T.N. Sriram

Robust Estimation of Mixture Complexity

In many applications, it is important to find the mixture with fewest number of components, known as the mixture complexity, that provides a satisfactory fit to the data. This article focuses on developing an estimator of mixture complexity that is consistent when the form of component densities are unknown but are postulated to be members of some parametric family and is simultaneously robust against model misspecification. We treat the estimation of mixture complexity as a model selection problem and construct an estimator of mixture complexity as a byproduct of minimizing a Hellinger information criterion. This estimator is shown to be consistent for any parametric family of mixtures. When the model is correctly specified, Monte Carlo simulations for a wide variety of normal mixtures show that our estimator is very competitive with several others in the literature in correctly identifying the true mixture complexity. The basic construction, being firmly rooted in the minimum Hellinger distance approach, enables our estimator to naturally inherit the property of robustness, which is examined, through simulations, under symmetric departures from postulated component normality. In terms of correctly identifying the mixture complexity under model misspecification, our estimator performs much better than an estimator based on the Kullback–Leibler distance due to James, Priebe, and Marchette. An example concerning hypertension is revisited to further illustrate the performance of our estimator.

Estimation for an adaptive allocation design

Abstract The generalized Polya urn model has been proposed as a class of adaptive allocation rules. Under a simple population model, maximum likelihood estimators can be derived. Previous work has shown them to be jointly asymptotically normal. We prove some additional results, including strong consistency of the estimators and a law of the iterated logarithm. We then derive the exact Fishers information matrix and construct fixed-size confidence regions for a fully sequential procedure.

Editor's Special Invited Paper: Sequential Estimation for Time Series Models

Abstract This article revisits sequential estimation of the autoregressive parameter β in a first-order autoregressive (AR(1)) model and construction of a sequential confidence region for a parameter vector θ in a first-order threshold autoregressive (TAR(1)) model. To resolve a theoretical conjecture raised in Sriram (1986), we provide a comprehensive numerical study that strongly suggests that the regret in using a sequential estimator of β can be significantly negative for many heavy-tailed error distributions and even for normal errors. Secondly, to investigate yet another conjecture about the limiting distribution of a sequential pivotal quantity for θ in a TAR(1) model, we conduct an extensive numerical study that strongly suggests that the sequential confidence region has much better coverage probability than that of a fixed sample counterpart, regardless of whether the θ values are inside or on or near the boundary of the ergodic region of the series. These highlight the usefulness of sequential sampling methods in fitting linear and nonlinear time series models.

Minimum Hellinger distance estimation for supercritical Galton-Watson processes

This paper studies the asymptotic behavior of the minimum Hellinger distance estimator of the underlying parameter in a supercritical branching process whose offspring distribution is known to belong to a parametric family. This estimator is shown to be asymptotically normal, efficient at the true model and robust against gross errors. These extend the results of Beran (Ann. Statist. 5, 445-463 (1977)) from an i.i.d., continuous setup to a dependent, discrete setup.

Multivariate Association and Dimension Reduction: A Generalization of Canonical Correlation Analysis

In this article, we propose a new generalized index to recover relationships between two sets of random vectors by finding the vector projections that minimize an L(2) distance between each projected vector and an unknown function of the other. The unknown functions are estimated using the Nadaraya-Watson smoother. Extensions to multiple sets and groups of multiple sets are also discussed, and a bootstrap procedure is developed to detect the number of significant relationships. All the proposed methods are assessed through extensive simulations and real data analyses. In particular, for environmental data from Los Angeles County, we apply our multiple-set methodology to study relationships between mortality, weather, and pollutants vectors. Here, we detect existence of both linear and nonlinear relationships between the dimension-reduced vectors, which are then used to build nonlinear time-series regression models for the dimension-reduced mortality vector. These findings also illustrate potential use of our method in many other applications. A comprehensive assessment of our methodologies along with their theoretical properties are given in a Web Appendix.

Central Mean Subspace in Time Series

We propose a notion of central mean dimension reduction subspace for time series {xt} which does not require specification of a model but seeks to find a p×d matrix Φd, d≤p, so that the d×1 vector ΦdTXt−1, where Xt−1=(xt−1, …, xt−p)T for some p≥1, includes all the information about xt that is available from E(xt|Xt−1). For known p and d, we estimate the mean central subspace through the Nadaraya–Watson kernel smoother and establish the strong consistency of our estimator. In addition, we propose estimation of d and p using a modified Schwarz Bayesian criterion, if either of d and p is unknown. Finally, we examine the performance of all the estimators extensively through a variety of simulations and provide a new analysis of the well-known Canadian lynx data. Supplemental materials for this article are available online.

A modified bootstrap for autoregression without stationarity

Abstract In this paper, we consider a bootstrap approximation to the distribution of the least squares estimator \ gb of the autoregressive parameter β in a first-order autoregressive process which may or may not have a stationary solution. Our bootstrap procedure is a modification of the standard bootstrap and employs a data based shrinkage towards the critical values β = ±1. The error in estimation of the sampling distribution of \ gb by the above procedure converges to zero as the sample size grows, irrespective of the value of the autoregressive parameter. This result is in sharp contrast with the behavior of the standard bootstrap for which a random limiting distribution emerges at the critical values β = ±1, resulting in a failure of the bootstrap procedure. The asymptotic validity of our procedure enables us, inter alia, to construct a confidence interval for β which will have the correct coverage probability (asymptotically) under any real β. The theoretical validity result for the modified bootstrap is supported by appropriate simulations. Finally, we also indicate an extension of the proposed modified bootstrap method to the pth order autoregressive processes.

Robust multivariate association and dimension reduction using density divergences

In this article, we introduce two new families of multivariate association measures based on power divergence and alpha divergence that recover both linear and nonlinear dependence relationships between multiple sets of random vectors. Importantly, this novel approach not only characterizes independence, but also provides a smooth bridge between well-known distances that are inherently robust against outliers. Algorithmic approaches are developed for dimension reduction and the selection of the optimal robust association index. Extensive simulation studies are performed to assess the robustness of these association measures under different types and proportions of contamination. We illustrate the usefulness of our methods in application by analyzing two socioeconomic datasets that are known to contain outliers or extreme observations. Some theoretical properties, including the consistency of the estimated coefficient vectors, are investigated and computationally efficient algorithms for our nonparametric methods are provided.

Fixed size confidence regions for parameters of threshold AR(1) models

For parameters of single and multiple threshold autoregressive models of order one, sequential procedures are proposed for constructing fixed size confidence ellipsoids. Sequential procedures are also proposed for constructing fixed proportional accuracy confidence ellipsoids and fixed width confidence intervals for linear combination of parameters. The confidence ellipsoids and intervals are shown to be asymptotically consistent and the associated stopping rules are shown to be asymptotically efficient as the size/width of the region becomes small.

An informational measure of association and dimension reduction for multiple sets and groups with applications in morphometric analysis

In this article we propose a new general index that measures relationships between multiple sets of random vectors. This index is based on Kullback–Leibler (KL) information, which measures linear or nonlinear dependence between multiple sets using joint and marginal densities of affine transformations of the random vectors. Estimates of the matrixes are obtained by maximizing the KL information and are shown to be consistent. The motivation for introducing such an index comes from morphological integration studies, a topic in biological science. As a special case of this index, we define an overall measure of association and two other measures for dimension reduction. The use of these measures is illustrated through real data analysis in morphometric studies and extensive simulations, and their performance is compared with that of approaches based on canonical correlation analysis. Extensions of the aforementioned measures to multiple groups are also discussed. In contrast to canonical correlation analysis, our general index not only provides an overall measure of association, but also determines nonlinear relationships, thereby making it useful in many other applications.