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Kolmogorov-Smirnov simultaneous confidence bands for time series distribution function. (English) Zbl 1505.62246

Summary: Claims about distributions of time series are often unproven assertions instead of substantiated conclusions for lack of hypotheses testing tools. In this work, Kolmogorov-Smirnov type simultaneous confidence bands (SCBs) are constructed based on simple random samples (SRSs) drawn from realizations of time series, together with smooth SCBs using kernel distribution estimator (KDE) instead of empirical cumulative distribution function of the SRS. All SCBs are shown to enjoy the same limiting distribution as the standard Kolmogorov-Smirnov for i.i.d. sample, which is validated in simulation experiments on various time series. Computing these SCBs for the standardized S&P 500 daily returns data leads to some rather unexpected findings, i.e., Student’s \(t\)-distributions with degrees of freedom no less than 3 and the normal distribution are all acceptable versions of the standardized daily returns series’ distribution, with proper rescaling. These findings present challenges to the long held belief that daily financial returns distribution is fat-tailed and leptokurtic.

MSC:

62-08 Computational methods for problems pertaining to statistics
62G08 Nonparametric regression and quantile regression
62G07 Density estimation
62G15 Nonparametric tolerance and confidence regions
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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References:

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