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.     

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.     

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.     

Semiparametric Efficient Distribution Free Estimation of Panel Models

This article generalizes results from Park et al. (1998 Park , B. U. , Sickles , R. C. , Simar , L. ( 1998 ). Stochastic frontiers: a semiparametric approach . J. Econometrics 84 : 273 – 301 .[Crossref], [Web of Science ®] , [Google Scholar]) and Adams et al. (1999 Adams , R. M. , Berger , A. N. , Sickles , R. C. ( 1999 ). Semiparametric approaches to stochastic panel frontiers with applications in the banking industry . J. Bus. Econ. Statist. 17 : 349 – 358 .[Taylor & Francis Online] , [Google Scholar]) on semiparametric efficient estimation of panel models. The form of semiparametric efficient estimators depends on the statistical assumptions imposed. Normality assumptions on the transitory error are sometimes inappropriate. We relax the normality assumption used in the articles above to derive more general semiparametric efficient estimators. These estimators are illustrated in a Monte Carlo simulation and an analysis of banking productivity.     

Bias-Corrected Maximum Likelihood Estimators in Nonlinear Heteroscedastic Models

Nonlinear heteroscedastic models are widely used in econometrics and statistical applications. We derive matrix formulae for the second-order biases of the maximum likelihood estimators of the parameters in the mean and variance response which generalize previous results by Cook et al. (1986 Cook , D. R. , Tsai , C. L. , Wei , B. C. ( 1986 ). Bias in nonlinear regression . Biometrika 73 : 615 – 623 .[Crossref], [Web of Science ®] , [Google Scholar]) and Cordeiro (1993 Cordeiro , G. M. ( 1993 ). Bartlett corrections and bias correction for two heteroscedastic regression models . Commun. Statist. Theor. Meth. 22 : 169 – 188 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The biases of the estimators are easily obtained as vectors of regression coefficients from suitable weighted linear regressions. The practical use of such biases is illustrated in a simulation study and in an application to a real data set.     

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.     

Estimating Incidence Rate on Current Status Data with Application to a Phase IV Cancer Trial

New drug discovery in the pediatrics has dramatically improved survival, but with long- term adverse events. This motivates the examination of adverse outcomes such as long-term toxicity in a phase IV trial. An ideal approach to monitor long-term toxicity is to systematically follow the survivors, which is generally not feasible. Instead, cross-sectional surveys are conducted in Hudson et al. (2007 Hudson , M. M. , Rai , S. N. , Nunez , C. , Merchant , T. E. , Marina , N. M. , Zalamea , N. , Cox , C. , Phipps , S. , Pompeu , R. , Rosenthal , D. ( 2007 ). Noninvasive evaluation of late anthracycline cardiac toxicity in childhood cancer survivors . J. Clin. Oncol. 25 : 3635 – 3643 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), with one of the objectives to estimate the cumulative incidence rates along with specific interest in fixed-term (5 or 10 year) rates. We present inference procedures based on current status data to our motivating example with very interesting findings.     

On Complete Convergence for Arrays of Dependent Random Variables

A complete convergence theorem for an array of rowwise independent random variables was established by Sung et al. (2005 Sung , S. H. , Volodin , A. I. , Hu , T.-C. ( 2005 ). More on complete convergence for arrays . Statist. Probab. Lett. 71 : 303 – 311 .[Crossref], [Web of Science ®] , [Google Scholar]). This result has been generalized and extended by Kruglov et al. (2006 Kruglov , V. M. , Volodin , A. I. , Hu , T.-C. ( 2006 ). On complete convergence for arrays . Statist. Probab. Lett. 76 : 1631 – 1640 .[Crossref], [Web of Science ®] , [Google Scholar]) and Chen et al. (2007 Chen , P. , Hu , T.-C. , Liu , X. , Volodin , A. ( 2007 ). On complete convergence for arrays of rowwise negatively associated random variables . Theor. Probab. Appl. 52 : 393 – 397 . [Google Scholar]). In this article, we extend the results of Sung et al. (2005 Sung , S. H. , Volodin , A. I. , Hu , T.-C. ( 2005 ). More on complete convergence for arrays . Statist. Probab. Lett. 71 : 303 – 311 .[Crossref], [Web of Science ®] , [Google Scholar]), Kruglov et al. (2006 Kruglov , V. M. , Volodin , A. I. , Hu , T.-C. ( 2006 ). On complete convergence for arrays . Statist. Probab. Lett. 76 : 1631 – 1640 .[Crossref], [Web of Science ®] , [Google Scholar]), and Chen et al. (2007 Chen , P. , Hu , T.-C. , Liu , X. , Volodin , A. ( 2007 ). On complete convergence for arrays of rowwise negatively associated random variables . Theor. Probab. Appl. 52 : 393 – 397 . [Google Scholar]) to an array of dependent random variables satisfying Hoffmann-Jørgensen type inequalities.     

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.”     

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.     

Nonparametric Knn estimation with monotone constraints

ABSTRACT

The K-nearest-neighbor (Knn) method is known to be more suitable in fitting nonparametrically specified curves than the kernel method (with a globally fixed smoothing parameter) when data sets are highly unevenly distributed. In this paper, we propose to estimate a nonparametric regression function subject to a monotonicity restriction using the Knn method. We also propose using a new convergence criterion to measure the closeness between an unconstrained and the (monotone) constrained Knn-estimated curves. This method is an alternative to the monotone kernel methods proposed by Hall and Huang (2001 Hall, P., Huang, L.-S. (2001). Nonparametric kernel regression subject to monotonicity constraints. The Annals of Statistics 29:624–647.[Crossref], [Web of Science ®] , [Google Scholar]), and Du et al. (2013 Du, P., Parmeter, C. F., Racine, J. S. (2013). Nonparametric kernel regression with multiple predictors and multiple shape constraints. Statistic Sinica 23:1347–1371.[Web of Science ®] , [Google Scholar]). We use a bootstrap procedure for testing the validity of the monotone restriction. We apply our method to the “Job Market Matching” data taken from Gan and Li (2016 Gan, L., Li, Q. (2016). Efficiency of thin and thick market. Journal of Econometrics 192(1):40–54.[Crossref], [Web of Science ®] , [Google Scholar]) and find that the unconstrained/constrained Knn estimators work better than kernel estimators for this type of highly unevenly distributed data.     

Combining Unbiased Ridge and Principal Component Regression Estimators

In the presence of multicollinearity problem, ordinary least squares (OLS) estimation is inadequate. To circumvent this problem, two well-known estimation procedures often suggested are the unbiased ridge regression (URR) estimator given by Crouse et al. (1995 Crouse , R. , Jin , C. , Hanumara , R. ( 1995 ). Unbiased ridge estimation with prior information and ridge trace . Commun. Statist. Theor. Meth. 24 : 2341 – 2354 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and the (r, k) class estimator given by Baye and Parker (1984 Baye , M. , Parker , D. ( 1984 ). Combining ridge and principal component regression: a money demand illustration . Commun. Statist. Theor. Meth. 13 : 197 – 205 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). In this article, we proposed a new class of estimators, namely modified (r, k) class ridge regression (MCRR) which includes the OLS, the URR, the (r, k) class, and the principal components regression (PCR) estimators. It is based on a criterion that combines the ideas underlying the URR and the PCR estimators. The standard properties of this new class estimator have been investigated and a numerical illustration is done. The conditions under which the MCRR estimator is better than the other two estimators have been investigated.     

The Restricted and Unrestricted Two-Parameter Estimators

In this article, we introduce a new two-parameter estimator by grafting the contraction estimator into the modified ridge estimator proposed by Swindel (1976 Swindel , B. F. ( 1976 ). Good ridge estimators based on prior information . Commun. Statist. Theor. Meth. A5 : 1065 – 1075 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). This new two-parameter estimator is a general estimator which includes the ordinary least squares, the ridge, the Liu, and the contraction estimators as special cases. Furthermore, by setting restrictions Rβ = r on the parameter values we introduce a new restricted two-parameter estimator which includes the well-known restricted least squares, the restricted ridge proposed by Groß (2003 Groß , J. ( 2003 ). Restricted ridge estimation . Statist. Probab. Lett. 65 : 57 – 64 .[Crossref], [Web of Science ®] , [Google Scholar]), the restricted contraction estimators, and a new restricted Liu estimator which we call the modified restricted Liu estimator different from the restricted Liu estimator proposed by Kaç?ranlar et al. (1999 Kaç?ranlar , S. , Sakall?o?lu , S. , Akdeniz , F. , Styan , G. P. H. , Werner , H. J. ( 1999 ). A new biased estimator in linear regression and a detailed analysis of the widely-analysed dataset on Portland cement . Sankhya Ser. B., Ind. J. Statist. 61 : 443 – 459 . [Google Scholar]). We also obtain necessary and sufficient condition for the superiority of the new two-parameter estimator over the ordinary least squares estimator and the comparison of the new restricted two-parameter estimator to the new two-parameter estimator is done by the criterion of matrix mean square error. The estimators of the biasing parameters are given and a simulation study is done for the comparison as well as the determination of the biasing parameters.     

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]).     

A Monte Carlo Study of Recent Ridge Parameters

A number of procedures have been developed for finding biased estimators of regression parameters. One of these procedures is the ridge regression. In this article, a new approach to obtain the ridge parameter K is suggested and then evaluated by Monte Carlo simulations. A number of different models are investigated for different number of observations, the strength of correlation between the explanatory variables, and distribution of the error terms. The mean squared error (MSE) criterion is used to examine the performance of the proposed estimators when compared with other well-known estimators. Under certain conditions, it is shown that at least one of the proposed estimators have a smaller MSE than the ordinary least squared estimator (OLS) and Hoerl and Kennard (1970a Hoerl , A. E. , Kennard , R. W. ( 1970a ). Ridge regression: biased estimation for non-orthogonal problems . Technometrics 12 : 55 – 67 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) estimator (HK).     

An Improved Class of Estimators in Two-Phase Sampling Using Two Auxiliary Variables

This article suggests an improved class of estimators under the general framework of two-phase sampling scheme in presence of two auxiliary variables. This class includes a large number of estimators (Chand, 1975 Chand , L. ( 1975 ). Some Ratio-Type Estimator Based on Two or More Auxiliary Variables. Unpublished Ph.D. dissertation, Iowa State University, Iowa . [Google Scholar]; Kiregyera, 1980 Kiregyera , B. ( 1980 ). A chain ratio-type estimator in finite population double sampling using two auxiliary variables . Metrika 27 : 217 – 223 .[Crossref] , [Google Scholar], 3; Mukharjee et al., 1987 Mukharjee , R. , Rao , T. J. , Vijayan , K. ( 1987 ). Regression-type estimators using multiple auxiliary information . Aust. J. Statist. 29 : 244 – 254 . [Google Scholar]) and also the class of estimators suggested by Sahoo et al. (1993 Sahoo , J. , Sahoo , L. N. , Mohanty , S. ( 1993 ). A regression approach to estimation in two phase sampling using two auxiliary variables . Curr. Sci. 65 ( 1 ): 73 – 75 . [Google Scholar]).     

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.     

A Generalization of the Skew-Normal Distribution: The Beta Skew-Normal

We consider a new generalization of the skew-normal distribution introduced by Azzalini (1985 Azzalini , A. ( 1985 ). A class of distributions which includes the normal ones . Scand. J. Statis. 12 ( 2 ): 171 – 178 .[Web of Science ®] , [Google Scholar]). We denote this distribution Beta skew-normal (BSN) since it is a special case of the Beta generated distribution (Jones, 2004 Jones , M. C. ( 2004 ). Families of distributions of order statistics . Test 13 ( 1 ): 1 – 43 .[Crossref], [Web of Science ®] , [Google Scholar]). Some properties of the BSN are studied. We pay attention to some generalizations of the skew-normal distribution (Bahrami et al., 2009 Bahrami , W. , Agahi , H. , Rangin , H. ( 2009 ). A two-parameter Balakrishnan skew-normal distribution . J. Statist. Res. Iran 6 : 231 – 242 . [Google Scholar]; Sharafi and Behboodian, 2008 Sharafi , M. , Behboodian , J. ( 2008 ). The Balakrishnan skew-normal density . Statist. Pap. 49 : 769 – 778 .[Crossref], [Web of Science ®] , [Google Scholar]; Yadegari et al., 2008 Yadegari , I. , Gerami , A. , Khaledi , M. J. ( 2008 ). A generalization of the Balakrishnan skew-normal distribution . Statist. Probab. Lett. 78 : 1165 – 1167 .[Crossref], [Web of Science ®] , [Google Scholar]) and to their relations with the BSN.     

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.     

A Class of Improved Heteroskedasticity-Consistent Covariance Matrix Estimators

Abstract

The heteroskedasticity-consistent covariance matrix estimator proposed by White [White, H. A. (1980 White, H. A. 1980. Heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48: 817–838. [Crossref], [Web of Science ®] , [Google Scholar]). Heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48:817–838], also known as HC0, is commonly used in practical applications and is implemented into a number of statistical software. Cribari–Neto et al. [Cribari–Neto, F., Ferrari, S. L. P., Cordeiro, G. M. (2000 Cribari–Neto, F., Ferrari, S. L. P. and Cordeiro, G. M. 2000. Improved heteroscedasticity–consistent covariance matrix estimators. Biometrika, 87: 907–918. [Crossref], [Web of Science ®] , [Google Scholar]). Improved heteroscedasticity–consistent covariance matrix estimators. Biometrika 87:907–918] have developed a bias-adjustment scheme that delivers bias-corrected White estimators. There are several variants of the original White estimator that are also commonly used by practitioners. These include the HC1, HC2, and HC3 estimators, which have proven to have superior small-sample behavior relative to White's estimator. This paper defines a general bias-correction mechamism that can be applied not only to White's estimator, but to variants of this estimator as well, such as HC1, HC2, and HC3. Numerical evidence on the usefulness of the proposed corrections is also presented. Overall, the results favor the sequence of improved HC2 estimators.