Estimators and Tests for Change in Variances

Abstract

We propose some testa for detecting possible change in the variance of independent observations. We obtain their asymptotic properties under the nochange null hypothesis. We also investigate the limit distributions of estimators for the time of change.

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“…Further, this new test can accommodate the presence of weight functions that can be tuned to have more power against specific alternatives, as for example in early/late detection. We begin by illustrating the U-statistic type process considered by Gombay et al (1996). These authors consider the following setting:…”

“…To test the null hypothesis Gombay et al (1996) use the change in mean framework to develop a statistic suited to testing for at most one change in the variance. Their process, reproduced below, compares two estimators of the variance before and after the change:…”

“…One estimator is fashioned from the first Ts observations and then compared to the estimator constructed from the last T À (T + 1)s observations. After some simple algebra, the above process can be re-expressed as: Gombay et al (1996) substitute the sample mean Y T of the series Y t for l, the population mean, to arrive at:…”

“…Further, Gombay et al (1996) explore the use of weight functions to improve the statistical power of related tests to detect changes in the parameters produced at specific subsamples of the evaluation period. In particular, they study the following family of weight functions to the process captured in Eq.…”

“…This statistic makes use of the family of weight functions introduced by Gombay et al (1996) to increase the power of the test against early and late detection. It does, however, depend on unknown slope parameters of the ECAPM which must be estimated in order for the test to be implemented.…”