Gray bootstrap method for estimating frequency-varying random vibration signals with small samples

During environment testing, the estimation of random vibration signals （RVS） is an important technique for the airborne platform safety and reliability. However, the available meth- ods including extreme value envelope method （EVEM）, statistical tolerances method （STM） and improved statistical tolerance method （ISTM） require large samples and typical probability distri- bution. Moreover, the frequency-varying characteristic of RVS is usually not taken into account. Gray bootstrap method （GBM） is proposed to solve the problem of estimating frequency-varying RVS with small samples. Firstly, the estimated indexes are obtained including the estimated inter- val, the estimated uncertainty, the estimated value, the estimated error and estimated reliability. In addition, GBM is applied to estimating the single flight testing of certain aircraft. At last, in order to evaluate the estimated performance, GBM is compared with bootstrap method （BM） and gray method （GM） in testing analysis. The result shows that GBM has superiority for estimating dynamic signals with small samples and estimated reliability is proved to be 100% at the given confidence level.

Stock profiling using time–frequency‑varying systematic risk measure

This study proposes a wavelets approach to estimating time–frequency-varying betas in the capital asset pricing model(CAPM)framework.The dynamic of systematic risk across time and frequency is analyzed to investigate stock risk-profile robustness.Furthermore,we emphasize the effect of an investor’s investment horizon on the robustness of portfolio characteristics.We use a daily panel of French stocks from 2012 to 2022.Results show that varying systematic risk varies in time and frequency,and that its short and long-run evolutions differ.We observe differences in short and long dynamics,indicating that a stock’s betas differently fluctuate to early announcements or signs of events.However,short-run and long-run betas exhibit similar dynamics during persistent shocks.Betas are more volatile during times of crisis,resulting in greater or lesser robustness of risk profiles.Significant differences exist in short-run and longrun risk profiles,implying a different asset allocation.We conclude that the standard CAPM assumes short-run investment.Then,investors should consider time–frequency CAPM to perform systematic risk analysis and portfolio allocation.