摘要:

A longstanding difficulty in multivariate statistics is identifying and evaluating nonnormal data structures in high dimensions with high statistical efficiency and low search effort. Here the possibilities of using sample moments to identify mixtures of multivariate normals are investigated. A particular system of moment equations is devised and then shown to be one that identifies the true mixing distribution, with some limitations (indicated in the text), and thus provides consistent estimates. Moreover, the estimates are shown to be quickly calculated in any dimension and to be highly efficient in the sense of being close to the values of the parameters that maximize the likelihood function. This is shown by simulation and the application of the method to Fisher's iris data. While establishing these results, we discuss certain limitations associated with moment methods with regard to uniqueness and equivariance and explain how we addressed these problems.

ISSN:0162-1459    

DOI:10.1080/01621459.1993.10476297

出版商:Taylor & Francis Group    

年代:1993

数据来源: Taylor

摘要:

We propose a technique for simulating the joint distribution of thejlargest order statistics of a very large sample. We assume that the parent population is in the domain of attraction of the Type 1 (Gumbel) extreme value distribution. The bootstrap variates are generated by resampling the normalized spacings of theklargest observed values in the original data wherekis larger thanj. We compare the bootstrap distribution to the fitted extremal distribution of Weissman. Both distributions have the same means, conditional on theklargest observed values in the data set. Ifkis large and the normalized spacings behave as independent and identically distributed exponential random variables then the bootstrap variates behave as though sampled from the extremal distribution. We propose several procedures for estimatingkand give a numerical example.

ISSN:0162-1459    

DOI:10.1080/01621459.1993.10476298

出版商:Taylor & Francis Group    

年代:1993

数据来源: Taylor

摘要:

We consider the problem of selecting a model having the best predictive ability among a class of linear models. The popular leave-one-out cross-validation method, which is asymptotically equivalent to many other model selection methods such as the Akaike information criterion (AIC), theCp, and the bootstrap, is asymptotically inconsistent in the sense that the probability of selecting the model with the best predictive ability does not converge to 1 as the total number of observationsn→ ∞. We show that the inconsistency of the leave-one-out cross-validation can be rectified by using a leave-nv-out cross-validation withnv, the number of observations reserved for validation, satisfyingnv/n→ 1 asn→ ∞. This is a somewhat shocking discovery, becausenv/n→ 1 is totally opposite to the popular leave-one-out recipe in cross-validation. Motivations, justifications, and discussions of some practical aspects of the use of the leave-nv-out cross-validation method are provided, and results from a simulation study are presented.

ISSN:0162-1459    

DOI:10.1080/01621459.1993.10476299

出版商:Taylor & Francis Group    

年代:1993

数据来源: Taylor

摘要:

As a sequel to an earlier article by Gu and Qiu, this article describes and illustrates a dimensionless automatic algorithm for nonparametric probability density estimation using smoothing splines. The algorithm is designed to calculate an adaptive finite dimensional solution to the penalized likelihood problem, which was shown by Gu and Qiu to share the same asymptotic convergence rates as the nonadaptive infinite dimensional solution. The smoothing parameter is updated jointly with the estimate in a performance-oriented iteration via a cross-validation performance estimate, where the performance is measured by proxies of the symmetrized Kullback-Leibler distance between the true density and the estimate. Simulations of limited scale are conducted to examine the relative effectiveness of the automatic smoothing parameter selection procedure and to assess the practical statistical performance of the methodology in general. The method is also applied to some real data sets. The algorithm is implemented in a few Ratfor routines, which will be included in later versions of RKPACK for public access.

ISSN:0162-1459    

DOI:10.1080/01621459.1993.10476300

出版商:Taylor & Francis Group    

年代:1993

数据来源: Taylor

摘要:

Robust location and covariance estimators are developed via generalMestimation for covariance matrix eigenvectors and eigenvalues. The solution to this GM estimation problem is obtained by transforming it into a series of robust regression problems based on a new algorithm for the singular value decomposition. It is shown here that the singular value decomposition can be represented as an iteration of two steps: a least squares regression fit of the data matrix followed by a rotation to the regression hyperplanes. An algorithm to obtain the solution to this GM estimation problem is presented, along with results of a Monte Carlo study and examples of its application. In addition, it is shown how the output of this algorithm can be used to numerically search for multivariate outliers, which is especially useful in exploratory data analysis with high-dimensional data and large sample sizes, where standard graphical techniques are difficult to implement. Because the algorithm computes robust estimates of the eigenvectors and eigenvalues of the covariance matrix, it can be used as a basis for other multivariate methods such as errors-in-variables regression, discriminant analysis, and principal components.

ISSN:0162-1459    

DOI:10.1080/01621459.1993.10476301

出版商:Taylor & Francis Group    

年代:1993

数据来源: Taylor

摘要:

Identification of multiple outliers and leverage points is difficult because of the masking effect. Recently, Rousseeuw and van Zomeren suggested using high-breakdown robust estimation methods—the least median of squares and minimum volume ellipsoid—for unmasking these observations. These methods tend to declare too many observations as extreme, however. A stepwise analysis is proposed here for confirmation of outliers and leverage points detected using the robust methods. Diagnostic measures are constructed for observations added back to the reduced sample. They are shown graphically. The complementary use of robust and diagnostic methods gives satisfactory results in analyzing two data sets. One data set consists often bad and four good leverage points. Four (or 10, using a different cutoff) extreme observations of the other data set (of size 28) are identified using the robust methods, but the stepwise analysis confirms only one. The limitations of Atkinson's confirmatory approach are discussed and illustrated.

ISSN:0162-1459    

DOI:10.1080/01621459.1993.10476302

出版商:Taylor & Francis Group    

年代:1993

数据来源: Taylor

摘要:

The basic kernel density estimator in one dimension has a single smoothing parameter, usually referred to as the bandwidth. For higher dimensions, however, there are several options for smoothing parameterization of the kernel estimator. For the bivariate case, there can be between one and three independent smoothing parameters in the estimator, which leads to a flexibility versus complexity trade-off when using this estimator in practice. In this article the performances of the different possible smoothing parameterizations are compared, using both the asymptotic and exact mean integrated squared error. Our results show that it is important to have independent smoothing parameters for each of the coordinate directions. Although this is enough for many situations, for densities with high amounts of curvature in directions different to those of the coordinate axes, substantial gains can be made by allowing the kernel mass to have arbitrary orientations. The “sphering” approaches to choosing this orientation are shown to be detrimental in general, however.

ISSN:0162-1459    

DOI:10.1080/01621459.1993.10476303

出版商:Taylor & Francis Group    

年代:1993

数据来源: Taylor

摘要:

Methods of adaptive smoothing of density estimates, where the amount of smoothing applied varies according to local features of the underlying density, are investigated. The difficulties of applying Taylor series arguments in this context are explored. Simple properties of the estimates are investigated by numerical integration and compared with the fixed kernel approach. Optimal smoothing strategies, based on the multivariate Normal distribution, are derived. As an application of these techniques, two tests of multivariate Normality—one based on integrated squared error and one on entropy—are developed, and some power calculations are carried out.

ISSN:0162-1459    

DOI:10.1080/01621459.1993.10476304

出版商:Taylor & Francis Group    

年代:1993

数据来源: Taylor

摘要:

Nonlinear experiments involve response and regressors that are connected through a nonlinear regression-type structure. Examples of nonlinear models include standard nonlinear regression, logistic regression, probit regression. Poisson regression, gamma regression, inverse Gaussian regression, and so on. The Fisher information associated with a nonlinear experiment is typically a complex nonlinear function of the unknown parameter of interest. As a result, we face an awkward situation. Designing an efficient experiment will require knowledge of the parameter, but the purpose of the experiment is to generate data to yield parameter estimates! Our principal objective here is to investigate proper designing of nonlinear experiments that will let us construct efficient estimates of parameters. We focus our attention on a very general nonlinear setup that includes many models commonly encountered in practice. The experiments considered have two fundamental stages: a static design in the initial stage, followed by a fully adaptive sequential stage in which the design points are chosen sequentially, exploiting aD-optimality criterion and using parameter estimates based on available data. We explore the behavior of the maximum likelihood estimate when observations are generated from such an experiment. Two major technical hurdles are (1) the dependent nature of the data obtained from an adaptive sequential experiment and (2) the randomness in the total Fisher information associated with the experiment. Our analysis exploits a martingale structure rooted in the likelihood. We derive sufficient conditions that will ensure convergence of the chosen design to aD-optimal one as the number of trials grows. Besides ensuring the large sample optimality of the design, the convergence of the average Fisher information provides an ergodicity condition related to the growth of the martingale processes intrinsically associated with the likelihood. This key observation eventually yields the first-order efficiency of the maximum likelihood estimate via martingale central limit theorem and confirms the asymptotic validity of statistical inference based on the likelihood.

ISSN:0162-1459    

DOI:10.1080/01621459.1993.10476305

出版商:Taylor & Francis Group    

年代:1993

数据来源: Taylor

摘要:

Variable selection techniques are often used in combination with multiple linear regression to produce a parsimonious model that fits the data well. It is clearly undesirable for the final model to depend strongly on the inclusion of a few influential cases in the data set. This article discusses a measure of influence of single cases on the final model, based on a similar measure used in ordinary multiple regression. When variables are selected objectively, deletion of individual cases can strongly affect the choice of model. The influence of individual cases on the parameters of the selected model are often assessed as part of the model building process. However, such conditional measures fail to evaluate the influence of the cases on the variable selection process. Modern computing environments make it feasible to use an unconditional criterion to determine the influence of each case on the selection procedure. A number of examples are discussed to illustrate the differences between these approaches. Heuristics are developed to explain the examples. We conclude that, although the conditional approach gives valuable information about the selected model, the use of the unconditional approach can lead to greater insight about the influence of individual observations on the process of model selection.

ISSN:0162-1459    

DOI:10.1080/01621459.1993.10476306

出版商:Taylor & Francis Group    

年代:1993

数据来源: Taylor