A numerical method for minimum distance estimation problems

This paper introduces a general method for the numerical derivation of a minimum distance (MD) estimator for the parameters of an unknown distribution. The approach is based on an active sampling of the space in which the random sample takes values and on the optimization of the parameters of a suitable approximating model. This allows us to derive the MD estimator function for any given distribution, by which we can immediately obtain the MD estimate of the unknown parameters in correspondence to any observed random sample. Convergence of the method is proved when mild conditions on the sampling process and on the involved functions are satisfied, and it is shown that favorable rates can be obtained when suitable deterministic sequences are employed. Finally, simulation results are provided to show the effectiveness of the proposed algorithm on two case studies.     

An asymptotic approximation for EPMC in linear discriminant analysis based on two-step monotone missing samples

In this paper, we consider the expected probabilities of misclassification (EPMC) in the linear discriminant function (LDF) based on two-step monotone missing samples and derive an asymptotic approximation for the EPMC with an explicit form for the considered LDF. For this purpose, we also provide some results of the expectations for the inverted Wishart matrices in this paper. Finally, we conduct the Monte Carlo simulation for evaluating our result.     

On spline regression under Gaussian subordination with long memory

Motivated by an example from neurobiology, we consider estimation in a spline regression model with long-range dependent errors that are generated by Gaussian subordination. Consistency and the asymptotic distribution are derived for general Hermite ranks. Simulations illustrate the asymptotic results and finite sample properties. The method is applied to optical measurements of calcium concentration in the antennal lobe of honey bees used in the study of olfactory patterns.     

An asymptotics look at the generalized inference

This paper provides an asymptotics look at the generalized inference through showing connections between the generalized inference and two widely used asymptotic methods, the bootstrap and plug-in method. A generalized bootstrap method and a generalized plug-in method are introduced. The generalized bootstrap method can not only be used to prove asymptotic frequentist properties of existing generalized confidence regions through viewing fiducial generalized pivotal quantities as generalized bootstrap variables, but also yield new confidence regions for the situations where the generalized inference is unavailable. Some examples are presented to illustrate the method. In addition, the generalized F-test (Weerahandi, 1995 [26]) can be derived by the generalized plug-in method, then its asymptotic validity is obtained.     

On linear models with long memory and heavy-tailed errors

We consider the robust estimation of regression parameters in linear models with long memory and heavy-tailed errors. Asymptotic Bahadur-type representations of robust estimates are developed and their limiting distributions are obtained. It is shown that the limiting distributions are very different from those obtained under short memory. A simulation study is carried out to compare the performance of various asymptotic representations.     

Local likelihood estimation for nonstationary random fields

We develop a weighted local likelihood estimate for the parameters that govern the local spatial dependency of a locally stationary random field. The advantage of this local likelihood estimate is that it smoothly downweights the influence of faraway observations, works for irregular sampling locations, and when designed appropriately, can trade bias and variance for reducing estimation error. This paper starts with an exposition of our technique on the problem of estimating an unknown positive function when multiplied by a stationary random field. This example gives concrete evidence of the benefits of our local likelihood as compared to unweighted local likelihoods. We then discuss the difficult problem of estimating a bandwidth parameter that controls the amount of influence from distant observations. Finally we present a simulation experiment for estimating the local smoothness of a local Matérn random field when observing the field at random sampling locations in [0,1]². The local Matérn is a fully nonstationary random field, has a closed form covariance, can attain any degree of differentiability or Hölder smoothness and behaves locally like a stationary Matérn. We include an appendix that proves the positive definiteness of this covariance function.     

A link-free method for testing the significance of predictors

One important step in regression analysis is to identify significant predictors from a pool of candidates so that a parsimonious model can be obtained using these significant predictors only. However, most of the existing methods assume linear relationships between response and predictors, which may be inappropriate in some applications. In this article, we discuss a link-free method that avoids specifying how the response depends on the predictors. Therefore, this method has no problem of model misspecification, and it is suitable for selecting significant predictors at the preliminary stage of data analysis. A test statistic is suggested and its asymptotic distribution is derived. Examples are used to demonstrate the proposed method.     

Principal points of a multivariate mixture distribution

A set of n-principal points of a distribution is defined as a set of n points that optimally represent the distribution in terms of mean squared distance. It provides an optimal n-point-approximation of the distribution. However, it is in general difficult to find a set of principal points of a multivariate distribution. Tarpey et al. [T. Tarpey, L. Li, B. Flury, Principal points and self-consistent points of elliptical distributions, Ann. Statist. 23 (1995) 103-112] established a theorem which states that any set of n-principal points of an elliptically symmetric distribution is in the linear subspace spanned by some principal eigenvectors of the covariance matrix. This theorem, called a “principal subspace theorem”, is a strong tool for the calculation of principal points. In practice, we often come across distributions consisting of several subgroups. Hence it is of interest to know whether the principal subspace theorem remains valid even under such complex distributions. In this paper, we define a multivariate location mixture model. A theorem is established that clarifies a linear subspace in which n-principal points exist.