The investigator studies three subjects: extreme value theory, ARCH/GARCH models, and nonparametric regression with censored data. The particular issues examined in this proposal include: 1) constructing confidence intervals for high quantiles in risk management; 2) constructing confidence intervals for the difference of two means with heavy tails; 3) constructing confidence intervals for parameters in ARCH/GARCH models; 4) constructing a confidence interval for the maximal moment exponent of a GARCH (1,1) sequence; 5) forecasting the 1-step ahead conditional Value-at-Risk based on GARCH models; 6) applying the data tilting method to obtain confidence intervals for the tail index of a double autoregressive model; and 7) applying the data tilting method to obtain confidence intervals for a conditional survival function with censored data. The proposed activity described above involves novel applications of data tilting methods. The research approach is a combination of theoretical asymptotic analysis, Monte Carlo simulation and real data analysis.
The problems studied in this project arise from real applications in various fields including meteorology, hydrology, insurance, and finance. Examples are: in the design of electrical distribution systems and buildings, the extremes of wind pressure loading must be accounted for; insurers who underwrite the financial risk associated with natural risks like floods, storms and earthquakes must have good estimates of the size and impact of extreme events in order to set their premiums at a profitable level. This project studies the following issues: modeling extreme events; predicting Value-at-Risk in risk management; estimating frontier functions for comparing the performance of different firms; estimating parameters of volatility models in financial time series; and estimating the conditional life time in assessing the influence of risk factors on survival. Progress in this project can enhance the interaction among several areas in statistical science, including extreme value theory, nonparametric smoothing, time series analysis, and survival analysis. The new methods to be developed can be applied to financial time series, sea level prediction, internet traffic data, medical data, to name a few. The theoretical asymptotic results to be developed are expected to have broader applications as well.
