Long-Memory Processes by Jan Beran Yuanhua Feng Sucharita Ghosh & Rafal Kulik
Author:Jan Beran, Yuanhua Feng, Sucharita Ghosh & Rafal Kulik
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg
where convergence is in D[α low,α up] equipped with the supremum norm and 0<α low<α up<1. This makes inference for quantiles rather simple. Because of convergence in the sup-norm, it is possible to define simultaneous confidence bands for an arbitrary (and even uncountable) number of quantiles. For instance, a 95 % confidence interval for all quantiles between α low=0.005 and α up=0.995 can be defined as
(5.116)
If c f and d have to be estimated, then exactly the same finite sample corrections as discussed in Sect. 5.2 can be applied, since the standardization is the same as for the sample mean. Formula (5.116) is very much in contrast to the case of i.i.d. observations (and also similar results under short memory) where the asymptotic distribution of Q n,X (α) depends on α and in particular p X (α). In particular, for i.i.d. observations the asymptotic variance is equal to . For instance, if the marginal distribution is standard normal, then under the i.i.d. assumption the asymptotic variance of the median is whereas for the 5 %-quantile it is about 4.47. In contrast, under long memory the asymptotic variance of both empirical quantiles is the same. It should be noted that the simplicity induced by Q n,X (α)−Q(α) converging to the same random variable is not necessarily good for statistical inference because it also means that for a given data set all quantiles simultaneously either under- or overestimate the corresponding true values. This is illustrated by the following example.
Example 5.17
Consider a FARIMA(0,d,0) process with , and simultaneous estimation of the 10 %- and 90 %-quantiles. Figures 5.9 and 5.10 display boxplots of observations from one simulated path of length n=400 and 1000, respectively, with d=0.1, 0.2, 0.3 and 0.4. The dashed horizontal lines represent the correct quantiles. The shaded areas are the corresponding 95 % confidence intervals based on the observed path (assuming that c f and d are known). Note that here the intervals are of the form
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