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Title: Kolmogorov-Smirnov simultaneous confidence bands for time series distribution function Authors:  Jie Li - Center for Statistical Science and Department of Industrial Engineering, Tsinghua University (China) [presenting]
Jiangyan Wang - Nanjing Audit University (China)
Lijian Yang - Tsinghua University (China)
Abstract: Claims about distribution functions of time series are more often folklores than substantiated conclusions, due to lack of hypotheses testing tools. Kolmogorov-Smirnov type simultaneous confidence bands (SCBs) are constructed based on a simple random sample (SRS) drawn from a realization of time series, together with smooth SCBs using kernel distribution estimator (KDE). All SCBs are shown to enjoy the same limiting distribution as the standard Kolmogorov-Smirnov SCB for i.i.d. sample. This theoretical fact has been validated in simulation experiments performed on various time series. Hypotheses testing based on these SCBs has led to the unexpected finding that with proper rescaling, Gaussian distribution and most student's t-distributions are all acceptable alternatives of the S\& P 500 daily returns' stationary distribution. This discovery challenges the long held belief that daily financial returns' distribution is fat-tailed and leptokurtic.