Pitman nearness comparisons of Stein-type estimators for regression coefficients in replicated experiments

This paper presents a comparative study of the performance properties of one unbiased and two Stein-type estimators for combining the estimates of coefficients in a linear regression model when data sets are available from replicated experiments conducted at possibly different stations.     

Comparison of some common ratio estimators using pitman measure of nearness

In this article we compare some common ratio estimators for estimating the population total of a given characteristic. The sampling schemes considered are simple random sampling (S.R.S.) and S.R.S.under stratification. The comparisons are made using the Pitman Nearness criterion under the model-based approach. The error term is assumed normal with mean zero and variance σg(x). The function g(x) is a known function of the auxiliary variable x. Special interest is on the cases of g(x) =l and x. The result is found the same as that using MSE criterion, although the PN is very different from the MSE intrinsically.     

Shrunken estimators via a pitman nearness criterion

Shrunken estimators have traditionally been developed and studied using mean square error (MSE). Recent research on Pitman nearness (PN), however, indicates that it is an interesting, “intrinsic”, alternative to the mean square error (MSE) criterion for investigating estimators. Thus, we develop a shrunken estimator for the mean of a multivariate normal distribution based on minimizing PN, instead of MSE, Further, since the shrinkage factor of this estimator depends on unknown parameters, we examine two approaches for determining this factor: (1) “plug-in” estimates, (2) a range of values for the factor based on an approximate cońfidence interval for the Pitman Nearness probability. A numerical example is given.     

A comparison of biased regression estimators using a pitman nearness criterion

Biased regression estimators have traditionally benn studied using the Mean Square Error (MSE) criterion. Usually these comparisons have been based on the sum of the MSE's of each of the individual parameters, i.e., a scaler valued measure that is the trace of the MSE matrix. However, since this summed MSE does not consider the covariance structure of the estimators, we propose the use of a Pitman Measure of Closeness (PMC) criterion (Keating and Gupta, 1984; Keating and Mason, 1985). In this paper we consider two versions of PMC. One of these compares the estimates and the other compares the resultant predicted values for 12 different regression estimators. These estimators represent three classes of estimators, namely, ridge, shrunken, and principal component estimators. The comparisons of these estimators using the PMC criteria are contrasted with the usual MSE criteria as well as the prediction mean square error. Included in the estimators is a relatively new estimator termed the generalized principal component estimator proposed by Jolliffe. This estimator has previously received little attention in the literature.     

Comparison of the Stein and the usual estimators for the regression error variance under the Pitman nearness criterion when variables are omitted

This paper compares the Stein and the usual estimators of the error variance under the Pitman nearness (PN) criterion in a regression model which is mis-specified due to missing relevant explanatory variables. The exact expression of the PN-probability is derived and numerically evaluated. Contrary to the well-known result under mean squared errors (MSE), with the PN criterion the Stein variance estimator is uniformly dominated by the usual estimator when no relevant variables are excluded from the model. With an increased degree of model mis-specification, neither estimator strictly dominates the other. The authors are grateful to two anonymous referees for their valuable comments. Also, the first author is grateful to the Japan Society for the Promotion of Science for partial financial support.     

On pitman nearness and variance of estimators

In this article, some conditions on variances are presented under which the (Generalized) Pitman Nearness Criterion Would Prefer one estimator over another. Results for univariate as well as multivariate cases are derived. An exact expression for a result of Rao, Keating and Mason (1985) is provided.     

Comparison of improved regression estimators with and without moments

For estimating the coefficients in a linear regression model, the double k–class estimators are considered and the small disturbance asymptotic approximation for their density function is obtained. Then employing the criterion of concentration probability around the true parameter values, a comparison is made between the estimators possessing finite moments and the estimators having no finite moments.     

Some properties of generalized ridge estimators

Exact expressions, in the form of infinite series expansions, are given for the first and second moments of two well known generalized ridge estimators. These series expansions are then evaluated using recursive formulas and computations are verified using approximations. Results are presented for the relative mean square error and bias of these estimators as well as their relative efficiency with respect to least squares.     

Comparison of the Iterative Stein-Rule and the Usual Estimators of the Disturbance Variance Under the Pitman Nearness Criterion in a Linear Regression Model with Proxy Variables

In a linear regression model with proxy variables, the iterative Stein-rule estimator and the usual estimator of the disturbance variance is compared under the Pitman Nearness Criterion. The exact expression of Pitman Nearness probability is derived and numerically evaluated.     

Comparison of preliminary test estimators based on generalized order statistics from proportional hazard family using Pitman measure of closeness

In this article, based on generalized order statistics from a family of proportional hazard rate model, we use a statistical test to generate a class of preliminary test estimators and shrinkage preliminary test estimators for the proportionality parameter. These estimators are compared under Pitman measure of closeness (PMC) as well as MSE criteria. Although the PMC suffers from non transitivity, in the first class of estimators, it has the transitivity property and we obtain the Pitman-closest estimator. Analytical and graphical methods are used to show the range of parameter in which preliminary test and shrinkage preliminary test estimators perform better than their competitor estimators. Results reveal that when the prior information is not too far from its real value, the proposed estimators are superior based on both mentioned criteria.     

A simulation study of five biased estimators for straight line regression

Five biased estimators of the slope in straight line regression are considered. For each, the estimate of the “bias parameter”, k, is a function of N, the number of observations, and [rcirc]² , the square of the least squares estimate of the standardized slope, β. The estimators include that of Farebrother, the ridge estimator of Hoerl, Kennard, and Baldwin, Vinod's shrunken estimators., and a new modification of one of the latter. Properties of the estimators are studied for 13 combinations of N and 3. Results of simulation experiments provide empirical evidence concerning the values of means and variances of the biased estimators of the slope and estimates of the “bias parameter”, the mean square errors of the estimators, and the frequency of improvement relative to least squares. Adjustments to degrees of freedom in the biased regression analysis of variance table are also considered. An extension of the new modification to the case of p> 1 independent variables is presented in an Appendix.     

Pitman closeness,monotonicity and consistency of best linear unbiased and invariant estimators for exponential distribution under Type II censoring

Comparisons of best linear unbiased estimators with some other prominent estimators have been carried out over the last 50 years since the ground breaking work of Lloyd [E.H. Lloyd, Least squares estimation of location and scale parameters using order statistics, Biometrika 39 (1952), pp. 88–95]. These comparisons have been made under many different criteria across different parametric families of distributions. A noteworthy one is by Nagaraja [H.N. Nagaraja, Comparison of estimators and predictors from two-parameter exponential distribution, Sankhyā Ser. B 48 (1986), pp. 10–18], who made a comparison of best linear unbiased (BLUE) and best linear invariant (BLIE) estimators in the case of exponential distribution. In this paper, continuing along the same lines by assuming a Type II right censored sample from a scaled-exponential distribution, we first compare BLUE and BLIE of the exponential mean parameter in terms of Pitman closeness (nearness) criterion. We show that the BLUE is always Pitman closer than the BLIE. Next, we introduce the notions of Pitman monotonicity and Pitman consistency, and then establish that both BLUE and BLIE possess these two properties.     

Exact moments of lawless and wang's operational ridge regression estimator

Assuming the disturbances are normally distributed, we derive expressions for, and simple conditions for the existence of the exact bias and matrix of second order moments of the Lawless and Wang Operational Ridge Regression estimator.     

Deficiencies of minimum discrepancy estimators

Suppose the multinomial parameters p_r (θ) are functions of a real valued parameter 0, r = 1,2, …, k. A minimum discrepancy (m.d.) estimator θ of θ is defined as one which minimises the discrepancy function D = Σ n_rf(p_r/n_r), for a suitable function f where n_r is the relative frequency in r-th cell, r = 1,2, …, k. All the usual estimators like maximum likelihood (m. l), minimum chi-square (m. c. s.)., etc. are m.d. estimators. All m.d. estimators have the same asymptotic (first order) efficiency. They are compared on the basis of their deficiencies, a concept recently introduced by Hodges and Lehmann [2]. The expression for least deficiency at any θ is derived. It is shown that in general uniformly least deficient estimators do not exist. Necessary and sufficient conditions on p_r (0) for m. t. and m. c. s. estimators to be uniformly least deficient are obtained.     

Measure of location-based estimators in simple linear regression

In this note we consider certain measure of location-based estimators (MLBEs) for the slope parameter in a linear regression model with a single stochastic regressor. The median-unbiased MLBEs are interesting as they can be robust to heavy-tailed samples and, hence, preferable to the ordinary least squares estimator (LSE). Two different cases are considered as we investigate the statistical properties of the MLBEs. In the first case, the regressor and error is assumed to follow a symmetric stable distribution. In the second, other types of regressions, with potentially contaminated errors, are considered. For both cases the consistency and exact finite-sample distributions of the MLBEs are established. Some results for the corresponding limiting distributions are also provided. In addition, we illustrate how our results can be extended to include certain heteroskedastic and multiple regressions. Finite-sample properties of the MLBEs in comparison to the LSE are investigated in a simulation study.     

Comparison of order statistics in two-sample problem in the sense of Pitman closeness

In this paper, we study the Pitman measure of closeness of order statistics of two independent samples from the same distribution to population quantiles. We then derive various exact expressions of the probability closeness of order statistics from the X and Y samples. Some distribution-free results for the median of the sampling distribution are obtained. Exact and explicit expressions are presented for Uniform(?1, 1) and exponential distributions. Numerical results for illustrative purposes are also provided.     

The exact risks of some pre-test and stein-type regression estimators umder balanced loss

In regression analysis we are often interested in using an estimator which is “precise” and which simultaneously provides a model with “good fit”, In this paper we consider the risk properties of several estimators of the regression coefficient vector "trader “balanced” loss, This loss function (Zellner, 1994) reflects both of the described attributes. Under a particular form of balanced loss, we derive the predictive risk of the pre-test estimator which results after a test for exact linear restrictions on the coefficient vector. The corresponding risks of Stein-rule and positive-part Stein-rale estimators are also established. The risks based on loss functions which allow only for estimation precision, or only for goodness of fit, are special cases of our results, and we draw appropriate comparisons, In particular, we show that some of the well-known results under (quadratic) precision-only loss are not robust to our generalization of the loss function     

Developing a restricted two-parameter Liu-type estimator: A comparison of restricted estimators in the binary logistic regression model

In the context of estimating regression coefficients of an ill-conditioned binary logistic regression model, we develop a new biased estimator having two parameters for estimating the regression vector parameter β when it is subjected to lie in the linear subspace restriction Hβ = h. The matrix mean squared error and mean squared error (MSE) functions of these newly defined estimators are derived. Moreover, a method to choose the two parameters is proposed. Then, the performance of the proposed estimator is compared to that of the restricted maximum likelihood estimator and some other existing estimators in the sense of MSE via a Monte Carlo simulation study. According to the simulation results, the performance of the estimators depends on the sample size, number of explanatory variables, and degree of correlation. The superiority region of our proposed estimator is identified based on the biasing parameters, numerically. It is concluded that the new estimator is superior to the others in most of the situations considered and it is recommended to the researchers.     

Comparison of estimators of the location parameter using pitman's closeness criterion

Necessary and sufficient conditions for a linear estimator to dominate another linear estimator of a location parameter under the Pitman's criterion of comparison are discussed. Consequently it is demonstrated that a linear biased estimator can not dominate a linear unbiased estimator under Pitman's criterion and that the sample mean is the Closest Linear Unbiased Estimator (CLUE). It is also shown that the ridge regression estimator with a known biasing constant can not dominate the ordinary least squares estimator. If an estimator δdominates an estimator δin the average loss sense then sufficient conditions are obtained under which δis also preferred over δunder Pitman's criterion. Further we obtain sufficient conditions under which preference under the Pitman's criterion will lead to preference under the mean squared error sense.