Ansgar Steland

On detecting jumps in time series: Nonparametric setting

Motivated by applications in statistical quality control and signal analysis, we propose a sequential detection procedure which is designed to detect structural changes, in particular jumps, immediately. This is achieved by modifying a median filter by appropriate kernel-based jump-preserving weights (shrinking) and a clipping mechanism. We aim at both robustness and immediate detection of jumps. Whereas the median approach ensures robust smooths when there are no jumps, the modification ensure immediate reaction to jumps. For general clipping location estimators, we show that the procedure can detect jumps of certain heights with no delay, even when applied to Banach space-valued data. For shrinking medians, we provide an asymptotic upper bound for the normed delay. The finite sample properties are studied by simulations which show that our proposal outperforms classical procedures in certain respects.

Nonparametric Sequential Change-Point Detection by a Vertically Trimmed Box Method

This paper examines a new method for sequential detection of a sudden and unobservable change in a sequence of independent observations with completely unspecified distribution functions. A nonparametric detection rule is proposed which relies on the concept of a moving vertically trimmed box. As such, it will be coined as the Vertical Box Control Chart (V-Box Chart). Its implementation requires merely to count the number of data points which fall into the box attached to the last available observation. No a priori knowledge of data distributions is required and proper tuning of the box size provides a quick detection technique. This is supported by establishing statistical properties of the method which explain the role of the tuning parameters used in the V-Box Chart. These theoretical results are verified by simulation studies which indicate that the V-Box Chart may provide quick detection with zero delay for jumps of moderate sizes. Its averaged run length to detection is more favorable than the one for the classical EWMA method. By comparison with the classical Shewhart chart, which was optimized for normal errors, our method provides comparable or better performance.

Sequential control of time series by functionals of kernal-weighted empirical processes under local alternatives

Motivated in part by applications in model selection in statistical genetics and sequential monitoring of financial data, we study an empirical process framework for a class of stopping rules which rely on kernel-weighted averages of past data. We are interested in the asymptotic distribution for time series data and an analysis of the joint influence of the smoothing policy and the alternative defining the deviation from the null model (in-control state). We employ a certain type of local alternative which provides meaningful insights. Our results hold true for short memory processes which satisfy a weak mixing condition. By relying on an empirical process framework we obtain both asymptotic laws for the classical fixed sample design and the sequential monitoring design. As a by-product we establish the asymptotic distribution of the Nadaraya-Watson kernel smoother when the regressors do not get dense as the sample size increases. Copyright Springer-Verlag 2004

MONITORING PROCEDURES TO DETECT UNIT ROOTS AND STATIONARITY

When analyzing time series an important issue is to decide whether the time series is stationary or a random walk. Relaxing these notions, we consider the problem to decide in favor of the I(0) or I(1) property. Fixed-sample statistical tests for that problem are well studied in the literature. In this paper we provide first results for the problem of monitoring sequentially a time series. Our stopping times are based on a sequential version of a kernel-weighted variance-ratio statistic. The asymptotic distributions are established for I(1) processes, a rich class of stationary processes, possibly affected by local nonparametric alternatives, and the localto-unity model. Further, we consider the two interesting change-point models where the time series changes its behavior after a certain fraction of the observations and derive the associated limiting laws. Our Monte Carlo studies show that the

New approaches to nonparametric density estimation and selection of smoothing parameters

The application of Singular Spectrum Analysis (SSA) to the empirical distribution function sampled at a grid of points spanning the range of the sample leads to a novel and promising method for the computer-intensive nonparametric estimation of both the distribution function and the density function. SSA yields a data-adaptive filter, whose length is a parameter that controls the smoothness of the filtered series. A data-adaptive algorithm for the automatic selection of a general smoothing parameter is introduced, which controls the number of modes of the estimated density. Extensive computer simulations demonstrate that the new automatic bandwidth selector improves on other popular methods for various densities of interest. A general uniform error bound is proved for the proposed SSA estimate of the distribution function, which ensures its uniform consistency. The simulation results indicate that the SSA density estimate with the automatic choice of the filter length outperforms the kernel density estimate in terms of the mean integrated squared error and the Kolmogorov-Smirnov distance for various density shapes. Two applications to problems arising in photovoltaic quality control and economic market research are studied to illustrate the benefits of SSA estimation.

Financial statistics and mathematical finance : methods, models and applications

Mathematical fi nance has grown into a huge area of research which requires a lot of care and a large number of sophisticated mathematical tools. Mathematically rigorous and yet accessible to advanced level practitioners and mathematicians alike, it considers various aspects of the application of statistical methods in fi nance and illustrates some of the many ways that statistical tools are used in fi nancial applications.

Sequentially Updated Residuals and Detection of Stationary Errors in Polynomial Regression Models

Abstract The question whether a time series behaves as a random walk or as a stationary process is an important and delicate problem, particularly arising in financial statistics, econometrics, and engineering. This article studies the problem to detect sequentially that the error terms in a polynomial regression model no longer behave as a random walk but as a stationary process. We provide the asymptotic distribution theory for a Monitoring procedure given by a control chart; i.e., a stopping time, which is related to a well-known unit root test statistic calculated from sequentially updated residuals. We provide a functional central limit theorem for the corresponding stochastic process that implies a central limit theorem for the control chart. The finite sample properties are investigated by a simulation study.

Multilocus statistics to uncover epistasis and heterogeneity in complex diseases: revisiting a set of multiple sclerosis data.

New statistics are developed to gather the contribution of many alleles at different loci to common diseases. Both inferential and descriptive statistics are included in order to uncover epistatic effects as well as heterogeneity. The problem of multiple testing is circumvented by considering a global null hypothesis. Global testing is supplemented by descriptive methods that make use of measures like odds ratio or the P-value of individually tested allele combinations. Visualization helps to reflect complex data sets. The methods described here have been scrutinized by statistical simulations, and we show that power gains can be substantial as compared to single locus statistics. Typing data of multiple sclerosis patients and controls are investigated, representing an example of larger scale information in screening candidate genes for their impact on complex diseases. New insights emerge from this data set demonstrating genetic heterogeneity and evidence for epistasis.

Bootstrapping Rank Statistics

The bootstrap, which provides powerful approximations for many classes of statistics, is studied for simple linear rank statistics employing bounded and smooth score functions. To verify consistency we view a rank statistic as a statistic induced by a statistical functional ψ which is evaluated at a pair of dependent signed measures. Thus, we can apply the von Mises method to verify asymptotic results for the bootstrap. The strong consistency of the bootstrap distribution estimator is derived for the bootstrap based on resampling from the original data. Further, the residual bootstrap is studied. The accuracy of the bootstrap approximations for small sample sizes is studied by simulations. The simulations indicate that the bootstrap provides better results than a normal approximation.

Nonparametric Sequential Signal Change Detection Under Dependent Noise

A nonparametric version of the sequential signal detection problem is studied. Our signal model includes a class of time-limited signals for which we collect data in the sequential fashion at discrete points in the presence of correlated noise. For such a setup we introduce a novel signal detection algorithm relying on the postfiltering smooth correction of the classical Whittaker-Shannon interpolation series. Given a finite frame of noisy samples of the signal, we design a detection algorithm being able to detect a departure from a reference signal as quickly as possible. Our detector is represented as a normalized partial-sum continuous time stochastic process, for which we obtain a functional central limit theorem under weak assumptions on the correlation structure of the noise. Particularly, our results allow for noise processes such as ARMA and general linear processes as well as α-mixing processes. The established limit theorems allow us to design monitoring algorithms with the desirable level of the probability of false alarm and able to detect a change with probability approaching one.