Local Power Analyses of Goodness-of-fit Tests for Copulas

Abstract.  The asymptotic behaviour of several goodness-of-fit statistics for copula families is obtained under contiguous alternatives. Many comparisons between a Cramér–von Mises functional of the empirical copula process and new moment-based goodness-of-fit statistics are made by considering their associated asymptotic local power curves. It is shown that the choice of the estimator for the unknown parameter can have a significant influence on the power of the Cramér–von Mises test and that some of the moment-based statistics can provide simple and efficient goodness-of-fit methods.     

Some results concerning off-training-set and IID error for the Gibbs and the Bayes optimal generalizers

In this paper we analyse the average behaviour of the Bayes-optimal and Gibbs learning algorithms. We do this both for off-training-set error and conventional IID (independent identically distributed) error (for which test sets overlap with training sets). For the IID case we provide a major extension to one of the better known results. We also show that expected IID test set error is a non-increasing function of training set size for either algorithm. On the other hand, as we show, the expected off-training-set error for both learning algorithms can increase with training set size, for non-uniform sampling distributions. We characterize the relationship the sampling distribution must have with the prior for such an increase. We show in particular that for uniform sampling distributions and either algorithm, the expected off-training-set error is a non-increasing function of training set size. For uniform sampling distributions, we also characterize the priors for which the expected error of the Bayes-optimal algorithm stays constant. In addition we show that for the Bayes-optimal algorithm, expected off-training-set error can increase with training set size when the target function is fixed, but if and only if the expected error averaged over all targets decreases with training set size. Our results hold for arbitrary noise and arbitrary loss functions.     

When is a curve an outlier? An account of a tricky problem

ABSTRACT The analysis of a set of data consisting of N short (≤20 observations each) multivariate time series, where the observations are irregularly spaced and where observations for the different components of each multivariate series are observed at different times, is discussed. With the increased use of automatic recording devices in many fields, data such as these, which are of course samples from smooth response curves, are becoming more common. In this application, which was a clinical trial comparing two cements for use in hip replacement surgery, the key to the analysis was in recognizing that the interest lay in the degree to which the five curves representing a patient's vital signs deviated from baseline (i.e., normal for that patient) during surgery. This enabled the statisticians to define appropriate response variables. The analysis included Rosseeuw's (1984) technique for the identification of multivariate outliers and logistic regressions to identify any effects on the process producing the outliers due to treatment or covariates.     

Sample Size and Power Calculation in Comparing Diagnostic Accuracy of Biomarkers with Pooled Assessments

When testing of a biomarker is costly, pooling of samples becomes a useful and efficient alternative (Faraggi et al., 2003). In this paper, we develop procedures for sample size and power calculations for planning a study comparing the accuracy of biomarkers in diagnosis of diseases with pooled samples. Explicit formulas are derived for several important pooling strategies. The effects of pooling samples on sample size and power of the test are also discussed.     

Gender and marital satisfaction early in marriage: A growth curve approach

The purpose of this study is to assess differences between husbands and wives (N= 526 couples at the first assessment) on (a) growth curves over the first 4 years of marriage for psychological distress, marriage‐specific appraisals, spousal interactions, social support, and marital satisfaction; (b) the strength of intraspouse links and cross‐spouse links involving the initial assessment of the first four variables and the growth curve for marital satisfaction; and (c) differences between spouses headed for divorce versus spouses in stable marriages on the growth curves for all five variables. On the basis of evidence that interspouse differences were largely nonsignificant, there was little support for the view that there are his and her versions of the processes that affect marital outcomes.     

Comparison of Separable Components in Different Samples

Abstract.  Imagine we have two different samples and are interested in doing semi- or non-parametric regression analysis in each of them, possibly on the same model. In this paper, we consider the problem of testing whether a specific covariate has different impacts on the regression curve in these two samples. We compare the regression curves of different samples but are interested in specific differences instead of testing for equality of the whole regression function. Our procedure does allow for random designs, different sample sizes, different variance functions, different sets of regressors with different impact functions, etc. As we use the marginal integration approach, this method can be applied to any strong, weak or latent separable model as well as to additive interaction models to compare the lower dimensional separable components between the different samples. Thus, in the case of having separable models, our procedure includes the possibility of comparing the whole regression curves, thereby avoiding the curse of dimensionality. It is shown that bootstrap fails in theory and practice. Therefore, we propose a subsampling procedure with automatic choice of subsample size. We present a complete asymptotic theory and an extensive simulation study.     

A comparison of alternative rate estimators in some frequently used growth models

This paper deals with the linear, exponential and Gompertz models and compares the bias and variance of a number of estimators of the growth-rate parameters of these models. Some of these estimators have been commonly recommended in the literature and others are most frequently used in empirical practice. The present analysis points to the strengths and weaknesses of the estimators in their usage.     

THE REDUCED-RANK GROWTH CURVE MODEL FOR DISCRIMINANT ANALYSIS OF LONGITUDINAL DATA

This paper presents a method of discriminant analysis especially suited to longitudinal data. The approach is in the spirit of canonical variate analysis (CVA) and is similarly intended to reduce the dimensionality of multivariate data while retaining information about group differences. A drawback of CVA is that it does not take advantage of special structures that may be anticipated in certain types of data. For longitudinal data, it is often appropriate to specify a growth curve structure (as given, for example, in the model of Potthoff & Roy, 1964). The present paper focuses on this growth curve structure, utilizing it in a model-based approach to discriminant analysis. For this purpose the paper presents an extension of the reduced-rank regression model, referred to as the reduced-rank growth curve (RRGC) model. It estimates discriminant functions via maximum likelihood and gives a procedure for determining dimensionality. This methodology is exploratory only, and is illustrated by a well-known dataset from Grizzle & Allen (1969).     

Local principal curves

Principal components are a well established tool in dimension reduction. The extension to principal curves allows for general smooth curves which pass through the middle of a multidimensional data cloud. In this paper local principal curves are introduced, which are based on the localization of principal component analysis. The proposed algorithm is able to identify closed curves as well as multiple curves which may or may not be connected. For the evaluation of the performance of principal curves as tool for data reduction a measure of coverage is suggested. By use of simulated and real data sets the approach is compared to various alternative concepts of principal curves.     

Robustness of Approaches to ROC Curve Modeling under Misspecification of the Underlying Probability Model

A variety of statistical regression models have been proposed for the comparison of ROC curves for different markers across covariate groups. Pepe developed parametric models for the ROC curve that induce a semiparametric model for the market distributions to relax the strong assumptions in fully parametric models. We investigate the analysis of the power ROC curve using these ROC-GLM models compared to the parametric exponential model and the estimating equations derived from the usual partial likelihood methods in time-to-event analyses. In exploring the robustness to violations of distributional assumptions, we find that the ROC-GLM provides an extra measure of robustness.