含漂移項(xiàng)分?jǐn)?shù)布朗運(yùn)動(dòng)方差估計(jì)量的Edgeworth展開(kāi)

司成杰

(武漢大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院, 湖北 武漢 430072)

含漂移項(xiàng)分?jǐn)?shù)布朗運(yùn)動(dòng)方差估計(jì)量的Edgeworth展開(kāi)

司成杰

(武漢大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院, 湖北 武漢 430072)

對(duì)含漂移項(xiàng)分?jǐn)?shù)布朗運(yùn)動(dòng)方差估計(jì)量進(jìn)行研究. 在胡耀忠等人已有工作的基礎(chǔ)上, 對(duì)方差的極大似然估計(jì)量的三、四階原點(diǎn)矩進(jìn)行計(jì)算, 并在矩的基礎(chǔ)上得到相應(yīng)的累積量, 最后給出標(biāo)準(zhǔn)化后的方差估計(jì)量所滿足的三項(xiàng)局部Edgeworth展開(kāi)項(xiàng).

極大似然估計(jì)；原點(diǎn)矩；累積量；Edgeworth展開(kāi)

0 引言

長(zhǎng)記憶過(guò)程在金融、生物、信號(hào)網(wǎng)絡(luò)等領(lǐng)域有著非常廣泛的應(yīng)用. 分?jǐn)?shù)布朗運(yùn)動(dòng)也是一類特殊的長(zhǎng)記憶過(guò)程(H>1/2). 關(guān)于分?jǐn)?shù)布朗運(yùn)動(dòng)的隨機(jī)積分也已經(jīng)有了較為完善的研究[1-2]. 當(dāng)利用長(zhǎng)記憶過(guò)程描述某些具體現(xiàn)象時(shí), 確定模型中相關(guān)參數(shù)就顯得尤為重要. 本文中主要研究含漂移項(xiàng)分?jǐn)?shù)布朗運(yùn)動(dòng)線性模型

Xt=μt+σBH(t)

(1)

其中μ和σ是需要通過(guò)過(guò)程X的離散觀測(cè)樣本來(lái)進(jìn)行估計(jì)的參數(shù), {BH(t),t≥0}是Hurst參數(shù)H∈(0,1)的分?jǐn)?shù)布朗運(yùn)動(dòng)(不考慮H=1/2情形). 假設(shè)過(guò)程經(jīng)由離散時(shí)間(t1,t2,…，t,tN)觀測(cè). 為方便起見(jiàn), 取tk=kh,k=1,2,…,N. 這樣觀測(cè)過(guò)程序列X=(Xt1,Xt2,…,XtN).

之所以選擇研究模型(1)是因?yàn)槠渚哂袑?shí)際意義和應(yīng)用性. 關(guān)于含漂移項(xiàng)時(shí)Hurst參數(shù)的參數(shù)估計(jì)在[3-5]中有了較為深入的研究, 其中最為著名的一種Hurst參數(shù)估計(jì)就是R/S估計(jì). 此外, 在金融中很熱門(mén)研究的幾何分?jǐn)?shù)布朗運(yùn)動(dòng), 也是來(lái)源于模型(1). 基于連續(xù)時(shí)間觀測(cè)樣本的分?jǐn)?shù)O-U過(guò)程中參數(shù)估計(jì)[6]也與模型(1)有著密不可分的關(guān)系.

胡耀忠等[7]給出了模型(1)在離散樣本下即觀測(cè)時(shí)間序列t=(h,2h,…,Nh)′和分?jǐn)?shù)布朗運(yùn)動(dòng)序列BH(t)=(BH(h),BH(2h),…,BH(Nh))′條件下, 參數(shù)σ2的極大似然估計(jì)[8]

(2)

其中MH是協(xié)方差矩陣, 具體形式

(3)

1 預(yù)備知識(shí)

定義1.1隨機(jī)變量X的特征函數(shù)定義為C(t)=E(eitX), 特征函數(shù)與矩存在如下關(guān)系

與矩類似,X的累積量Γk是特征函數(shù)取對(duì)數(shù)后的展開(kāi)式系數(shù)

前4階累積量與矩對(duì)應(yīng)關(guān)系如下

引理1.1假定{Zi}是一組均值和方差分別為μ和σ2. 記Xn為其標(biāo)準(zhǔn)和

(4)

若記Fn為隨機(jī)變量Xn的分布函數(shù). 對(duì)序列Xi, 其均值、方差、r階累積量依次為μ、σ2和σrΓr. 則其前幾項(xiàng)Edgeworth展開(kāi)項(xiàng)滿足

(5)

其中Φ(x)表示標(biāo)準(zhǔn)正態(tài)分布的分布函數(shù),Φ(3)(x)表示Φ(x)關(guān)于x的三階求導(dǎo).

引理1.1是最為常見(jiàn)的Edgeworth展開(kāi)項(xiàng)形式, 由Cramér[9]于1928年證明展開(kāi)項(xiàng)級(jí)數(shù)關(guān)于x一致存在, 且前三項(xiàng)的局部展開(kāi)誤差為ox(n-1).

2 方差估計(jì)量的三、四階原點(diǎn)矩

(6)

(7)

(8)

引理2.1的證明利用胡耀忠[7]通過(guò)高斯構(gòu)造法得到的等式(α≠1/2)

(9)

將(9)式左右兩端分別作泰勒展開(kāi)并比較系數(shù), 有

(10)

即證明了(7)式成立. 再將(10)式結(jié)果帶入到RN的三階原點(diǎn)矩計(jì)算, 即有(8)式成立. 證畢.

(11)

(12)

引理2.2的證明與引理2.1證明過(guò)程類似, 只需計(jì)算得到

將上述結(jié)果帶入相應(yīng)的原點(diǎn)矩計(jì)算公式即可. 證畢.

3 方差估計(jì)量的三、四階累積量與Edgeworth展開(kāi)

(13)

(14)

(15)

(16)

證畢.

(17)

(18)

E(RN)3-3E(RN)E(RN)2+2(E(RN))3=

(19)

E(RN)4-4E(RN)E(RN)3+6E(RN)2(E(RN))2-3(E(RN))4=

(20)

證畢.

(21)

(22)

致謝：非常感謝審稿人的審稿意見(jiàn), 也非常感謝導(dǎo)師高付清老師的選題與指導(dǎo).

[1] Biagini F, Hu Yaozhong, ksendal B, et al. Stochastic calculus for fractional brownian motion and applications[M]. New York: Spring, 2008.

[2] Feller W. An introduction to probability theory and its applications[M].3rd ed. New York: Wiley, 1969.

[3] Beran J. Statistics for long-memory processes[M]. New York: Chapman and Hall, 1994.

[4] Fox R, Taqqu M S. Large-sample properties of parameter estimates for strongly dependent stationary gaussian time series[J]. Ann Statist, 1986, 14(2): 517-532.

[5] Hannan E J. The asymptotic theory of linear time-series models[J]. J Appl Probability, 1973, 10(1): 130-145.

[6] Hu Yaozhong, David N. Parameter estimation for fractional Ornstein-Uhlenbeck processes[J]. Statistics Probability Letters, 2010, 80(11/12): 1030-1038.

[7] Hu Yaozhong, David N, Xiao Weilin, et al. Exact maximum likelihood estimator for drift fractional Brownian motion at discrete observation[J]. Acta Math Sc, 2011, 31(5): 1851-1859.

[8] Shao Jun. Mathematical statistics[M].2nd ed. New York: Springer, 2003.

[9] Cramér H. On the composition of elementary errors[J]. Skand Aktuarietidskr,1928(11): 13-74.

[10] David N, Salvador O L. Central limit theorems for multiple stochastic integrals and Malliavin calculus[J]. Stoc Proc Appl, 2008, 118(4): 614-628.

[11] Fujikoshi Y, Ulyanov V V, Shimizu R. Multivariate statistics: high-dimensional and large-sample approximations[M]. New York: Wiley, 2009.

Edgeworthexpansionofvarianceestimatorfordriftfractionalbrownianmotion

SI Chengjie

(School of Mathematics and Statistics, Wuhan University, Wuhan 430072,China)

This article is mainly about the variance estimation for drift fractional brownian motion. Based on maximum likelihood estimator from the work of Hu Yaozhong, etc., we computed the third and forth moments of the estimator, and also it was cumulants, then we claimed the Edgeworth expansion series for the variance estimation.

maximum likelihood estimator; moment; cumulant; Edgeworth expansion series

2017-05-15

司成杰(1992-)，男，碩士生，E-mail:740909911@qq.com

1000-2375(2017)06-0563-04

X36

A

10.3969/j.issn.1000-2375.2017.06.001

(責(zé)任編輯 趙燕)