Types of laws of random vectors and best estimators

Optimality of estimators of a vector parameter in terms of the probability of the estimators being contained in suitable region(s) around the parameter point is defined. Conditions under which optimal estimators in the usual senses are also optimal in the above sense are investigated.     

Moderate Deviations and Large Deviations for Kernel Density Estimators

Let f _n be the non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in ^d . It is proved that if the kernel function is an integrable function with bounded variation, and the common density function f of the random variables is continuous and f(x) 0 as |x| , then the moderate deviation principle and large deviation principle for hold.     

A Large Sample Approximation to the Profile Plug-in Upper Confidence Limit

We consider the problem of constructing a 1– upper confidence limit for the scalar parameter ₀ in the presence of the nuisance parameter vector ₀, when the data are discrete. The 'profile plug-in' upper confidence limit is introduced by Kabaila and Lloyd. This confidence limit is based on computing a P-value from an estimator of ₀, replacing the nuisance parameter by the profile maximum likelihood estimate for known, and equating to . Theoretical and numerical evidence for the good coverage properties of this confidence limit is presented by Kabaila and Lloyd. An upper confidence limit should be assessed not only by its coverage properties but also by how large this confidence limit is. We measure how large the profile plug-in upper limit is by using a large sample approximation to it. This large sample approximation is used to delineate further the good properties of this confidence limit.     

On Uniform Laws of Large Numbers for Ergodic Diffusions and Consistency of Estimators

Consider a regular diffusion process X with finite speed measure m. Denote the normalized speed measure by μ. We prove that the uniform law of large numbers holds if the class has an envelope function that is μ-integrable, or if is bounded in L _p(μ) for some p>1. In contrast with uniform laws of large numbers for i.i.d. random variables, we do not need conditions on the ‘size’ of the class in terms of bracketing or covering numbers. The result is a consequence of a number of asymptotic properties of diffusion local time that we derive. We apply our abstract results to improve consistency results for the local time estimator (LTE) and to prove consistency for a class of simple M-estimators. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bayes estimation with spherically symmetric,convex loss

Summary It is desired to estimate a parameter with the loss function of the formL(θ, a)=W(‖θ−a‖), where is convex, differentiable, and non-decreasing. With this structure a characterization of Bayes estimators is given. Also it is noted that if the sample space, , for the observation,X, is a complete separable metric space then a Bayes estimator exists.     

基于多元局部多项式方法的混沌时间序列预测

根据Takens定理，把混沌时间序列构造为一组序列对，然后用多元局部多项式方法来预测其序列.这种核估计方法可以结合局域法与全局法的优点，使得预测的精度更高.仿真结果表明，该方法非常有效.     

Use of a tree‐structured hierarchical model for estimation of location and uncertainty in multivariate spatial data

Analysis and modeling of spatial data are of considerable interest in many applications. However, the prediction of geographical features from a set of chemical measurements on a set of geographically distinct samples has never been explored. We report a new, tree‐structured hierarchical model for the estimation of geographical location of spatially distributed samples from their chemical measurements. The tree‐structured hierarchical modeling used in this study involves a set of geographic regions stored in a hierarchical tree structure, with each nonterminal node representing a classifier and each terminal node representing a regression model. Once the tree‐structured model is constructed, given a sample with only chemical measurements available, the predicted regional location of the sample is gradually restricted as it is passed through a series of classification steps. The geographic location of the sample can be predicted using a regression model within the terminal subregion. We show that the tree‐structured modeling approach provides reasonable estimates of geographical region and geographic location for surface water samples taken across the entire USA. Further, the location uncertainty, an estimate of a probability that a test sample could be located within a pre‐estimated, joint prediction interval that is much smaller than the terminal subregion, can also be assessed. Copyright © 2014 John Wiley & Sons, Ltd.     

Evaluation of the use of consensus values in proficiency testing programmes

Proficiency testing (PT) is an essential tool used by laboratory accreditation bodies to assess the competency of laboratories. Because of limited resources of PT providers or for other reasons, the assigned reference value used in the calculation of z-score values has usually been derived from some sort of consensus value obtained by central tendency estimators such as the arithmetic mean or robust mean. However, if the assigned reference value deviates significantly from the ‘true value’ of the analyte in the test material, laboratories’ performance will be evaluated incorrectly. This paper evaluates the use of consensus values in proficiency testing programmes using the Monte Carlo simulation technique. The results indicated that the deviation of the assigned value from the true value could be as large as 40%, depending on the parameters of the proficiency testing programmes under investigation such as sample homogeneity, number of participant laboratories, concentration level, method precision and laboratory bias. To study how these parameters affect the degree of discrepancy between the consensus value and the true value, a fractional factorial design was also applied. The findings indicate that the number of participating laboratories and the distribution of laboratory bias were the prime two factors affecting the deviation of the consensus value from the true value.     

Stability of principal components

In this article we deal with the problem of stability of the conclusions from principal components analysis over repeated samples. We define a measure of stability for each component and investigate some of the measures properties. We then obtain the maximum likelihood estimators (MLEs) of the measures, and derive their joint limiting distributions. The MLEs of the measures turn out to be asymptotically unbiased and jointly have the multivariate normal distribution. Modified estimators are also found to reduce the amount of bias in the MLEs. To facilitate interpretation of the measures we define stability confidence level as coverage probability, and associate with each measure a stability confidence level to describe the measure in terms of probability. Finally, we investigate the stability of the components via a simulation study and compare the performance of the MLEs and the modified estimators in terms of bias and precision. This work was sponsored by a grant from the Office of Vice-President for Research at Kuwait University under project number SS049.     

Estimation of a parameter of Morgenstern type bivariate exponential distribution by ranked set sampling

Ranked set sampling is applicable whenever ranking of a set of sampling units can be done easily by a judgement method or based on the measurement of an auxiliary variable on the units selected. In this work, we consider ranked set sampling, in which ranking of units are done based on measurements made on an easily and exactly measurable auxiliary variable X which is correlated with the study variable Y. We then estimate the mean of the study variate Y by the BLUE based on the measurements made on the units of the ranked set sampling regarding the study variable Y, when (X ,Y) follows a Morgenstern type bivariate exponential distribution. We then consider unbalanced multistage ranked set sampling and estimate the mean of the study variate Y by the BLUE based on the observations made on the units of multistage ranked set sample regarding the study variable Y. Efficiency comparison is also made on all estimators considered in this work.