A Class of Estimators Using Auxiliary Information for Estimating Finite Population Variance in Presence of Measurement Errors

This article addresses the problem of estimating the population variance using auxiliary information in the presence of measurement errors. When the measurement error variance associated with study variable is known, a class of estimators of the population variance using auxiliary information has been proposed. We obtain the bias and mean squared errors of the suggested class of estimators upto the terms of order n ^?1, and also optimum estimators in asymptotic sense of the class with approximate mean squared error formula.     

Smooth tests of fit for censored gmma samples

In this paper properties of two estimators of Cpm are investigated in terms of changes in the process mean and variance. The bias and mean squared error of these estimators are derived. It can be shown that the estimate of Cpm proposed by Chan, Cheng and Spiring (1988) has smaller bias than the one proposed by Boyles (1991) and also has a smaller mean squared error under certain conditions. Various approximate confidence intervals for Cpm are obtained and are compared in terms of coverage probabilities, missed rate and average interval width.     

Robust model-based sampling designs

We investigate methods for the design of sample surveys, and address the traditional resistance of survey samplers to the use of model-based methods by incorporating model robustness at the design stage. The designs are intended to be sufficiently flexible and robust that resulting estimates, based on the designer’s best guess at an appropriate model, remain reasonably accurate in a neighbourhood of this central model. Thus, consider a finite population of N units in which a survey variable Y is related to a q dimensional auxiliary variable x. We assume that the values of x are known for all N population units, and that we will select a sample of n≤N population units and then observe the n corresponding values of Y. The objective is to predict the population total $T=\sum_{i=1}^{N}Y_{i}$ . The design problem which we consider is to specify a selection rule, using only the values of the auxiliary variable, to select the n units for the sample so that the predictor has optimal robustness properties. We suppose that T will be predicted by methods based on a linear relationship between Y—possibly transformed—and given functions of x. We maximise the mean squared error of the prediction of T over realistic neighbourhoods of the fitted linear relationship, and of the assumed variance and correlation structures. This maximised mean squared error is then minimised over the class of possible samples, yielding an optimally robust (‘minimax’) design. To carry out the minimisation step we introduce a genetic algorithm and discuss its tuning for maximal efficiency.     

A comparison of estimators for the proportion in the tail of a normal distribution

Three estimators of the proportion in a tail of the normal distribution are compared using the criteria of mean squared error and mean absolute error. The estimators that we compare are the maximum likelihood estimator, the minimum variance unbiased estimator, and an intuitive estimator that is frequently used in practice. The intuitive estimator is similar to the MLE but uses the usual unbiased estimator of σ² rather than the MLE of σ². We show that the intuitive estimator has low efficiency, and for this reason it is not recommended. For very smallp and for largep the MVUE has the highest efficiency. The MLE is best for moderate values ofp.     

Estimating the variance of the sample median

The small-sample bias and root mean squared error of several distribution-free estimators of the variance of the sample median are examined. A new estimator is proposed that is easy to compute and tends to have the smallest bias and root mean squared error.     

Alternative moment approximations for berkson's minimum logit chi-squared estimator

Expressions are derived for the bias to order J^-1 , the variance to order J^-2 and the mean squared error to order J^-2 of Berkson's minimum logit chi-squared estimator where J is the number of distinct design points. These moment approximations are numerically compared to Monte Carlo estimates of the true moments and the moment approximations of Amemiya (1980) which are appropriate when the “average” number of observations per design point is large. They are used to compare the mean squared error of the minimum logit chi-squared estimator to that of the maximum likelihood estimator and to investigate the effect of bias on confidence intenrals constructed using the minimum logit chi-squared estimator.     

An empirical investigation of different operating characteristics of several estimators of the intraclass correlation in the analysis of binary data

This paper is concerned with properties (bias, standard deviation, mean square error and efficiency) of twenty six estimators of the intraclass correlation in the analysis of binary data. Our main interest is to study these properties when data are generated from different distributions. For data generation we considered three over-dispersed binomial distributions, namely, the beta-binomial distribution, the probit normal binomial distribution and a mixture of two binomial distributions. The findings regarding bias, standard deviation and mean squared error of all these estimators, are that (a) in general, the distributions of biases of most of the estimators are negatively skewed. The biases are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution; (b) the standard deviations are smallest when data are generated from the beta-binomial distribution; and (c) the mean squared errors are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution. Of the 26, nine estimators including the maximum likelihood estimator, an estimator based on the optimal quadratic estimating equations of Crowder (1987), and an analysis of variance type estimator is found to have least amount of bias, standard deviation and mean squared error. Also, the distributions of the bias, standard deviation and mean squared error for each of these estimators are, in general, more symmetric than those of the other estimators. Our findings regarding efficiency are that the estimator based on the optimal quadratic estimating equations has consistently high efficiency and least variability in the efficiency results. In the important range in which the intraclass correlation is small (≤0 5), on the average, this estimator shows best efficiency performance. The analysis of variance type estimator seems to do well for larger values of the intraclass correlation. In general, the estimator based on the optimal quadratic estimating equations seems to show best efficiency performance for data from the beta-binomial distribution and the probit normal binomial distribution, and the analysis of variance type estimator seems to do well for data from the mixture distribution.     

Computer experiment designs for accurate prediction

Computer experiments using deterministic simulators are sometimes used to replace or supplement physical system experiments. This paper compares designs for an initial computer simulator experiment based on empirical prediction accuracy; it recommends designs for producing accurate predictions. The basis for the majority of the designs compared is the integrated mean squared prediction error (IMSPE) that is computed assuming a Gaussian process model with a Gaussian correlation function. Designs that minimize the IMSPE with respect to a fixed set of correlation parameters as well as designs that minimize a weighted IMSPE over the correlation parameters are studied. These IMSPE-based designs are compared with three widely-used space-filling designs. The designs are used to predict test surfaces representing a range of stationary and non-stationary functions. For the test conditions examined in this paper, the designs constructed under IMSPE-based criteria are shown to outperform space-filling Latin hypercube designs and maximum projection designs when predicting smooth functions of stationary appearance, while space-filling and maximum projection designs are superior for test functions that exhibit strong non-stationarity.     

Zero-Bias Locally Adaptive Density Estimators

Strategies for improving fixed non-negative kernel estimators have focused on reducing the bias, either by employing higher-order kernels or by adjusting the bandwidth locally. Intuitively, bandwidths in the tails should be relatively larger in order to reduce wiggles since there is less data available in the tails. We show that in regions where the density function is convex, it is theoretically possible to find local bandwidths such that the pointwise bias is exactly zero. The corresponding pointwise mean squared error converges at the parametric rate of O ( n ⁻¹ ) rather than the slower O ( n ^−4/5). These so-called zero-bias bandwidths are constant and are usually orders of magnitude larger than the optimal locally adaptive bandwidths predicted by asymptotic mean squared error analysis. We describe data-based algorithms for estimating zero-bias bandwidths over intervals where the density is convex. We find that our particular density estimator attains the usual O ( n ^−4/5) rate. However, we demonstrate that the algorithms can provide significant improvement in mean squared error, often clearly visually superior curves, and a new operating point in the usual bias-variance tradeoff.     

An investigation of a quadratic design criterion for estimation of the hemodynamic response function

Optimal experimental design for estimation of the hemodynamic response function (HRF) is investigated using a nonlinear model with a quadratic mean squared error design criterion. This criterion is used, along with a genetic algorithm, to select locally optimal designs that are shown to be, in most cases, more efficient than designs selected with the more commonly used linear expansion criterion. These designs are also shown to result in lower overall asymptotic estimator variance and bias. The investigation focuses on a single stimulus type, but the criterion can also be used with multiple stimulus types.     

Extrapolation designs with constraints

The author considers (asymptotically) minimax extrapolation designs for an approximately multiple linear model with the model contaminant f being restricted only by its L₂ norm. He splits the integrated mean squared prediction error (IMSPE) of the fitted value over the extrapolation space into two parts, namely the integrated prediction variance (IPV) and the integrated prediction bias (IPB). For a spherical design space and an annular extrapolation space, he constructs the design that minimizes the maximum value, over f, of IPB subject to bounding IPV. He also constructs the design that minimizes IPV subject to bounding the maximum IPB.     

A method of selecting the levels of regressor variables in lognormal regression model

Kupper and Meydrech and Myers and Lahoda introduced the mean squared error (MSE) approach to study response surface designs, Duncan and DeGroot derived a criterion for optimality of linear experimental designs based on minimum mean squared error. However, minimization of the MSE of an estimator maxr renuire some knowledge about the unknown parameters. Without such knowledge construction of designs optimal in the sense of MSE may not be possible. In this article a simple method of selecting the levels of regressor variables suitable for estimating some functions of the parameters of a lognormal regression model is developed using a criterion for optimality based on the variance of an estimator. For some special parametric functions, the criterion used here is equivalent to the criterion of minimizing the mean squared error. It is found that the maximum likelihood estimators of a class of parametric functions can be improved substantially (in the sense of MSE) by proper choice of the values of regressor variables. Moreover, our approach is applicable to analysis of variance as well as regression designs.     

Asymptotic Theory for the Multiscale Wavelet Density Derivative Estimator

Abstract

In this paper we consider the wavelet-based estimation of density derivatives. The multiscale density derivative estimator is proposed which is constructed by using a number of scaling functions. Asymptotic theory is developed in which asymptotic expressions for the bias, the variance and the mean integrated squared error are included. In addition, asymptotic normality of the proposed estimator is proved. Theoretical and numerical comparisons with the usual kernel-based estimators are also reported.     

Two Non-Negative Estimators for the Model With a Common Mean

Recently C. R. Rao (1984) suggested two modifications to the MINQUE, called MINQUE(S.D.) and MINQUE(C.P.), for estimating the variance components in a linear model. These modifications provide non-negative estimates unlike the MINQUE. In this article we shall compare the performance of these two estimators with some of the other existing modifications of MINQUE in terms of standard criteria such as the mean squared error and total squared bias.     

A New Characterization and Estimation of the Zero–bias Bandwidth

It is well established that bandwidths exist that can yield an unbiased non–parametric kernel density estimate at points in particular regions (e.g. convex regions) of the underlying density. These zero–bias bandwidths have superior theoretical properties, including a 1/n convergence rate of the mean squared error. However, the explicit functional form of the zero–bias bandwidth has remained elusive. It is difficult to estimate these bandwidths and virtually impossible to achieve the higher–order rate in practice. This paper addresses these issues by taking a fundamentally different approach to the asymptotics of the kernel density estimator to derive a functional approximation to the zero–bias bandwidth. It develops a simple approximation algorithm that focuses on estimating these zero–bias bandwidths in the tails of densities where the convexity conditions favourable to the existence of the zerobias bandwidths are more natural. The estimated bandwidths yield density estimates with mean squared error that is O(n^–4/5), the same rate as the mean squared error of density estimates with other choices of local bandwidths. Simulation studies and an illustrative example with air pollution data show that these estimated zero–bias bandwidths outperform other global and local bandwidth estimators in estimating points in the tails of densities.     

A family of shrinkage estimators for Weibull shape parameter in censored sampling

In this paper a new class of shrinkage estimators has been introduced for the shape parameter in an independently identically distributed two-parameterWeibull model under censored sampling. The main idea is to incorporate the prior guessed value by correcting the standard estimator, which is essentially an unbiased estimator, with optimally weighted ratios of the guessed value and the standard estimator, instead of considering a convex combination of the standard estimator and the difference of the guessed value and the standard estimator. The resulting estimator dominates the standard estimator in a surprisingly large neighborhood of the guessed value. The suggested estimator has also been compared with the minimum mean squared error estimator and a class of estimators suggested by Singh and Shukla in IAPQR Trans 25(2), 107–118, 2000. It is found that the suggested class of estimators has lesser bias as well as lesser mean squared error than its competitors subject to certain conditions.      

A new non parametric estimator for Pdf based on inverse gamma distribution

ABSTRACT

The non parametric approach is considered to estimate probability density function (Pdf) which is supported on(0, ∞). This approach is the inverse gamma kernel. We show that it has same properties as gamma, reciprocal inverse Gaussian, and inverse Gaussian kernels such that it is free of the boundary bias, non negative, and it achieves the optimal rate of convergence for the mean integrated squared error. Also some properties of the estimator were established such as bias and variance. Comparison of the bandwidth selection methods for inverse gamma kernel estimation of Pdf is done.     

Parametrically Guided Non-parametric Regression

We present a new approach to regression function estimation in which a non-parametric regression estimator is guided by a parametric pilot estimate with the aim of reducing the bias. New classes of parametrically guided kernel weighted local polynomial estimators are introduced and formulae for asymptotic expectation and variance, hence approximated mean squared error and mean integrated squared error, are derived. It is shown that the new classes of estimators have the very same large sample variance as the estimators in the standard non-parametric setting, while there is substantial room for reducing the bias if the chosen parametric pilot function belongs to a wide neighbourhood around the true regression line. Bias reduction is discussed in light of examples and simulations.     

Secondary design considerations for minimum bias estimation

Several authors have suggested the method of minimum bias estimation for estimating response surfaces. The minimum bias estimation procedure achieves minimum average squared bias of the fitted model without depending on the values of the unknown parameters of the true surface. The only requirement is that the design satisfies a simple estimability condition. Subject to providing minimum average squared bias, the minimum bias estimator also provides minimum average variance of ?(x) where ?(x) is the estimate of the response at the point x.

To support the estimation of the parameters in the fitted model, very little has been suggested in the way of experimental designs except to say that a full rank matrix X of independent variables should be used. This paper presents a closer look at the estimability conditions that are required for minimum bias estimation, and from the form of the matrix X, a formula is derived which measures the amount of design flexibility available. The design flexibility is termed “the degrees of freedom” of the X matrix and it is shown how the degrees of freedom can be used to decide if other design optimality criteria might be considered along with minimum bias estimation. Several examples are provided.     

Response probability estimation

This paper extends the ideas in Giommi (Proc. 45th Session of the Internat. Statistical Institute, Vol. 2 (1985) 577–578; Techniques d'enquête 13(2) (1987) 137–144) and, in Särndal and Swenson (Bull. Int. Statist. Inst. 15(2) (1985) 1–16; Int. Statist. Rev. 55(1987) 279–294). Given the parallel between a ‘three-phase sampling’ and a ‘sampling with subsequent unit and item nonresponse’, we apply results from three-phase sampling theory to nonresponse situation. To handle the practical problem of unknown distributions at the second and the third phases of selection (the response mechanisms) in the nonresponse case, we use two approaches of response probability estimation: response homogeneity groups (RHG) model (Särndal and Swenson, 1985, 1987) and the nonparametric estimation (Giommi, 1985, 1987). To motivate the three-phase selection, imputation procedures for item nonresponse are used with the RHG model for unit nonresponse. By means of a Monte Carlo study, we find that the regression-type estimators are the most precise of those studied under the two approaches of response probability estimation in terms of lower bias, mean square error and variance; variance estimator close to the true variance and achieved coverage rates closer to the nominal levels. The simulation study shows how poor the variance estimators are under the single imputation approach currently used to handle the problem of missing values.