An alternative stochastic restricted Liu estimator in linear regression

In this paper, we introduce an alternative stochastic restricted Liu estimator for the vector of parameters in a linear regression model when additional stochastic linear restrictions on the parameter vector are assumed to hold. The new estimator is a generalization of the ordinary mixed estimator (OME) (Durbin in J Am Stat Assoc 48:799–808, 1953; Theil and Goldberger in Int Econ Rev 2:65–78, 1961; Theil in J Am Stat Assoc 58:401–414, 1963) and Liu estimator proposed by Liu (Commun Stat Theory Methods 22:393–402, 1993). Necessary and sufficient conditions for the superiority of the new stochastic restricted Liu estimator over the OME, the Liu estimator and the estimator proposed by Hubert and Wijekoon (Stat Pap 47:471–479, 2006) in the mean squared error matrix (MSEM) sense are derived. Furthermore, a numerical example based on the widely analysed dataset on Portland cement (Woods et al. in Ind Eng Chem 24:1207–1241, 1932) and a Monte Carlo evaluation of the estimators are also given to illustrate some of the theoretical results.     

Ratio estimator for the population mean using ranked set sampling

We adapt the ratio estimation using ranked set sampling, suggested by Samawi and Muttlak (Biometr J 38:753–764, 1996), to the ratio estimator for the population mean, based on Prasad (Commun Stat Theory Methods 18:379–392, 1989), in simple random sampling. Theoretically, we show that the proposed ratio estimator for the population mean is more efficient than the ratio estimator, in Prasad (1989), in all conditions. In addition, we support this theoretical result with the aid of a numerical example.      

A partially adaptive estimator for the censored regression model based on a mixture of normal distributions

The goal of this paper is to introduce a partially adaptive estimator for the censored regression model based on an error structure described by a mixture of two normal distributions. The model we introduce is easily estimated by maximum likelihood using an EM algorithm adapted from the work of Bartolucci and Scaccia (Comput Stat Data Anal 48:821–834, 2005). A Monte Carlo study is conducted to compare the small sample properties of this estimator to the performance of some common alternative estimators of censored regression models including the usual tobit model, the CLAD estimator of Powell (J Econom 25:303–325, 1984), and the STLS estimator of Powell (Econometrica 54:1435–1460, 1986). In terms of RMSE, our partially adaptive estimator performed well. The partially adaptive estimator is applied to data on wife’s hours worked from Mroz (1987). In this application we find support for the partially adaptive estimator over the usual tobit model.     

Variable selection for recurrent event data via nonconcave penalized estimating function

Variable selection is an important issue in all regression analysis and in this paper, we discuss this in the context of regression analysis of recurrent event data. Recurrent event data often occur in long-term studies in which individuals may experience the events of interest more than once and their analysis has recently attracted a great deal of attention (Andersen et al., Statistical models based on counting processes, 1993; Cook and Lawless, Biometrics 52:1311–1323, 1996, The analysis of recurrent event data, 2007; Cook et al., Biometrics 52:557–571, 1996; Lawless and Nadeau, Technometrics 37:158-168, 1995; Lin et al., J R Stat Soc B 69:711–730, 2000). However, it seems that there are no established approaches to the variable selection with respect to recurrent event data. For the problem, we adopt the idea behind the nonconcave penalized likelihood approach proposed in Fan and Li (J Am Stat Assoc 96:1348–1360, 2001) and develop a nonconcave penalized estimating function approach. The proposed approach selects variables and estimates regression coefficients simultaneously and an algorithm is presented for this process. We show that the proposed approach performs as well as the oracle procedure in that it yields the estimates as if the correct submodel was known. Simulation studies are conducted for assessing the performance of the proposed approach and suggest that it works well for practical situations. The proposed methodology is illustrated by using the data from a chronic granulomatous disease study.     

The Minimum Density Power Divergence Estimation for the Lognormal Density

In this article, we implement the minimum density power divergence estimation for estimating the parameters of the lognormal density. We compare the minimum density power divergence estimator (MDPDE) and the maximum likelihood estimator (MLE) in terms of robustness and asymptotic distribution. The simulations and an example indicate that the MDPDE is less biased than MLE and is as good as MLE in terms of the mean square error under various distributional situations.     

Robust second-order least-squares estimator for regression models

The second-order least-squares estimator (SLSE) was proposed by Wang (Statistica Sinica 13:1201–1210, 2003) for measurement error models. It was extended and applied to linear and nonlinear regression models by Abarin and Wang (Far East J Theor Stat 20:179–196, 2006) and Wang and Leblanc (Ann Inst Stat Math 60:883–900, 2008). The SLSE is asymptotically more efficient than the ordinary least-squares estimator if the error distribution has a nonzero third moment. However, it lacks robustness against outliers in the data. In this paper, we propose a robust second-order least squares estimator (RSLSE) against X-outliers. The RSLSE is highly efficient with high breakdown point and is asymptotically normally distributed. We compare the RSLSE with other estimators through a simulation study. Our results show that the RSLSE performs very well.     

A generalization of Turnbull’s estimator for nonparametric estimation of the conditional survival function with interval-censored data

Simple nonparametric estimates of the conditional distribution of a response variable given a covariate are often useful for data exploration purposes or to help with the specification or validation of a parametric or semi-parametric regression model. In this paper we propose such an estimator in the case where the response variable is interval-censored and the covariate is continuous. Our approach consists in adding weights that depend on the covariate value in the self-consistency equation proposed by Turnbull (J R Stat Soc Ser B 38:290–295, 1976), which results in an estimator that is no more difficult to implement than Turnbull’s estimator itself. We show the convergence of our algorithm and that our estimator reduces to the generalized Kaplan–Meier estimator (Beran, Nonparametric regression with randomly censored survival data, 1981) when the data are either complete or right-censored. We demonstrate by simulation that the estimator, bootstrap variance estimation and bandwidth selection (by rule of thumb or cross-validation) all perform well in finite samples. We illustrate the method by applying it to a dataset from a study on the incidence of HIV in a group of female sex workers from Kinshasa.     

A note on GMM estimation of probit models with endogenous regressors

Dagenais in (Econ Lett 63:19–21, 1999) and Lucchetti in (Econ Lett 75:179–185, 2002) have demonstrated that the naive GMM estimator of Grogger in (Econ Lett 33:329–332, 1990) for the probit model with an endogenous regressor is not consistent. This paper completes their discussion by explaining the reason for the inconsistency and presenting a natural solution. Furthermore, the resulting GMM estimator is analyzed in a Monte-Carlo simulation and compared with alternative estimators.     

On the Breslow–Holubkov estimator

Breslow and Holubkov (J Roy Stat Soc B 59:447–461 1997a) developed semiparametric maximum likelihood estimation for two-phase studies with a case–control first phase under a logistic regression model and noted that, apart for the overall intercept term, it was the same as the semiparametric estimator for two-phase studies with a prospective first phase developed in Scott and Wild (Biometrica 84:57–71 1997). In this paper we extend the Breslow–Holubkov result to general binary regression models and show that it has a very simple relationship with its prospective first-phase counterpart. We also explore why the design of the first phase only affects the intercept of a logistic model, simplify the calculation of standard errors, establish the semiparametric efficiency of the Breslow–Holubkov estimator and derive its asymptotic distribution in the general case.     

On MSE of EBLUP

We consider Best Linear Unbiased Predictors (BLUPs) and Empirical Best Linear Unbiased Predictors (EBLUPs) under the general mixed linear model. The BLUP was proposed by Henderson (Ann Math Stat 21:309–310, 1950). The formula of this BLUP includes unknown elements of the variance-covariance matrix of random variables. If the elements in the formula of the BLUP proposed by Henderson (Ann Math Stat 21:309–310, 1950) are replaced by some type of estimators, we obtain the two-stage predictor called the EBLUP which is model-unbiased (Kackar and Harville in Commun Stat A 10:1249–1261, 1981). Kackar and Harville (J Am Stat Assoc 79:853–862, 1984) show an approximation of the mean square error (the MSE) of the predictor and propose an estimator of the MSE. The MSE and estimators of the MSE are also studied by Prasad and Rao (J Am Stat Assoc 85:163–171, 1990), Datta and Lahiri (Stat Sin 10:613–627, 2000) and Das et al. (Ann Stat 32(2):818–840, 2004). In the paper we consider the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976. Ża̧dło (On unbiasedness of some EBLU predictor. Physica-Verlag, Heidelberg, pp 2019–2026, 2004) shows that the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976) may be treated as a generalisation of the BLUP proposed by Henderson (Ann Math Stat 21:309–310, 1950) and proves model unbiasedness of the EBLUP based on the formula of the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976) under some assumptions. In this paper we derive the formula of the approximate MSE of the EBLUP and its estimators. We prove that the approximation of the MSE is accurate to terms o(D ⁻¹) and that the estimator of the MSE is approximately unbiased in the sense that its bias is o(D ⁻¹) under some assumptions, where D is the number of domains. The proof is based on the results obtained by Datta and Lahiri (Stat Sin 10:613–627, 2000). Using our results we show some EBLUP based on the special case of the general linear model. We also present the formula of its MSE and estimators of its MSE and their performance in Monte Carlo simulation study.      

Efficiency improvement in a class of survival models through model-free covariate incorporation

In randomized clinical trials, we are often concerned with comparing two-sample survival data. Although the log-rank test is usually suitable for this purpose, it may result in substantial power loss when the two groups have nonproportional hazards. In a more general class of survival models of Yang and Prentice (Biometrika 92:1–17, 2005), which includes the log-rank test as a special case, we improve model efficiency by incorporating auxiliary covariates that are correlated with the survival times. In a model-free form, we augment the estimating equation with auxiliary covariates, and establish the efficiency improvement using the semiparametric theories in Zhang et al. (Biometrics 64:707–715, 2008) and Lu and Tsiatis (Biometrics, 95:674–679, 2008). Under minimal assumptions, our approach produces an unbiased, asymptotically normal estimator with additional efficiency gain. Simulation studies and an application to a leukemia study show the satisfactory performance of the proposed method.     

On an exponential bound for the Kaplan–Meier estimator

We review limit theory and inequalities for the Kaplan–Meier Kaplan and Meier (J Am Stat Assoc 53:457–481, 1958) product limit estimator of a survival function on the whole line . Along the way we provide bounds for the constant in an interesting inequality due to Biotouzé et al. (Ann Inst H Poincaré Probab Stat 35:735–763, 1999), and provide some numerical evidence in support of one of their conjectures. Supported in part by NSF grant DMS-0503822 and by NI-AID grant 2R01 AI291968-04.     

Estimating a sensitive proportion through randomized response procedures based on auxiliary information

Randomized response techniques are widely employed in surveys dealing with sensitive questions to ensure interviewee anonymity and reduce nonrespondents rates and biased responses. Since Warner’s (J Am Stat Assoc 60:63–69, 1965) pioneering work, many ingenious devices have been suggested to increase respondent’s privacy protection and to better estimate the proportion of people, π _A , bearing a sensitive attribute. In spite of the massive use of auxiliary information in the estimation of non-sensitive parameters, very few attempts have been made to improve randomization strategy performance when auxiliary variables are available. Moving from Zaizai’s (Model Assist Stat Appl 1:125–130, 2006) recent work, in this paper we provide a class of estimators for π _A , for a generic randomization scheme, when the mean of a supplementary non-sensitive variable is known. The minimum attainable variance bound of the class is obtained and the best estimator is also identified. We prove that the best estimator acts as a regression-type estimator which is at least as efficient as the corresponding estimator evaluated without allowing for the auxiliary variable. The general results are then applied to Warner and Simmons’ model.     

Spectral estimation for locally stationary time series with missing observations

Time series arising in practice often have an inherently irregular sampling structure or missing values, that can arise for example due to a faulty measuring device or complex time-dependent nature. Spectral decomposition of time series is a traditionally useful tool for data variability analysis. However, existing methods for spectral estimation often assume a regularly-sampled time series, or require modifications to cope with irregular or ‘gappy’ data. Additionally, many techniques also assume that the time series are stationary, which in the majority of cases is demonstrably not appropriate. This article addresses the topic of spectral estimation of a non-stationary time series sampled with missing data. The time series is modelled as a locally stationary wavelet process in the sense introduced by Nason et al. (J. R. Stat. Soc. B 62(2):271–292, 2000) and its realization is assumed to feature missing observations. Our work proposes an estimator (the periodogram) for the process wavelet spectrum, which copes with the missing data whilst relaxing the strong assumption of stationarity. At the centre of our construction are second generation wavelets built by means of the lifting scheme (Sweldens, Wavelet Applications in Signal and Image Processing III, Proc. SPIE, vol. 2569, pp. 68–79, 1995), designed to cope with irregular data. We investigate the theoretical properties of our proposed periodogram, and show that it can be smoothed to produce a bias-corrected spectral estimate by adopting a penalized least squares criterion. We demonstrate our method with real data and simulated examples.     

More Powerful Control of the False Discovery Rate Under Dependence

In a breakthrough paper, Benjamini and Hochberg (J Roy Stat Soc Ser B 57:289–300, 1995) proposed a new error measure for multiple testing, the FDR; and developed a distribution-free procedure to control it under independence among the test statistics. In this paper we argue by extensive simulation and theoretical considerations that the assumption of independence is not needed. Along the lines of (Ann Stat 32:1035–1061, 2004b), we moreover provide a more powerful method, that exploits an estimator of the number of false nulls among the tests. We propose a whole family of iterative estimators that prove robust under dependence and independence between the test statistics. These estimators can be used to improve also classical multiple testing procedures, and in general to estimate the weight of a known component in a mixture distribution. Innovations are illustrated by simulations.     

Estimators in capture–recapture studies with two sources

This paper investigates the applications of capture–recapture methods to human populations. Capture–recapture methods are commonly used in estimating the size of wildlife populations but can also be used in epidemiology and social sciences, for estimating prevalence of a particular disease or the size of the homeless population in a certain area. Here we focus on estimating the prevalence of infectious diseases. Several estimators of population size are considered: the Lincoln–Petersen estimator and its modified version, the Chapman estimator, Chao’s lower bound estimator, the Zelterman’s estimator, McKendrick’s moment estimator and the maximum likelihood estimator. In order to evaluate these estimators, they are applied to real, three-source, capture-recapture data. By conditioning on each of the sources of three source data, we have been able to compare the estimators with the true value that they are estimating. The Chapman and Chao estimators were compared in terms of their relative bias. A variance formula derived through conditioning is suggested for Chao’s estimator, and normal 95% confidence intervals are calculated for this and the Chapman estimator. We then compare the coverage of the respective confidence intervals. Furthermore, a simulation study is included to compare Chao’s and Chapman’s estimator. Results indicate that Chao’s estimator is less biased than Chapman’s estimator unless both sources are independent. Chao’s estimator has also the smaller mean squared error. Finally, the implications and limitations of the above methods are discussed, with suggestions for further development. We are grateful to the Medical Research Council for supporting this work.     

Sampling design proportional to order statistic of auxiliary variable

The sampling designs dependent on sample moments of auxiliary variables are well known. Lahiri (Bull Int Stat Inst 33:133–140, 1951) considered a sampling design proportionate to a sample mean of an auxiliary variable. Sing and Srivastava (Biometrika 67(1):205–209, 1980) proposed the sampling design proportionate to a sample variance while Wywiał (J Indian Stat Assoc 37:73–87, 1999) a sampling design proportionate to a sample generalized variance of auxiliary variables. Some other sampling designs dependent on moments of an auxiliary variable were considered e.g. in Wywiał (Some contributions to multivariate methods in, survey sampling. Katowice University of Economics, Katowice, 2003a); Stat Transit 4(5):779–798, 2000) where accuracy of some sampling strategies were compared, too.These sampling designs cannot be useful in the case when there are some censored observations of the auxiliary variable. Moreover, they can be much too sensitive to outliers observations. In these cases the sampling design proportionate to the order statistic of an auxiliary variable can be more useful. That is why such an unequal probability sampling design is proposed here. Its particular cases as well as its conditional version are considered, too. The sampling scheme implementing this sampling design is proposed. The inclusion probabilities of the first and second orders were evaluated. The well known Horvitz–Thompson estimator is taken into account. A ratio estimator dependent on an order statistic is constructed. It is similar to the well known ratio estimator based on the population and sample means. Moreover, it is an unbiased estimator of the population mean when the sample is drawn according to the proposed sampling design dependent on the appropriate order statistic.     

On pseudo-values for regression analysis in competing risks models

For regression on state and transition probabilities in multi-state models Andersen et al. (Biometrika 90:15–27, 2003) propose a technique based on jackknife pseudo-values. In this article we analyze the pseudo-values suggested for competing risks models and prove some conjectures regarding their asymptotics (Klein and Andersen, Biometrics 61:223–229, 2005). The key is a second order von Mises expansion of the Aalen-Johansen estimator which yields an appropriate representation of the pseudo-values. The method is illustrated with data from a clinical study on total joint replacement. In the application we consider for comparison the estimates obtained with the Fine and Gray approach (J Am Stat Assoc 94:496–509, 1999) and also time-dependent solutions of pseudo-value regression equations.     

Estimation of the scale parameter of the half-logistic distribution under progressively type II censored sample

Based on a progressively type II censored sample, the maximum likelihood and Bayes estimators of the scale parameter of the half-logistic distribution are derived. However, since the maximum likelihood estimator (MLE) and Bayes estimator do not exist in an explicit form for the scale parameter, we consider a simple method of deriving an explicit estimator by approximating the likelihood function and derive the asymptotic variances of MLE and approximate MLE. Also, an approximation based on the Laplace approximation (Tierney and Kadane in J Am Stat Assoc 81:82–86, 1986) and importance sampling methods are used for obtaining the Bayes estimator. In order to compare the performance of the MLE, approximate MLE and Bayes estimates of the scale parameter, we use Monte Carlo simulation.     

Assessing the contribution of R&;D to total factor productivity—a Bayesian approach to account for heterogeneity and heteroskedasticity

This paper proposes a hierarchical Bayes estimator for a panel data random coefficient model with heteroskedasticity to assess the contribution of R&D capital to total factor productivity. Based on Hall (1993) data for 323 US firms over 1976–1990, we find that there appear to have substantial unobserved heterogeneity and heteroskedasticity across firms and industries that support the use of our Bayes inference procedure. We find much higher returns to R&D capital and a more pronounced downswing for the 1981–1985 period, followed by a more pronounced upswing than those yielded by the conventional feasible generalized least squares estimators or other estimates. The estimated elasticities of R&D capital are 0.062 for 1976–1980, 0.036 for 1981–1985 and 0.081 for 1986–1990, while the estimated elasticities of ordinary capital are much more stable over these periods.