Confidence bands from censored samples

Several types of asymptotic confidence bands have been proposed in the literature when the data are randomly censored on the right. Introducing new classes of bands, we place the old bands and their relationship to one another within a comprehensive theory of bands. A thorough analysis yields narrower bands and two kinds of modifications which are asymptotically distribution- and censor-free. One of these is useful when the interval on which the bands are constructed is predetermined and the width of the bands is random; the other, when there is a predetermined bound on the width and the interval is random. We illustrate our bands on the Szeged pacemaker data. These methods also provide a general modification of the Kolmogorov band in the uncensored case.     

Darling–Erdős limit results for change-point detection in panel data

We wish to test the null hypothesis if the means of N panels remain the same during the observation period of length T. A quasi-likelihood argument leads to self-normalized statistics whose limit distribution under the null hypothesis is double exponential. The main results are derived assuming that the each panel is based on independent observations and then extended to linear processes. The proofs are based on an approximation of the sum of squared CUSUM processes using the Skorokhod embedding scheme. A simulation study illustrates that our results can be used in case of small and moderate N and T. We apply our results to detect change in the “corruption index”.     

On the reaction time of moving sum detectors

In this paper, we quantify the reaction time of on-line monitoring schemes for changes in the mean based on moving sums. The corresponding sequential test procedure requires a historical sample of size m   as a baseline, while decisions are made based on a window of size h=h(m)$h=h(m)$ containing the h most recent observations. Perhaps surprisingly, the limit distribution (obtained as m tends to infinity) of the associated stopping time crucially depends on the asymptotic relation of these two quantities, posing potential problems in applications. In the empirical part of the paper, we study therefore the finite sample behavior of the monitoring schemes. We provide tables of critical values for the various limit distributions and guidelines for practitioners via a simulation study.     

Detecting changes in the mean of functional observations

Summary.  Principal component analysis has become a fundamental tool of functional data analysis. It represents the functional data as X _i ( t )= μ ( t )+Σ_{1≤ l <∞} η _{i ,  l} +  v _l ( t ), where μ is the common mean, v _l are the eigenfunctions of the covariance operator and the η _{i ,  l} are the scores. Inferential procedures assume that the mean function μ ( t ) is the same for all values of i . If, in fact, the observations do not come from one population, but rather their mean changes at some point(s), the results of principal component analysis are confounded by the change(s). It is therefore important to develop a methodology to test the assumption of a common functional mean. We develop such a test using quantities which can be readily computed in the R package fda. The null distribution of the test statistic is asymptotically pivotal with a well-known asymptotic distribution. The asymptotic test has excellent finite sample performance. Its application is illustrated on temperature data from England.     

Two sample inference in functional linear models

We propose a method of comparing two functional linear models in which explanatory variables are functions (curves) and responses can be either scalars or functions. In such models, the role of parameter vectors (or matrices) is played by integral operators acting on a function space. We test the null hypothesis that these operators are the same in two independent samples. The complexity of the test statistics increases as we move from scalar to functional responses and relax assumptions on the covariance structure of the regressors. They all, however, have an asymptotic chi‐squared distribution with the number of degrees of freedom which depends on a specific setting. The test statistics are readily computable using the R package fda , and have good finite sample properties. The test is applied to egg‐laying curves of Mediterranean flies and to data from terrestrial magnetic observatories. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Estimating the Risk of a Down's Syndrome Term Pregnancy Using Age and Serum Markers: Comparison of Various Methods

The risk of an individual woman having a pregnancy associated with Down's syndrome is estimated given her age, α-fetoprotein, human chorionic gonadotropin, and pregnancy-specific β1-glycoprotein levels. The classical estimation method is based on discriminant analysis under the assumption of lognormality of the marker values, but logistic regression is also applied for data classification. In the present work, we compare the performance of the two methods using a dataset containing the data of almost 89,000 unaffected and 333 affected pregnancies. Assuming lognormality of the marker values, we also calculate the theoretical detection and false positive rates for both the methods.     

Confidence bands for ROC curves

We develop two methods to construct confidence bands for the receiver operating characteristic (ROC) curve without estimating the densities of the underlying distributions. The first method is based on the smoothed bootstrap while the second method uses the Bonferroni inequality. As an illustration, we provide confidence bands for the ROC curve using data on Duchanne Muscular Dystrophy.     

Testing for changes in the covariance structure of linear processes

We consider several procedures to detect changes in the mean or the covariance structure of a linear process. The tests are based on the weighted CUSUM process. The limit distributions of the test statistics are derived under the no change null hypothesis. We develop new strong and weak approximations for the sample mean as well as the sample correlations of linear processes. A small Monte Carlo simulation illustrates the applicability of our results.