Invariant methods for estimating variance components in mixed linear models 2

The subject of this contribution is to present a survey on new methods for variance component estimation, which appeared in the literature in recent years. Starting from mixed models treated in analysis of variance research work on this field turned over to a more general approach in which the covariance matrix of the vector of observations is assumed to be a unknown linear combination of known symmetric matrices. Much interest has been shown in developing some kinds op optimal estimators for the unknown parameters and most results were obtained for estimators being invariant with respect to a certain group of translations. Therefore we restrict attention to this class of estimates. We will deal with minimum variance unbiased estimators, least squared errors estimators, maximum likelihood estimators. Bayes quadratic estimators and show some relations to the mimimum norm quadratic unbiased estimation principle (MINQUE) introduced by C. R. Rao [20]. We do not mention the original motivation of MINQUE since the otion of minimum norm depends on a measure that is not accepted by all statisticians. Also we do‘nt deal with other approaches like the BAYEsian and fiducial methods which were successfully applied by S. Portnoy [18], P. Rusolph [22], G. C. Tiao, W. Y. Tan [28], M. J. K. Healy [9] and others, although in very special situations, only. Additionally we add some new results and also new insight in the properties of known estimators. We give a new characterization of MINQUE in the class of all estimators, extend explicite expressions for locally optimal quadratic estimators given by C. R. Rao [22] to a slightly more general situation and prove complete class theorems useful for the computation of BAYES quadratic estimators. We also investigate situations in which BAYES quadratic unbiased estimators do'nt change if the distribution of the error terms differ from the normal distribution.     

Large deviations for estimators of some threshold parameters

In this paper, we consider a class of statistical models with a real-valued threshold parameter, which is either the minimum or the maximum of the support of the sampling distribution. We prove large deviation principles for sequences of estimators (maximum likelihood estimators and posterior distributions) as the sample size goes to infinity. Furthermore we illustrate some connections with the analogous large deviation results for the natural exponential families.     

Estimating the minimum and maximum location parameters for two ig-exponential scale mixtures

Suppose that we have two components, each having a two-parameter exponential distribution. Suppose further that these components are conditionally independent, sharing a common random hazard rate and possessing unequal, fixed, unknown location parameters. We develop estimators for the minimum and maximum of these location parameters when the random hazard rate has an inverse Gaussian distribution. Performance comparisons are made among the proposed estimators. Maximum likelihood estimators are shown to be inadmissible.     

Estimating the Mean with Known Coefficient of Variation

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.     

On the efficiencies of some common quick estimators

The quick estimators of location and scale have broad applications and are widely used. For a variety of symmetric populations we obtain the quantiles and the weights for which the asymptotic variances of the quick estimators are minimum. These optimal quick estimators are then used to obtain the asymptotic relative efficiencies of the commonly used estimators such as trimean. gastwirth. median, midrange. and interquartile range with respect to the optimal quick estimators in order to determine a choice among them and to check whether they are unacceptably poor. In the process it is seen that the interquartile range is the optimal quick estimator of scale for Cauchy populations; but the interdecile range is in general preferable. Also the optimal estimator of the location for the logistic distribution puts weights 0.3 on each of the two quartiles and 0.4 on the median. It is shown that for the symmetric distributions, such as the beta and Tukey- lambda with [d] > 0, which have finite support and short tails, i.e. the tail exponents (Parzen, 1979) satisfy [d] < 1, the midrange and the range are the optimal quick estimators of location and scale respectively if [d] < 1/2. The class of such distributions Include the distributions with high discontinuous tails, e.g. Tukey-lambda with [d] > 1, as well as some distributions with p.d.f.'s going to zero at the ends of the finite support, such as Tukey-lambda with 1/2 < [d] < 1. As a byproduct an interesting tail correspondence between beta and Tukey-lambda distributions is seen.     

Estimation of parameters and the mean life of a mixed failure time distribution

This paper deals with estimation of parameters and the mean life of a mixed failure time distribution that has a discrete probability mass at zero and an exponential distribution with mean O for positive values. A new sampling scheme similar to Jayade and Prasad (1990) is proposed for estimation of parameters. We derive expressions for biases and mean square errors (MSEs) of the maximum likelihood estimators (MLEs). We also obtain the uniformly minimum variance unbiased estimators (UMVUEs) of the parameters. We compare the estimator of O and mean life fj based on the proposed sampling scheme with the estimators obtained by using the sampling scheme of Jayade and Prasad (1990).     

On the umvu estimators after using a normalizing transformation

The conditional distribution given complete sufficient statistics is used along with the Rao-Blackwell theorem to obtain uniformly minimum variance unbiased (UMVU) estimators after a transformation to normality has been applied to data. The estimators considered are for the mean, the variance and the cumulative distribution of the original non-normal data. Previous procedures to obtain UMVU estimators have used Laplace transforms, Taylor expansions and the jackknife. An integration method developed in this paper requires only integrability of the normalizing transformation function. This method is easy to employ and it is always possible to obtain a numerical result.     

Approximate mean squared errors of estimations of reliability in the two-parameter exponential case

Combining the method used i n Chao (1981) and conditional procedure, we extend our previous results for one-parameter exponential to two-parameter case, i .e. we provide simple approximation formulas for the mean squared errors of the maximum likelihood and minimum variance unbiased estimators of reliability of general k-out-of-m systems when the component lifetimes are independent and follow a two-parameter exponential distribution.     

Estimation of parameters for the truncated exponential distribution

We consider the right truncated exponential distribution where the truncation point is unknown and show that the ML equation has a unique solution over an extended parameter space. In the case of the estimation of the truncation point T we show that the asymptotic distribution of the MLE is not centered at T. A modified MLE is introduced which outperforms all other considered estimators including the minimum variance unbiased estimator. Asymptotic as well as small sample properties of different estimators are investigated and compared. The truncated exponential distribution has an increasing failure rate, ideally suited for use as a survival distribution for biological and industrial data.     

Estimation of finite population variance in successive sampling using multi-auxiliary variables

ABSTRACT

In this paper, a general class of estimators for estimating the finite population variance in successive sampling on two occasions using multi-auxiliary variables has been proposed. The expression of variance has also been derived. Further, it has been shown that the proposed general class of estimators is more efficient than the usual variance estimator and the class of variance estimators proposed by Singh et al. (2011) when we used more than one auxiliary variable. In addition, we support this with the aid of numerical illustration.     

Asymptotically optimum estimation of a probability in inverse binomial sampling under general loss functions

The optimum quality that can be asymptotically achieved in the estimation of a probability p using inverse binomial sampling is addressed. A general definition of quality is used in terms of the risk associated with a loss function that satisfies certain assumptions. It is shown that the limit superior of the risk for p asymptotically small has a minimum over all (possibly randomized) estimators. This minimum is achieved by certain non-randomized estimators. The model includes commonly used quality criteria as particular cases. Applications to the non-asymptotic regime are discussed considering specific loss functions, for which minimax estimators are derived.     

Concentration Ellipsoids,Their Planes of Support,and the Linear Regression Model

The relationship between the concentration ellipsoid of a random vector and its planes of support is exploited to provide a geometric derivation and interpretation of existing results for a general form of the linear regression model. In particular, the planes of support whose points of tangency to the ellipsoid are contained in the range (or column space) of the design matrix are the source of all linear unbiased minimum variance estimators. The connection between this idea and estimators based on projections is explored, as is also its use in obtaining and interpreting some existing relative efficiency results.     

Variance reduction of estimators arising from Metropolis–Hastings algorithms

The Metropolis–Hastings algorithm is one of the most basic and well-studied Markov chain Monte Carlo methods. It generates a Markov chain which has as limit distribution the target distribution by simulating observations from a different proposal distribution. A proposed value is accepted with some particular probability otherwise the previous value is repeated. As a consequence, the accepted values are repeated a positive number of times and thus any resulting ergodic mean is, in fact, a weighted average. It turns out that this weighted average is an importance sampling-type estimator with random weights. By the standard theory of importance sampling, replacement of these random weights by their (conditional) expectations leads to more efficient estimators. In this paper we study the estimator arising by replacing the random weights with certain estimators of their conditional expectations. We illustrate by simulations that it is often more efficient than the original estimator while in the case of the independence Metropolis–Hastings and for distributions with finite support we formally prove that it is even better than the “optimal” importance sampling estimator.     

On the improved estimation of a function of the scale parameter of an exponential distribution based on doubly censored sample

In the present article, we have studied the estimation of entropy, that is, a function of scale parameter ln⁡σ of an exponential distribution based on doubly censored sample when the location parameter is restricted to positive real line. The estimation problem is studied under a general class of bowl-shaped non monotone location invariant loss functions. It is established that the best affine equivariant estimator (BAEE) is inadmissible by deriving an improved estimator. This estimator is non-smooth. Further, we have obtained a smooth improved estimator. A class of estimators is considered and sufficient conditions are derived under which these estimators improve upon the BAEE. In particular, using these results we have obtained the improved estimators for the squared error and the linex loss functions. Finally, we have compared the risk performance of the proposed estimators numerically. One data analysis has been performed for illustrative purposes.     

Application of Bernstein Polynomials for smooth estimation of a distribution and density function

The empirical distribution function is known to have optimum properties as an estimator of the underlying distribution function. However, it may not be appropriate for estimating continuous distributions because of its jump discontinuities. In this paper, we consider the application of Bernstein polynomials for approximating a bounded and continuous function and show that it can be naturally adapted for smooth estimation of a distribution function concentrated on the interval [0,1] by a continuous approximation of the empirical distribution function. The smoothness of the approximating polynomial is further used in deriving a smooth estimator of the corresponding density. The asymptotic properties of the resulting estimators are investigated. Specifically, we obtain strong consistency and asymptotic normality under appropriate choice of the degree of the polynomial. The case of distributions with other compact and non-compact support can be dealt through transformations. Thus, this paper gives a general method for non-parametric density estimation as an alternative to the current estimators. A small numerical investigation shows that the estimator proposed here may be preferable to the popular kernel-density estimator.     

Minimax estimation of a probability of success under LINEX loss

Minimax estimation of a binomial probability under LINEX loss function is considered. It is shown that no equalizer estimator is available in the statistical decision problem under consideration. It is pointed out that the problem can be solved by determining the Bayes estimator with respect to a least favorable distribution having finite support. In this situation, the optimal estimator and the least favorable distribution can be determined only by using numerical methods. Some properties of the minimax estimators and the corresponding least favorable prior distributions are provided depending on the parameters of the loss function. The properties presented are exploited in computing the minimax estimators and the least favorable distributions. The results obtained can be applied to determine minimax estimators of a cumulative distribution function and minimax estimators of a survival function.     

Variance components estimation for the balanced random effects model under mixed prior distributions

For the balanced random effects models, when the variance components are correlated either naturally or through common prior structures, by assuming a mixed prior distribution for the variance components, we propose some new Bayesian estimators. To contrast and compare the new estimators with the minimum variance unbiased (MVUE) and restricted maximum likelihood estimators (RMLE), some simulation studies are also carried out. It turns out that the proposed estimators have smaller mean squared errors than the MVUE and RMLE.     

Logistic regression in meta-analysis using aggregate data

We derived two methods to estimate the logistic regression coefficients in a meta-analysis when only the 'aggregate' data (mean values) from each study are available. The estimators we proposed are the discriminant function estimator and the reverse Taylor series approximation. These two methods of estimation gave similar estimators using an example of individual data. However, when aggregate data were used, the discriminant function estimators were quite different from the other two estimators. A simulation study was then performed to evaluate the performance of these two estimators as well as the estimator obtained from the model that simply uses the aggregate data in a logistic regression model. The simulation study showed that all three estimators are biased. The bias increases as the variance of the covariate increases. The distribution type of the covariates also affects the bias. In general, the estimator from the logistic regression using the aggregate data has less bias and better coverage probabilities than the other two estimators. We concluded that analysts should be cautious in using aggregate data to estimate the parameters of the logistic regression model for the underlying individual data.     

JAMES-STEIN RULE ESTIMATORS IN LINEAR REGRESSION MODELS WITH MULTIVARIATE-t DISTRIBUTED ERROR

This paper considers estimation of β in the regression model y =Xβ+μ, where the error components in μ have the jointly multivariate Student-t distribution. A family of James-Stein type estimators (characterised by nonstochastic scalars) is presented. Sufficient conditions involving only X are given, under which these estimators are better (with respect to the risk under a general quadratic loss function) than the usual minimum variance unbiased estimator (MVUE) of β. Approximate expressions for the bias, the risk, the mean square error matrix and the variance-covariance matrix for the estimators in this family are obtained. A necessary and sufficient condition for the dominance of this family over MVUE is also given.     

Estimation of population distribution function involving measurement error in the presence of non response

Abstract

This article addresses the problem of estimating population distribution function for simple random sampling in the presence of non response and measurement error together. We suggest a general class of estimators for estimating the cumulative distribution function using the auxiliary information. The expressions for the bias and mean squared error are derived up to the first order of approximation. The performance of the proposed class of estimators is compared with considered estimators both theoretically and numerically. A real data set is used to support the theoretical findings.