Asymptotic Behavior Of Estimators Of The Risk Measures Based On Relative Entropy

Risk measures refer to the identification of risk on the basis of risk quantitative analysis and description,using the method of probability statistics to quantitatively analyze and predict the probability of occurrence of risk accidents and the severity of losses.The risk measures based on relative entropy are to introduce "entropy" into the field of financial investment to measure the risk of investment.This paper focuses on the asymptotic behavior of entropic risk measures,including asymptotic normality,the large deviation principles and the moderate deviation principlesThis paper is mainly divided into the following chapters:The first chapter mainly introduces the background and development of risk measures based on relative entropy,and gives the main results of this paper.The second chapter mainly describes the relative basic knowledge.The concepts and theorems mainly include risk measures theory,large deviation principles and moderate deviation principles,central limit theorems and kernel density estimationsThe third and fourth chapters are the main results of this paper.Firstly,we consider the parametric estimators of the convex entropic risk measure and the coherent entropic risk measure under the normal distribution and the exponential distribution by using the method of maximum likelihood estimation,the asymptotic normality,the large deviation principles and the moderate deviation principles of the estimators are proved in this paper.Furthermore,we also give the estimators of the convex entropic risk measure and the coherent entropic risk measure by means of the empirical distribution function and the kernel density estimation.Concretely,we prove that these nonparametric estimators satisfy asymptotic normality,the large deviation principles and the moderate deviation principles.Chapter 5 contains numerical simulations.By generating pseudo-random numbers,we draw the standardized histograms,kernel density curve,and the probability density function curves of standard normal distribution.For the moderate deviation principle,the tail probabilities convergence rates and rate functions are mainly compared.Similarly,with the increase of the sample size,the tail probability gradually tends to zero,and it also fits with the rate function well,which verifies the moderate deviation principles of each estimator.The sixth chapter mainly summarizes and looks forward to the thesis.