Aalen's linear model for doubly censored data

Double censoring often occurs in registry studies when left censoring is present in addition to right censoring. In this work, we examine estimation of Aalen's nonparametric regression coefficients based on doubly censored data. We propose two estimation techniques. The first type of estimators, including ordinary least squared (OLS) estimator and weighted least squared (WLS) estimators, are obtained using martingale arguments. The second type of estimator, the maximum likelihood estimator (MLE), is obtained via expectation-maximization (EM) algorithms that treat the survival times of left censored observations as missing. Asymptotic properties, including the uniform consistency and weak convergence, are established for the MLE. Simulation results demonstrate that the MLE is more efficient than the OLS and WLS estimators.     

A modified estimator of population mean using power transformation

Summary In this paper we have suggested two modified estimators of population mean using power transformation. It has been shown that the modified estimators are more efficient than the sample mean estimator, usual ratio estimator, Sisodia and Dwivedi’s (1981) estimator and Upadhyaya and Singh’s (1999) estimator at their optimum conditions. Empirical illustrations are also given for examining the merits of the proposed estimators. Following Kadilar and Cingi (2003) the work has been extended to stratified random sampling, and the same data set has been studied to examine the performance in stratified random sampling.     

Distribution estimation with auxiliary information for missing data

There is much literature on statistical inference for distribution under missing data, but surprisingly very little previous attention has been paid to missing data in the context of estimating distribution with auxiliary information. In this article, the auxiliary information with missing data is proposed. We use Zhou, Wan and Wang's method (2008) to mitigate the effects of missing data through a reformulation of the estimating equations, imputed through a semi-parametric procedure. Whence we can estimate distribution and the τth quantile of the distribution by taking auxiliary information into account. Asymptotic properties of the distribution estimator and corresponding sample quantile are derived and analyzed. The distribution estimators based on our method are found to significantly outperform the corresponding estimators without auxiliary information. Some simulation studies are conducted to illustrate the finite sample performance of the proposed estimators.     

A note on the greenwood variance estimator in cohort life tables: Isolating the effect of one source of bias

A modification of the Greenwood variance estimator is defined and shown to be free of bias whenever its constitu­ent interval estimators are conditionally unbiased, given the sample size at the start of the interval. Using the modified estimator as a standard of comparison, the original Greenwood estimator is seen to have an intrinsic positive bias.Under­estimation of variances through the use of Greenwood's formula must be due to bias in the constituent interval estimators and/or, with fixed interval bounds, due to disregarding the random character of the total number of life table intervals to exhaustion of ttje sample. Some easy to prove properties of the modified and the original Greenwood estimators are stated that apply in the absence of censoring. A suggest­ion is made for reducing the bias of the interval variance estimators.     

The small sample performance of some limited information estimators of a dynamic structural equation with autocorrelated errors †

This paper examines the small sample properties of the following seven estimators of a dynamic structural equation with autocorrelated errors: (1) 2SLS; (2) Fair’s modification of Sargan’s 2SLS; (3) the Dhrymes, Berner and Cummins (1974) variant of 2SLS; (4) a modified Theil's (1958) generalized 2SLS; (5) three two-step estimators proposed by Hatanaka (1976). Our principal results are that for low degrees of autocorrelation 2SLS performs well whereas for high degrees of autocorrelation the Theil and Dhrymes estimators are best with two of Hatanaka’s estimators close behind. The Fair and the remaining Hatanaka estimator are always dominated by the others. This is of some practical interest because the Fair estimator is a standard option in some software packages.     

Estimation of population coefficient of variation in simple and stratified random sampling under two-phase sampling scheme when using two auxiliary variables

We propose an improved class of exponential ratio type estimators for coefficient of variation (CV) of a finite population in simple and stratified random sampling using two auxiliary variables under two-phase sampling scheme. We examine the properties of the proposed estimators based on first order of approximation. The proposed class of estimators is more efficient than the usual sample CV estimator, ratio estimator, exponential ratio estimator, usual difference estimator and modified difference type estimator. We also use real data sets for numerical comparisons.     

A ridge-type estimator and good prior means

Swindel (1976) introduced a modified ridge regression estimator based on prior information. A necessary and sufficient condition is derived for Swindel's proposed estimator to have lower risk than the conventional ordinary ridge regression estimator when both estimators are computed using the same value of k.     

A comparison study of nonparametric imputation methods

Consider estimation of a population mean of a response variable when the observations are missing at random with respect to the covariate. Two common approaches to imputing the missing values are the nonparametric regression weighting method and the Horvitz-Thompson (HT) inverse weighting approach. The regression approach includes the kernel regression imputation and the nearest neighbor imputation. The HT approach, employing inverse kernel-estimated weights, includes the basic estimator, the ratio estimator and the estimator using inverse kernel-weighted residuals. Asymptotic normality of the nearest neighbor imputation estimators is derived and compared to kernel regression imputation estimator under standard regularity conditions of the regression function and the missing pattern function. A comprehensive simulation study shows that the basic HT estimator is most sensitive to discontinuity in the missing data patterns, and the nearest neighbors estimators can be insensitive to missing data patterns unbalanced with respect to the distribution of the covariate. Empirical studies show that the nearest neighbor imputation method is most effective among these imputation methods for estimating a finite population mean and for classifying the species of the iris flower data.     

Some General Comparative Points on Chao's and Zelterman's Estimators of the Population Size

Abstract. Two simple and frequently used capture–recapture estimates of the population size are compared: Chao's lower‐bound estimate and Zelterman's estimate allowing for contaminated distributions. In the Poisson case it is shown that if there are only counts of ones and twos, the estimator of Zelterman is always bounded above by Chao's estimator. If counts larger than two exist, the estimator of Zelterman is becoming larger than that of Chao's, if only the ratio of the frequencies of counts of twos and ones is small enough. A similar analysis is provided for the binomial case. For a two‐component mixture of Poisson distributions the asymptotic bias of both estimators is derived and it is shown that the Zelterman estimator can experience large overestimation bias. A modified Zelterman estimator is suggested and also the bias‐corrected version of Chao's estimator is considered. All four estimators are compared in a simulation study.     

Consistent estimators of the variance-covariance matrix of the gmanova model with missing data

A common problem in multivariate general linear models is partially missing response data. The simplest method of analysis in the presence of missing data has been to delete all observations on any individual with any missing data(listwise deletion) and utilize a traditional complete data approach. However: this can result in a great loss of information: and perhaps inconsistencies in the estimation of the variance-covariance matrix. In the generalized multivariate analysis of variance(GMANOVA) model with missing data: Kleinbaum(1973) proposed an estimated generalized least squares approach. In order to apply this: however: a consistent estimate of the variance-covariance matrix is needed. Kleinbaum proposed an estimator which is unbiased and consistent: but it does not take advantage of the fact that the underlying model is GMANOVA and not MANOVA. Using the fact that the underlying model is GMANOVA we have constructed four other con¬sistent estimators. A Monte Carlo simulation experiment is conducted tto further examine how well these estimators compare to the estimator proposed by Kleinbaum.     

Estimating variances of strata in ranked set sampling

In RSS, the variance of observations in each ranked set plays an important role in finding an optimal design for unbalanced RSS and in inferring the population mean. The empirical estimator (i.e., the sample variance in a given ranked set) is most commonly used for estimating the variance in the literature. However, the empirical estimator does not use the information in the entire data over different ranked sets. Further, it is highly variable when the sample size is not large enough, as is typical in RSS applications. In this paper, we propose a plug-in estimator for the variance of each set, which is more efficient than the empirical one. The estimator uses a result in order statistics which characterizes the cumulative distribution function (CDF) of the rth order statistics as a function of the population CDF. We analytically prove the asymptotic normality of the proposed estimator. We further apply it to estimate the standard error of the RSS mean estimator. Both our simulation and empirical study show that our estimators consistently outperform existing methods.     

Amputation versus imputation of missing values through ratio method in sample surveys

In this article, we consider the estimation of a population mean when some observations on the study characteristic are missing in the bivariate sample data. In all, five estimators are presented and their efficiency properties are discussed. One estimator arises from the the amputation of incomplete observations while the remaining four estimators are formulated using imputed values obtained by the ratio method of estimation. This work was carried out before Professor V.K. Srivastava passed away in 2001.     

A Class of Estimators of Population Variance Using Auxiliary Information in Sample Surveys

This article advocates the problem of estimating the population variance of the study variable using information on certain known parameters of an auxiliary variable. A class of estimators for population variance using information on an auxiliary variable has been defined. In addition to many estimators, usual unbiased estimator, Isaki's (1983), Upadhyaya and Singh's (1999), and Kadilar and Cingi's (2006) estimators are shown as members of the proposed class of estimators. Asymptotic expressions for bias and mean square error of the proposed class of estimators have been obtained. An empirical study has been carried out to judge the performance of the various estimators of population variance generated from the proposed class of estimators over usual unbiased estimator, Isaki's (1983), Upadhyaya and Singh's (1999) and Kadilar and Cingi's (2006) estimators.     

Performance of nonparametric maximum likelihood estimator in a

When the probability of selecting an individual in a population is propor­tional to its lifelength, it is called length biased sampling. A nonparametric maximum likelihood estimator (NPMLE) of survival in a length biased sam­ple is given in Vardi (1982). In this study, we examine the performance of Vardi's NPMLE in estimating the true survival curve when observations are from a length biased sample. We also compute estimators based on a linear combination (LCE) of empirical distribution function (EDF) estimators and weighted estimators. In our simulations, we consider observations from a mix­ture of two different distributions, one from F and the other from G which is a length biased distribution of F. Through a series of simulations with vari­ous proportions of length biasing in a sample, we show that the NPMLE and the LCE closely approximate the true survival curve. Throughout the sur­vival curve, the EDF estimators overestimate the survival. We also consider a case where the observations are from three different weighted distributions, Again, both the NPMLE and the LCE closely approximate the true distribu­tion, indicating that the length biasedness is properly adjusted for. Finally, an efficiency study shows that Vardi's estimators are more efficient than the EDF estimators in the lower percentiles of the survival curves.     

The stochastic approximation method for the estimation of a multivariate probability density

We apply the stochastic approximation method to construct a large class of recursive kernel estimators of a probability density, including the one introduced by Hall and Patil [1994. On the efficiency of on-line density estimators. IEEE Trans. Inform. Theory 40, 1504–1512]. We study the properties of these estimators and compare them with Rosenblatt's nonrecursive estimator. It turns out that, for pointwise estimation, it is preferable to use the nonrecursive Rosenblatt's kernel estimator rather than any recursive estimator. A contrario, for estimation by confidence intervals, it is better to use a recursive estimator rather than Rosenblatt's estimator.     

Usefulness of robust estimators in sample survey

It Is illustrated that the adaptation of robust estimators, Tiku's (1967, 1978, 1980) modified maximum likelihood estimators for example, in sample survey can lead to considerable improvements In efficiencies in estimating the mean of a finite population (a random sample of Independent observations from a super-popularlon). In this paper, however, we confine our attention to symmetric super-populations     

Simulated deficiencies of robust estimators of a proportionality constant

In linear regression, robust methods are at the beginning of their use in practice. In the small sample case, such robust methods provide a necessary measure of protection against deviations from the assumed error distribution. This paper studies through simulation the deficiencies of bioptimal estimators and compares them with more common methods like Huber's estimator or Tukey's estimator. Polyoptimal estimators are convex combinations of Pitman estimators and are optimally robust for a confrontation containing several shapes. The word confrontation is due to J.W. Tukey. It expresses the situation when compromising two or several error distributions. The paper uses the confrontation containing the Gaussian distribution along with a symmetric heavy-tailed distribution having a tail of order 0(t^-2) as t→ ±∞.     

Wavelet estimation of density for censored data with censoring indicator missing at random

In this paper, we define the nonlinear wavelet estimator of density for the right censoring model with the censoring indicator missing at random (MAR), and develop its asymptotic expression for mean integrated squared error (MISE). Unlike for kernel estimator, the MISE expression of the estimator is not affected by the presence of discontinuities in the curve. Meanwhile, asymptotic normality of the estimator is established. The proposed estimator can reduce to the estimator defined by Li [Non-linear wavelet-based density estimators under random censorship. J Statist Plann Inference. 2003;117(1):35–58] when the censoring indicator MAR does not occur and a bandwidth in non-parametric estimation is close to zero. Also, we define another two nonlinear wavelet estimators of the density. A simulation is done to show the performance of the three proposed estimators.     

Unbiased Estimation of the Distribution Function of an Exponential Population Using Order Statistics with Application in Ranked Set Sampling

In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.     

On bias reduction estimators of skew-normal and skew-t distributions

A particular concerns of researchers in statistical inference is bias in parameters estimation. Maximum likelihood estimators are often biased and for small sample size, the first order bias of them can be large and so it may influence the efficiency of the estimator. There are different methods for reduction of this bias. In this paper, we proposed a modified maximum likelihood estimator for the shape parameter of two popular skew distributions, namely skew-normal and skew-t, by offering a new method. We show that this estimator has lower asymptotic bias than the maximum likelihood estimator and is more efficient than those based on the existing methods.