9 research outputs found

### Inference Under Convex Cone Alternatives for Correlated Data

In this research, inferential theory for hypothesis testing under general
convex cone alternatives for correlated data is developed. While there exists
extensive theory for hypothesis testing under smooth cone alternatives with
independent observations, extension to correlated data under general convex
cone alternatives remains an open problem. This long-pending problem is
addressed by (1) establishing that a "generalized quasi-score" statistic is
asymptotically equivalent to the squared length of the projection of the
standard Gaussian vector onto the convex cone and (2) showing that the
asymptotic null distribution of the test statistic is a weighted chi-squared
distribution, where the weights are "mixed volumes" of the convex cone and its
polar cone. Explicit expressions for these weights are derived using the
volume-of-tube formula around a convex manifold in the unit sphere.
Furthermore, an asymptotic lower bound is constructed for the power of the
generalized quasi-score test under a sequence of local alternatives in the
convex cone. Applications to testing under order restricted alternatives for
correlated data are illustrated.Comment: 31 page

### On large-sample estimation and testing via quadratic inference functions for correlated data

Hansen (1982) proposed a class of "generalized method of moments" (GMMs) for
estimating a vector of regression parameters from a set of score functions.
Hansen established that, under certain regularity conditions, the estimator
based on the GMMs is consistent, asymptotically normal and asymptotically
efficient. In the generalized estimating equation framework, extending the
principle of the GMMs to implicitly estimate the underlying correlation
structure leads to a "quadratic inference function" (QIF) for the analysis of
correlated data. The main objectives of this research are to (1) formulate an
appropriate estimated covariance matrix for the set of extended score functions
defining the inference functions; (2) develop a unified large-sample
theoretical framework for the QIF; (3) derive a generalization of the QIF test
statistic for a general linear hypothesis problem involving correlated data
while establishing the asymptotic distribution of the test statistic under the
null and local alternative hypotheses; (4) propose an iteratively reweighted
generalized least squares algorithm for inference in the QIF framework; and (5)
investigate the effect of basis matrices, defining the set of extended score
functions, on the size and power of the QIF test through Monte Carlo simulated
experiments.Comment: 32 pages, 2 figure

### A New Technique for Finding Needles in Haystacks: A Geometric Approach to Distinguishing Between a New Source and Random Fluctuations

We propose a new test statistic based on a score process for determining the
statistical significance of a putative signal that may be a small perturbation
to a noisy experimental background. We derive the reference distribution for
this score test statistic; it has an elegant geometrical interpretation as well
as broad applicability. We illustrate the technique in the context of a model
problem from high-energy particle physics. Monte Carlo experimental results
confirm that the score test results in a significantly improved rate of signal
detection.Comment: 5 pages, 4 figure

### Testing for order-restricted hypotheses in longitudinal data

In many biomedical studies, we are interested in comparing treatment effects with an inherent ordering. We propose a "quadratic score test" (QST) based on a quadratic inference function for detecting an order in treatment effects for correlated data. The quadratic inference function is similar to the negative of a log-likelihood, and it provides test statistics in the spirit of a "Ï‡"-super-2-test for testing nested hypotheses as well as for assessing the goodness of fit of model assumptions. Under the null hypothesis of no order restriction, it is shown that the QST statistic has a Wald-type asymptotic representation and that the asymptotic distribution of the QST statistic is a weighted "Ï‡"-super-2-distribution. Furthermore, an asymptotic distribution of the QST statistic under an arbitrary convex cone alternative is provided. The performance of the QST is investigated through Monte Carlo simulation experiments. Analysis of the polyposis data demonstrates that the QST outperforms the Wald test when data are highly correlated with a small sample size and there is a significant amount of missing data with a small number of clusters. The proposed test statistic accommodates both time-dependent and time-independent covariates in a model. Copyright 2006 Royal Statistical Society.

### Semiparametric Mixtures of Generalized Exponential Families

A semiparametric mixture model is characterized by a non-parametric mixing distribution Q (with respect to a parameter "Î¸" ) and a structural parameter "Î²" common to all components. Much of the literature on mixture models has focused on fixing "Î²" and estimating Q . However, this can lead to inconsistent estimation of both Q and the order of the model "m". Creating a framework for consistent estimation remains an open problem and is the focus of this article. We formulate a class of generalized exponential family (GEF) models and establish sufficient conditions for the identifiability of finite mixtures formed from a GEF along with sufficient conditions for a nesting structure. Finite identifiability and nesting structure lead to the central result that semiparametric maximum likelihood estimation of Q and "Î²" fails. However, consistent estimation is possible if we restrict the class of mixing distributions and employ an information-theoretic approach. This article provides a foundation for inference in semiparametric mixture models, in which GEFs and their structural properties play an instrumental role. Copyright 2007 Board of the Foundation of the Scandinavian Journal of Statistics..