In many applications, there are multiple time series that are hierarchically organized and can be aggregated at several different levels in groups based on products, geography or some other features. We call these "hierarchical time series". They are commonly forecast using either a "bottom-up" or a "top-down" method. In this paper we propose a new approach to hierarchical forecasting which provides optimal forecasts that are better than forecasts produced by either a top-down or a bottom-up approach. Our method is based on independently forecasting all series at all levels of the hierarchy and then using a regression model to optimally combine and reconcile these forecasts. The resulting revised forecasts add up appropriately across the hierarchy, are unbiased and have minimum variance amongst all combination forecasts under some simple assumptions. We show in a simulation study that our method performs well compared to the top-down approach and the bottom-up method. We demonstrate method by forecasting Australian tourism demand where the data are disaggregated by purpose of visit and geographical region.

We propose new tools for visualizing large numbers of functional data in the form of smooth curves or surfaces. The proposed tools include functional versions of the bagplot and boxplot, and make use of the first two robust principal component scores, Tukey's data depth and highest density regions. By-products of our graphical displays are outlier detection methods for functional data. We compare these new outlier detection methods with existing methods for detecting outliers in functional data and show that our methods are better able to identify the outliers.

a b s t r a c tWe propose forecasting functional time series using weighted functional principal component regression and weighted functional partial least squares regression. These approaches allow for smooth functions, assign higher weights to more recent data, and provide a modeling scheme that is easily adapted to allow for constraints and other information. We illustrate our approaches using age-specific French female mortality rates from 1816 to 2006 and age-specific Australian fertility rates from 1921 to 2006, and show that these weighted methods improve forecast accuracy in comparison to their unweighted counterparts. We also propose two new bootstrap methods to construct prediction intervals, and evaluate and compare their empirical coverage probabilities.

Using the age-and sex-specific data of 14 developed countries, we compare the point and interval forecast accuracy and bias of ten principal component methods for forecasting mortality rates and life expectancy. The ten methods are variants and extensions of the Lee-Carter method. Based on one-step forecast errors, the weighted HyndmanUllah method provides the most accurate point forecasts of mortality rates and the LeeMiller method is the least biased. For the accuracy and bias of life expectancy, the weighted Hyndman-Ullah method performs the best for female mortality and the LeeMiller method for male mortality. While all methods underestimate variability in mortality rates, the more complex Hyndman-Ullah methods are more accurate than the simpler methods. The weighted Hyndman-Ullah method provides the most accurate interval forecasts for mortality rates, while the robust Hyndman-Ullah method provides the best interval forecast accuracy for life expectancy. IntroductionIn recent years, the rapid ageing of the population has been a growing concern for governments and societies. In many developed countries, the concerns are concentrated on the sustainability of pensions and health and aged care systems, especially given increased longevity. This has resulted in a surge of interest among government policy makers and planners in accurately modeling and forecasting age-specific mortality rates. Any improvements in the forecast accuracy of mortality rates would be beneficial for policy decisions regarding the allocation of current and future resources. In particular, future mortality rates are of great interest to the insurance and pension industries. Several authors have proposed new approaches for forecasting mortality rates and life expectancy using statistical modeling (see Booth 2006; Booth and Tickle 2008, for reviews). Of these, a significant milestone in demographic forecasting was the work of Lee and Carter (1992). They used a principal component method to extract a single timevarying index of the level of mortality rates, from which the forecasts are obtained using a random walk with drift. Since then, this method has been widely used for forecasting mortality rates in various countries, including Australia The strengths of the Lee-Carter (LC) method are its simplicity and robustness in situations where age-specific log mortality rates have linear trends . A weakness of the LC method is that it attempts to capture the patterns of mortality rates using only one principal component and its scores. To address this, Hyndman and Ullah (2007) propose a model that utilizes second and higher order principal components to capture additional dimensions of change in mortality rates. Although other methods have been developed (e.g., Renshaw and Haberman 2003a,b,c;Currie, Durban, and Eilers 2004;Bongaarts 2005;Girosi and King 2008;Renshaw and Haberman 2006;Haberman and Renshaw 2008;Ediev 2008), the LC method is often considered as the benchmark method. For example, the LC method is compared with other approach...

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