a general framework and then specialized to the case of a periodic Gaussian vector autoregressive moving-average (PVARMA) model.     

Improved Estimation of Finite Population Variance Using Auxiliary Information

This article addresses the problem of estimating of finite population variance using auxiliary information in simple random sampling. A ratio-cum-difference type class of estimators for population variance has been suggested with its properties under large sample approximation. It has been shown that the suggested class of estimators is more efficient than usual unbiased, difference, Das and Tripathi (1978 Das , A. K. , Tripathi , T. P. ( 1978 ). Use of auxiliary information in estimating the finite population variance . Sankhya C 40 : 139 – 148 . [Google Scholar]), Isaki (1983 Isaki , C. T. ( 1983 ). Variance estimation using auxiliary information . J. Amer. Statist. Assoc. 78 : 117 – 123 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Singh et al. (1988 Singh , H. P. , Upadhyaya , L. N. , Namjoshi , U. D. ( 1988 ). Estimation of finite population variance . Curr. Sci. 57 : 1331 – 1334 .[Web of Science ®] , [Google Scholar]), Kadilar and Cingi (2006 Kadilar , C. , Cingi , H. ( 2006 ). Ratio estimators for the population variance in simple and stratified random sampling . Appl. Math. Comp. 173 ( 2 ): 1047 – 1059 .[Crossref], [Web of Science ®] , [Google Scholar]), and other estimators/classes of estimators. In addition, we support this theoretical result with the aid of a empirical study.