General Class Of Estimators Research Articles

Estimating population proportions in the presence of missing data

This paper discusses the estimation of a population proportion in the presence of missing data and using auxiliary information at the estimation stage. A general class of estimators, which make efficient use of the available information, are proposed. Some theoretical properties of the proposed estimators are analyzed, and they allow us to find the optimal value for the proposed class in the sense of minimal variance. The optimal estimator is thus more efficient than the customary estimator. Results derived from a simulation study indicate that the proposed optimal estimator gives desirable results in comparison to alternative estimators.

Utilization of Some Known Population Parameters for Estimating Population Mean in Presence of Non-Response

In presence of non-response, a general class of estimators is proposed for estimating the population mean of the study variable with known value of some population parameter(s) of the auxiliary variable. The bias and mean squared error (MSE) of the proposed strategy are derived under simple random sampling without replacement (SRSWOR). Some estimators are also obtained from proposed strategy by giving suitable values to the constants used. Comparisons of the proposed strategy with the usual unbiased estimator and other estimators have been made. An empirical study is carried out to demonstrate the performance of the proposed estimator.

A General Class of Estimators for the Linear Regression Model Affected by Collinearity and Outliers

In this article, a general class of estimators for the linear regression model affected by outliers and collinearity is introduced and studied in some detail. This class of estimators combines the theory of light, maximum entropy, and robust regression techniques. Our theoretical findings are illustrated through a Monte Carlo simulation study.

A general procedure for estimating the population mean in stratified sampling using auxiliary information

In this paper we have suggested a general procedure for estimating the population mean in stratified sampling using auxiliary information. A general class of estimators is defined with its properties under large sample approximation. In particular, various classes of estimators are identified as particular member of the suggested class. The correct version of the mean squared error of Kadilar and Cingi (2005) class of estimators is derived. It has been shown that the proposed class of estimators is better than Kadilar and Cingi (2005) estimator, usual unbiased estimator, usual combined ratio estimator yRC, combined product estimator yPC, Gangde et al. (1993) type estimator and Searls (1964) type estimator. Numerical illustration is given in support of present study.

A Generalized Method of Estimation for Two-Stage Sampling Using Two Auxiliary Variables

Abstract This paper is aimed at developing an estimation methodology on gathering auxiliary information available at different stages of a survey under a two-stage sampling set-up. With this objective, we develop a general class of estimators for the population mean with the aid of two auxiliary variables when their means for the selected first stage units as well as for the overall population are known. Numerical studies have also been provided to evaluate the gain in efficiency of the suggested methodology.

HIGHER ORDER ASYMPTOTIC THEORY WHEN A PARAMETER IS ON A BOUNDARY WITH AN APPLICATION TO GARCH MODELS

Andrews (1999, Econometrica 67, 1341–1383) derived the first-order asymptotic theory for a very general class of estimators when a parameter is on a boundary. We derive the second-order asymptotic theory in this setting in some special cases. We focus on the behavior of the quasi maximum likelihood estimator (QMLE) in stationary and nonstationary generalized autoregressive conditionally heteroskedastic (GARCH) models when constraints are imposed in the maximization procedure. We show how in this case both a first- and a second-order bias appear in the estimator and how the bias can be quite large. We provide two types of bias correction mechanisms for the researcher to choose in practice: either to bias correct only for a first-order bias or for a first- and second-order bias. We show that when some constraints are imposed, it is advisable to bias correct not only for the first-order bias but also for the second-order bias.We thank Bruce Hansen and two referees for helpful comments. The first author gratefully acknowledges financial support from the MSU Intramural Research Grants Program. The second author gratefully acknowledges financial support from the ESRC.

A general class of estimators with auxiliary information based on available units

This paper proposes some estimators for the finite population mean when observations on some units (on either the auxiliary variable or the main variable) are missing. The optimum estimator in the proposed class is derived. Asymptotic properties of optimum estimator are studied. A brief simulation study shows goods properties for moderate sample sizes.

Asymptotics of Multivariate Regression with Consecutively Added Dependent Variables

We consider multivariate regression where new dependent variables are consecutively added during the experiment (or in time).So, viewed at the end of the experiment, the number of observations decreases with each added variable. The explanatory variables are observed throughout.In a previous paper we determined the least squares and maximum likelihood estimators for the parameters in this model.In this paper we discuss the estimation technique of iterative least squares to calculate the maximum likelihood estimates and we prove the consistency of the estimators in each iteration.Moreover, we introduce a general class of estimators for the regression parameters based on arbitrary starting estimators for the covariance matrix.We prove the consistency of these new estimators and - for sake of completeness - of the previously obtained least squares and maximum likelihood estimators as well.

Revisiting the Role of the Jackknife Methodology in the Estimation of a Positive Tail Index

In this article, and in a context of regularly varying tails, we analyze a generalization of the classical Hill estimator of a positive tail index. The members of this general class of estimators are not asymptotically more efficient than the original one, the Hill estimator. We thus propose a class of generalized Jackknife estimators associated with any two members of the first class. They enable the reduction of the main component of bias of the Hill estimator and are dependent of a tuning parameter, which is adequately chosen through an asymptotic variance minimization criterion. These new estimators are compared with the Hill estimator, both asymptotically and for finite samples and, when the underlying distribution is in Hall's class of models, they really improve on the well-known, bias-variance, trade-off characteristic of the Hill estimator.

General class of estimators in multi-character surveys

In this paper, a general class of estimators for the estimation of a finite population total in multi-character surveys is proposed. It is shown that the estimators proposed by Arnab (2002), Amahiaet al. (1989) and Bansal and Singh (1985) are the special cases of the proposed class of estimators. The proposed class of estimators is always more efficient than the estimator proposed by Rao (1966).

Optimal estimation for finite population mean in two-phase sampling

Using two-phase sampling scheme, we propose a general class of estimators for finite population mean. This class depends on the sample means and variances of two auxiliary variables. The minimum variance bound for any estimator in the class is provided (up to terms of ordern −1). It is also proved that there exists at least a chain regression type estimator which reaches this minimum. Finally, it is shown that other proposed estimators can reach the minimum variance bound, i.e. the optimal estimator is not unique.

Comparison of maximum entropy and higher-order entropy estimators

We show that generalized maximum entropy (GME) is the only estimation method that is consistent with a set of five axioms. The GME estimator can be nested using a single parameter, α, into two more general classes of estimators: GME- α estimators. Each of these GME- α estimators violates one of the five basic axioms. However, small-sample simulations demonstrate that certain members of these GME- α classes of estimators may outperform the GME estimation rule.

Multivariate Density Estimation with General Flat-Top Kernels of Infinite Order

The problem of nonparametric estimation of a multivariate density function is addressed. In particular, a general class of estimators with favorable asymptotic performance (bias, variance, rate of convergence) is proposed. The proposed estimators are characterized by the flatness near the origin of the Fourier transform of the kernel and are actually shown to be exactlyN-consistent provided the density is sufficiently smooth.

Sequential bounded risk point estimation of parameters of a negative exponential distribution

In this paper we consider the probier11 of estirrlatiiig with given constants a and b > 0, based on t,he sample from an exponentiai distribution with location and scale pararrleters µ and δ. both unknov:n. For a preassigned quantity w > 0 we want to estimate θby using the sampie of the smallest size such that the risk associated with an estimator is not greater than w. We propose an estimator and a slight more general class of estimators and define a stopping rule. The asymptotic expansions of the risks associated wit11 these estimators are obtained. \\le also compare them from the point of view of risk.

A Generalized Formula for Variance in Unequal Probability Sampling

A variance formula is given for a general class of estimators for any fixed sample size design. A formula for an unbiased estimator of the variance is also provided. These results generalize the Yates-Grundy variance formula of the Horvitz-Thompson estimator.

Robust Estimation for Grouped Data

Here we investigate the robustness properties of the class of minimum power divergence estimators for grouped data. This class contains the classical maximum likelihood estimators for grouped data. We find that the bias of these estimators due to deviations from the assumed underlying model can be large. Therefore, we propose a more general class of estimators that allows us to construct robust procedures. By analogy with Hampel's theorem, we define optimal bounded influence function estimators, and by a simulation study, we show that under small model contaminations, these estimators are more stable than the classical estimators for grouped data. Finally, we apply our results to a particular real example.

Robust Estimation for Grouped Data

Abstract Here we investigate the robustness properties of the class of minimum power divergence estimators for grouped data. This class contains the classical maximum likelihood estimators for grouped data. We find that the bias of these estimators due to deviations from the assumed underlying model can be large. Therefore, we propose a more general class of estimators that allows us to construct robust procedures. By analogy with Hampel's theorem, we define optimal bounded influence function estimators, and by a simulation study, we show that under small model contaminations, these estimators are more stable than the classical estimators for grouped data. Finally, we apply our results to a particular real example.

Measurement Error in the Response in the General Linear Model

Abstract Problems where there is measurement error in the response variable in a general linear model are considered. With Y denoting the true value and U the observed/surrogate value, a rich class of measurement error models is considered in which (a) the mean of U given Y = y can be either linear or nonlinear in y, (b) the variance of U given Y = y can depend on y, and (c) the parameters in the model for U given Y can change across observations in the study. This could occur, for example, when using a radioimmunoassay to measure the outcome variable with recalibration for each batch of reagents. The parameters in the measurement error models are estimated using independent “calibration” data. Full and pseudo—maximum likelihood estimators (MLE's) and their asymptotic properties are described under distributional assumptions. The pseudo-MLE's and naive estimates using imputed values are special cases of a general class of estimators. The asymptotic properties of such estimators are given and then used to ...

Estimating the Mean with Known Coefficient of Variation

Abstract Searls in 1964 showed that when the coefficient of variation is known, the sample mean is dominated with respect to mean squared error by an improved estimator that makes use of that coefficient. In this article we illustrate that this is true for a general class of estimators. Expressions for the minimum mean squared error and the relative efficiency are given for general distributions. The improvement, as measured by relative efficiency, is seen to be independent of the form of the distribution.

Asymptotics of the Repeated Median Slope Estimator

The influence function is determined for (twice) repeated median estimators with arbitrary kernel functions, and more generally in the case where the two medians are replaced by a general class of estimators. Asymptotic normality is then established for the repeated median estimator of the slope parameter in simple linear regression. In this case the influence function is bounded. For bivariate Gaussian data the efficiency becomes $4/\pi^2 \approx 40.5{\tt \%},$ which is the square of the efficiency of the univariate median. The asymptotic results are compared with finite-sample efficiencies. It turns out that the convergence to the asymptotic behavior is extremely slow.