Explicit Expressions For A Class Of Asymptotic Statistics Based On Bell Polynomials

The normal distribution is the most common one of all distributions.The Three Sigma principle,derived from the normal distribution,states that the probability of deviation from the mean by three standard deviations or more is no more than 0.28%.In many cases,however,individual scales vary over an extremely wide range,often spanning orders of magnitude,such as the GDP of countries around the world.This extremely unbalanced data is often described by the heavy-tailed distribution,and the power-law distribution is the most frequently used heavy-tailed distribution.Many data in reality have the characteristics of power-law distribution,and the Paretotype distribution which is widely used is essentially a power-law distribution.Like other forms of heavy-tailed distribution,it is very suitable for risk management.For example,the stock return rate and stock transaction in finance,the loss amount and number of losses in actuarial insurance,the network traffic volume in communications,and the temperature and precipitation in hydrometeorology are generally non-normal heavy-tailed.In many cases,Pareto-type distribution can be used for statistical analysis.The object of this thesis is a class of asymptotic statistics in Pareto-type distribution.Let be a sequence of positive i.i.d random variables with regularly varying distribution tail of index 0<α<1 and distribution function F.Assume the survival function F(x)satisfies F(x):=1-F(x)～x-αl(x),x→∞,where l(x)is slowly varying,i.e.,(?)t>0.Such an F is a Pareto-type distribution with index a.Define Tn:X12+X22+…Xn2/(X1+X2+…+Xn)2 and τk:=(?)E(Tnk).Then τk is the class of asymptotic statistics to be studied.There have been many previous studies on Pareto-type distribution.The expression of the asymptotic statistic τk has been widely studied,but the results obtained in the literature are not perfect.For the Pareto-type distribution with index a,it is proved that the asymptotic statistic τk is a polynomial of degree k of a and an implicit expression of τk is given.According to this conclusion,an implicit representation is obtained by using the generating function method.The method is to express τk as a polynomial of degree k by expanding a two-variable generating function into a continued fraction,and its coefficient depends on the expansion of the continued fraction.Obviously,this result is not straightforward and unsatisfactory,because the expression is still complex and implicit.After reading literature and diligent research,we fortunately found that the asymptotic statistic τk is closely related to a counting problem and is hence related to the formal power series.So,we look for a combinatorial approach to solve the problems.Finally,we use Bell polynomials to solve these problems.Because Bell polynomials are a key fundamental representation of numerical-combinatorics and they are closely related to formal power series.The asymptotic statistic τk is a polynomial and linked to the counting problem,indicating that the explicit expression of τk is a polynomial in combinatorics.Based on the implicit expressions obtained in the literature,Bell polynomials defined in series expansion form are used for derivation and calculation.Thus,the asymptotic statistic τk are clearly expressed with Bell polynomial.One of the results of this thesis isτk=-1/αΓ(2k)Lk(<-α>2,<-α>4,…,<-α>2kl),where Lk is the logarithmic polynomial,and Γ(·)is the Euler gamma function and(·)is the rising factorial.The class of asymptotic statistics explicitly expressed by the Bell polynomial is concise and easy to calculate with symbolic computing software such as Mathematica.This thesis uses combinatorial approach to solve statistical problems,which is a new idea to provide a new scheme and train of thought for solving statistical problems.The structure of this thesis is organized as follows.The first chapter introduces the background knowledge of Pareto-type distribution,including Pareto distribution,power law distribution,etc.,and introduces the research status of asymptotic statistics of the Pareto-type distribution.The second chapter introduces the statistical knowledge involved in the study of the Pareto-type distribution and combinatorial knowledge related to Bell polynomials.The third chapter is the core part of this thesis,which uses the combinatorial approach to prove a proposition,and then the conclusion of the proposition is used to derive the explicit expression of the asymptotic statistic with the exponent part Bell polynomial,and this in turn is used to obtain the more concise expression of the logarithmic Bell polynomials.