摘要:

Local least squares kernel regression provides an appealing solution to the nonparametric regression, or “scatterplot smoothing,” problem, as demonstrated by Fan, for example. The practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter. In this article we apply the ideas of plug-in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators. Our results are applicable to odd-degree local polynomial fits and can be extended to other settings, such as derivative estimation and multiple nonparametric regression. An implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice. A by-product of our work is the development of a class of nonparametric variance estimators, based on local least squares ideas, and plug-in rules for their implementation.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476630

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

We investigate linear approximation (LA) confidence intervals for functionsg(θ) of the parametersθin a nonlinear regression model. These intervals are almost universally used and generally perform well, but at times have poor coverage probabilities. Using gradient direction plots, we identify transformations ofg(θ) that lead to more accurate LA intervals. These include power transformations, whose effectiveness is demonstrated in a variety of nonlinear regression problems via a simulation study. Finally, we show how to use profiletplots and bias indices to suggest transforms to improve LA intervals. The idea is to find a monotone transformationTsuch that the linearization confidence interval forT(g(θ)) has coverage probability close to its nominal value and then invert this interval to give an accurate interval forg(θ). The transformationTis obtained from a gradient direction plot that may be thought of as an attempt to view the graph of the estimatorĝofg(θ) in two dimensions. We develop theory to motivate this procedure and identify conditions under which the intervals produced are exact. We use the simulation study to demonstrate that the intervals also work well when the conditions of the theory are not satisfied, as is the usual case in practice. By showing how the profiletplot and bias indices are related to the gradient direction plot, we show how these may also be used to suggest approximate transformationsT. Because, as is shown, linear approximation intervals are invariant under reparameterization, our development is based on invariant constructs such asg, the solution locusMfor the regression model, and the gradient direction plot.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476631

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

The periodogram for a spatial process observed on a lattice is often used to estimate the spectral density. The bases for such estimators are two asymptotic properties that periodograms commonly possess: (1) the periodogram at a particular frequency is approximately unbiased for the spectral density, and (2) the correlation of the periodogram at distinct frequencies is approximately zero. For spatial data, it is often appropriate to use fixed-domain asymptotics in which the observations get increasingly dense in some fixed region as their number increases. Using fixed-domain asymptotics, this article shows that standard asymptotic results for periodograms do not apply and that using the periodogram of the raw data can yield highly misleading results. But by appropriately filtering the data before computing the periodogram, it is possible to obtain results similar to the standard asymptotic results for spatial periodograms.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476632

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

In this article we consider statistical inference for the autoregressive parameter of a first-order autoregressive sequence with positive innovations via an extreme value estimator ϕ. We show that a bootstrap procedure correctly estimates the sampling distribution of an asymptotically pivotal quantity (whose distribution depends only on the exponent of regular variation of the innovation distribution) based on ϕ, provided that the ratio of the bootstrap sample sizemand the original sample sizenconverges to zero. This result enables us to construct a totally nonparametric confidence interval for the autoregressive parameter. We also consider bootstrapping a normalized version of ϕ with an application toward bias correction. To obtain the bootstrap validity results, we develop a continuous convergence result for certain associated point processes. We also present results of simulation studies and a numerical example.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476633

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

Dynamic linear models (DLM's) with time-varying cyclical components are developed for the analysis of time series with persistent though time-varying cyclical behavior. The development covers inference on wavelengths of possibly several persistent cycles in nonstationary time series, permitting explicit time variation in amplitudes and phases of component waveforms, decomposition of stochastic inputs into purely observational noise and innovations that impact on the waveform characteristics, with extensions to incorporate ranges of (time-varying) time series and regression terms wihin the standard DLM context. Bayesian inference via iterative stochastic simulation methods is developed and illustrated. Some indications of model extensions and generalizations are given. In addition to the specific focus on cyclical component models, the development provides the basis for Bayesian inference, via stochastic simulation, for state evolution matrix parameters and variance components in DLM's, building on recent work on Gibbs sampling for state vectors in such models by other authors.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476634

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

In the context of Bayes estimation via Gibbs sampling, with or without data augmentation, a simple approach is developed for computing the marginal density of the sample data (marginal likelihood) given parameter draws from the posterior distribution. Consequently, Bayes factors for model comparisons can be routinely computed as a by-product of the simulation. Hitherto, this calculation has proved extremely challenging. Our approach exploits the fact that the marginal density can be expressed as the prior times the likelihood function over the posterior density. This simple identity holds for any parameter value. An estimate of the posterior density is shown to be available if all complete conditional densities used in the Gibbs sampler have closed-form expressions. To improve accuracy, the posterior density is estimated at a high density point, and the numerical standard error of resulting estimate is derived. The ideas are applied to probit regression and finite mixture models.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476635

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

This article explores numerical methods for stochastic optimization, with special attention to Bayesian design problems. A common and challenging situation occurs when the objective function (in Bayesian applications, the expected utility) is very expensive to evaluate, perhaps because it requires integration over a space of very large dimensionality. Our goal is to explore a class of optimization algorithms designed to gain efficiency in such situations, by exploiting smoothness of the expected utility surface and borrowing information from neighboring design points. The central idea is that of implementing stochastic optimization by curve fitting of Monte Carlo samples. This is done by simulating draws from the joint parameter/sample space and evaluating the observed utilities. Fitting a smooth surface through these simulated points serves as estimate for the expected utility surface. The optimal design can then be found deterministically. In this article we introduce a general algorithm for curve-fitting-based optimization, discuss implementation options, and present a consistency property for one particular implementation of the algorithm. To illustrate the advantages and limitations of curve-fitting-based optimization, and to compare it with some of the alternatives, we consider in detail two important practical applications: an information theoretical stopping rule for a clinical trial, with an objective function based on the expected amount of information acquired about a subvector of parameters of interest, and the design of exploratory shock levels in the implantation of heart defibrillators. This latter example is also used for comparison with some of the alternative schemes. One of the main attractions of efficient optimization algorithms in design is the application to sequential problems. We conclude with an outlook on how the ideas presented here can be extended to solve stochastic dynamic programming problems such as those occurring in Bayesian sequential design.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476636

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

A new class of continuous multivariate distributions on × ∈ ℜnis proposed. We define these so-called υ-spherical distributions through properties of the density function in a location-scale context. We derive conditions for properness of υ-spherical distributions and discuss how to generate them in practice. The name “υ-spherical” is motivated by the fact that these distributions generalize the classes of spherical (when υ(·) is thel2norm) andlq-spherical (when υ(·) is thelqnorm) distributions. Isodensity sets are still always situated around the location parameter μ, but exchangeability and axial symmetry are no longer imposed, as is illustrated in some examples. As an important special case, we define a class of distributions suggested by independent sampling from a generalization of exponential power distributions. This allows us to model skewness. Interestingly, all the robustness results found previously for spherical andlq-spherical models carry over directly to υ-spherical models. In particular, it is shown that under a common improper prior on the scale parameter τ−1, any υ-spherical distribution with the same isodensity sets will lead to the same densityp(x, μ). Under proper priors on τ, we can still find some robustness results, although of lesser generality.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476637

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

Consider non-Bayesian inference about a parameter of interest β in the presence of a nuisance parameter λ based on a statisticS= (T, A). If the conditional distribution ofTgivenAdoes not depend on λ and ifAcontains no information about β in the presence of λ, thenAis said to be ancillary for θ in the presence of λ, and inference about θ is to be based on the conditional distribution ofTgivenA. This approach has been the basis for several definitions of ancillarity in the presence of a nuisance parameter; the definitions vary in how the phrase “no information about θ in the presence of λ” is interpreted. In this article a definition of ancillarity in the presence of a nuisance parameter is proposed using a definition of “no information about θ in the presence of λ” based on Bayesian inference.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476638

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

Recently, Consonni and Veronese have shown that the form of the standard conjugate distribution for the mean parameter μ of a univariate natural exponential familyFcoincides with that of the distribution induced on μ by the standard conjugate distribution for the canonical parameter if and only ifFhas a quadratic variance function. In this article we present significant extensions of this result, identifying conditions under which transformations of the canonical or mean parameter preserve the form of the standard conjugate family. Generalizations to the multivariate case are also considered, and results relating Jeffreys's prior to the standard conjugate family are presented. The variance function is seen to play an important role throughout.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476639

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor