數(shù)學(xué)專(zhuān)業(yè)畢業(yè)論文外文翻譯---混合整值arch模型（英文）

1、A mixture integer-valued ARCH modelFukang Zhu, Qi Li, Dehui Wang ?School of Mathematics, Jilin University, Changchun, Jilin 130012, PR Chinaa r t i c l e i n f oArticle history:Received 1 June 2009Received in revised for

2、m22 October 2009Accepted 26 January 2010 Available online 4 February 2010Keywords:AutocorrelationEM algorithmInteger-valued time seriesMixture modelModel selectionStationaritya b s t r a c tWe propose a mixture integer-v

3、alued ARCH model for modeling integer-valued timeseries with overdispersion. The model consists of a mixture of K stationary or non-stationary integer-valued ARCH components. The advantages of the mixture model overthe s

4、ingle-component model include the ability to handle multimodality and non-stationary components. The necessary and sufficient first- and second-order stationarityconditions, the necessary arbitrary-order stationarity con

5、ditions, and the autocorrela-tion function are derived. The estimation of parameters is done through an EMalgorithm, and the model is selected by three information criterions, whoseperformances are studied via simulation

6、s. Finally, the model is applied to a real dataset.p1,y,pK) model is first-order stationary, then from (2.2) we havem ¼: EðXtÞ ¼ EðmtÞ ¼PK k ¼ 1 akbk0 1? PK k ¼ 1 Ppk i ¼

7、 1 akbki :Theorem 2. Suppose that the process Xt following a MINARCH(K;p,y,p) model is first-order stationary. A necessary and sufficient condition for the process to be second-order stationary is that all roots of 1?C1z

8、?1? ? ? ? ?Cpz?p ¼ 0 lie inside the unit circle, where for u,l=1,y,p-1,Cu ¼ X Kk ¼ 1 ak b2 ku? X p?1v ¼ 1Xji?jj ¼ v bkibkjbvuyu00@1A; Cp ¼ X Kk ¼ 1 akb2 kp;yl0 ¼ X Kk ¼ 1 akbk

9、l; yll ¼ X Kk ¼ 1 ak Xji?lj ¼ l bki0@1A?1 and ylu ¼ X Kk ¼ 1 ak Xji?lj ¼ u bki; ual;where B and B?1 are ðp?1Þ ? ðp?1Þ matrices such that B ¼ ðyijÞp?1 i;j &

10、#188; 1 and B?1=(bij)i,j=1 p-1 .In the following, we give two special cases of Theorem 2.Corollary 1. Suppose that the process Xt following a MINARCH(K;p,y,p) model is first-order stationary. When p=1, the second- order

11、stationarity condition is PK k ¼ 1 akb2 k1 o1. When p=2, the second-order stationarity condition is d2 þd1 o1, whered1 ¼ X Kk ¼ 1 akb2 k1 þ2PK k ¼ 1 akbk1bk2? ? PK k ¼ 1 akbk1? ?1? PK k

12、 ¼ 1 akbk2 ; d2 ¼ X Kk ¼ 1 akb2 k2:As an illustration, consider the MINARCH(2;1,1) model. The conditions for first- and second-order stationarities are a1b11 þa2b21 o1 and a1b2 11 þa2b2 21 o1, re

13、spectively. Note that it is possible for one of the components to be a non- stationary INARCH process yet for the time series to still be second-order stationary. We illustrate this point with a simulated series from the

14、 following MINARCH(2;1,1) model (A): a1 ¼ 0:75;a2 ¼ 0:25; l1t ¼ 1þ0:5Xt?1; l2t ¼ 2þ1:5Xt?1. The time plots, the sample autocorrelation functions, the sample partial autocorrelation functions

15、, and a histogram for the simulated series are shown in Figs. 1 and 2, respectively. The simulated time series appears to be stationary and bimodal. As the MINARCH model is a mixture of INARCH models, the range of autoco

16、rrelations of the time series generated by the MINARCH model should be similar to that of an INARCH model. For a second-order stationary process Xt following a MINARCH model, we haveCovðXt?mt; Xt?jÞ ¼ E

17、89;ðXt?mtÞðXt?j?mÞ? ¼ E½ðXt?j?mÞEððXt?mtÞjF t?1Þ? ¼ 0thenCovðXt; Xt?jÞ ¼ Covðmt; Xt?jÞ ¼ Cov X pi ¼ 1X Kk ¼ 1 akbki

18、!Xt?i; Xt?j!sorj ¼ X pi ¼ 1X Kk ¼ 1 akbki!rji?jj; j ¼ 1; . . . ; p;where rj is the lag j autocorrelation. Note that these equations are similar to the Yule–Walker equations for the ordinaryINARCH(p) p

19、rocess (see Weiß, 2009), where the coefficient PK k ¼ 1 akbki replaces the lag i coefficient of the INARCH(p)process. Specially, for the MINARCH(K;1,y,1) model, rj ¼ ? PK k ¼ 1 akbk1?j .The range of p

20、ossible autocorrelations is as great as that of the standard INARCH process, since the INARCH model is just a limiting case of the MINARCH model. It is not difficult to modify the MINARCH model (2.1) to handle non-statio

21、nary time series. This can be done by restricting one of the roots of the equation 1?bk1z?1? ? ? ? ?bkpkz?pk ¼ 0 to be 1 for each of the K components. However, it is equivalent to the fitting of a stationary MINARCH

22、 model for the difference series Xt ?Xt?1. The necessary and sufficient condition for the process Xt following a MINARCH(K;p1,y,pK) model to be m th order stationary is difficult to be given. Next, we will derive a neces

23、sary condition for the process Xt that is m th order stationary.Theorem 3. Suppose that the process Xt following a MINARCH(K;p1,y,pK) model is mth order stationary, then Q o1, whereQ ¼ X Kk ¼ 1 ak X pki ¼