Testing for a jump in the periodogram

Abstract

We consider the problem of testing for the presence of compound periodicities using properties of the periodogram over [0,π] rather than (2/j/T,j = l,…,[T/2]). We propose a number of new procedures based on the 'size' of the periodogram lying above a relatively high threshold. Our first suggestion is based on positive differences between the values of the periodogram evaluated at relative maxima and the threshold. In the second we use the area above the threshold and beneath the periodogram. Both tests are generalizations of Siegel (1980), which is based on the periodogram ordinates. Our aim is to construct tests which have good performance against compound periodicities lying outside (2π/j/T,j = l,...,[T/2]). We compare the performance of our approach with a number of competitors by simulation. These include tests based on the supremum of the periodogram, see Turkman and Walker (1984) and Chiu (1989). The latter uses robust variance estimates to give improved performance against compound periodicities. Our results for the distribution and performance of Chiu's approach may be of independent interest as there appears to be no work on its finite sample properties. Our experiments show that tests based on the periodogram over [0,π] give enhanced performance, although this is accompanied by a small reduction in relative performance against ordinate based methods for frequencies in (2π/T,j = l,...,[T/2]). There appears to be no uniformly best test among those considered, although our suggestions generally give improved performance over Chiu's approach. Selected percentage points for the periodogram based tests are presented and used to assess the value of an asymptotic approximation due to Turkman and Walker (1984).

Keywords:

• Detecting periodic components
• exceedance
• non-fourier frequencies
• periodogram
• relative maxima
• robust estimation

^*To whom correspondence should be addressed.

^*To whom correspondence should be addressed.

Notes

^*To whom correspondence should be addressed.