Detection of change-points in normal means is a well-studied problem. The parallel problem of detecting changes in variance has had less attention. The form of the generalized likelihood ratio test statistic has long been known, but its null distribution resisted exact analysis. In this paper, we formulate the change-point problem for a sequence of chi-square random variables. We describe a procedure that is exact for the distribution of the likelihood ratio statistic for all even degrees of freedom, and gives upper and lower bounds for odd (and also for non-integer) degrees of freedom. Both the liberal and conservative bounds for X21 degrees of freedom are shown through simulation to be reasonably tight. The important problem of testing for change in the normal variance of individual observations corresponds to the X21 case. The non-null case is also covered, and confidence intervals for the true change point are derived. The methodology is illustrated with an application to quality control in a deep level gold mine. Other applications include ambulatory monitoring of medical data and econometrics.
- Likelihood ratio test
- Quality improvement