Shrinkage and Pretest Nonparametric Estimation of Regression Parameters from Censored Data with Multiple Observations at Each Level of Covariate

A simple linear regression model is considered where the independent variable assumes only a finite number of values and the response variable is randomly right censored. However, the censoring distribution may depend on the covariate values. A class of noniterative estimators for the slope parameter, namely, the noniterative unrestricted estimator, noniterative restricted estimator and noniterative improved pretest estimator are proposed. The asymptotic bias and mean squared errors of the proposed estimators are derived and compared. The relative dominance picture of the estimators is investigated. A simulation study is also performed to asses the properties of the various estimators for small samples.     

Improved Generalized Ratio-Cum-Product Estimation

In this paper, a generalized ratio-cum-product estimator for estimating the ratio (product) of two population means using auxiliary information on two other variables is given of which the estimators by SINGH (1969) and SHAH and SHAH (1978) are particular cases. The estimator is regeneralized when the covariance between two auxiliary variables is known.     

利用多重抽样估计稀有物种的种类数

利用矩估计和二个稳健估计方法（jackknife估计,bootstrap估计）来处理野外生态学工作者的调查数据,在假定已经发现一些稀有物种的情形下,通过统计推断得到那些未被发现的物种的种类数。利用本文所提出的方法调查水稻水稻田的昆虫群落和林地的在面植被群落的稀有种是十分有效的。     

A comparison of two bias-corrected covariance estimators for generalized estimating equations

Mancl and DeRouen (2001, Biometrics57, 126-134) and Kauermann and Carroll (2001, JASA96, 1387-1398) proposed alternative bias-corrected covariance estimators for generalized estimating equations parameter estimates of regression models for marginal means. The finite sample properties of these estimators are compared to those of the uncorrected sandwich estimator that underestimates variances in small samples. Although the formula of Mancl and DeRouen generally overestimates variances, it often leads to coverage of 95% confidence intervals near the nominal level even in some situations with as few as 10 clusters. An explanation for these seemingly contradictory results is that the tendency to undercoverage resulting from the substantial variability of sandwich estimators counteracts the impact of overcorrecting the bias. However, these positive results do not generally hold; for small cluster sizes (e.g., <10) their estimator often results in overcoverage, and the bias-corrected covariance estimator of Kauermann and Carroll may be preferred. The methods are illustrated using data from a nested cross-sectional cluster intervention trial on reducing underage drinking.     

Cross-trait prediction accuracy of summary statistics in genome-wide association studies

In the era of big data, univariate models have widely been used as a workhorse tool for quickly producing marginal estimators; and this is true even when in a high-dimensional dense setting, in which many features are “true,” but weak signals. Genome-wide association studies (GWAS) epitomize this type of setting. Although the GWAS marginal estimator is popular, it has long been criticized for ignoring the correlation structure of genetic variants (i.e., the linkage disequilibrium [LD] pattern). In this paper, we study the effects of LD pattern on the GWAS marginal estimator and investigate whether or not additionally accounting for the LD can improve the prediction accuracy of complex traits. We consider a general high-dimensional dense setting for GWAS and study a class of ridge-type estimators, including the popular marginal estimator and the best linear unbiased prediction (BLUP) estimator as two special cases. We show that the performance of GWAS marginal estimator depends on the LD pattern through the first three moments of its eigenvalue distribution. Furthermore, we uncover that the relative performance of GWAS marginal and BLUP estimators highly depends on the ratio of GWAS sample size over the number of genetic variants. Particularly, our finding reveals that the marginal estimator can easily become near-optimal within this class when the sample size is relatively small, even though it ignores the LD pattern. On the other hand, BLUP estimator has substantially better performance than the marginal estimator as the sample size increases toward the number of genetic variants, which is typically in millions. Therefore, adjusting for the LD (such as in the BLUP) is most needed when GWAS sample size is large. We illustrate the importance of our results by using the simulated data and real GWAS.     

On Second Order Efficiency of Estimators Derived by Generalised χ2 Distance Functions

Taylor (1953) proposed a distance function in connection with the logit χ² estimator. For product (associated) multinomial distributions, he showed that minimization of the distance function yields BAN estimators. Aithal (1986) and Rao (1989) considered a modified version of Taylor's distance function and showed that a member belonging to this class leads to a second order efficient estimator. In this paper we consider Taylor's distance function and show that a member belonging to this class produces a second order efficient estimator. In addition to the above two, the m.l. estimator is also second order efficient. In order to compare these three second order efficient estimators, the small sample variances of the estimators are estimated through a simulation study. The results indicate that the variance of the m.l. estimator is the smallest in most of the cases.     

Five interval estimators for proportion ratio under a stratified randomized clinical trial with noncompliance

It is not uncommon that we may encounter a randomized clinical trial (RCT) in which there are confounders which are needed to control and patients who do not comply with their assigned treatments. In this paper, we concentrate our attention on interval estimation of the proportion ratio (PR) of probabilities of response between two treatments in a stratified noncompliance RCT. We have developed and considered five asymptotic interval estimators for the PR, including the interval estimator using the weighted-least squares (WLS) estimator, the interval estimator using the Mantel-Haenszel type of weight, the interval estimator derived from Fieller's Theorem with the corresponding WLS optimal weight, the interval estimator derived from Fieller's Theorem with the randomization-based optimal weight, and the interval estimator based on a stratified two-sample proportion test with the optimal weight suggested elsewhere. To evaluate and compare the finite sample performance of these estimators, we apply Monte Carlo simulation to calculate the coverage probability and average length in a variety of situations. We discuss the limitation and usefulness for each of these interval estimators, as well as include a general guideline about which estimators may be used for given various situations.     

On a Class of Almost Unbiased Ratio Estimators

A class of almost unbiased ratio estimators for population mean σ is derived by weighting sample σ = (1/n) σ y_i, ratio estimators σ and an estimator, σ (y_i/x_i). It is shown that NIETO DE PASCUAL (1961) estimator is a particular member of the class and an optimum estimator in the class (in the minimum variance sense) is identified. The results are illustrated through two numerical examples.     

Interval Estimation of Simple Difference under Independent Negative Binomial Sampling

When the sample size is not large or when the underlying disease is rare, to assure collection of an appropriate number of cases and to control the relative error of estimation, one may employ inverse sampling, in which one continues sampling subjects until one obtains exactly the desired number of cases. This paper focuses discussion on interval estimation of the simple difference between two proportions under independent inverse sampling. This paper develops three asymptotic interval estimators on the basis of the maximum likelihood estimator (MLE), the uniformly minimum variance unbiased estimator (UMVUE), and the asymptotic likelihood ratio test (ALRT). To compare the performance of these three estimators, this paper calculates the coverage probability and the expected length of the resulting confidence intervals on the basis of the exact distribution. This paper finds that when the underlying proportions of cases in both two comparison populations are small or moderate (≤0.20), all three asymptotic interval estimators developed here perform reasonably well even for the pre-determined number of cases as small as 5. When the pre-determined number of cases is moderate or large (≥50), all three estimators are essentially equivalent in all the situations considered here. Because application of the two interval estimators derived from the MLE and the UMVUE does not involve any numerical iterative procedure needed in the ALRT, for simplicity we may use these two estimators without losing efficiency.     

Analysis of longitudinal marginal structural models

In this article we construct and study estimators of the causal effect of a time-dependent treatment on survival in longitudinal studies. We employ a particular marginal structural model (MSM), proposed by Robins (2000), and follow a general methodology for constructing estimating functions in censored data models. The inverse probability of treatment weighted (IPTW) estimator of Robins et al. (2000) is used as an initial estimator and forms the basis for an improved, one-step estimator that is consistent and asymptotically linear when the treatment mechanism is consistently estimated. We extend these methods to handle informative censoring. The proposed methodology is employed to estimate the causal effect of exercise on mortality in a longitudinal study of seniors in Sonoma County. A simulation study demonstrates the bias of naive estimators in the presence of time-dependent confounders and also shows the efficiency gain of the IPTW estimator, even in the absence such confounding. The efficiency gain of the improved, one-step estimator is demonstrated through simulation.     

Relative Efficiency of Estimators of the Mean of a Normal Distribution when Coefficient of Variation is Known

A biased but simple and consistent estimator of the parameter ? has been obtained for the normal distribution N(?, a?₂), ?>0 where a is a known constant. It is shown that the estimator is more efficient than the sample mean or any suitably chosen constant multiple of the sample standard deviation. It is also proved to be more efficient than the mimumum variance unbiased estimator among a typical class of unbiased estimators derived by RASUL KHAN (1968).     

Buckley-James-type estimator with right-censored and length-biased data

We present a natural generalization of the Buckley-James-type estimator for traditional survival data to right-censored length-biased data under the accelerated failure time (AFT) model. Length-biased data are often encountered in prevalent cohort studies and cancer screening trials. Informative right censoring induced by length-biased sampling creates additional challenges in modeling the effects of risk factors on the unbiased failure times for the target population. In this article, we evaluate covariate effects on the failure times of the target population under the AFT model given the observed length-biased data. We construct a Buckley-James-type estimating equation, develop an iterative computing algorithm, and establish the asymptotic properties of the estimators. We assess the finite-sample properties of the proposed estimators against the estimators obtained from the existing methods. Data from a prevalent cohort study of patients with dementia are used to illustrate the proposed methodology.     

A Class of Almost Unbiased Estimators for Finite Populations Mean Using Two Auxiliary Variables

For estimating finite population mean -Y₀ of study character y₀, a class of almost unbiased estimators applying jackknife technique envisaged by Quenouille (1956) is derived. Optimum unbiased estimator (OUE) is also investigated with its variance formula. An empirical study is carried out to demonstrate the performance of the constructed estimator over the usual unbiased estimator, Srivastava (1965), Singh (1967), Singh and Biradar (1992), Tracy , Singh , and Singh (1996) and other almost unbiased estimators.     

Estimation of Several Missing Values in F-Square Designs

The present study deals with the estimation of several missing values in F-square designs. The estimating equations for the non-iterative least squares estimation of Missing Values and explicit expressions for the estimators of the particular patterns of Missing Values are presented. This procedure is illustrated with the help of a numerical example.     

A Monte Carlo Study of Robustness of Pretest and Shrinkage Estimators in Pooling Coefficients of Variation

Pooling data, when justified, is advantageous for estimating the true parameter. In this paper the problem of estimating the coefficient of variation is considered when it is a priori suspected that two coefficients of variation are the same. Various estimators based on pretest and shrinkage rules are considered. A comparison through the Simulated Mean Squared Error (SMSE) criterion is carried out among various proposed estimators of the target coefficient of variation. The relative simulated efficiencies of the restricted, shrinkage restricted and shrinkage pretest estimators are studied. It is found that the proposed estimators are quite robust when the sample sizes are not too large. The result of Monte Carlo study indicates that the proposed shrinkage pretest estimator is efficient than the usual estimator in a wider range.     

A Note on an Optimality Property of the Regression Estimator

This note points out the derivation of regression estimator through an optimality consideration over a class of estimators generating Generalised Product and dual to ratio estimators.     

On the mean square error associated with adaptive generalized ridge regression

Some numerical results are presented for generalized ridge regression where the additive constants are based on the data. The adaptive estimator so obtained is compared with the least-squares estimator on the basis of mean square error (MSE). It is shown that the MSE of each component of the vector of ridge estimators may be as low as 47.1% of the variance of the corresponding component of the least squares vector or as high as 125.2%.     

On the inadmissibility of Watterson's estimator

We consider the estimation of the scaled mutation parameter θ, which is one of the parameters of key interest in population genetics. We provide a general result showing when estimators of θ can be improved using shrinkage when taking the mean squared error as the measure of performance. As a consequence, we show that Watterson’s estimator is inadmissible, and propose an alternative shrinkage-based estimator that is easy to calculate and has a smaller mean squared error than Watterson’s estimator for all possible parameter values 0<θ<∞. This estimator is admissible in the class of all linear estimators. We then derive improved versions for other estimators of θ, including the MLE. We also investigate how an improvement can be obtained both when combining information from several independent loci and when explicitly taking into account recombination. A simulation study provides information about the amount of improvement achieved by our alternative estimators.     

On the potential for illogic with logically defined outcomes

Logically defined outcomes are commonly used in medical diagnoses and epidemiological research. When missing values in the original outcomes exist, the method of handling the missingness can have unintended consequences, even if the original outcomes are missing completely at random. In this note, we consider 2 binary original outcomes, which are missing completely at random. For estimating the prevalence of a logically defined "or" outcome, we discuss the properties of 4 estimators: the complete-case estimator, the available-case estimator, the maximum likelihood estimator (MLE), and a moment-based estimator. With the exception of the available-case case estimator, all the estimators are consistent. The MLE exhibits superior performance and should be generally adopted.     

On an Efficient Regression Type Estimator

The regression type estimator proposed by KAUR (1985) is considered. Another expression for the approximated mean square error (AMSE), to a first degree of approximation, is obtained. This AMSE is also minimized with respect to a parameter α. Three numerical examples are included. These numerical examples show that this estimator is not significantly more efficient than regression estimator and with respect to ratio and sample mean estimators, it does not always exhibit a high efficiency, as was contended by KAUR (1985). Moreover, an upper bound for the relative precision of the proposed estimator with respect to linear regression estimator is derived.