Fractional order estimation and testing, application to Swedish temperature data

Here we introduce a concept of periodogram roughness, which we use for estimating the roughness (variation) of spectral density. Periodogram roughness is used for several estimation and testing purposes. With the help of a spectral compensator factor it can be used to estimate fractional order as well as the autoregressive parameters or the moving average. The periodogram roughness is asymptotically normal when sample size increases. We applied our methods to temperature measurements in 10 network stations throughout Sweden to estimate the fractional order. The cycle pattern in temperature measurements has been tested. It shows long range dependency structure.