  This paper explores seasonal and long-memory time series properties by using
the seasonal fractional ARIMA model when the seasonal data has one and two
seasonal periods and short-memory counterparts. The stationarity and
invertibility parameter conditions are established for the model studied. To
estimate the memory parameters, the method given in Reisen, Rodrigues and Palma
(2006 a,b) is generalized here to deal with a time series with two seasonal
fractional long-memory parameters. The asymptotic properties are established
and the accuracy of the method is investigated through Monte Carlo experiments.
The good performance of the estimator indicates that it can be an alternative
competitive procedure to estimate seasonal long-memory time series data.
Artificial and PM10 series were considered as examples of applications of the
proposed estimation method.
