Optimal Design and Analysis of the Exponentially Weighted Moving Average Chart for Exponential Data

Aleksey S Polunchenko,Grigory Sokolov,Alexander G Tartakovsky

doi:10.4038/sljastats.v5i4.7784

Abstract

We study optimal design of the ExponentiallyWeighted Moving Average (EWMA) chart by a proper choice of the smoothing factor and the initial value (headstart) of the decision statistic. The particular problem addressed is that of quickest detection of an abrupt change in the parameter of a discrete-time exponential model. Both pre- and post-change parameter values are assumed known, but the change-point is not known. For this change-point detection scenario, we examine the performance of the conventional one-sided EWMA chart with respect to two optimality criteria: Pollak’s minimax criterion associated with the maximal conditional expected delay to detection and Shiryaev’s multi-cyclic setup associated with the stationary expected delay to detection. Using the integral-equations approach, we derive the exact closed-form formulae for all of the required performance measures. Based on these formulae we find the optimal smoothing factor and headstart by solving the corresponding two bivariate constraint optimization problems. Finally, the performance of the optimized EWMA chart is compared against that of the Shiryaev–Roberts–r procedure in the minimax setting, and against that of the original Shiryaev–Roberts procedure in the multi-cyclic setting. The main conclusion is that the EWMA chart, when fully optimized, turns out to be a very competitive procedure, with performance nearly indistinguishable from that of the known-to-be best Shiryaev–Roberts–r and Shiryaev–Roberts procedures. DOI: http://dx.doi.org/10.4038/sljastats.v5i4.7784

Highlights

Quickest change-point detection is concerned with the design and analysis of procedures for on-line detection of possible changes in the characteristics of an observed random process
A sequential change-point detection procedure is defined as a stopping time adapted to the observed data, at which one stops and declares that apparently a change is in effect
Page’s (1954) Cumulative Sum (CUSUM) “inspection scheme,” a popular detection procedure, which is based on the maximum likelihood ratio (LR) argument

Summary

Introduction

Quickest change-point detection is concerned with the design and analysis of procedures for on-line detection of possible changes in the characteristics of an observed random process. Optimal Design and Analysis of the Exponentially Weighted Moving Average Chart used as a change-point detection procedure. Hunter observed that, on the one hand, if λ = 1, the EWMA chart is “memoryless” and uses only the most recent observation, Xn, ignoring all prior data This is no different from the X –chart. Srivastava and Wu (1997) considered EWMA to detect a change in the drift of Brownian motion and obtained asymptotically optimal value for the smoothing factor, but their EWMA procedure is slightly different from the conventional one considered in our paper. We apply the obtained formulae to optimize the EWMA chart simultaneously with respect to the smoothing factor and the headstart. This is done for the minimax problem and for the multi-cyclic problem separately. The popular CUSUM scheme lacks either of these properties and for this reason is not considered

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EWMA Performance Evaluation and Optimization

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