Kolmogorov-Smirnov Simultaneous Confidence Bands for Time Series Distribution Function

• Activity Number: 35 - Applications of Nonparametric Methods
• Type: Contributed
• Date/Time: Sunday, July 28, 2019 : 2:00 PM to 3:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Jie Li* and Jiangyan Wang and Lijian Yang
• Companies: Tsinghua University and Nanjing Audit University and Tsinghua University
• Keywords: Bandwidth; Brownian bridge; Kernel; Kolmogorov distribution; Simple Random Sample; Stationarity

Abstract

Claims about distributions of time series are more often than not folklores 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 (ECDF) 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 solid findings challenge the long held belief that daily financial returns distribution is fat-tailed and leptokurtic.