In recent work on time series analysis considerable interest has been focused on series having the property of long memory. Long memory is a characteristic of time series in which the dependence between distant observations is not negligible. The model that has been most frequently studied, which in some situations shows properties of long memory, is based on the autoregressive integrated movingaverage ARIMA(p, d , q ) process. Hosking (Fractional differencing, Biornetrika 68 (1) (1981), 165-76) generalized this model by permitting the degree of differencing d to take fractional values. He then demonstrated that for d in the range 0 < d < 0.5 the process is stationary and possesses the long memory property. Our study is based on the ARIMA(p,d,q) model when d takes any real non-integer value in the interval (-0.5,0.5). The main aim of our study is to examine methods for estimating the parameters of this model. For estimating d we suggest an estimator based on the smoothed periodogram. Using an empirical approach we compare this estimator with other which are well known in the literature of long memory models, e.g. the raw periodogram regression method and the Hurst coefficient method.

A desirable property of an autocovariance estimator is to be robust to the presence of additive outliers. It is well known that the sample autocovariance, being based on moments, does not have this property. Hence, the use of an autocovariance estimator which is robust to additive outliers can be very useful for time-series modelling. In this article, the asymptotic properties of the robust scale and autocovariance estimators proposed by Rousseeuw and Croux (1993) and Ma and Genton (2000) are established for Gaussian processes, with either short-or long-range dependence. It is shown in the short-range dependence setting that this robust estimator is asymptotically normal at the rate ffiffiffi n p , where n is the number of observations. An explicit expression of the asymptotic variance is also given and compared with the asymptotic variance of the classical autocovariance estimator. In the long-range dependence setting, the limiting distribution displays the same behaviour as that of the classical autocovariance estimator, with a Gaussian limit and rate ffiffiffi n p when the Hurst parameter H is less than 3/4 and with a non-Gaussian limit (belonging to the second Wiener chaos) with rate depending on the Hurst parameter when H 2 (3/4,1). Some Monte Carlo experiments are presented to illustrate our claims and the Nile River data are analysed as an application. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the dependence structure of Gaussian processes.

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Environmental epidemiological studies of the health effects of air pollution frequently utilize the generalized additive model (GAM) as the standard statistical methodology, considering the ambient air pollutants as explanatory covariates. Although exposure to air pollutants is multi‐dimensional, the majority of these studies consider only a single pollutant as a covariate in the GAM model. This model restriction may be because the pollutant variables do not only have serial dependence but also interdependence between themselves. In an attempt to convey a more realistic model, we propose here the hybrid generalized additive model–principal component analysis–vector auto‐regressive (GAM–PCA–VAR) model, which is a combination of PCA and GAMs along with a VAR process. The PCA is used to eliminate the multicollinearity between the pollutants whereas the VAR model is used to handle the serial correlation of the data to produce white noise processes as covariates in the GAM. Some theoretical and simulation results of the methodology proposed are discussed, with special attention to the effect of time correlation of the covariates on the PCA and, consequently, on the estimates of the parameters in the GAM and on the relative risk, which is a commonly used statistical quantity to measure the effect of the covariates, especially the pollutants, on population health. As a main motivation to the methodology, a real data set is analysed with the aim of quantifying the association between respiratory disease and air pollution concentrations, especially particulate matter PM10, sulphur dioxide, nitrogen dioxide, carbon monoxide and ozone. The empirical results show that the GAM–PCA–VAR model can remove the auto‐correlations from the principal components. In addition, this method produces estimates of the relative risk, for each pollutant, which are not affected by the serial correlation in the data. This, in general, leads to more pronounced values of the estimated risk compared with the standard GAM model, indicating, for this study, an increase of almost 5.4% in the risk of PM10, which is one of the most important pollutants which is usually associated with adverse effects on human health.