Class Of Linear Unbiased Estimators Research Articles

Aitken’s Generalized Least Square Method for Estimating Parameter of Demand Function of Animal Protein In Indonesia

Ordinary least square (OLS) method is the most popular and commonly technique used to estimate numerical values of parameters of selected regression models. This was due to its unbiasedness property. However, when elements of dependent variable have unequal variances and/or correlated each other, there is no guarantee that the OLS estimator will show the most efficient within the class of linear unbiased estimators. For conditions generally encountered, GLS method is proposed an estimation procedure which yields coefficient estimators at least asymptotically more efficient than single equation OLS estimator. This method is derived by Aitken and it’s named Aitken GLS. This paper reported a study of application of Aitken’s GLS method for estimating parameter of demand function of animal protein in Indonesia of which have a system equation. This system of equation causes violation of the assumptions of homoscedasticity and independency of estimated parameters. Secondary data obtained from the Central Bureau of Statistics 2016 in 34 provinces in Indonesia was used in this study. Animal food was grouped in terms of fish, meat, eggs and milk. Results showed that error of those three equations were correlated. It suggests that GLS method should be used. Normality and homogenity assumptions were not violated. The determination coefficient was 99%, indicating that the method was very good.

On the Loss Robustness of Least-Square Estimators

The article revisits univariate and multivariate linear regression models. It is shown that least-square estimators (LSEs) are minimum risk estimators in general class of linear unbiased estimators under some general divergence loss. This amounts to the loss robustness of LSEs.

The best linear unbiased estimator in a singular linear regression model

In this paper, the best linear unbiased estimator of regression coefficients in the singular linear model was considered. Under the weighted balanced loss function the minimum risk properties of linear estimators of regression coefficients in the class of linear unbiased estimators are derived. Some kinds of relative efficiencies of the best linear unbiased estimator are given, and the lower bounds or upper bounds of these relative efficiencies are also presented.

Optimal designs for comparing regression models with correlated observations

The problem under investigation is that of efficient statistical inference for comparing two regression curves estimated from two samples of dependent measurements. Based on a representation of the best pair of linear unbiased estimators in continuous time models as a stochastic integral, a pair of linear unbiased estimators with corresponding optimal designs for finite sample size is constructed. This pair minimises the width of the confidence band for the difference between the estimated curves in a class of linear unbiased estimators approximating the stochastic integrals and is very close to the pair of weighted least squares estimators with corresponding optimal design. Thus results readily available in the literature are extended to the case of correlated observations and an easily implementable solution is provided which is practically non distinguishable from the weighted least squares estimators. The advantages of using the proposed pairs of estimators with corresponding optimal designs for the comparison of regression models are illustrated via several numerical examples.

Theory of integer equivariant estimation with application to GNSS

Carrier phase ambiguity resolution is the key to high-precision global navigation satellite system (GNSS) positioning and navigation. It applies to a great variety of current and future models of GPS, modernized GPS and Galileo. The so-called ‘fixed’ baseline estimator is known to be superior to its ‘float’ counterpart in the sense that its probability of being close to the unknown but true baseline is larger than that of the ‘float’ baseline, provided that the ambiguity success rate is sufficiently close to its maximum value of one. Although this is a strong result, the necessary condition on the success rate does not make it hold for all measurement scenarios. It is discussed whether or not it is possible to take advantage of the integer nature of the ambiguities so as to come up with a baseline estimator that is always superior to both its ‘float’ and its ‘fixed’ counterparts. It is shown that this is indeed possible, be it that the result comes at the price of having to use a weaker performance criterion. The main result of this work is a Gauss–Markov-like theorem which introduces a new minimum variance unbiased estimator that is always superior to the well-known best linear unbiased (BLU) estimator of the Gauss–Markov theorem. This result is made possible by introducing a new class of estimators. This class of integer equivariant estimators obeys the integer remove–restore principle and is shown to be larger than the class of integer estimators as well as larger than the class of linear unbiased estimators. The minimum variance unbiased estimator within this larger class is referred to as the best integer equivariant (BIE) estimator. The theory presented applies to any model of observation equations having both integer and real-valued parameters, as well as for any probability density function the data might have.

Efficient Non-Iterative and Nonparametric Estimation of Heterogeneity Variance for the Standardized Mortality Ratio

In this paper the situation of extra population heterogeneity in the standardized mortality ratio is discussed from the point-of-view of an analysis of variance. First, some simple non-iterative ways are provided to estimate the variance of the heterogeneity distribution without estimating the heterogeneity distribution itself. Next, a wider class of linear unbiased estimators is introduced and their properties investigated. Consistency is shown for a wide sub-class of estimators charactererized by the fact that the associated linear weights are within some positive, finite bounds. Furthermore, it is shown that an efficient estimator is often provided when the weights are proportional to the expected counts.

Minimal Complete Class of Linear Unbiased Estimators of Population Mean for a Special Sampling Design1

In this paper we give a complete description of the minimal complete subclass of C the class of all homogeneous linear unbiased estimators of a finite population mean for the extremely special case of taking sample of size 2 units from a population of size 4, where only samples containing units ( U1, Ui+ 1) have equal positive probability.

Reduction of Linear Models Using Correct or Incorrect Prior Information

AbstractWhen there is no prior knowledge on the parameter vector β of the linear model then the LSE is most favourable at least in the class of linear unbiased estimators. On the other hand, in many practical problems one has some guessed estimate of β. Using this information leads to two‐step estimation procedures which may or may not dominate the LSE with respect to MSE. The dominance depends on the degree of incorrectness of the guessed parameter and is analyzed numerically for the case of sample reduction.

Testing for incremental information content in the presence of collinearity

A number of recent research papers use two-stage procedures in lieu of a single multiple regression, in some cases purportedly as a solution to colinearity among independent variables. We demonstrate that, since collinearity is inherently a data problem rather than a statistical problem, no partitions of dependent or independent variables, orthogonal or otherwise, can provide insights into the relative influence of collinear variables. For the class of linear unbiased estimators this follows directly from the Gauss-Markov Theorem, but we demonstrate some of the results in detail as an aid to interpreting particular papers.

On Admissibility and Completeness of Linear Unbiased Estimators in a General Linear Model

Abstract Admissibility and completeness of linear unbiased estimators in a linear model with general covariance structure are examined. Under rather unrestrictive conditions, a minimal complete class of linear unbiased estimators is described.

Invariant Quadratic Unbiased Estimation for Two Variance Components

For a normally distributed mixed model with two unknown variance components $\theta_1$ and $\theta_2$, a tractable characterization is given for the admissible estimators within the class $\tilde{\mathscr{N}}_\delta$ of invariant quadratic unbiased estimators for $\delta_1\theta_1 + \delta_2\theta_2$. Here the term admissible is used with reference only to the class $\tilde{\mathscr{N}}_\delta$. This characterization is based on a result for general linear models which characterizes the admissible estimators within the class of linear unbiased estimators. The admissibility of MINQUE estimators and the usual analysis of variance estimators is considered.

Post Data Model Evaluation

JHE PURPOSE of this article is to bring to the attention of the readers of this journal a number of related and important estimators that are currently being discussed in the statistical literature which have implications for applied work since rules are employed which seek to improve the performance of conventional estimators. In spite of the rapid advances, over the last three decades, of economic theory, econometric procedures, and data relating to economic processes and institutions, the search for quantitative economic knowledge still remains to some extent an essay in persuasion. In the process of nonexperimental model building there are typically many admissible economic and statistical models which do not contradict our perceived knowledge of human behavior. Thus, in model specification there is usually uncertainty, for example, relative to the algebraic form, classification, number and timing of variables to be included in the behavioral and technical relations, and the corresponding stochastic assumptions. When econometric models are correctly specified, statistical theory provides procedures for obtaining point and interval estimates and evaluating the performance of various linear and usually unbiased (at least asymptotically) estimators. But, the applied worker must inevitably work with false models, where the true specification of the sampling model is unknown. Furthermore, the statistical model employed is usually determined by some preliminary testing of hypotheses using the data at hand. This search procedure, involving two-stage or repeated significance test procedures applied to the same set of data and yielding an estimate after the preliminary test(s) if significance, is often used in applied work in economics with little or no information on the sampling properties of the resulting estimator and with little or no consideration to the possible distortion of subsequent inferences. Within this context, we, seek to generalize and extend the results of Wallace and Ashar (1972) relative to preliminary test or two-stage estimating procedures and call attention to another important class of estimators and estimator comparisons. In particular, we review the possible statistical consequences of using preliminary test or sequential estimators in the search process and suggest old and new estimators, that are superior, under a squared error loss measure for gauging estimator performance, to the conventional estimators usually employed. We also note that conventional estimating procedures currently used in applied work may not be appropriate for the problem at hand and, perhaps more importantly for the researcher, we show that better alternative estimators exist. Perhaps it is appropriate at this point to note that, when making a choice between estimators, the traditional solution is to restrict consideration to the class of linear unbiased estimators and hope that among the estimators in the restricted class, one has uniformly smallest risk. Fortunately for many problems a best linear unbiased estimate exists. In this paper, in discussing the estimators that are alternatives to the conventional least squares estimator, we will leave the class of linear unbiased estimators. The notion of unbiasedness which has been accepted by or perhaps forced on applied workers, although intuitively plausible, is an arbitrary restriction or property and has no direct connection with the loss due to incorrect decisions. The economist who is interested in parameter estimates or predictions appropriate for choice purposes, may not care if he is right on the average, and thus the unbiasedness property may be unsatisfactory from a decision point of view. In any event our purpose, which is to some extent expository in nature, is to focus on point estimation under a squared error loss measure of goodness and bring the statistical consequences of making use of conventional and Received for publication October 2, 1972. Revision accepted for publication April 3, 1973. *Arnold Zellner read an early draft of this paper and made many helpful comments.

Regression with Non-Gaussian Stable Disturbances: Some Sampling Results

IN THEIR PAPER [1] with the above title, Blattberg and Sargent have suggested a method for estimating regression parameters when disturbances are generated by a symmetric stable law. This procedure was originally suggested by John Wise [4]. Wise arrived at this estimator by confining attention to the class of linear unbiased estimators and minimizing the scale parameter. He also suggested that his procedure might be considered as a generalization of the classical least squares procedure which fails when the second moments of the disturbances do not exist. Blattberg and Sargent have compared various estimators by using simulated data. The main purpose of this note is to show that the procedure suggested by Wise and Blattberg and Sargent has an optimal property, namely that it yields an estimator that has minimum mean absolute error among the class of all linear unbiased estimators. The model is the well-known linear regression model given by

A Comparison of Maximum Likelihood Versus Blue Estimators

T HE most frequently used estimating technique for applied economic research has been ordinary least squares (OLS). There are two theoretical justifications for its use. First, the Gauss-Markov theorem suggests that OLS estimators will outperform all other techniques in the class of linear unbiased estimators.1 Secondly, under the assumption that the error terms are normally distributed, OLS estimators can be derived as the maximum likelihood estimates. Frequently when OLS is introduced in elementary texts, the method of minimizing the sum of absolute deviations (MAD) is presented for comparison purposes, but rarely is it given serious consideration for applied uses.) Certainly one reason for such behavior stems from previous evaluations of OLS versus MAD estimators. Asher and Wallace (1963) found that the use of MAD meant one should be prepared to give up considerable efficiency 3 More recently Glahe and Hunt (1970), suggested several estimators derived under the general minimization criterion. In contrast to the Asher-Wallace study, the Glahe-Hunt model was a two-equation linear simultaneous system. Their results show that neither form of absolute deviation estimator outperformed either OLS or two-stage least squares.4 The purpose of this paper is to suggest that in at least one aspect these comparisons have given OLS a differential advantage. That is, both of these studies employed errors for their hypothesized models which were drawn from normal distributions. Consequently the OLS estimators are both maximum likelihood and best linear unbiased estimators (BLUE) under such circumstances. Since recently published works by Zeckhauser and Thompson (1970), Fama (1965) and others have called into question the assumption of normally distributed error terms, attention has begun to shift to other alternatives.5 Blattberg and Sargent (1971) have examined three techniques including both OLS and MAD when the errors are drawn from a stable Paretian distribution. Their findings indicate that the MAD estimator . performs sufficiently well that it deserves further study and elaboration. G Accordingly we have chosen to explore the relative merits of OLS and MAD for a single equation model whose errors are drawn from a double exponential parent distribution. This distribution was chosen because both estimators will exhibit theoretically desirable properties. OLS remains the BLUE estimator, while MAD is the maximum likelihood estimator. Furthermore, this distribution is one member of the power distribution suggested by Zeckhauser and Thompson as an alternative to the normal. This paper is divided into, three sections. The first describes the design of the experiments. Section II presents the empirical results and the last summarizes the primary findings of the paper.

Commutativity of the operations of optimum estimator and optimum sampling procedure for Horvitz and Thompson’sT 1 class of linear unbiased estimators

Commutativity of the operations of optimum estimator and optimum sampling procedure for Horvitz and Thompson’sT 1 class of linear unbiased estimators

On a Complete Class of Linear Unbiased Estimators for Randomized Factorial Experiments

Consider a factorial system of order $N = p^m$, which consists of $m$ factors each at $p$ levels. The factorial model relates the expected yield to the various treatment combinations in terms of a linear function of $N = p^m$ parameters $(\beta_0, \beta_1, \cdots, \beta_{N-1})$, which represent the main effects and interactions. A subset of $S = p^s(s < m)$ preassigned parameters is specified for estimation and testing of hypotheses. The other $N - S$ parameters are considered as nuisance ones. Unbiased estimates of the $S = p^s$ preassigned parameters may be obtained by different procedures of balanced random allocation designs (see Dempster [2], [3]). In the present paper we consider unbiased estimators with two types of randomized fractional replication, studied previously by Ehrenfeld and Zacks [4]. These procedures, designated by R.P.I. and R.P.II, are based on orthogonal fractional replication designs. As will be shown in a subsequent paper, both procedures have some optimal properties in cases where the nuisance parameters may assume arbitrary values. R.P.I. consists of choosing at random, with or without replacement, $n$ blocks of treatment combinations from the set of $M = p^{m-s}$ blocks, constructed by confounding the nuisance parameters. R.P.II. consists of choosing at random $n$ treatment combinations independently from each one of the $p^s$ blocks, constructed by confounding the pre-assigned parameters. The estimator of the pre-assigned parameters studied in the previous paper is the "least squares estimator", commonly applied in fractional replication procedures (see Kempthorne [5], Cochran and Cox [1]). When all the nuisance parameters are zero then this estimator is the least-squares estimator, and thus the best unbiased linear estimator. However, if the nuisance parameters are not zero there is no uniformly best unbiased estimator. The first question raised is whether unbiased estimates of the pre-assigned parameters can be attained by choosing a block of treatment combinations, in a similar manner to R.P.I., but with unequal probabilities, (unbalanced designs). As proven in the present paper, unbiased estimates cannot be attained with a procedure that assigns unequal probabilities to different blocks. Thus, the class of linear unbiased estimators is studied with respect to R.P.I. The same estimators can be applied to R.P.II., or to any balanced allocation design that yields a factorial model with similar properties to those of R.P.I. The statistical model adopted in the present paper is the orthogonal factorial model. Accordingly, the main effects and interactions are represented by orthogonal contrasts of the expected values of the observed random variable. This is a common practice for measuring main-effects and interactions (see Kempthorne [5]). The resulting algebra and numerical procedures are relatively simple and free of the difficult problems associated with the approach of Dempster [2], [3]. Basic notions as well as the required algebra for the factorial model are represented in Section 2. Theorems concerning the specification of the class of linear unbiased estimators, and the characterization of the subclass of conditional least squares estimators are given in Section 3. In Section 4 we prove that the subclass of conditional least squares estimators is complete. For this purpose we first derive an explicit formula for the variance-covariance matrix of any linear unbiased estimator with R.P.I.; and then we show that to any linear unbiased estimator not in the subclass of the conditional least-squares estimators one can find a conditional least-squares estimator so that the difference between the corresponding variance-covariance matrices is positive definite. For simplifying the arguments and notation, the definitions and theorems in Sections 3 and 4 are given relative to a particular choice of the pre-assigned parameters for special defining parameters, by which the classification of the treatment combinations into blocks in R.P.I. is carried out. In Section 5 the results are generalized for arbitrary sets of pre-assigned and defining parameters. In Section 6 the results are extended for R.P.II.

Optimal Strategies in Factorial Experiments

The present study is an extension of the previous work of the authors [3], [5], in which two randomization procedures in fractional factorial experiments were investigated. The general problem is to choose, in an optimal manner, a fractional replication of a full factorial system and an estimator, for the purpose of making inferences concerning a subset of pre-assigned parameters. We consider factorial systems of order $2^m$, which consist of $m$ factors each at 2 levels. The results can be generalized for factorial systems of order $p^m$, where $p > 2$. The factorial model adopted for the present study is the same as that adopted in the previous work, and the reader is referred to the preceeding papers [3], [5] for details and properties. The first two sections of [5] are essential for the present paper. The statistical properties of the conditional least-squares estimators (c.l.s.e.) were studied in [5] with respect to two specified randomization procedures (R.P.I. and R.P.II.), which are particular types of random allocation designs. It was shown that the c.l.s.e.'s constitute a complete class of linear unbiased estimators. The c.l.s.e. can be characterized as least-square estimators adjusted for the block of treatment combinations chosen and the information available on the nuisance parameters. In the present study we extend the investigation into the comparison of different randomization procedures. We consider a general class of procedures, characterized as follows: By some confounding method we construct $M = 2^{m-s}$ blocks, each one containing $S = 2^s$ treatment combinations. We choose one of the blocks with an arbitrary probability vector, $\xi$, and observe the associated random variables. R.P.I. is the special case where every block has the same probability of being chosen. A fixed fractional replication procedure is the special case where one of the blocks is chosen with probability one. Each randomization procedure is represented (uniquely) by a probability vector, $\xi$, and each c.l.s.e. is represented by a vector $\gamma$ in $(2^m - 2^s)$-dimensional Euclidean space. A strategy of the Statistician is thus represented by a pair of vectors $(\xi, \gamma)$. Minimax strategies are studied in the present paper for two states of information concerning the nuisance parameters: (i) all the nuisance parameters are bounded; (ii) all the nuisance parameters are bounded and all their signs are known. As proven in the present paper, the minimax strategy corresponding to Case (i) consists of R.P.I. with an unadjusted c.l.s.e. and is thus independent of the actual bounds of the nuisance parameters. On the other hand, the minimax strategy for Case (ii) consists of some fixed fractional replication with an adjusted c.l.s.e. which depends on the actual bounds of the nuisance parameters. These minimax theorems are proven with respect to a mean-square-error loss function, defined as the trace of the mean square-error matrix. A closeness loss function is considered too. The closeness of an estimator is defined as the probability that the values of the estimator will lie in a prescribed neighborhood of the true values of the parameters. In Section 2 the mean-square-error matrix and the closeness of a c.l.s.e., under an arbitrary randomization procedure, $\xi$, are derived. The mean-square-error and the closeness loss functions are defined in Section 3. Formulae for mean-square-error Bayes strategies, against any given a priori distribution of the nuisance parameters, are then derived. The closeness risk function is approximated by a similar function which has the same Bayes strategies as the mean-square-error loss function. It is shown that when the size of the experiment, $S = 2^s$, grows the closeness risk function and the approximative function converge. The minimax theorems are stated and proved in Section 4 only with respect to the mean-square-error loss function. It can be shown that the minimax closeness strategies, for the states of information studied, are the same as those for the mean-square-error loss function. This can be concluded also from the results of Section 3. In Section 5 we present a numerical example to illustrate the results and the computations involved.