Self-weighted least absolute deviation estimation for infinite variance autoregressive models

Transcription

1 J. R. Statist. Soc. B (25) 67, Part 3, pp Self-weighted least absolute deviatio estimatio for ifiite variace autoregressive models Shiqig Lig Hog Kog Uiversity of Sciece ad Techology, People s Republic of Chia [Received Jauary 24. Revised December 24] Summary. How to udertake statistical iferece for ifiite variace autoregressive models has bee a log-stadig ope problem. To solve this problem, we propose a self-weighted least absolute deviatio estimator ad show that this estimator is asymptotically ormal if the desity of errors ad its derivative are uiformly bouded. Furthermore, a Wald test statistic is developed for the liear restrictio o the parameters, ad it is show to have o-trivial local power. Simulatio experimets are carried out to assess the performace of the theory ad method i fiite samples ad a real data example is give. The results are etirely differet from other published results ad should provide ew isights for future research o heavy-tailed time series. Keywords: Autoregressive model; Heavy-tailed time series; Ifiite variace; Least absolute deviatio estimatio; Self-weighted least absolute deviatio 1. Itroductio Cosider statioary autoregressive (AR(p)) time series {y t } which are geerated by y t = φ + φ 1 y t φ p y t p + " t,.1:1/ where {" t } is a sequece of idepedet ad idetically distributed errors ad φ =.φ, φ 1,...,φ p / is a ukow parameter vector with its true value φ. Whe E." 2 t / is fiite, it is well kow that various estimators of φ are asymptotically ormal ad several methods are available for statistical iferece. Whe E." 2 t / is ifiite, model (1.1) is called the ifiite variace autoregressive (IVAR) model. Such heavy-tailed models are ecoutered i several fields, such as teletraffic egieerig (see Duffy et al. (1994)), hydrology (see Castillo (1988)) ad ecoomics ad fiace (see Koedijk et al. (199) ad Jase ad de Vries (1991)). A comprehesive review ad more refereces ca be foud i Resick (1997). The statistical theory of the IVAR model is fudametally differet from that of AR models with fiite variaces. Kater ad Steiger (1974) showed weak cosistecy of the least squares estimator of φ. Furthermore, Haa ad Kater (1977) proved its strog cosistecy with a coverget rate 1=δ, where is the sample size, δ > α ad α., 2/ is the tail idex of " t (see also Kight (1987)). The limitig distributio of the least squares estimator was ot available util Davis ad Resick (1985, 1986). O the basis of a poit process techique, they showed that the least squares estimator coverges weakly to a ratio of two stable radom variables with the rate 1=α L./, where L./ is a slowly varyig fuctio. Gross ad Steiger (1979) cosidered Address for correspodece: Shiqig Lig, Departmet of Mathematics, Hog Kog Uiversity of Sciece ad Techology, Clear Water Bay, Hog Kog, People s Republic of Chia. malig@ust.hk 25 Royal Statistical Society /5/67381

2 382 S. Lig the least absolute deviatio (LAD) estimator ad proved its strog cosistecy. A ad Che (1982) showed that the covergece rate of the LAD estimator is 1=δ. The asymptotic theory of the LAD ad M-estimators of φ was completely established by Davis et al. (1992). They showed that these estimators coverge weakly to the miimum of a stochastic process with rate a = if{x : P. " t >x/ 1 }. Mikosch et al. (1995) studied the Whittle estimator for IVAR movig average models ad showed that this estimator coverges to a fuctio of a sequece of stable radom variables. This result was exteded by Kokoszka ad Taqqu (1996) for log memory autoregressive fractioally itegrated movig average models. However, oe of the limitig distributios i the above has a closed form. We therefore caot use them for statistical iferece i practice. This has bee a log-stadig ope problem for IVAR models. To solve this problem, we propose a self-weighted LAD estimator ad show that this estimator is asymptotically ormal. Furthermore, a Wald test statistic is developed for testig liear restrictios o the parameters ad it is show to have o-trivial local power. Simulatio experimets are udertake to assess the performace of the theory ad method i fiite samples ad a real data example is give to illustrate its practicality. We orgaize the paper as follows. Sectio 2 presets the mai results. Sectio 3 reports the simulatio results, ad Sectio 4 presets the real data example. Cocludig remarks are give i Sectio 5 ad all proofs are give i Appedix A. 2. The self-weighted least absolute deviace estimator ad the mai results I the regressio set-up, oe advatage of LAD estimatio is that it does ot require ay momet coditio o the errors to obtai asymptotic ormality (see Koeker ad Bassett (1978)). However, whe we use this method for model (1.1), such as i Koul ad Saleh (1995), Koeker ad Zhao (1996), Davis ad Dusmuir (1997), Mukherjee (1999), Koul (22) ad Lig ad McAleer (24), this advatage disappears. As discussed by Davis et al. (1992), a ituitive reaso is that the large positive or egative values of " t i IVAR models produce y t which appear to be outliers, ad the same " t produces may leverage poits (i.e. the large positive or egative values of y s as s>t) such that {y t } itself has heavy tails. Compared with the least squares estimator, the usual LAD estimator gives less weight to the outliers but it gives essetially the same weight to the leverage poits. Thus, the LAD estimator caot reduce the effect of these poits o the covariace matrix ad so asymptotic ormality o loger holds. From the view of robust estimators, this property may be recovered if we ca reduce the effect of these leverage poits. This motivates us to defie the objective fuctio L.φ/ = w t y t X t φ, where X t =.1, y t 1,...,y t p / ad w t is a give fuctio of {y t 1,...,y t p }. Let ˆφ = arg mi φ Θ {L.φ/}, where Θ is the parameter space. ˆφ is called the self-weighted LAD (SLAD) estimator of the true value φ i Θ. We give the strictly statioary ad ergodic coditio of model (1.1) as follows (see propositio i Brockwell ad Davis (1996)). Assumptio 1. The characteristic polyomial 1 φ 1 z... φ p z p has all roots outside the uit circle ad {" t } are idepedet ad idetically distributed with E " t δ < for some δ >.

3 Self-weighted Least Absolute Deviatio Estimatio 383 The purpose of the weight w t is first to dowweight the leverage poits i X t such that the covariace matrices Ω ad Σ i theorem 1 below are fiite. Secod, the w t allow us to approximate L. ˆφ / by a quadratic form. From the discussio i Davis et al. (1992), we ca see that it caot be approximated by a quadratic form whe w t = 1 sice the remaider term caot be igored. A key techical part i our method is to make this term disappear asymptotically by usig the coditio E{.w t + wt 2/ X t 3 } < ad sup x R f.x/ < (see equatios (A.4) ad (A.5) i Appedix A). Formally, we state these coditios as follows. Assumptio 2. w t = g.y t 1,...,y t p / ad g.x 1,...,x p / is a real ad positive fuctio o R p such that E{.w t + wt 2/. X t 2 + X t 3 /} <. Assumptio 3. The errors {" t } have zero media ad a differetiable desity f.x/ everywhere i R with f./> ad sup x R f.x/ <. Here ad i what follows, L deotes covergece i distributio as. The followig is our mai result. Theorem 1. If assumptios 1 3 hold, the it follows that { } 1=2 1. ˆφ φ / N, L 4 f 2./ Σ 1 ΩΣ 1, where Σ = E.w t X t X t / ad Ω = E.w2 t X tx t /. This result is ovel ad etirely differet from the results that are available i the literature for the IVAR model. To use it for statistical iferece, we eed to estimate f./, Σ ad Ω. Usig the logistic kerel K.x/ = exp. x/={1 + exp. x/} 2 ad the badwidth b = c= ν with ν., 2 1 / ad a costat c>, we ca estimate f./ by fˆ./ = 1 ( yt ˆφ w t K X ) t,.2:1/ ˆσ w b where ˆσ w = 1 Σ w t. Σ ad Ω ca be estimated respectively by ˆΣ = 1.w t X t X t /, ˆΩ = 1.2:2/.wt 2 X tx t /: Theorem 1 ad equatios (2.1) ad (2.2) ca ow be used to udertake statistical iferece for the IVAR model. Here, we cosider p 1 liear hypotheses of the form H : Γφ = γ, i the usual otatio. The Wald test statistic for the hypothesis H is defied as W.p 1 / = 4 fˆ 2./.Γ ˆφ γ/.γ ˆΣ 1 ˆΩ ˆΣ 1 Γ / 1.Γ ˆφ γ/: The followig theorem gives the limitig distributio of W.p 1 /. Theorem 2. If assumptios 1 3 hold with E.wt 2/<, ad b = O.1= ν / with ν., 2 1 /, the uder hypothesis H it follows that W.p 1 / L χ 2 p 1. Whe testig a order p agaist a higher order p, we ca take a Γ such that Γφ =.φ,p +1,...,φ p / ad let γ =.,...,/ p1 1 with p 1 = p p. As poited out by a referee, this b

4 384 S. Lig is very useful i model buildig. The Wald statistic ca be used to fashio a test for idepedet white oise agaist a AR alterative, i.e. test whether φ = φ 1 =...= φ p =, by usig W.p + 1/ = 4 fˆ 2./ ˆφ. ˆΣ 1 ˆΩ ˆΣ 1 / 1 ˆφ : We reject the idepedece hypothesis whe W.p + 1/ is large. This also provides a model selectio tool. Sice the residuals are approximately idepedet ad idetically distributed for a well-fitted model, we may use W.p + 1/ to test the idepedece of the residuals {ˆ" t }, where ˆ" t = y t X t ˆφ. If we caot reject the idepedece hypothesis, the the fitted model is cosidered to be adequate for the data. A atural questio is whether or ot W.p 1 / has a local power. For this, we cosider the local alterative hypothesis H 1 : Γφ = γ, where φ = φ + ν= 1=2 ad ν R p+1 is a costat vector. The stadard method for this is to show that the probability measures of.y 1,...,y / uder hypotheses H ad H 1 are cotiguous ad the to use Le Cam s third lemma. It is ot clear whether cotiguity holds for the IVAR model. Eve if cotiguity holds, it would be difficult to prove that i the usual method as i Lig ad McAleer (23). I Appedix A, we prove the followig result by a direct method. This result implies that W.p 1 / has a o-trivial local power. Theorem 3. If the assumptios of theorem 2 hold ad g.x 1,...,x p / is almost everywhere cotiuous o R p, the uder hypotheses H 1 { }.a/ 1=2 1. ˆφ φ / N ν, L 4 f 2./ Σ 1 ΩΣ 1,.b/ W.p 1 / χ 2 p L 1.µ/, where µ = 4 f 2./ν Γ.ΓΣ 1 ΩΣ 1 Γ / 1 Γν is a o-cetral parameter. To use the results i this sectio, we eed to select a weight w t. It seems reasoable to use the followig weight aalogue to Huber s (1977) ifluece fuctio: { 1 if at =, w t = C 3 =at 3.2:3/ if a t, where a t = Σ p i=1 y t i. y t i C/ ad C> is a costat. This weight satisfies assumptio 2. It dowweights the covariace matrices with leverage poits but takes full advatage of all matrices without leverage poits. Similar to Huber s estimator for regressio models, we eed to choose C. Whe P." t >x/= P." t < x/ = x α with < α < 2forx 1, by expressio (2.7) i Davis ad Resick (1985), we ca show that Σ 1 ΩΣ 1 = O.1/[E{y 2 t 1 I. y t 1 C/} + C 2 α ] 1, as p = 1 ad C. Thus, the larger C is, the smaller the asymptotic variace is. However, for a large C, the distributio of ˆφ may ot be well approximated by its limitig distributio as is small. We still have o theory to support the choice of C. But our simulatio results i Sectio 3 show that it works well whe C is the 9% or 95% quatile of data {y 1,...,y }. Also, we ote that a small-tailed idex α results i a small asymptotic variace. Obviously, there

5 Self-weighted Least Absolute Deviatio Estimatio 385 are may other weights, such as w t =.1 + C X t 2 / 3=2 ad I.max 1 i p y t 1 C/, that satisfy assumptio 2. However, our simulatio results which are ot reported i this paper show that the SLAD estimator based o w t i expressio (2.3) is much more efficiet tha that based o these weights. 3. Simulatio studies This sectio examies the performace of the asymptotic results i fiite samples through Mote Carlo experimets. I all experimets, we used the weight w t i expressio (2.3) with C beig the 95% quatile of data {y 1,...,y }. To estimate f./, the optimal badwidth b thatisgive i Silverma (1986), page 4, was used, which is automatically selected from the data. We first study the meas ad stadard deviatios of the SLAD estimator. Data are geerated through the AR(1) model y t = φ + φ 1 y t 1 + " t. The true parameters are take to be.φ, φ 1 / =., :5/,., :5/,., :8/. Three error distributios, Cauchy, t 2 ad N., 1/, are cosidered. Here, N., 1/ has a fiite variace ad is give for referece. The sample sizes are =2 ad =4. The umber of replicatios is 1. Table 1 summarizes the empirical meas, empirical stadard deviatios SD ad asymptotic stadard deviatios AD of the SLAD estimators. The ADs are calculated by usig the estimated covariaces i equatios (2.1) ad (2.2). Table 1 shows that all the biases are very small ad all the SDs ad ADs are very close, particularly whe = 4. As icreases from 2 to 4, all the SDs ad ADs become smaller. Whe the tail idex becomes smaller, ote that the N., 1/, t 2 - ad Cauchy distributios are i order ad all the SDs ad ADs become smaller. This cofirms our discussio i Sectio 2. Table 1. Meas ad stadard deviatios of the SLAD estimators for AR(1) models φ φ 1 Results for Results for Results for " t Cauchy " t t 2 " t N(, 1) ˆφ ˆφ 1 ˆφ ˆφ 1 ˆφ ˆφ 1 = 2..5 Mea SD AD Mea SD AD Mea SD AD = 4..5 Mea SD AD Mea SD AD Mea SD AD replicatios.

6 386 S. Lig Fig. 1. Desity curves of N., 1/ (- -), N c ( ) ad N t ( ) To give a overall view o the distributio of ˆφ 1 ad its limitig distributio, we simulate 2 replicatios for the case with φ 1 = :5 ad = 4 whe the error distributios are Cauchy ad t 2. Deote 1=2. ˆφ 1 :5/= ˆσ SLAD by N c ad N t whe the distributio is Cauchy ad t 2 respectively, where ˆσ SLAD are the ADs of ˆφ 1. Fig. 1 shows the desity curves of N c, N t ad N., 1/. The desity curve of N c, deoted by f c.x/, is approximated by f c.x i / 2 I.x i 1 N rc x i /=2 r=1 with x = 6 ad x i =x i 1 +1, where i=1,...,12adn rc is the value of N c at its rth replicatio, ad similarly for N t. Fig. 1 shows that the desity of N., 1/ approximates reasoably well those of N c ad N t for all x. I particular, they are very close to each other whe x > 1, which is importat i hypothesis testig ad i costructig cofidece itervals. We ow ivestigate the size ad power of W.1/ for testig the AR(1) agaist the AR(2) model whe.φ, φ 1 / =., :5/. W.1/ gives its sizes whe φ 2 = uder hypothesis H ad its powers whe φ 2 =±:1 ad φ 2 =±2. Agai, the sample sizes are = 2 ad = 4 ad the umber of replicatios is 1. Cauchy, t 2 - ad N., 1/ distributios are used. Table 2 summarizes the results at sigificace levels.1,.5 ad.1. From Table 2, we see that all the sizes are close to the omial sigificace levels whe = 4 ad reasoable whe = 2. The powers are icreased whe becomes large or whe the distace betwee the alterative ad the ull H becomes large as expected. These simulatio results idicate that the Wald test works well i fiite samples. We also carried out experimets for all the cases i Tables 1 ad 2 usig the weight w t i equatio (2.3) with C beig the 9% quatile of data {y 1,...,y }. The estimators are very similar to those i Table 1, but a little less efficiet. The sizes of the Wald test are similar to those i Table 2 whe the error distributio is t 2 ad N., 1/. Whe the error distributio is Cauchy, the sizes whe = 2 are a little better tha those i Table 2. However, they are similar to those i Table2whe = 4. All the powers are a little lower tha those i Table 2. These results are available from the author o request.

7 Self-weighted Least Absolute Deviatio Estimatio 387 Table 2. Sizes ad powers of the Wald test for ull hypothesis H : φ 2 D : at sigificace levels α i autoregressive models φ φ 1 φ 2 Results for the Cauchy Results for the t 2 - Results for the N(, 1) distributio ad the distributio ad the distributio ad the followig values of α: followig values of α: followig values of α: = = replicatios; umbers i italics deote size. 4. Example This sectio examies the Hag Seg Idex (HSI) i the Hog Kog stock-market. It is oe of the most importat idices i the Asia fiacial markets ad has bee extesively ivestigated i the literature. We cosider the HSI daily closig idex from Jue 3rd, 1996, to May 31st, 1998, which has a total of 497 observatios. Let x t deote the origial data ad y t deote the log-retur data, i.e. y t = log.x t =x t 1 /. Figs 2 ad 3 are time plots of x t ad 1y t respectively. They display some drastic shocks, which were caused by the South-east Asia ecoomic ad fiacial crisis i We first study whether or ot {y t } has a ifiite variace. For this, we use the Hill estimator for the tail idex of y t. Let y.1/ >y.2/ >... >y./ be the order statistic of {y t :1 t }. The estimators of the right- ad left-tailed idices are defied as Fig. 2. Time plot of the origial HSI (x t )

8 388 S. Lig Fig. 3. Time plot of the log-retur of the HSI (1y t ) Fig. 4. Hill estimators of the left-had tail idex H 1k ( ) ad the right-had tail idex H 2k (- -) { 1 H 1k = k { 1 H 2k = k k i=1 ( )} 1 y.i/ log, y.k+1/ k log i=1 ( )} 1 y. i+1/ y. k/ respectively. The Hill estimator is cosistet whe, k ad k= ; see for example Resick (1997). They may ot be stable i terms of k selected. Fig. 4 graphs {.k, H 1k / :1 k 2} ad {.k, H 2k / :1 k 2}. From Fig. 4, it seems difficult to obtai a accurate estimator for the tail idex. But it is clear that the right-had tail idex is less tha 2 ad the left-had tail idex is most likely less tha 2. This meas that y t should have a ifiite variace ad hece it seems reasoable to study {y t } by usig the results i Sectio 2. We ow use W.8/ ad W.12/ to test whether or ot {y t } is idepedet white oise. Their values are ad respectively. Both reject the idepedece hypothesis at the.5-level. The, we try to fit a AR(7) model to the data. The estimators are. ˆφ, ˆφ 1, ˆφ 2, ˆφ 3, ˆφ 4, ˆφ 5, ˆφ 6, ˆφ 7 / =.:66, :65, :, :15, :29, :84, :24, :9/, ad their stadard deviatios are.58,.43,.44,.44,.44,.44,.43 ad.43

9 Self-weighted Least Absolute Deviatio Estimatio 389 respectively. Agai, usig idepedece tests W.8/ ad W.12/ for the residuals, we obtai that W.8/=1:689 ad W.12/=13:36. The idepedece hypothesis caot ow be rejected ad hece a AR(7) process should be adequate for the data. For each i, weusew.1/ to test H : φ i =. Their values are as follows: i W.1/ 1:266 2:281 : 5:71 :431 3:637 :316 4:33 W.1/ rejects oly φ 3 = ad φ 7 = at the.5-level. Furthermore, we cosider the followig model for the data: y t = φ 3 y t 3 + φ 7 y t 7 + " t :.4:1/ The estimators are ˆφ 3 =:135 ad ˆφ 7 = :51 with stadard deviatios.44 ad.43 respectively. The values of idepedece tests W.8/ ad W.12/ for the residuals are 7.19 ad respectively, which caot reject the idepedece hypothesis ad hece model (4.1) should also be adequate for the data. The values of W.1/ are ad respectively, for testig φ 3 = ad φ 7 =. It oly rejects φ 7 = at the.5-level. Fially, we use the followig model to fit the data: y t = φ 3 y t 3 + " t :.4:2/ We obtai that ˆφ 3 = :123 ad its stadard deviatio is.43, ad W.1/ = 8:53, which rejects φ 3 = at the.5-level. Usig idepedece tests for the residuals, we obtai that W.8/=9:442 ad W.12/ = 17:152. They caot reject the idepedece hypothesis ad hece model (4.2) should be suitable as a fial model for the data. A possible iterpretatio for model (4.2) is that the Hog Kog stock-market has a delayed 3-day reactio to dyamic movemets. This may arise because ivestors i the Hog Kog stock-market prefer to observe other ews before makig tradig decisios. 5. Cocludig remarks This paper has proposed a self-weighted LAD estimator for IVAR models. This estimator was show to be asymptotically ormal ad it was used to costruct a Wald test for testig the liear restrictio o the parameters. The simulatio study ad a real data example showed that our theory ad method should be useful i practice. The self-weight priciple ca be used for other ifiite variace time series models, such as autoregressive movig average ad threshold autoregressive models (see Lig (24) for a self-weighted maximum likelihood estimator of autoregressive movig average geeralized autoregressive coditioal heteroscedastic models). Our results ca be exteded to other estimators, such as the M-, quatile, R- ad L-estimators i Koul (22). This paper provides a ew way to hadle heavy-tailed time series data ad should have a large applicable area i the future. Ackowledgemets The author greatly appreciates two referees, the Joit Editor, H. Koul, W. K. Li, M. McAleer ad V. A. Ukefer for their very helpful commets ad suggestios, ad thaks the Hog Kog Research Grats Coucil for fiacial support through Competitive Earmarked Research Grat HKUST4765/3H.

10 39 S. Lig Appedix A: Proofs The proof of theorem 1 is similar to those i Davis ad Dusmuir (1997), Kight (1998) ad Peg ad Yao (23). We deote the Euclidea orm by ad a radom sequece covergig to i probability by o p.1/, ad let F t = σ{" t, " t 1,...}. A.1. Proof of theorem 1 Deote ˆθ = 1=2. ˆφ φ / ad L.u/ = ( w t " t 1 ) 1=2 u X t " t, where u R p+1. The, ˆθ is the miimizer of L.u/ o R p+1. Usig the idetity x y x = y{i.x > / I.x < /} + 2 y {I.x s/ I.x /} ds, which holds whe x (see Kight (1998)), it follows that L.u/ = u T + 2 ξ t.u/,.a:1/ where T = Σ w tx t {I." t > / I." t < /}= 1=2 ad u X t=1=2 ξ t.u/ = w t {I." t s/ I." t /} ds: Deote the distributio of " t by F.x/. By Taylor s expasio, we have E{ξ t.u/ F t 1 } = u X t=1=2 w t {F.s/ F./} ds = u X t=1=2 w t = u { f./ 2 { sf./ s2 f.s Å } / ds } w t X t X t u u X t=1=2 w t s 2 f.s Å / ds, where s Å., s/. Thus, by equatios (A.1) ad (A.2), it follows that ( ) 1 L.u/ = u T + f./u w t X t X t u + R.u/,.A:2/.A:3/ where R.u/ = u X t=1=2 w t s 2 f.s Å / ds + 2 [ξ t.u/ E{ξ t.u/ F t 1 }]: By assumptios 2 ad 3, for each u, it follows that u X t=1=2 w t s 2 f.s Å / ds sup f.x/ u 3 w x 1=2 t X t 3,.A:4/ almost surely as. As for equatio (A.2), we ca show that

11 Self-weighted Least Absolute Deviatio Estimatio 391 { E{ξ 2 1 u X t = 1=2 } t.u/} E w 2 1=2 t u X t I." t s/ I." t / ds 1 [ u X t = 1=2 ] E w 2 1=2 t u X t {F.s/ F./} 1=2 ds 1 ( u X t = 1=2 ) max{f.x/ 1=2 } E w 2 1=2 t x u X t s 1=2 ds u 2:5 max{f.x/ 1=2 } E.w 2 1+1=4 t X t 2+1=2 / x u 2:5 max{f.x/ 1=2 } E{w 2 1+1=4 t. X t 2 + X t 3 /}: x By assumptios 2 ad 3, for each u, it follows that ( ) 2 = E [ξ t.u/ E{ξ t.u/ F t 1 }] E[ξ t.u/ E{ξ t.u/ F t 1 }] 2 2 E{ξ 2 t.u/},.a:5/ as. By iequalities (A.4) ad (A.5), we kow that R.u/ = o p.1/ for each u. By assumptios 1 ad 2 ad the ergodic theorem, we have 1 1.w t X t X t / Σ almost surely,.a:6/.w 2 t X tx t / Ω almost surely as. By the secod part of expressio (A.6) ad the cetral limit theorem, we readily show that T L Φ, where Φ N., Ω/. Thus, by equatio (A.3), for each u, wehave L.u/ u Φ + f./u Σu: L The limit u Φ + f./u Σu has a uique miimum at u = {2 f./} 1 Σ 1 Φ almost surely. As for corollary 2 i Kight (1998), by the covexity of L.u/ for each, we ca show that { } ˆθ = 1=2 1. ˆφ φ / N, L 4f 2./ Σ 1 ΩΣ 1 : This completes the proof. A.2. Proof of theorem 2 Deote A t.u/ = K.y/ f.b y + u X t = 1=2 / dy, where u R p+1. The, we have { ( ) 1 "t u X t = 1=2 ( ) E K F t 1} = 1b x u X t = 1=2 K f.x/ dx = A t.u/: b b b Usig this with K.x/ dx = 1 ad K 2.x/ dx<, for each u, it follows that [ { ( ) }] 1 1 "t u X t = 1=2 2 E w t K A t.u/ b b = 1 [ { ( ) }] 1 E w 2 "t u X t = 1=2 2 2 t K A t.u/ b b 2 { ( )} w 2 E t K 2 "t u X t = 1=2 + 2 E{w 2 2 b b 2 t A2 t.u/} b

12 392 S. Lig 2 2 b { E w 2 t 2 max x{f.x/}e.w 2 t / b ( ) } K 2.y/f b y + u X t dy =2 E{w 2 t A2 t.u/} K 2.y/ dy + 2 max x{f.x/}e.wt 2/ = o.1/,.a:7/ by assumptio 3. By iequality (A.7) ad the cotiuity of f, we ca further show that { ( ) sup 1 1 "t u X t = 1=2 w t K A t.u/} = o p.1/, b u M b for ay fixed costat M>. Let ˆθ = 1=2. ˆφ φ /, which is bouded i probability. By the precedig equatio, it follows that { ( 1 1 "t ˆθ w t K X ) } t= 1=2 A t. ˆθ / = o p.1/: b b Sice y K.y/ dy<, by Taylor s expasio ad assumptios 2 ad 3, we have 1 w t A t.ˆθ / f./ = 1 w t K.y/{f.b y + ˆθ X t= 1=2 / f./} dy = 1 w t K.y/.b y + ˆθ X t= 1=2 /f.ξ Å t / dy { max f b.x/ w t y K.y/ dy + ˆθ } w x 3=2 t X t ( 1 ) = O p.b / + O p = o 1=2 p.1/, where ξ Å t lies betwee ad b y + ˆθ X t. By the precedig two iequalities, we ca readily show that fˆ./ = f./ + o p.1/. Fially, by theorem 1 ad expressio (A.6), the coclusio holds. This completes the proof. A.3. Proof of theorem 3 Note that y t uder hypothesis H 1 depeds o. To emphasize this, we deote y t by y t uder H 1. Here, y t is a fuctio of, φ, ν ad {" t } ad y t = y t whe ν =, where y t is defied by model (1.1) uder H. Similarly defie w t ad X t. It is easy to see that y t y t almost surely. Sice g is cotiuous almost everywhere, w t w t almost surely whe. Now, uder H 1, ˆφ = arg mi φ Θ ( ) w t y t X t 1 φ : Let a t = w t X t {I." t > / I." t < /}. By Markov s iequality, the domiated covergece theorem ad the ergodic theorem, we ca show that 1 a t a t = Ω + o p.1/,.a:8/ 1 E.a t a t F t 1/ = Ω + o p.1/: Sice a t is a martigale differece i terms of F t, by the cetral limit theorem for martigale differeces ad expressio (A.8), it follows that 1 a 1=2 t N., Ω/:.A:9/ L Usig expressios (A.8) ad (A.9) ad a similar method as for theorem 1, we ca show that part (a) holds. Furthermore, we ca show that ˆΣ = Σ + o p.1/, ˆΩ = Ω + o p.1/ ad ˆ f./ = f./ + o p.1/ uder hypothesis

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