Use of Cumulative Sums of Squares for Retrospective Detection of Changes of Variance

Abstract This article studies the problem of multiple change points in the variance of a sequence of independent observations. We propose a procedure to detect variance changes based on an iterated cumulative sums of squares (ICSS) algorithm. We study the properties of the centered cumulative sum of squares function and give an intuitive basis for the ICSS algorithm. For series of moderate size (i.e., 200 observations and beyond), the ICSS algorithm offers results comparable to those obtained by a Bayesian approach or by likelihood ratio tests, without the heavy computational burden required by these approaches. Simulation results comparing the ICSS algorithm to other approaches are presented.

[1]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[2]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[3]  D. Hinkley.  Inference about the change-point from cumulative sum tests , 1971 .

[4]  R. Miller,et al.  On the Stable Paretian Behavior of Stock-Market Prices , 1974 .

[5]  J. Durbin,et al.  Techniques for Testing the Constancy of Regression Relationships Over Time , 1975 .

[6]  A. F. Smith.  A Bayesian approach to inference about a change-point in a sequence of random variables , 1975 .

[7]  George E. P. Box,et al.  Time Series Analysis: Forecasting and Control , 1977 .

[8]  Dean W. Wichern,et al.  Changes of Variance in First-Order Autoregressive Time Series Models -With an Application , 1976 .

[9]  D. Hsu.  Tests for Variance Shift at an Unknown Time Point , 1977 .

[10]  G. Cobb.  The problem of the Nile: Conditional solution to a changepoint problem , 1978 .

[11]  D. A. Hsu,et al.  Detecting Shifts of Parameter in Gamma Sequences with Applications to Stock Price and Air Traffic Flow Analysis , 1979 .

[12]  Adrian F. M. Smith.  Change-Point problems: approaches and applications , 1980 .

[13]  U. Menzefricke.  A Bayesian Analysis of a Change in the Precision of a Sequence of Independent Normal Random Variables at an Unknown Time Point , 1981 .

[14]  N. B. Booth,et al.  A Bayesian approach to retrospective identification of change-points , 1982 .

[15]  D. A. Hsu,et al.  A Bayesian Robust Detection of Shift in the Risk Structure of Stock Market Returns , 1982 .

[16]  Linus Schrage,et al.  A guide to simulation , 1983 .

[17]  Paul Bratley,et al.  A guide to simulation , 1983 .

[18]  B. Abraham,et al.  Inferences about the parameters of a time series model with changing variance , 1984 .

[19]  Daniel Commenges,et al.  Inference about a change point in experimental neurophysiology , 1986 .

[20]  K. Worsley.  Confidence regions and tests for a change-point in a sequence of exponential family random variables , 1986 .

[21]  Paul Bratley,et al.  A guide to simulation (2nd ed.) , 1986 .

[22]  S. Geer,et al.  The likelihood ratio test for the change point problem for exponentially distributed random variables , 1987 .

[23]  R. Tsay.  Outliers, Level Shifts, and Variance Changes in Time Series , 1988 .

[24]  Patsy Haccou,et al.  Testing for the number of change points in a sequence of exponential random variables , 1988 .

[25]  G. C. Tiao,et al.  Estimation of time series parameters in the presence of outliers , 1988 .

[26]  J. Richard,et al.  Econometrics and Structural Change. , 1988 .

[27]  Dag Tjøstheim,et al.  An autoregressive model with suddenly changing parameters and an application to stock market prices , 1988 .

[28]  J. Q. Smith,et al.  1. Bayesian Statistics 4 , 1993 .