A simple test for white noise in functional time series
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Date
2016
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Abstract
We propose a new procedure for white noise testing of a functional time series.
Our approach is based on an explicit representation of the L2-distance between the
spectral density operator and its best (L2-)approximation by a spectral density operator
corresponding to a white noise process. The estimation of this distance can be
easily accomplished by sums of periodogram kernels and it is shown that an appropriately
standardized version of the estimator is asymptotically normal distributed
under the null hypothesis (of functional white noise) and under the alternative. As a
consequence we obtain a very simple test (using the quantiles of the normal distribution)
for the hypothesis of a white noise functional process. In particular the test
does neither require the estimation of a long run variance (including a fourth order
cumulant) nor resampling procedures to calculate critical values. Moreover, in contrast
to all other methods proposed in the literature our approach also allows to test
for "relevant" deviations from white noise and to construct confidence intervals for
a measure which measures the discrcepancy of the underlying process from a functional
white noise process.