Random walks - a sequential approach

In this paper sequential monitoring schemes to detect nonparametric drifts are studied for the random walk case. The procedure is based on a kernel smoother. As a by-product we obtain the asymptotics of the Nadaraya-Watson estimator and its as- sociated sequential partial sum process under non-standard sampling. The asymptotic behavior differs substantially from the stationary situation, if there is a unit root (random walk component). To obtain meaningful asymptotic results we consider local nonpara- metric alternatives for the drift component. It turns out that the rate of convergence at which the drift vanishes determines whether the asymptotic properties of the monitoring procedure are determined by a deterministic or random function. Further, we provide a theoretical result about the optimal kernel for a given alternative

A procedure for the change point problem in parametric models based on phi-divergence test-statistics

This paper studies the change point problem for a general parametric, univariate or multivariate family of distributions. An information theoretic procedure is developed which is based on general divergence measures for testing the hypothesis of the existence of a change. For comparing the accuracy of the new test-statistic a simulation study is performed for the special case of a univariate discrete model. Finally, the procedure proposed in this paper is illustrated through a classical change-point example

Retrospective change detection for binary time series models

Detection of changes in health care performance, financial markets, and industrial processes have recently gained momentum due to the increased availability of complex data in real-time. As a consequence, there has been a growing demand in developing statistically rigorous methodologies for change-point detection in various types of data. In many practical situations, the data being monitored for the purpose of detecting changes are autocorrelated binary time series. We propose a new statistical procedure based on the partial likelihood score process for the retrospective detection of change in the coefficients of a logistic regression model with AR(p)-type autocorrelations. We carry out some Monte Carlo experiments to evaluate the power of the detection procedure as well as its probability of false alarm (type I error). We illustrate the utility using data on 30-day mortality rates after cardiac surgery and to data on IBM share transactions

Change-point detection in multinomial data using phi-divergence test statistics

We propose two families of maximally selected phi-divergence tests to detect a change in the probability vectors of a sequence of multinomial random variables with possibly different sizes. In addition, the proposed statistics can be used to estimate the location of the change-point. We derive the limit distributions of the proposed statistics under the no change null hypothesis. One of the families has an extreme value limit. The limit of the other family is the maximum of the norm of a multivariate Brownian bridge. We check the accuracy of these limit distributions in case of finite sample sizes. A Monte Carlo analysis shows the possibility of improving the behavior of the test statistics based on the likelihood ratio and chi-square tests introduced in Horvath and Serbinowska [7]. The classical Lindisfarne Scribes problem is used to demonstrate the applicability of the proposed statistics to real life data sets