Application Of Kernel Density Estimation Under Transformation And Else

ARCH model (Auto Regression Conditional Heteroscedasticity)has extensive use in econometric study. ARCH model can express the time varying volatility of economic data fluctuation ratio. But general ARCH model further describe the properties of cluster of fluctuation ratio.At present, Var risk estimation technology is attached great importance by people in financial market. It is regarded a effective risk monitor means,the crucial link is the conditional distribution of rate of return on investment ,which is decided by the conditional distribution of error.Because we often assump the normal model as before,it is diffcult to express the properties of data facing to the heavy-tail data.Therefore,we use the result of Ling et al and semiparametric transformation of Wand et al (1991)in order to search the kernel density estimation of 2-dimentional security index.We transfer each variable respectively, and give a symmetric distribution through the transformation of adjustment. It is importance to decide the bandwidth in kernel density estimation.In this paper we obtain the optimal bandwidth through minimizing MISE of estimator. We obtain the kernel density estimation of 2-dimentional security index later.The nonparametric kernel density estimation transformation used by Wand et al(1991) isbutions).The parameters (λ₁,λ₂)is called the transformation parameters. We give the kernel density estimation of Y space with the form ofthen the density estimation of fx is inverse transformation with the form oft=lLet Xis the estimator of the parameter \,h*is the optimal bandwidth we obtain the kernel density estimation of fx with the form ofWe define the skewness , For the sample Y\, Y%,... ,Ynwe force the estimated skevmess to be zero, then we obtain the estimator Ai, AVWe obtain the optimal bandwidth ft* through minimizing MISE of estimator. Forwe define: MISEf(x) = E J[f(x) — f(x)}2 dx, where the kernel function K(.) satisfy : / K(t) dt = 1, / tK{t) dt = O,J t2K(t) = k2. Then we haveMISEf(x) = -h*kl f f"(xf dx+\ [K{tf dt + 0(h4 +\), 4 J nh J nhLetAMISEf(x) = \hAk% f f"(x)2dx + J- f K(t)2dt, 4 J nh JMinimizing MISEf(x) is equivalent to minimizing AMISEf(x). we obtaim the optimal bandwidth with the form ofK=\kUf"ifd) n "_'2 /? e -â– ur- - uo ?v x -^ uo. ana wte Kernel junction is J\ (I) = —f= ^TftenLet f{x) = J- e 2^ — 00 < x < 00. and the kernel function is K(t) = -±=e^<sup>2'.K{t)2 dt)* ( f f"{x)2 dx)ⁱ =1.059 Y2J, where gi\(xj)(i — l,2)have the same form as before. We give the kernel density transformation of Y space with the form offor the given parameter X,the density estimation of X space is its inverse transformation with the form of u) - g(xi) g(X2i) - g(x2) ^ , ^--------)Kh(*-------^--------)9ifx(xi,x2;h,X) ^J^and the bandwidth has the same form as before.In this paper, we give the historic data of shangzheng index and shenzhengzongzhi throught internet ,and we fit with ARCH(1,1) model:Xt =et = Dt =h2t — W2 where Xt = (Xu,X2t)',ï¿¡t = (ï¿¡u,ï¿¡2t,Y,Vt = (Vlt,V2tYhft 0 1 0 h%\we obtain the estimator Dt of Dt throught maximuming the likelihood function ,where we use the quasi Newton algorithm. We find the distribution is symmetric from the figure,namely, we obtain a smooth estimator, and we comet the original distribution through the semiparameter transformation.Parameter Estimation of Exponential Distribution Under Q-Symmetric Entropy Loss FunctionExponential distribution is very important in life testing and reliability study. In history ,it is the most early life model.Development about its statistical method is very extensive.For a exponential population of its density is f(x) = Ae^*z(x > 0, A > 0),we often are interesting in its unknovm parameterX. The parameter A appears as failure rate or loss rate when we use it in partical,so people care about it very much.Hence ,all kinds of estimation about the parameter X appear. There is a result when loss function is square loss and entropy loss function , and there is a result too when symmetric entropy loss function is j + j — 2 and (|)2 + (f)2 — 2. Now we study the loss function with the form ofL(A,*) = (ï¿¡)? + (ï¿¡)?-2 (?>0),which is called the q-symmetric entropy loss function , and we discuss the parameter estimation under the scale parameter.If the distribution density of random variable X is Xe^Xx(x > 0, A > 0),X\,X2, â–  â– â– , Xnis the sample that comes from the population,the joint density of Xi, X2, â–  â–  â– , Xn isf(x1,x2,...,xn;X)=Xne-xZ'Li*i (A > 0).We can obtain following results,and we omit the proofs.Theorem 2.1 1/ the density function ofX = (Xi, X2, â–  â–  â– , Xn) isZ = {Z\,Z2,...,Zn) ,where Ziisr^h, i = l,2,...,n. n > 2. Under the loss function with the form of) = (±)? + (j)?-2(q>0), we assume that 6o(X)is the equivalent estimation of the parameter X , and has the finite risk . Then the minimum risk equivalent estimation of the parameterX isand it is uniqe almost everywhere,where E\ express mathematical expection when the parameter X is one.Following ,we study the exact expression of 8*(X).IfSo(X) = r-n1 x., it is an equivalent estimation of the parameter X, and T = J27=i Xi,which has density function with the form of tn¹Xne^*t/T(n) .Then the minimum risk equivalent estimation of the parameter X isTheorem 2.2 Assume X = (.X"i, X2,.. ?, Xn) , under the loss function L(X, 6) = (j)g + (j)q - 2 (q > 0), for an prior distribution, the Bayes estimation of the parameter X is 8B(X) = [E(Xq\X)/E(±\X)]^. and if exsits 5', its Bayes risk satisfies R(5') < 00, then the Bayes estimation is unique.If the prior distribution of the parameter X is Gamma distribution, which is conjugate distribution of exponential distrubution ,its parameter is a, 0, Then the Bayes estimation of the parameter X issB(x) = [If &' = Yl?=i Xi,its Bayes risk satisfies R(5') < +00. Then the Bayes estimation is unique.Basing on the above discuss,we know that the Bayes estimation has the form of[cT+d\^l under the Gamma distribution , The admissibility and inadmissibility of estimator with the form of [cT + d]'1 is related to c and d. We discuss it when c and d have different value andwe assume that c* = , , 1, .where n> q .V("+9-1)-("-?) Theorem 2.3 When 0 ^ c < c*,d > O,the estimator with the form of [cT + d}^r isadmissibility.Lemma 2.1 Under the loss function with the form of (|)g + (j)g - 2 (q > 0), if...