From Ordinary to Shrinkage Square-Root Estimators

Consider a skewed population. Suppose an intelligent guess could be made about an interval that contains the population mean. There may exist biased estimators with smaller mean squared error than the arithmetic mean within such an interval. This article indicates when it is advisable to shrink the arithmetic mean towards a guessed interval using root estimators. The goal is to obtain an estimator that is better near the average of natural origins. An estimator proposed. This estimator contains the Thompson (1968 Thompson , J. R. ( 1968 ). Accuracy borrowing in the estimation of the mean by shrinkage towards an interval . J. Amer. Statist. Assoc. 63 : 953 – 963 . [CSA] [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) ordinary shrinkage estimator, the Jenkins et al. (1973 Jenkins , O. C. , Ringer , L. J. , Hartley , H. O. ( 1973 ). Root estimators . J Amer. Statist. Assoc. 68 : 414 – 419 . [CSA] [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) square-root estimator, and the arithmetic sample mean as special cases. The bias and the mean squared error of the proposed more general estimator is compared with the three special cases. Shrinkage coefficients that yield minimum mean squared error estimators are obtained. The proposed estimator is considerably more efficient than the three special cases. This remains true for highly skewed populations. The merits of the proposed shrinkage square-root estimator are supported by the results of numerical and simulation studies.     

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]).     

Efficient Ratio and Product Estimators in Stratified Random Sampling

This paper suggests an efficient class of ratio and product estimators for estimating the population mean in stratified random sampling using auxiliary information. It is interesting to mention that, in addition to many, Koyuncu and Kadilar (2009 Koyuncu , N. , Kadilar , C. ( 2009 ). Ratio and product estimators in stratified random sampling . J. Statist. Plann. Infer. 139 : 2552 – 2558 .[Crossref], [Web of Science ®] , [Google Scholar]), Kadilar and Cingi (2003 Kadilar , C. , Cingi , H. ( 2003 ). Ratio estimator in stratified sampling . Biometr. J. 45 : 218 – 225 .[Crossref], [Web of Science ®] , [Google Scholar], 2005 Kadilar , C. , Cingi , H. ( 2005 ). A new estimator in stratified random sampling . Commun. Statist. Theor. Meth. 34 : 597 – 602 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), and Singh and Vishwakarma (2007 Singh , H. P. , Vishwakarma , G. K. ( 2007 ). Modified exponential ratio and product estimators for finite population mean in double sampling . Austr. J. Statist. 36 ( 3 ): 217 – 225 . [Google Scholar]) estimators are identified as members of the proposed class of estimators. The expressions of bias and mean square error (MSE) of the proposed estimators are derived under large sample approximation in general form. Asymptotically optimum estimator (AOE) in the class is identified alongwith its MSE formula. It has been shown that the proposed class of estimators is more efficient than combined regression estimator and Koyuncu and Kadilar (2009 Koyuncu , N. , Kadilar , C. ( 2009 ). Ratio and product estimators in stratified random sampling . J. Statist. Plann. Infer. 139 : 2552 – 2558 .[Crossref], [Web of Science ®] , [Google Scholar]) estimator. Moreover, theoretical findings are supported through a numerical example.     

Estimators for the Population Mean in the Case of Missing Data

We propose a class of estimators for the population mean when there are missing data in the data set. Obtaining the mean square error equations of the proposed estimators, we show the conditions where the proposed estimators are more efficient than the sample mean, ratio-type estimators, and the estimators in Singh and Horn (2000 Singh , S. , Horn , S. ( 2000 ). Compromised imputation in survey sampling . Metrika 51 : 267 – 276 .[Crossref], [Web of Science ®] , [Google Scholar]) and Singh and Deo (2003 Singh , S. , Deo , B. (2003). Imputation by power transformation. Statist. Pap. 44:555–579.[Crossref], [Web of Science ®] , [Google Scholar]) in the case of missing data. These conditions are also supported by a numerical example.     

On Estimating the Finite Population Mean Using Two Auxiliary Variables in Two Phase Sampling in the Presence of Non Response

This article considers the problem of estimating the population mean using information on two auxiliary variables in the presence of non response under two-phase sampling. Some improved ratio-in-regression type estimators have been proposed in four different situations of non response along with their properties under large sample approximation. Efficiency comparisons of the proposed estimators with the usual unbiased estimator by Hansen and Hurwitz (1946 Hansen , M. H. , Hurwitz , W. N. ( 1946 ). The problem of non response in sample surveys . J. Amer. Statist. Assoc. 41 : 517 – 529 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), conventional ratio and regression estimators using single auxiliary variable and Singh and Kumar (2010b Singh , H. P. , Kumar , S. ( 2010b ). Improved estimation of population mean under double sampling with sub-sampling the non-respondents . J. Statist. Plann. Infer. 140 ( 9 ): 2536 – 2550 .[Crossref], [Web of Science ®] , [Google Scholar]) estimators using two auxiliary variables have been made. Finally, these theoretical findings are illustrated by a numerical example.     

Generalized Scrambling in Quantitative Optional Randomized Response Models

Huang (2010 Huang , K. C. ( 2010 ). Unbiased estimators of mean, variance and sensitivity level for quantitative characteristics in finite population sampling . Metrika 71 : 341 – 352 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed an optional randomized response model using a linear combination scrambling which is a generalization of the multiplicative scrambling of Eichhorn and Hayre (1983 Eichhorn , B. H. , Hayre , L. S. ( 1983 ). Scrambled randomized response methods for obtaining sensitive quantitative data . J. Statist. Plann. Infer. 7 : 307 – 316 .[Crossref], [Web of Science ®] , [Google Scholar]) and the additive scrambling of Gupta et al. (2006, 2010). In this article, we discuss two main issues. (1) Can the Huang (2010 Huang , K. C. ( 2010 ). Unbiased estimators of mean, variance and sensitivity level for quantitative characteristics in finite population sampling . Metrika 71 : 341 – 352 .[Crossref], [Web of Science ®] , [Google Scholar]) model be improved further by using a two-stage approach?; (2) Does the linear combination scrambling provide any benefit over the additive scrambling of Gupta et al. (2010 Gupta , S. N. , Shabbir , J. , Sehra , S. ( 2010 ). Mean and sensitivity estimation in optional randomized response models . J. Statist. Plann. Infer. 140 : 2870 – 2874 .[Crossref], [Web of Science ®] , [Google Scholar])? We will note that the answer to the first question is “yes” but the answer to the second question is “no.”     

MSE Performance and Minimax Regret Significance Points for a HPT Estimator when each Individual Regression Coefficient is Estimated

In this article, we consider a heterogeneous preliminary test (HPT) estimator whose components are the OLS and feasible ridge regression (FRR) estimators, and derive the exact formulae for the moments of the HPT estimator using mathematical method. Since we cannot examine the MSE of the HPT estimator analytically, we execute the numerical evaluation to investigate the MSE performance of the HPT estimator, and compare the MSE performance of the HPT estimator with those of the FRR estimator and the usual OLS estimator. Furthermore, using the minimax regret criterion proposed by Sawa and Hiromatsu (1973 Sawa , T. , Hiromatsu , T. ( 1973 ). Minimax regret significance points for a preliminary test in regression analysis . Econometrica 41 : 1093 – 1101 .[Crossref], [Web of Science ®] , [Google Scholar]), we derive the optimal critical points of the preliminary F test. Our results show that the optimal significance points are greater than 19% and the optimal signicance points decrease as the denominator degrees of freedom of the preliminary F test statistic increases.     

Additive Mixed Effect Model for Clustered Doubly Censored Data

Cai and Zeng (2011 Cai, J. and Zeng, D. 2011. Additive mixed effect model for clustered failure time data. Biometrics, 67(4): 1340–1351. [Crossref], [PubMed] , [Google Scholar]) proposed an additive mixed effect model to analyze clustered right-censored data. In this article, we demonstrate that the approach of Cai and Zeng (2011 Cai, J. and Zeng, D. 2011. Additive mixed effect model for clustered failure time data. Biometrics, 67(4): 1340–1351. [Crossref], [PubMed] , [Google Scholar]) can be extended to clustered doubly censored data. Furthermore, when both left- and right-censoring variables are always observed, we propose alternative estimators using the approach of Cai and Cheng (2004 Cai, T. and Cheng, S. C. 2004. Semiparametric regression analysis for doubly censored data. Biometrika, 91: 277–290. [Crossref], [Web of Science ®] , [Google Scholar]). A simulation study is conducted to investigate the performance of the proposed estimators.     

Regression M-estimators with One Modified Form of Doubly Censored Data

In this article, we consider the M-estimators for the linear regression model when both response and covariate variables are subject to double censoring. The proposed estimators are constructed as some functional of three types of estimators for a bivariate survival distribution. The first two estimators are the generalizations of the Campbell and Földes (1982 Campbell, G. and Földes, A. 1982. “Large sample properties of nonparametric statistical inference”. In Nonparametric Statistical Inference., Edited by: Gnredenko, B. V., Puri, M. L. and Vineze, I. 103–122. Amsterdam: North-Holland.  [Google Scholar]) and Dabrowska (1988 Dabrowska, D. M. 1988. Kaplan-Meier estimate on the plane. Annals of Statistics, 18: 1475–1489. [Crossref], [Web of Science ®] , [Google Scholar]) estimators proposed by Shen (2009 Shen, P. S. 2009. Nonparametric estimation of the bivariate survival function one modified form of doubly censored data. Computational Statistics, 25: 203–313. [Crossref], [Web of Science ®] , [Google Scholar]). The third estimator is the generalization of the Prentice and Cai (1992 Prentice, R. L. and Cai, J. 1992. Covariance and survivor function estimation using censored multivariate failure time data. Biometrika, 79: 495–512. [Crossref], [Web of Science ®] , [Google Scholar]) estimator. The consistency of the proposed M-estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators. Furthermore, the simple bootstrap methods are used to estimate standard deviations and construct interval estimators.     

A Class of Improved Estimators for the Scale Parameter of a Mixture Model of Exponential Distribution with Unknown Location

Under Stein's loss, a class of improved estimators for the scale parameter of a mixture of exponential distribution with unknown location is constructed. The method is analogous to Maruyama's (1998 Maruyama , Y. ( 1998 ). Minimax estimators of a normal variance . Metrika 48 : 209 – 214 .[Crossref], [Web of Science ®] , [Google Scholar]) construction for the variance of a normal distribution and also an extension of the result produced in Petropoulos and Kourouklis (2002 Petropoulos , C. , Kourouklis , S. ( 2002 ). A class of improved estimators for the scale parameter of an exponential distribution with unknown location . Commun. Statist. Theor. Meth. 31 : 325 – 335 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Also, robustness properties are considered.     

Using the Score Test to Identify the Longitudinal Biomarker Considering Accelerate Failure Time Model with the Frailty in Survival Analysis

Quality of life (QOL) is looked upon as a multidimensional entity comprising physical, psychological, social, and medical parameters. QOL is a good prognostic factor for the cancer patients. In this article, we want to determine if QOL is a good biomarker as a surrogate to indicate the survival time of gastric cancer patients. We conducted a single institutional trial and examines QOL of gastric cancer patients receiving the different surgery. In this trial, QOL is a longitudinal measurement. The accelerated failure time model can be used to deal with survival data when the proportionality assumption fails to capture the relationship between the survival time and covariates. In this article, similar to Henderson et al. (2000 Henderson , R. , Diggle , P. , Dobson , A. ( 2000 ). Joint modelling of longitudinal measurements and event time data . Biostatistics 1 ( 4 ): 465 – 480 .[Crossref], [PubMed] , [Google Scholar], 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 survival times under the accelerated failure time assumption. We introduce a method employing a frailty model to identify longitudinal biomarkers or surrogates for a time to event outcome. We allow random effects to be present in both the longitudinal biomarker and underlying survival function. The random effects in the biomarker are introduced via an explicit term while the random effect in the underlying survival function is introduced by the inclusion of frailty into the model. We will introduce a method to identify longitudinal biomarkers or surrogates for a time to event outcome based on the accelerated failure time assumption.     

An Asymmetric Kernel Estimator of Density Function for Stationary Associated Sequences

Here, we apply the smoothing technique proposed by Chaubey et al. (2007 Chaubey , Y. P. , Sen , A. , Sen , P. K. ( 2007 ). A new smooth density estimator for non-negative random variables. Technical Report No. 1/07. Department of Mathematics and Statistics, Concordia University, Montreal, Canada . [Google Scholar]) for the empirical survival function studied in Bagai and Prakasa Rao (1991 Bagai , I. , Prakasa Rao , B. L. S. ( 1991 ). Estimation of the survival function for stationary associated processes . Statist. Probab. Lett. 12 : 385 – 391 .[Crossref], [Web of Science ®] , [Google Scholar]) for a sequence of stationary non-negative associated random variables.The derivative of this estimator in turn is used to propose a nonparametric density estimator. The asymptotic properties of the resulting estimators are studied and contrasted with some other competing estimators. A simulation study is carried out comparing the recent estimator based on the Poisson weights (Chaubey et al., 2011 Chaubey , Y. P. , Dewan , I. , Li , J. ( 2011 ). Smooth estimation of survival and density functions for a stationary associated process using poisson weights . Statist. Probab. Lett. 81 : 267 – 276 .[Crossref], [Web of Science ®] , [Google Scholar]) showing that the two estimators have comparable finite sample global as well as local behavior.     

On the Preliminary Test Backfitting and Speckman Estimators in Partially Linear Models and Numerical Comparisons

Przystalski and Krajewski (2007 Przystalski , M. , Krajewski , P. ( 2007 ). Constrained estimators of treatment parameters in semiparametric models . Statist. Probab. Lett. 77 : 914 – 919 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed the restricted backfitting (RBCF) estimator and restricted Speckman (RSPC) estimator for the treatment effects in a partially linear model when some additional exact linear restrictions are assumed to hold. In this article, we introduce the preliminary test backfitting (PTBCF) estimator and preliminary test Speckman (PTSPC) estimator when the validity of the restrictions is suspected. Performances of the proposed estimators are examined with respect to the mean squared error (MSE) criterion. In addition, numerical behaviors of the proposed estimators are illustrated and compared via a Monte Carlo simulation study.     

Minimum Divergence,Generalized Empirical Likelihoods,and Higher Order Expansions

This article studies the minimum divergence (MD) class of estimators for econometric models specified through moment restrictions. We show that MD estimators can be obtained as solutions to a tractable lower dimensional optimization problem. This problem is similar to the one solved by the generalized empirical likelihood estimators of Newey and Smith (2004 Newey , W. K. , Smith , R. J. ( 2004 ). Higher order properties of GMM and Generalized Empirical Likelihood estimators . Econometrica 72 : 219 – 255 .[Crossref], [Web of Science ®] , [Google Scholar]), but it is equivalent to it only for a subclass of divergences. The MD framework provides a coherent testing theory: tests for overidentification and parametric restrictions in this framework can be interpreted as semiparametric versions of Pearson-type goodness of fit tests. The higher order properties of MD estimators are also studied and it is shown that MD estimators that have the same higher order bias as the empirical likelihood (EL) estimator also share the same higher order mean square error and are all higher order efficient. We identify members of the MD class that are not only higher order efficient, but also, unlike the EL estimator, well behaved when the moment restrictions are misspecified.     

Nonparametric Estimation Methods of Integrated Multivariate Volatilities

Estimation of integrated multivariate volatilities of an Itô process is an interesting and important issue in finance, for example, in order to evaluate portfolios. New non-parametric estimators have been recently proposed by Malliavin and Mancino (2002 Malliavin , P. , Mancino , M. ( 2002 ). Fourier series method for measurement of multivariate volatilities . Finance and Stochastics 6 : 49 – 61 .[Crossref], [Web of Science ®] , [Google Scholar]) and Hayashi and Yoshida (2005a Hayashi , T. , Yoshida , N. (2005a). On covariance estimation of nonsynchronously observed diffusion processes. Bernoulli 11(2):359–379.[Crossref], [Web of Science ®] , [Google Scholar]) as alternative methods to classical realized quadratic covariation. The purpose of this article is to compare these alternative estimators both theoretically and empirically, when high frequency data is available. We found that the Hayashi–Yoshida estimator performs the best among the alternatives in view of the bias and the MSE. The other estimators are shown to have possibly heavy bias mostly toward the origin. We also applied these estimators to Japanese Government Bond futures to obtain the results consistent with our simulation.     

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.     

Inverse Sampling with Unequal Selection Probabilities

In some real situations the population of interest is divided into two groups, of which one contains only a few units. In other cases, the population may be considered as subdivided into two group', for example, if only a few units display a value of the variable of interest which is highly different from zero, while all the other units show a value equal to or near zero. In both cases, inverse sampling is more efficient than classical fixed sample-size designs to obtain the parameter estimators for the whole population as well as for its groups (e.g., Salehi and Seber, 2004 Salehi , M. M. , Seber , G. A. F. ( 2004 ). A general inverse sampling scheme and its application to adaptive cluster sampling . Austral. NZ J. Statist. 46 : 483 – 494 .[Crossref], [Web of Science ®] , [Google Scholar]). In fact, in this design the procedure selection continues until a prefixed number of units with the characteristic of interest is sampled. Since it is not known a priori to which group the population units belong, the sample size is a random variable. Christman and Lan (2001 Christman , M. C. , Lan , F. ( 2001 ). Inverse adaptive cluster sampling . Biometrics 57 : 1096 – 1105 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Salehi and Seber (2001 Salehi , M. M. , Seber , G. A. F. ( 2001 ). A new proof of Murthy's estimator which applies to sequential sampling . Austral. NZ J. Statist. 43 : 281 – 286 . [Google Scholar] 2004 Salehi , M. M. , Seber , G. A. F. ( 2004 ). A general inverse sampling scheme and its application to adaptive cluster sampling . Austral. NZ J. Statist. 46 : 483 – 494 .[Crossref], [Web of Science ®] , [Google Scholar]) considered inverse sampling designs when all the population units have equal selection probabilities. In this article, we consider the general case in which the units may have unequal probabilities of being included in the sample. In fact, in many real situations different units may have different selection probabilities because of some inherent features of the sampling procedure, or in order to obtain better estimates. We derive unbiased estimators of the totals of the two groups, their variance and the corresponding unbiased variance estimators in inverse sampling with replacement. Finally, we derive similar results for more complex designs, where the selection procedure stops before observing the prefixed number of units from the rare group.     

The Product-Limit Estimate as an Inverse-Probability-Weighted Average

Abstract

For randomly censored data, (Satten, G. A., Datta S. (2001 Satten, G. A. and Datta, S. 2001. The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. Ass., 55: 207–210. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. Ass. 55:207–210) showed that the Kaplan–Meier estimator (product-limit estimator (PLE)) can be expressed as an inverse-probability-weighted average. In this article, we consider the other two PLEs: the truncation PLE and the censoring-truncation PLE. For the data subject to left-truncation or both left-truncation and right-censoring, it is shown that these two PLEs can be expressed as inverse-probability-weighted averages.     

Independent Quadratic Forms in 3-Variance-Component Models

This article addresses derivation and existence of quadratic forms that were suggested by Burch (2007 Burch , B. D. ( 2007 ). Generalized confidence intervals for proportions of total variance in mixed linear models . J. Statist. Plann. Infer. 137 : 2394 – 2404 .[Crossref], [Web of Science ®] , [Google Scholar]) for procedures for inference on variance components in mixed linear models in combination with generalized fiducial inference. A relatively simple algorithm leading to the required quadratic forms in a general 3-variance-component model is stated and designs for two-way ANOVA models without interactions that permit Burch's procedure are characterized. This complements developments in the original article by Burch.     

Relative Errors of Difference-Based Variance Estimators in Nonparametric Regression

Difference-based estimators for the error variance are popular since they do not require the estimation of the mean function. Unlike most existing difference-based estimators, new estimators proposed by Müller et al. (2003 Müller , U. , Schick , A. , Wefelmeyer , W. ( 2003 ). Estimating the error variance in nonparametric regression by a covariate-matched U-statistic . Statistics 37 : 179 – 188 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Tong and Wang (2005 Tong , T. , Wang , Y. ( 2005 ). Estimating residual variance in nonparametric regression using least squares . Biometrika 92 : 821 – 830 .[Crossref], [Web of Science ®] , [Google Scholar]) achieved the asymptotic optimal rate as residual-based estimators. In this article, we study the relative errors of these difference-based estimators which lead to better understanding of the differences between them and residual-based estimators. To compute the relative error of the covariate-matched U-statistic estimator proposed by Müller et al. (2003 Müller , U. , Schick , A. , Wefelmeyer , W. ( 2003 ). Estimating the error variance in nonparametric regression by a covariate-matched U-statistic . Statistics 37 : 179 – 188 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), we develop a modified version by using simpler weights. We further investigate its asymptotic property for both equidistant and random designs and show that our modified estimator is asymptotically efficient.