Bounds on the efficiency of unbalanced ranked-set sampling

Abstract

Takahasi and Wakimoto (1968 Takahasi, K., and K. Wakimoto. 1968. On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the Institute of Statistical Mathematics 20:1–31.[Crossref], [Web of Science ®] , [Google Scholar]) derived a sharp upper bound on the efficiency of the balanced ranked-set sampling (RSS) sample mean relative to the simple random sampling (SRS) sample mean under perfect rankings. The bound depends on the set size and is achieved for uniform distributions. Here we generalize the Takahasi and Wakimoto (1968 Takahasi, K., and K. Wakimoto. 1968. On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the Institute of Statistical Mathematics 20:1–31.[Crossref], [Web of Science ®] , [Google Scholar]) result by finding a sharp upper bound in the case of unbalanced RSS. The bound depends on the particular unbalanced design, and the distributions where the bound is achieved can be highly nonuniform. The bound under perfect rankings can be exceeded under imperfect rankings.     

Interaction Terms in Distance-Based Regression

We propose a method of including polynomial and interaction terms in Distance-Based Regression (Cuadras and Arenas, 1990 Cuadras , C. M. , Arenas , C. ( 1990 ). A distance based regression model for prediction with mixed data . Commun. Statist. A Theor. Meth. 19 : 2261 – 2279 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), relying on properties of a semi-Hadamard or Khatri-Rao product of matrices. We demonstrate its application to real data examples.     

Tests of Multivariate Independence for Ordinal Data

Population and sample versions of Kendall and Spearman measures of association suitable for multivariate ordinal data are defined. The latter generalize the indices of dependence of Ruymgaart and van Zuijlen (1978 Ruymgaart , F. H. , van Zuijlen , M. C. A. ( 1978 ). Asymptotic normality of multivariate linear rank statistics in the non-i.i.d. case . Ann. Statist. 6 : 588 – 602 .[Crossref], [Web of Science ®] , [Google Scholar]), Joe (1990 Joe , H. ( 1990 ). Multivariate concordance . J. Multivariate Anal. 35 : 12 – 30 .[Crossref], [Web of Science ®] , [Google Scholar]), and Schmid and Schmidt (2007 Schmid , F. , Schmidt , R. ( 2007 ). Multivariate extensions of Spearman's rho and related statistics . Statist. Probab. Lett. 77 : 407 – 416 .[Crossref], [Web of Science ®] , [Google Scholar]) by allowing atoms in the underlying distribution. The representation of the proposed empirical measures as U-statistics enables to establish their asymptotic normality under general distributions. A special attention is given to tests of independence for multivariate ordinal data, where the power of the new methodologies are investigated under fixed and contiguous alternatives.     

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.     

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.     

Least Median of Squares Solution of Multiple Linear Regression Models Through the Origin

Barreto and Maharry (2006 Barreto , H. , Maharry , D. ( 2006 ). Least median of squares and regression through the origin . Comput. Statist. Data Anal. 50 : 1391 – 1397 .[Crossref], [Web of Science ®] , [Google Scholar]) showed that PROGRESS algorithm fails to find a correct minimum “Least Median of Squares/LMS” estimate for bivariate regression models which have no intercept. Kayhan and Gunay (2008 Kayhan , Y. , Gunay , S. ( 2008 ). A new approach to least median of squares and regression through the origin . Commun. Statist. Theor. Meth. 37 ( 5 ): 773 – 781 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) presented a different approach for the regression models through the origin which includes at most two unknown parameters. However, LMS estimate for multiple linear regression models still remains an open issue. The aim of this study is to show that finding true LMS estimate for zero intercept multiple linear regression models can be treated as a convex optimization problem and to provide a more general algorithm for any dimensional linear regression models.     

Calibration of Proportional Hazards and Accelerated Failure Time Models

Given a prognostic model based on one population, one may ask: Can this model be used to accurately predict disease in a different population? When the underlying rate of disease differs in the new population, the model must be calibrated. van Houwelingen (2000 van Houwelingen , H. ( 2000 ). Validation, calibration, revision and combination of prognostic survival models . Statistics in Medicine 19 : 3401 – 3415 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) considered this calibration problem focusing on proportional hazards models. We extend the validation by calibration to the log-logistic accelerated failure time model. We use calibration of proportional hazards models and log-logistic accelerated failure time models to examine whether a survival model based on the Framingham Heart Study can be applied to diverse studies around the world.     

Bayesian Inference for the Precision Matrix for Scale Mixtures of Normal Distributions

Baysian inference is considered for the precision matrix of the multivariate regression model with distribution of the random responses belonging to the multivariate scale mixtures of normal distributions. The posterior distribution and some identities involving expectations taken with respect to this posterior distribution are derived when the prior distribution of the parameters is from the conjugate family. The results are specialized to the case where the random responses have a matrix-t distribution and thus generalizing the results of Zellner (1976 Zellner , A. ( 1976 ). Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error terms . J. Amer. Statist. Assoc. 71 : 400 – 405 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Muirhead (1986 Muirhead , R. J. ( 1986 ). A note on some Wishart expectations . Metrika 33 : 247 – 251 .[Crossref] , [Google Scholar]).     

Covariate-Adjusted Partially Linear Regression Models

Motivated by covariate-adjusted regression (CAR) proposed by Sentürk and Müller (2005 Sentürk , D. , Müller , H. G. ( 2005 ). Covariate-adjusted regression . Biometrika 92 : 75 – 89 .[Crossref], [Web of Science ®] , [Google Scholar]) and an application problem, in this article we introduce and investigate a covariate-adjusted partially linear regression model (CAPLM), in which both response and predictor vector can only be observed after being distorted by some multiplicative factors, and an additional variable such as age or period is taken into account. Although our model seems to be a special case of covariate-adjusted varying coefficient model (CAVCM) given by Sentürk (2006 Sentürk , D. ( 2006 ). Covariate-adjusted varying coefficient models . Biostatistics 7 : 235 – 251 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), the data types of CAPLM and CAVCM are basically different and then the methods for inferring the two models are different. In this article, the estimate method motivated by Cui et al. (2008 Cui , X. , Guo , W. S. , Lin , L. , Zhu , L. X. ( 2008 ). Covariate-adjusted nonlinear regression . Ann. Statist. 37 : 1839 – 1870 . [Google Scholar]) is employed to infer the new model. Furthermore, under some mild conditions, the asymptotic normality of estimator for the parametric component is obtained. Combined with the consistent estimate of asymptotic covariance, we obtain confidence intervals for the regression coefficients. Also, some simulations and a real data analysis are made to illustrate the new model and methods.     

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.     

Bayesian Capture-Recapture Analysis: An Application in Modeling Semelparity of a Neotropical Didelphid Marsupial

We develop a series of Bayesian statistical models for estimating survival of a neotropic didelphid marsupial, the Brazilian gracile mouse opossum (Gracilinanus microtarsus). These models are based on the Cormack–Jolly–Seber model (Cormack, 1964 Cormack , R. M. ( 1964 ). Estimates of survival from the sighting of marked animals . Biometrika 51 : 429 – 438 .[Crossref], [Web of Science ®] , [Google Scholar]; Jolly 1965 Jolly , G. M. ( 1965 ). Explicit estimates from capture-recapture data with both death and immigration stochastic model . Biometrika 52 : 225 – 247 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Seber 1965 Seber , G. A. F. ( 1965 ). A note on the multiple recapture census . Biometrika 52 : 249 – 259 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) with both survival and recapture rates expressed as a function of covariates using a logit link. The proposed models allow taking into account heterogeneity in capture probability caused by the existence of different groups of individuals in the population. The models were applied to two cohorts (Cohort, 2000, 2001) with the first one including 14 and the second one 15 sampling occasions. The best models for each of the cohorts indicate that G. microtarsus is best described as partially semelparous, a condition in which mortality after the first mating is high but graded over time, with a fraction of males surviving for a second breeding season (Boonstra, 2005 Boonstra , R. ( 2005 ). Equipped for life: the adaptive role of the stress axis in male mammals . Journal of Mammalogy 86 : 236 – 247 .[Crossref], [Web of Science ®] , [Google Scholar]).     

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.     

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.     

A Broad Class of Partially Specified Autoregressions on Multi-Casting Data

Extending the bifurcating autoregressive (BAR) process (cf. Cowan and Staudte, 1986 Cowan , R. , Staudte , R. G. ( 1986 ). The bifurcating autoregression model in cell lineage studies . Biometrics 42 : 769 – 783 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to multi-casting (multi-splitting) data, Hwang and Choi (2009 Hwang , S. Y. , Choi , M. S. ( 2009 ). Modeling and large sample estimation for multi-casting autoregression . Statist. Prob. Lett. 79 : 1943 – 1950 .[Crossref], [Web of Science ®] , [Google Scholar]) introduced multi-casting autoregression (MCAR, for short) defined on multi-casting tree structured data. This article is concerned with the case when the MCAR model is partially specified only through conditional mean and variance without directly imposing autoregressive (AR) structure. The resulting class of models will be referred to as P-MCAR (partially specified MCAR). The P-MCAR considerably enlarges the class of multi-casting models including (as special cases) MCAR, random coefficient MCAR, conditionally heteroscedastic multi-casting models and binomial-thinning processes. Moment structures for this broad P-MCAR class are investigated. Least squares (LS) estimation method is discussed and asymptotic relative efficiency (ARE) of the generalized-LS over ordinary-LS is obtained in a closed form. A simulation study is conducted to illustrate results.     

Tuning Parameter Selection and Various Good Fitting Characteristics for the Liu-Type Estimator in Linear Regression

Liu (2003 Liu , K. ( 2003 ). Using Liu-Type estimator to combat collinearity . Commun. Statist. Theor. Meth. 32 ( 5 ): 1009 – 1020 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) proposed the Liu-Type estimator (LTE) to combat the well-known multicollinearity problem in linear regression. In this article, various better fitting characteristics of the LTE than those of the ordinary ridge regression estimator (Hoerl and Kennard, 1970 Hoerl , A. E. , Kennard , R. W. ( 1970 ). Ridge regression: Biased estimation for non-orthogonal problems . Technometrics 12 : 55 – 67 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are considered. In particular, we derived two methods to determine the parameter d for the LTE and find that the ridge parameter k could serve for regularization of an ill-conditioned design matrix, while the other parameter d could be used for tuning the fit quality. In addition, the coefficients of regression, coefficient of multiple determination, residual error variance, and generalized cross validation (GCV) of the prediction quality are very stable, and as the ridge parameter increases they eventually reach asymptotic levels, which produces robust regression models. Furthermore, a Monte Carlo evaluation of these features is also given to illustrate some of the theoretical results.     

The Alpha-power Tobit Model

The main object of this article is to propose an extension of the tobit model for which the error distribution follows the power-normal distribution (Gupta and Gupta, 2008 Gupta , D. , Gupta , R. C. ( 2008 ). Analyzing skewed data by power normal model . Test 17 : 197 – 210 .[Crossref], [Web of Science ®] , [Google Scholar]). Inference is dealt with by using the likelihood approach. Simulation studies and application to a real data set are used to demonstrate the usefulness of the extension.     

On Testing Sample Selection Bias Under the Multicollinearity Problem

ABSTRACT

This paper reviews and extends the literature on the finite sample behavior of tests for sample selection bias. Monte Carlo results show that, when the “multicollinearity problem” identified by Nawata (1993 Nawata , K. ( 1993 ). A note on the estimation of models with sample-selection biases . Economics Letters 42 : 15 – 24 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) is severe, (i) the t-test based on the Heckman–Greene variance estimator can be unreliable, (ii) the Likelihood Ratio test remains powerful, and (iii) nonnormality can be interpreted as severe sample selection bias by Maximum Likelihood methods, leading to negative Wald statistics. We also confirm previous findings (Leung and Yu, 1996 Leung , S. F. , Yu , S. ( 1996 ). On the choice between sample selection and two-part models . Journal of Econometrics 72 : 197 – 229 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) that the standard regression-based t-test (Heckman, 1979 Heckman , J. J. ( 1979 ). Sample selection bias as a specification error . Econometrica 47 : 153 – 161 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) and the asymptotically efficient Lagrange Multiplier test (Melino, 1982 Melino , A. ( 1982 ). Testing for sample selection bias . Review of Economic Studies 49 : 151 – 153 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), are robust to nonnormality but have very little power.     

A Euclidean Likelihood Estimator for Bivariate Tail Dependence

The spectral measure plays a key role in the statistical modeling of multivariate extremes. Estimation of the spectral measure is a complex issue, given the need to obey a certain moment condition. We propose a Euclidean likelihood-based estimator for the spectral measure which is simple and explicitly defined, with its expression being free of Lagrange multipliers. Our estimator is shown to have the same limit distribution as the maximum empirical likelihood estimator of Einmahl and Segers (2009 Einmahl , J. H. J. , Segers , J. ( 2009 ). Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution . Ann. Statist. 37 ( 5B ): 2953 – 2989 .[Crossref], [Web of Science ®] , [Google Scholar]). Numerical experiments suggest an overall good performance and identical behavior to the maximum empirical likelihood estimator. We illustrate the method in an extreme temperature data analysis.     

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.”     

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.