Jackknife Estimation for Smooth Functions of the Parametric Component in Partially Linear Regression Models

Abstract

It is known that due to the existence of the nonparametric component, the usual estimators for the parametric component or its function in partially linear regression models are biased. Sometimes this bias is severe. To reduce the bias, we propose two jackknife estimators and compare them with the naive estimator. All three estimators are shown to be asymptotically equivalent and asymptotically normally distributed under some regularity conditions. However, through simulation we demonstrate that the jackknife estimators perform better than the naive estimator in terms of bias when the sample size is small to moderate. To make our results more useful, we also construct consistent estimators of the asymptotic variance, which are robust against heterogeneity of the error variances.     

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.     

Estimation of the parameters of the gamma distribution by means of a maximal invariant

The use of a scale invariance criterion allows estimation of the shape parameter of the two parameter gamma distribution without estimating the scale parameter. Simulation experiments are used to show that the resulting estimators of both parameters are better than the usual maximum likelihood estimators in terms of both bias and mean square error. Approximately unbiased versions of the maximal invariant based estimators are derived and are shown to be as good as approximately unbiased versions of the usual maximum likelihood estimators     

On the failure rate estimation of the inverse gaussian distribution

New estimators of the inverse Gaussian failure rate are proposed based on the maximum likelihood predictive densities derived by Yang (1999). These estimators are compared, via Monte Carlo simulation, with the usual maximum likelihood estimators of the failure rate and found to be superior in terms of bias and mean squared error. Sensitivity of the estimators against the departure from the inverse Gaussian distribution is studied.     

Assessing accuracy of a continuous screening test in the presence of verification bias

Summary.  In studies to assess the accuracy of a screening test, often definitive disease assessment is too invasive or expensive to be ascertained on all the study subjects. Although it may be more ethical or cost effective to ascertain the true disease status with a higher rate in study subjects where the screening test or additional information is suggestive of disease, estimates of accuracy can be biased in a study with such a design. This bias is known as verification bias. Verification bias correction methods that accommodate screening tests with binary or ordinal responses have been developed; however, no verification bias correction methods exist for tests with continuous results. We propose and compare imputation and reweighting bias-corrected estimators of true and false positive rates, receiver operating characteristic curves and area under the receiver operating characteristic curve for continuous tests. Distribution theory and simulation studies are used to compare the proposed estimators with respect to bias, relative efficiency and robustness to model misspecification. The bias correction estimators proposed are applied to data from a study of screening tests for neonatal hearing loss.     

Parametric estimation of mixtures of two uniform distributions

In this paper, we consider a mixture of two uniform distributions and derive L-moment estimators of its parameters. Three possible ways of mixing two uniforms, namely with neither overlap nor gap, with overlap, and with gap, are studied. The performance of these L-moment estimators in terms of bias and efficiency is compared to that obtained by means of the conventional method of moments (MM), modified maximum likelihood (MML) method and the usual maximum likelihood (ML) method. These intensive simulations reveal that MML estimators are the best in most of the cases, and the L-moment estimators are less subject to bias in estimation for some mixtures and more efficient in most of the cases than the conventional MM estimators. The L-moment estimators are, in some cases, more efficient than the ML and MML estimators.     

Interval estimation of the mean in a two-stage nested model

The problem of constructing confidence intervals to estimate the mean in a two-stage nested model is considered. Several approximate intervals, which are based on both linear and nonlinear estimators of the mean are investigated. In particular, the method of bootstrap is used to correct the bias in the ‘usual’ variance of the nonlinear estimators. It is found that the intervals based on the nonlinear estimators did not achieve the nominal confidence coefficient for designs involving a small number of groups. Further, it turns out that the intervals are generally conservative, especially at small values of the intraclass correlation coefficient, and that the intervals based on the nonlinear estimators are more conservative than those based on the linear estimators. Compared with the others, the intervals based on the unweighted mean of the group means performed well in terms of coverage and length. For small values of the intraclass correlation coefficient, the ANOVA estimators of the variance components are recommended, otherwise the unweighted means estimator of the between groups variance component should be used. If one is fortunate enough to have control over the design, he is advised to increase the number of groups, as opposed to increasing group sizes, while avoiding groups of size one or two.     

Estimation in a Two Variance Components Model When one Component is Known

We study a mixed linear model with two variance components. We suppose that one component is known. The objective of the paper is the estimation of the unknown component. The usual MINQE estimators seem to be unadapted to the problem. So we propose a new family of quadratic estimators, based on a natural class of estimators and the idea upon which the MINQE theory is built. All the estimators are compared on simulated data.     

Preliminary test estimators in double sampling with two auxiliary variables

An attempt has been mads to suggest some estimators for population mean in double sampling with two auxiliary variables., alternative to the usual regression estimator. When the experimenter has partial Information about the mean of the auxiliary variable or variables, preliminary test estimators can be used. The bias, mean square error, relative efficiency and optimum allocation of sample sizes are obtained for the suggested estimators.     

Estimation of parameters in heavy-tailed distribution when its second order tail parameter is known

Estimating parameters in heavy-tailed distribution plays a central role in extreme value theory. It is well known that classical estimators based on the first order asymptotics such as the Hill, rank-based and QQ estimators are seriously biased under finer second order regular variation framework. To reduce the bias, many authors proposed the so-called second order reduced bias estimators for both first and second order tail parameters. In this work, estimation of parameters in heavy-tailed distributions are studied under the second order regular variation framework when the second order parameter in the distribution tail is known. This is motivated in large part by a recent work by the authors showing that the second order tail parameter is known for a large class of popular random difference equations (for example, ARCH models). The focus is on least squares estimators that generalize rank-based and QQ estimators. Though other possible estimators are also briefly discussed, the least squares estimators are most simple to use and perform best for finite samples in Monte Carlo simulations.     

Estimation of Bowley's Coefficient of Skewness in the Presence of Auxiliary Information

In this paper, we suggest regression-type estimators for estimating the Bowley's coefficient of skewness using auxiliary information. To the first degree of approximation, the bias and mean-squared error expressions of the regression-type estimators are obtained, and the regions under which these estimators are more efficient than the conventional estimator are also determined. Further, a general class of estimators of the Bowley's coefficient of skewness is defined along with its properties. A class of estimators based on estimated optimum values is also defined. It is shown to the first degree of approximations that the variance of the class of estimators based on estimated optimum values is the same as that of the minimum variance of the proposed class of estimators. A simulation study is carried out to demonstrate the performance of the proposed difference estimator over the usual estimator.     

Bayesian shrinkage estimators for a measure of dispersion of an inverse gaussian distribution

The Bayesian shrinkage estimation for a measure of dispersion with known mean is studied for the inverse Gaussian distribution. An optimum choice of the shrinkage factor and the properties of the proposed Bayesian shrinkage estimators are being studied. It is shown that these estimators have smaller risk than the usual estimator of the reciprocal measure of dispersion.     

A new class of unbiased estimators of the variance of the systematic sample mean

We propose a class of estimators of the variance of the systematic sample mean, which is unbiased under the assumption that the population follows a superpopulation model that satisfies some mild conditions. The approach is based on the separate estimation of the portion of the variance due to the systematic component of the model and that due to the stochastic component. In particular, we deal with two estimators belonging to the proposed class that are based on moving averages and local polynomials to estimate the systematic component of the model. The latter estimators are unbiased under the assumption that the population follows a linear trend and the errors are homoscedastic and uncorrelated. Through a simulation study we show that these estimators generally outperform, in terms of bias and mean square error, the usual estimator based on the first differences also when the superpopulation model departs significantly from linearity and the errors are heteroscedastic.     

On difference-based variance estimation in nonparametric regression when the covariate is high dimensional

Summary.  We consider the problem of estimating the noise variance in homoscedastic nonparametric regression models. For low dimensional covariates t  ∈  R ^d ,  d =1, 2, difference-based estimators have been investigated in a series of papers. For a given length of such an estimator, difference schemes which minimize the asymptotic mean-squared error can be computed for d =1 and d =2. However, from numerical studies it is known that for finite sample sizes the performance of these estimators may be deficient owing to a large finite sample bias. We provide theoretical support for these findings. In particular, we show that with increasing dimension d this becomes more drastic. If d 4, these estimators even fail to be consistent. A different class of estimators is discussed which allow better control of the bias and remain consistent when d 4. These estimators are compared numerically with kernel-type estimators (which are asymptotically efficient), and some guidance is given about when their use becomes necessary.     

Equivariant estimation of a normal mean using a normal concomitant variable for covariance adjustment

Equivariant point estimators of one component of a bivariate normal mean vector are considered when the second component is known. Equivariant point estimators are characterized and compared in terms of their risk functions with respect to a normalized squared-error loss function. Specific point estimators that dominate the usual estimator when the squared correlation coefficient is sufficiently large are provided.     

Performance of coefficient of variation estimators in ranked set sampling

In this paper, we propose and evaluate the performance of different parametric and nonparametric estimators for the population coefficient of variation considering Ranked Set Sampling (RSS) under normal distribution. The performance of the proposed estimators was assessed based on the bias and relative efficiency provided by a Monte Carlo simulation study. An application in anthropometric measurements data from a human population is also presented. The results showed that the proposed estimators via RSS present an expressively lower mean squared error when compared to the usual estimator, obtained via Simple Random Sampling. Also, it was verified the superiority of the maximum likelihood estimator, given the necessary assumptions of normality and perfect ranking are met.     

Construction of improved estimators of multinomial proportions

The improved large sample estimation theory for the probabilities of multi¬nomial distribution is developed under uncertain prior information (UPI) that the true proportion is a known quantity. Several estimators based on pretest and the Stein-type shrinkage rules are constructed. The expressions for the bias and risk of the proposed estimators are derived and compared with the maximum likelihood (ml) estimators. It is demonstrated that the shrinkage estimators are superior to the ml estimators. It is also shown that none of the preliminary test and shrinkage estimators dominate each other, though they perform y/ell relative to the ml estimators. The relative dominance picture of the estimators is presented. A simulation study is carried out to assess the performance of the estimators numerically in small samples.     

Hierarchic predictive ratio-based and product-based estimators and their efficiency

SUMMARY Invoking the predictive approach with a fixed population set-up, and employing initially the customary ratio and product estimators as potential predictors for the non-surveyed part of the population, we have generated sequences of ratio-based and product-based estimators. The proposed ratio-based and product-based estimators of order k are-under some practical conditions-found to be more efficient than the customary ratio and product estimators and the usual simple mean when k is chosen optimally. Under the optimal value of k, the kth-order ratio-based and product-based estimators are found to be as efficient as the linear regression estimator. We have used real population data to illustrate the efficacy of the proposed ratio-based and product-based estimators relative to the usual simple mean and the customary ratio and product 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.     

Bayes methods in the ecological fallacy cntext:estimation of individual correlation from aggregate data

The ecological fallacy is related to Simpson's paradox (1951) where relationships among group means may be counterintuitive and substantially different from relationships within groups, where the groups are usually geographic entities such as census tracts. We consider the problem of estimating the correlation between two jointly normal random variables where only ecological data (group means) are available. Two empirical Bayes estimators and one fully Bayesian estimator are derived and compared with the usual ecological estimator, which is simply the Pearson correlation coefficient of the group sample means. We simulate the bias and mean squared error performance of these estimators, and also give an example employing a dataset where the individual level data are available for model checking. The results indicate superiority of the empirical Bayes estimators in a variety of practical situations where, though we lack individual level data, other relevant prior information is available.