Statistical Inference Based on Data Tilting

研究者研究了三个主题：极值理论、ARCH/GARCH模型和带中心数据的非参数回归。本文所研究的具体问题包括：1)为风险管理中的高分数点构造置信度区间；2)为重尾均值的差值构造置信度区间；3)ARCH/GARCH模型中参数的置信度区间；4)GARCH(1，1)序列最大矩指数的置信度区间；5)基于GARCH模型预测超前一步的条件风险值；6)应用数据倾斜法得到双自回归模型尾部指数的置信度区间；以及7)应用数据倾斜方法获得截尾数据条件生存函数的置信度区间。上述活动涉及数据倾斜方法的新应用。本研究采用理论渐近分析、蒙特卡罗模拟和实际数据分析相结合的方法，研究的问题来源于气象、水文、保险、金融等多个领域的实际应用。例如：在配电系统和建筑物的设计中，必须考虑极端的风压负载；承保与洪水、风暴和地震等自然风险相关的金融风险的保险公司必须对极端事件的大小和影响有良好的估计，以便将保费设定在有利可图的水平。本项目研究以下问题：极端事件的建模；风险管理中的风险值预测；估计用于比较不同公司业绩的前沿函数；估计金融时间序列中波动模型的参数；以及在评估风险因素对生存的影响时估计条件寿命。这个项目的进展可以加强统计科学中几个领域之间的相互作用，包括极值理论、非参数平滑、时间序列分析和生存分析。即将开发的新方法可以应用于金融时间序列、海平面预测、互联网交通数据、医疗数据等。有待发展的理论渐近结果也有望得到更广泛的应用。

The investigator studies three subjects: extreme valuetheory, ARCH/GARCH models, and nonparametric regression with censoreddata. The particular issues examined in this proposal include:1) constructing confidence intervals for high quantilesin risk management; 2) constructing confidence intervalsfor the difference of two means with heavy tails; 3)constructing confidence intervals for parameters in ARCH/GARCHmodels; 4) constructing a confidence interval for the maximalmoment exponent of a GARCH (1,1) sequence; 5) forecasting the1-step ahead conditional Value-at-Risk based on GARCH models;6) applying the data tilting method to obtain confidenceintervals for the tail index of a double autoregressive model;and 7) applying the data tilting method to obtain confidenceintervals for a conditional survival function with censored data.The proposed activity described above involvesnovel applications of data tilting methods. The researchapproach 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 includingmeteorology, hydrology, insurance, andfinance. Examples are: in the design of electricaldistribution 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 inorder to set their premiums at a profitable level.This project studies the following issues: modeling extremeevents; predicting Value-at-Risk in riskmanagement; estimating frontier functions for comparing the performance of different firms; estimating parametersof volatility models in financial time series; and estimating theconditional life time in assessing the influence of risk factorson survival. Progress in this project can enhancethe interaction among several areas in statistical science, includingextreme value theory, nonparametric smoothing, time series analysis,and survival analysis. The new methods to be developedcan be applied to financial time series, sea level prediction,internet traffic data, medical data, to name a few. Thetheoretical asymptotic results to be developed are expected to havebroader applications as well.