Improvement of the Liu estimator in linear regression model

In the presence of stochastic prior information, in addition to the sample, Theil and Goldberger (1961) introduced a Mixed Estimator for the parameter vector β in the standard multiple linear regression model (T,Xβ,σ² I). Recently, the Liu estimator which is an alternative biased estimator for β has been proposed by Liu (1993). In this paper we introduce another new Liu type biased estimator called Stochastic restricted Liu estimator for β, and discuss its efficiency. The necessary and sufficient conditions for mean squared error matrix of the Stochastic restricted Liu estimator to exceed the mean squared error matrix of the mixed estimator will be derived for the two cases in which the parametric restrictions are correct and are not correct. In particular we show that this new biased estimator is superior in the mean squared error matrix sense to both the Mixed estimator and to the biased estimator introduced by Liu (1993).     

Biased estimation of a variance

Summary Let , whereX _i are i.i.d. random variables with a finite variance σ² and is the usual estimate of the mean ofX _i. We consider the problem of finding optimal α with respect to the minimization of the expected value of |S ²(σ)−σ²|^k for variousk and with respect to Pitman's nearness criterion. For the Gaussian case analytical results are obtained and for some non-Gaussian cases we present Monte Carlo results regarding Pitman's criteron. This research was supported by Science Fund of Serbia, grant number 04M03, through Mathematical Institute, Belgrade.     

Conditional estimation of average on the basis of weighting data

LetF(x,y) be a distribution function of a two dimensional random variable (X,Y). We assume that a distribution functionF _x(x) of the random variableX is known. The variableX will be called an auxiliary variable. Our purpose is estimation of the expected valuem=E(Y) on the basis of two-dimensional simple sample denoted by:U=[(X ₁, Y₁)…(X_n, Y_n)]=[X Y]. LetX=[X ₁…X _n]andY=[Y ₁…Y _n].This sample is drawn from a distribution determined by the functionF(x,y). LetX _(k)be the k-th (k=1, …,n) order statistic determined on the basis of the sampleX. The sampleU is truncated by means of this order statistic into two sub-samples: % MathType!End!2!1! and % MathType!End!2!1!.Let % MathType!End!2!1! and % MathType!End!2!1! be the sample means from the sub-samplesU _k,1 andU _k,2, respectively. The linear combination % MathType!End!2!1! of these means is the conditional estimator of the expected valuem. The coefficients of this linear combination depend on the distribution function of auxiliary variable in the pointx _(k).We can show that this statistic is conditionally as well as unconditionally unbiased estimator of the averagem. The variance of this estimator is derived. The variance of the statistic % MathType!End!2!1! is compared with the variance of the order sample mean. The generalization of the conditional estimation of the mean is considered, too.     

Estimation of a normal mean relative to balanced loss functions

LetX ₁,…,X _nbe a random sample from a normal distribution with mean θ and variance σ². The problem is to estimate θ with Zellner's (1994) balanced loss function, % MathType!End!2!1!, where 0<ω<1. It is shown that the sample mean % MathType!End!2!1!, is admissible. More generally, we investigate the admissibility of estimators of the form % MathType!End!2!1! under % MathType!End!2!1!. We also consider the weighted balanced loss function, % MathType!End!2!1!, whereq(θ) is any positive function of θ, and the class of admissible linear estimators is obtained under such loss withq(θ) =e ^θ .     

Efficiency comparison of M-estimates for scale at t-distributions

Using Fisher's information fort-distributions, the absolute asymptotic efficiency of some M-estimates for scale with known location parameter is calculated and graphically illustrated. The compared estimators are the standard deviationS _*, the mean absolute deviation, called mean deviationD _*, the median absolute deviation, called MAD_*, and some M-estimates for scale, one, which is very robust, and another one with high asymptotic efficiency fort-distributions close to the normal. The last one is considered with monotone (in the positive field) and with very late redescending χ-function too. Also the , an alternative and generalized excess measure defined as the double relative asymptotic variance of the underlying scale estimator in the previous paper, is calculated fort-distributions and graphically illustrated, because there is the relation that the higher the asymptotic efficiency of is, the lower is the corresponding .     

Optimal testing for equicorrelated linear regression models

Recently, Knautz and Trenkler (1993) considered Christensen’s (1987) equicorrelated linear regression model as an example to show that S² and are independent even though the disturbances are equicorrelated. This paper addresses the issue of testing for the equicorrelation coefficient in the linear regression model based on survey data. It computes exact and approximate critical values using Point optimal and F-test statistics, respectively. An empirical comparison of these critical values at five percent nominal level are presented to demonstrate the performance of the new tests.     

Estimation of the exponential mean time to failure under a weighted balanced loss function

This paper considers estimation of a exponential mean time to failure using a loss function that reflects both goodness of fit and precision of estimation. The admissibility and inadmissibility of a class of linear estimators of the form are studied.     

Approximation of density functions by orthogonal series with grouped data

Summary Letg(x) andf(x) be continuous density function on (a, b) and let {ϕ_j} be a complete orthonormal sequence of functions onL ₂(g), which is the set of squared integrable functions weighted byg on (a, b). Suppose that over (a, b). Given a grouped sample of sizen fromf(x), the paper investigates the asymptotic properties of the restricted maximum likelihood estimator of density, obtained by setting all but the firstm of the ϑ_j’s equal to0. Practical suggestions are given for performing estimation via the use of Fourier and Legendre polynomial series. Research partially supported by: CNR grant, n. 93. 00837. CT10.     

Generating random numbers of prescribed distribution using physical sources

When constructing uniform random numbers in [0, 1] from the output of a physical device, usually n independent and unbiased bits B _j are extracted and combined into the machine number . In order to reduce the number of data used to build one real number, we observe that for independent and exponentially distributed random variables X _n (which arise for example as waiting times between two consecutive impulses of a Geiger counter) the variable U _n : = X _{2n – 1}/(X _{2n – 1} + X _2n ) is uniform in [0, 1]. In the practical application X _n can only be measured up to a given precision (in terms of the expectation of the X _n ); it is shown that the distribution function obtained by calculating U _n from these measurements differs from the uniform by less than /2.We compare this deviation with the error resulting from the use of biased bits B _j with P {B _j = 1{ = (where ] – [) in the construction of Y above. The influence of a bias is given by the estimate that in the p-total variation norm Q ^TV _p = ( |Q()| ^p )^1/p (p 1) we have P _Y – P ⁰ _Y ^TV _p (c _n · )^1/p with c _n p for n . For the distribution function F _Y – F ⁰ _Y 2(1 – 2^–n )|| holds.     

Improved estimation of the mean in one-parameter exponential families with known coefficient of variation

The value for which the mean square error of a biased estimatoraT for the mean μ is less than the variance of an unbiased estimatorT is derived by minimizingMSE(aT). The resulting optimal value is 1/[1+c(n)v ²], wherev=σ/μ, is the coefficient of variation. WhenT is the UMVUE , thenc(n)=1/n, and the optimal value becomes 1/(n+v ²) (Searls, 1964). Whenever prior information about the size ofv is available the shrinkage procedure is useful. In fact for some members of the one-parameter exponential families it is known that the variance is at most a quadratic function of the mean. If we identify the pertinent coefficients in the quadratic function, it becomes easy to determinev.     

Estimating the expected value of fuzzy random variables in random samplings from finite populations

In this paper we consider the problem of estimating the expected value of a fuzzy-valued random element in random samplings from finite populations. To this purpose, we quantify the associated sampling error by means of a parameterized measure we have introduced in a previous paper. Keywords: Aumann's integral, expected value of a fuzzy random variable, fuzzy random variable, -mean squared dispersion, random samplings, random set.     

Improved estimators of common variance of <Emphasis Type="Italic">p</Emphasis>-populations when Kurtosis is known

The pooled variance of p samples presumed to have been obtained from p populations having common variance σ², has invariably been adopted as the default estimator for σ². In this paper, alternative estimators of the common population variance are developed. These estimators are biased and have lower mean-squared error values than . The comparative merit of these estimators over the unbiased estimator is explored using relative efficiency (a ratio of mean-squared error values).     

Comparison of location parameters of two exponential distributions when scale parameters are different and unknown

Letx _i(1)≤x _i(2)≤…≤x _i(r_i) be the right-censored samples of sizesn _i from theith exponential distributions $\sigma _i^{ - 1} exp\{ - (x - \mu _i )\sigma _i^{ - 1} \} ,i = 1,2$ where μ_i and σ_i are the unknown location and scale parameters respectively. This paper deals with the posteriori distribution of the difference between the two location parameters, namely μ₂-μ₁, which may be represented in the form $\mu _2 - \mu _1 \mathop = \limits^\mathcal{D} x_{2(1)} - x_{1(1)} + F_1 \sin \theta - F_2 \cos \theta $ where $\mathop = \limits^\mathcal{D} $ stands for equal in distribution,F _i stands for the central F-variable with [2,2(r _i?1)] degrees of freedom and $\tan \theta = \frac{{n_2 s_{x1} }}{{n_1 s_{x2} }}, s_{x1} = (r_1 - 1)^{ - 1} \left\{ {\sum\limits_{j = 1}^{r_i - 1} {(n_i - j)(x_{i(j + 1)} - x_{i(j)} )} } \right\}$ The paper also derives the distribution of the statisticV=F ₁ sin σ?F ₂ cos σ and tables of critical values of theV-statistic are provided for the 5% level of significance and selected degrees of freedom.     

Constructing an unbiased estimator of population mean in finite populations using auxiliary information

The objective of this paper is to construct an unbiased estimator (up to order 0(1/n)) of the population mean of the study variatey which is more efficient than the sample mean of the ‘n’ obsrvedy-values. In particular, the unbiased estimators are discussed for the cases of positive and negative correlations of the study variatey and the auxiliary variatex.     

Perfect simulation of positive Gaussian distributions

We provide an exact simulation algorithm that produces variables from truncated Gaussian distributions on ( ₊) ^p via a perfect sampling scheme, based on stochastic ordering and slice sampling, since accept-reject algorithms like the one of Geweke (1991) and Robert (1995) are difficult to extend to higher dimensions.     

Sequential experimental design and response optimisation

We consider the situation where one wants to maximise a functionf(θ,x) with respect tox, with θ unknown and estimated from observationsy _k . This may correspond to the case of a regression model, where one observesy _k =f(θ,x _k )+ε _k , with ε _k some random error, or to the Bernoulli case wherey _k ∈{0, 1}, with Pr[y _k =1|θ,x _k |=f(θ,x _k ). Special attention is given to sequences given by , with an estimated value of θ obtained from (x₁, y₁),...,(x _k ,y _k ) andd _k (x) a penalty for poor estimation. Approximately optimal rules are suggested in the linear regression case with a finite horizon, where one wants to maximize ∑ _i=1 ^N w _i f(θ, x _i ) with {w _i } a weighting sequence. Various examples are presented, with a comparison with a Polya urn design and an up-and-down method for a binary response problem.     

Likelihood as an index of fit

Summary In this paper likelihood is characterized as an index which measures how much a model fits a sample. Some properties required to an index of fit are introduced and discussed, while stressing how they describe aspects inner to idea of fit. Finally we prove that, if an index of fit is maximal when the model reaches the distribution of the sample, then such an index is an increasing continuous transform of , where thep _i's are the theoretical relative frequencies provided by the model and theq _i's are the actual relative frequencies of the sample.     

Convexity-based clustering criteria: theory,algorithms, and applications in statistics

This paper deals with the construction of optimum partitions of for a clustering criterion which is based on a convex function of the class centroids as a generalization of the classical SSQ clustering criterion for n data points. We formulate a dual optimality problem involving two sets of variables and derive a maximum-support-plane (MSP) algorithm for constructing a (sub-)optimum partition as a generalized k-means algorithm. We present various modifications of the basic criterion and describe the corresponding MSP algorithm. It is shown that the method can also be used for solving optimality problems in classical statistics (maximizing Csiszárs -divergence) and for simultaneous classification of the rows and columns of a contingency table.     

Comparing several exponential populations with more than one control

Suppose there are k ₁ (k ₁ ≥ 1) test treatments that we wish to compare with k ₂ (k ₂ ≥ 1) control treatments. Assume that the observations from the ith test treatment and the jth control treatment follow a two-parameter exponential distribution and , where θ is a common scale parameter and and are the location parameters of the ith test and the jth control treatment, respectively, i = 1, . . . ,k ₁; j = 1, . . . ,k ₂. In this paper, simultaneous one-sided and two-sided confidence intervals are proposed for all k ₁ k ₂ differences between the test treatment location and control treatment location parameters, namely , and the required critical points are provided. Discussions of multiple comparisons of all test treatments with the best control treatment and an optimal sample size allocation are given. Finally, it is shown that the critical points obtained can be used to construct simultaneous confidence intervals for Pareto distribution location parameters.