Estimate Copula Function By Nonparametric Method

Copulas are general tools to construct multivatiate distribution and to investigate dependence structure between random variables. However,the concept of copula is not popular in Finance. In this paper, we estimate the copula by the method of kernels.In section one,we present copula functions and some related fields,in particular the concept of dependence.We then consider the problem of estimation of copulas in section two.In section three,we analyse the dependence of stock market of Shanghai and Shenzhen by the copula of estimation.In section one,we introduce the following properties.Definition N-dimcnsional copula is a function C with the following properties:1. Dom.C = I^N = [0,1]^N;2. C is grounded and N-increasing;3. C has margins C_n which satisfyC_n(u) = C(1,..., 1, u, 1..., 1) = u for all u in I.Theorem (Sklar's theorem) Let F be an n-dimensional distribution function with continuous margins F₁,... ,F_n.Then F has a unique copula representation:The Sklar'stheorem is very important,because it provides a way to analyse the dependence structure of multivariate distributions without studying margins distribution.Remark: The density c associate to the copula is given byTo obtain the density /of the n-dimensional distribution F,we use the following relationship:where /, is the density of the margin distribution Fi.Dependence relations between random variables is one of the most widely studied subjects in probability and statistics .Since the most widely known scale-invariant measures of association are the population versions of Kendall's r and Spearman's p,they play important roles in non-parametric statistics.There are some relationships between the measures of concordance and the copula.= 4 f I C{u,v)dC{u,v)-l = 12 / !uvd.C{u,v) -3In section two.we introduce the method of kernels.Many people suggest, a class of estimation of the formwhere hn —> 0 as n —* oo,and K is a suitable density function.This is called kernel-type estimator.The estimation has many properties.Theorem Suppose K(-) is of bounded variation and the series ]T exp(—7n/i,J71=1converges for every 7 > O.ThenVn = sup|/n(a0-/(a;)|—>0nwith probability one a.s n —?? 00 if and only if the density / is uniformly continuous.Let A'i, X'2: â–  â–  â– , Xn be independent identically distributed p-dirnensional random vectors with unknown density function f(x) — f(x\,..., xv). Motivated by the discussion,we consider estimators /7,,(x) of the density f(x) of the following form' 1 n x - X-fn{x) = ^J^)^1K{^h{nf)where withCondition 1 (i) supK(x) < 00,(ii) snp \\x\\pK(x) < oo,and...