Testing for Periodicity and Trend in Long-Memory Processes

Sammanfattning: This thesis presents methods of testing the periodicity and trend for the time series, which exhibit dependence over long periods of time. Many such processes can be modeled by a class of models called fractionally differenced processes. The first approach we consider to analyse such processes is by using the band periodogram, which divides the periodogram into different intervals or bands, matching the band-pass of the discrete wavelet transform (DWT). In other words, we have a number of high frequency bands and one low frequency band. We investigate the distribution of statistics based on these bands of the long memory processes at high frequency bands, and we show that these distributions are similar to those derived from white noise processes. The next approach for analysing long memory processes is by using the DWT. For long memory processes, the properties of the periodogram of the wavelet scales are similar to those of the band periodograms. We propose two tests for testing the periodicity in the case of long memory processes. The two tests are based on the two methods mentioned above. We develop the Fisher test in the case of simple periodicity, and the Siegel test when we have multiple periodicities. We use simulations to study properties of the tests; in particular we provide results on size and power of the tests. We also investigate the testing of trends in long memory processes. We suggest two test statistics. The first one is based on the quotient of the low frequency band in the periodogram and the high frequency bands. The second one is based on the ratio of low frequency band periodogram to the sum of other ordinates in the periodogram. We also use the DWT method for testing the trend. By the use of simulation we compare our tests with alternative tests. We further investigate the impact of periodicity and trend on different methods of estimating the long memory parameter. We provide a simulation study that shows how the periodicity and trend may affect such methods. We apply our methodology to several environmental time series.