Asymptotic Distribution of the Delay Time in Page's Sequential Procedure

In this paper the asymptotic distribution of the stopping time in Page's sequential cumulative sum (CUSUM) procedure is presented. Page as well as ordinary cumulative sums are considered as detectors for changes in the mean of observations satisfying a weak invariance principle. The main results on the stopping times derived from these detectors extend a series of results on the asymptotic normality of stopping times of CUSUM-type procedures. In particular the results quantify the superiority of the Page CUSUM procedure to ordinary CUSUM procedures in late change scenarios. The theoretical results are illustrated by a small simulation study, including a comparison of the performance of ordinary and Page CUSUM detectors

Asymptotic Methods in Change-Point Analysis

This thesis is focussed on two areas of statistics, change-point analysis and functional data analysis, and the intersection of these two areas, i.e., the detection of structural breaks in functional stochastic models. The considered problems from (scalar) change-point analysis result from criticism of already existing sequential change-point procedures. The subject of this criticism was a lack of stability of these procedures regarding the time of occurrence of a change. As a possible solution to this criticism sequential methods are presented in this thesis in the framework of a linear regression model on the basis of the so-called Page CUSUM. To prove the desired properties of these procedures theoretically the asymptotic distribution of the delay time in the detection of structural breaks is derived in the special case of a location model. The notion "functional data analysis" represents an area of statistics that considers functions (in general defined on a compact interval) as "data points". Examples for such data are temperature curves or the path of a stock price on one trading day. To derive statistical procedures for this class of data dimension reduction techniques play a key role. The methods presented in this thesis are based on one of those techniques, the functional principal component analysis. They illustrate the construction of two-sample tests as well as change-point tests exploiting the properties of these functional principal components. In particular the problem of an adequate choice of the dimension of the space to project on in order to reduce the dimension is addressed and included in the construction of the respective testing procedures

Reaction times of monitoring schemes for ARMA time series

This paper is concerned with deriving the limit distributions of stopping times devised to sequentially uncover structural breaks in the parameters of an autoregressive moving average, ARMA, time series. The stopping rules are defined as the first time lag for which detectors, based on CUSUMs and Page's CUSUMs for residuals, exceed the value of a prescribed threshold function. It is shown that the limit distributions crucially depend on a drift term induced by the underlying ARMA parameters. The precise form of the asymptotic is determined by an interplay between the location of the break point and the size of the change implied by the drift. The theoretical results are accompanied by a simulation study and applications to electroencephalography, EEG, and IBM data. The empirical results indicate a satisfactory behavior in finite samples.Comment: Published at http://dx.doi.org/10.3150/14-BEJ604 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Functional Data Analysis with Increasing Number of Projections

Functional principal components (FPC's) provide the most important and most extensively used tool for dimension reduction and inference for functional data. The selection of the number, d, of the FPC's to be used in a specific procedure has attracted a fair amount of attention, and a number of reasonably effective approaches exist. Intuitively, they assume that the functional data can be sufficiently well approximated by a projection onto a finite-dimensional subspace, and the error resulting from such an approximation does not impact the conclusions. This has been shown to be a very effective approach, but it is desirable to understand the behavior of many inferential procedures by considering the projections on subspaces spanned by an increasing number of the FPC's. Such an approach reflects more fully the infinite-dimensional nature of functional data, and allows to derive procedures which are fairly insensitive to the selection of d. This is accomplished by considering limits as d tends to infinity with the sample size. We propose a specific framework in which we let d tend to infinity by deriving a normal approximation for the two-parameter partial sum process of the scores \xi_{i,j} of the i-th function with respect to the j-th FPC. Our approximation can be used to derive statistics that use segments of observations and segments of the FPC's. We apply our general results to derive two inferential procedures for the mean function: a change-point test and a two-sample test. In addition to the asymptotic theory, the tests are assessed through a small simulation study and a data example

Analysing forensic entomology data using additive mixed effects modelling

© Springer Science+Business Media B.V. 2010. Forensic pathologists and entomologists estimate the minimum post-mortem interval since a long time by describing the stage of succession and development of the necrophagous fauna (Amendt et al. 2004). From very simple calculations at the beginning, (Bergeret, see also Smith 1986) the discipline has evolved into a more mathematical one (e.g. Marchenko 2001; Grassberger and Reiter 2001, 2002) and tries to implement concepts like probabilities and confidence intervals (Lamotte and Wells 2000; Donovan et al. 2006; Tarone and Foran 2008, see also Villet et al. this book Chapter7). As pointed out by Tarone and Foran (2008) and Van Laerhoven (2008), the latter is one of the major tenets of the Daubert Standard (Daubert et al. v. Merrell Dow Pharmaceuticals (509 U.S. 579 (1993))

Microbial effects on the development of forensically important blow fly species

Colonisation times and development rates of specific blow fly species are used to estimate the minimum Post Mortem Interval (mPMI). The presence or absence of bacteria on a corpse can potentially affect the development and survival of blow fly larvae. Therefore an understanding of microbial-insect interactions is important for improving the interpretation of mPMI estimations. In this study, the effect of two bacteria (Escherichia coli and Staphylococcus aureus) on the growth rate and survival of three forensically important blow fly species (Lucilia sericata, Calliphora vicina and Calliphora vomitoria) was investigated. Sterile larvae were raised in a controlled environment (16:8 h day: night light cycle, 23:21 °C day: night temperature cycle and a constant 35% relative humidity) on four artificial diets prepared with 100 μl of 105 CFU bacterial solutions as follows: (1) E. coli, (2) S. aureus, (3) a 50:50 E. coli:S. aureus mix and (4) a sterile bacteria-free control diet. Daily measurements (length, width and weight) were taken from first instar larvae through to the emergence of adult flies. Survival rates were also determined at pupation and adult emergence. Results indicate that bacteria were not essential for the development of any of the blow fly species. However, larval growth rates were affected by bacterial diet, with effects differing between blow fly species. Peak larval weights also varied according to species-diet combination; C. vomitoria had the largest weight on E. coli and mixed diets, C. vicina had the largest weight on S. aureus diets, and treatment had no significant effect on the peak larval weight of L. sericata. These results indicate the potential for the bacteria that larvae are exposed to during development on a corpse to alter both developmental rates and larval weight in some blow fly species.Daphne Jackson Trus

Page's sequential procedure for change-point detection in time series regression

In a variety of different settings cumulative sum (CUSUM) procedures have been applied for the sequential detection of structural breaks in the parameters of stochastic models. Yet their performance depends strongly on the time of change and is best under early change scenarios. For later changes their finite sample behavior is rather questionable. We therefore propose modified CUSUM procedures for the detection of abrupt changes in the regression parameter of multiple time series regression models, that show a higher stability with respect to the time of change than ordinary CUSUM procedures. The asymptotic distributions of the test statistics and the consistency of the procedures are provided. In a simulation study it is shown that the proposed procedures behave well in finite samples. Finally the procedures are applied to a set of capital asset pricing data related to the Fama-French extension of the CAPM

SFB 823 Reaction times of monitoring schemes for ARMA time series Discussion Paper Reaction times of monitoring schemes for ARMA time series *

Abstract This paper is concerned with deriving the limit distributions of stopping times devised to sequentially uncover structural breaks in the parameters of an autoregressive moving average, ARMA, time series. The stopping rules are defined as the first time lag for which detectors, based on CUSUMs and Page's CUSUMs for residuals, exceed the value of a prescribed threshold function. It is shown that the limit distributions crucially depend on a drift term induced by the underlying ARMA parameters. The precise form of the asymptotic is determined by an interplay between the location of the break point and the size of the change implied by the drift. The theoretical results are accompanied by a simulation study and applications to electroencephalography, EEG, and IBM data. The empirical results indicate a satisfactory behavior in finite samples