A Class of Convolution‐Based Models for Spatio‐Temporal Processes with Non‐Separable Covariance Structure

Abstract. In this article, we propose a new parametric family of models for real‐valued spatio‐temporal stochastic processes S ( x , t ) and show how low‐rank approximations can be used to overcome the computational problems that arise in fitting the proposed class of models to large datasets. Separable covariance models, in which the spatio‐temporal covariance function of S ( x , t ) factorizes into a product of purely spatial and purely temporal functions, are often used as a convenient working assumption but are too inflexible to cover the range of covariance structures encountered in applications. We define positive and negative non‐separability and show that in our proposed family we can capture positive, zero and negative non‐separability by varying the value of a single parameter.     

Cross-Validated Assessments of Test Scoring Models

On a multiple choice test in which each item has r alternative options, a given number c of which are correct, various scoring models have been proposed. In one case the test-taker is allowed to choose any size solution subset and he/she is graded according to whether the subset is small and according to how many correct answers the subset contains. In a second case the test-taker is allowed to select only solution subsets of a prespecified maximum size and is graded as above. The first case is analogous to the situation where the test-taker is given a set of r options with each question; each question calls for a solution which consists of selecting that subset of the r responses which he/she believes to be correct. In the second case, when the prespecified solution subset is restricted to be of size at most one, the resulting scoring model corresponds to the usual model, referred to below as standard. The number c of correct options per item is usually known to the test-taker in this case.

Scoring models are evaluated according to how well they correctly identify the total scores of the individuals in the class of test-takers. Loss functions are constructed which penalize scoring models resulting in student scores which are not associated with the students true (or average) total score on the exam. Scoring models are compared on the basis of cross-validated assessments of the loss incurred by using each of the given models. It is shown that in many cases the assessment of the loss for scoring models which allow students the opportunity to choose more than one option for each question are smaller than the assessment of the loss for the standard scoring model.     

Nonparametric Methods in Continuous Time Model Specification

Some popular parametric diffusion processes have been assumed as such underlying diffusion processes. This paper considers an important case where both the drift and volatility functions of the underlying diffusion process are unknown functions of the underlying process, and then proposes using two novel testing procedures for the parametric specification of both the drift and diffusion functions. The finite-sample properties of the proposed tests are assessed through using data generated from four popular parametric models. In our implementation, we suggest using a simulated critical value for each case in addition to the use of an asymptotic critical value. Our detailed studies show that there is little size distortion when using a simulated critical value while the proposed tests have some size distortions when using an asymptotic critical value in each case.     

A stepwise approach for the purely periodic interaction spline model

The basic idea of an interaction spline model was presented in Barry (1983). The general interaction spline models were proposed by Wahba (1986). The purely periodic spline model, a special case of the general interaction spline models, is considered in this paper. A stepwise approach using generalized cross validation (GCV) for fitting the model is proposed. Based on the nice orthogonality properties of the purely periodic functions, the stepwise approach is a promising method for the interaction spline model. The approach can also be generalized to the non-purely-periodic spline models. But this is no done here.     

Considerations for optimal nonparametric regression under a generalizederror structure

Nonparametric smoothing, such as kernel or spline estimation, has been examined extensively under the assumption of uncorrelated errors. This paper addresses the effects of potential correlation on consistency and other asymptotic properties in a repeated-measures model, using directly optimized linear smoothers of the replicate means. Unrestricted optimal weights, with respect to squared error loss, are used to confirm a lack of consistency for all linear estimators in an autocorrelated errors model. The results indicate kernel methods that work well for an uncorrelated errors model may not have the ability to perform satisfactorily when correlation is introduced, due to an asymmetry in the optimal weights, which disappears for an uncorrelated errors model. These would include data-driven bandwidth selection methods, adjustments of the bandwidth to accommodate correlation, higher-order kernels, and related bias reduction techniques. The analytic results suggest alternative approaches, not considered here in detail, which have shown merit.     

Bayesian Approaches to Nonparametric Estimation of Densities on the Unit Interval

This paper investigates nonparametric estimation of density on [0, 1]. The kernel estimator of density on [0, 1] has been found to be sensitive to both bandwidth and kernel. This paper proposes a unified Bayesian framework for choosing both the bandwidth and kernel function. In a simulation study, the Bayesian bandwidth estimator performed better than others, and kernel estimators were sensitive to the choice of the kernel and the shapes of the population densities on [0, 1]. The simulation and empirical results demonstrate that the methods proposed in this paper can improve the way the probability densities on [0, 1] are presently estimated.     

Variance bounds for estimators in autoregressive models with constraints

We consider nonlinear and heteroscedastic autoregressive models whose residuals are martingale increments with conditional distributions that fulfil certain constraints. We treat two classes of constraints: residuals depending on the past through some function of the past observations only, and residuals that are invariant under some finite group of transformations. We determine the efficient influence function for estimators of the autoregressive parameter in such models, calculate variance bounds, discuss information gains, and suggest how to construct efficient estimators. Without constraints, efficient estimators can be given by weighted least squares estimators. With the constraints considered here, efficient estimators are obtained differently, as one-step improvements of some initial estimator, similarly as in autoregressive models with independent increments.     

An efficient cross-validation algorithm for window width selection for nonparametric kernel regression

This paper presents an approach to cross-validated window width choice which greatly reduces computation time, which can be used regardless of the nature of the kernel function, and which avoids the use of the Fast Fourier Transform. This approach is developed for window width selection in the context of kernel estimation of an unknown conditional mean.     

Analysis of Competing Risks by Using Bayesian Smoothing

We consider the competing risks set-up. In many practical situations, the conditional probability of the cause of failure given the failure time is of direct interest. We propose to model the competing risks by the overall hazard rate and the conditional probabilities rather than the cause-specific hazards. We adopt a Bayesian smoothing approach for both quantities of interest. Illustrations are given at the end.     

Uniform in Bandwidth Estimation of Integral Functionals of the Density Function

Abstract. We apply recent results on local U‐statistics to obtain uniform in bandwidth consistency and central limit theorems for some commonly used estimators of integral functionals of density functions.