Estimation Of Periodically Varying Means And Standard Deviations In Time Series Data

. Models which account for seasonal changes in mean and standard deviation in time series which are serially correlated are discussed. Full Gaussian maximum likelihood estimation of the parameters specifying the mean function, of the parameters specifying the standard deviation function, and of the parameters specifying the stationary serial correlation structure is discussed and the asymptotic distribution derived without the assumption of Gaussian data. A commonly used method which estimates the three components separately, and uses Fourier series parameterisation, is also reviewed and the asymptotic distribution for these estimates is also derived. Both of these have asymptotic covariances which depend upon third and fourth cumulants of the underlying distribution. When these vanish (e.g., Gaussian case) then a simple revision of the inefficient estimates yields estimates which are asymptotically equivalent to the Gaussian estimates. An application to a series of monthly Atlantic Ocean Sea surface temperatures is given to illustrate this last method.