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

Abstract

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 2004Control chart, Finance, Microarrays, Sequential test, Smoothing, Statistical genetics,

A binary control chart to detect small jumps

“…To simplify exposition, we slightly change the setting: We confine our study to a truncated version of the one-sided control chart which gives a signal if J n exceeds UCL for some 1 ≤ n ≤ N . However, our results can be extended to deal with the general case as outlined in Steland (2008). The small jump setting will be modelled by an appropriate asymptotic change-point model assuming a local alternative for the probabilities resp.…”

“…It is well known, the Shewart chart is well tuned to detect rather quickly large shifts, while EWMA and CUSUM charts are faster in detecting smaller shifts of the order 0.5σ. If the aim is to detect moderate to large jumps so called jump-preserving procedures are attractive, which are special cases of the unifying vertically weighted regression approach studied by Pawlak and Rafaj lowicz (1999), Steland (2005), and Steland (2004, 2008). Nonparametric kernel control charts and the optimization for certain out-of-control models covering mixing processes have been studied in Steland (2004) and Steland (2005).…”

“…If the aim is to detect moderate to large jumps so called jump-preserving procedures are attractive, which are special cases of the unifying vertically weighted regression approach studied by Pawlak and Rafaj lowicz (1999), Steland (2005), and Steland (2004, 2008). Nonparametric kernel control charts and the optimization for certain out-of-control models covering mixing processes have been studied in Steland (2004) and Steland (2005). Further, Wu and Spedding (2000) combined a classic Shewhart chart and a conforming run length chart yielding smaller ARLs for shifts larger than 0.8σ, but that method is inferior to the EWMA chart for smaller shifts.…”

NP-Optimal Kernels for Nonparametric Sequential Detection Rules

“…For simple shift functions this problem has been studied by Brodsky & Darkhovsky (1993). Results about its asymptotic distribution have been established by Steland (2003b). Results about its asymptotic distribution have been established by Steland (2003b).…”

On detecting jumps in time series: nonparametric setting

“…Sequential procedures based on related kernel smoothers and optimal kernels have been studied by Schmid and Steland (2000) and Steland (2003aSteland ( , 2003c. Asymptotic distribution theory for dependent time series can be found in Steland (2003b). For an application to sequential control of credit risk management see Steland (2002b).…”