SFB 823 Robust estimatio of chage-poit locatio Discussio Paper Caria Gersteberger Nr. 2/207
Robust Estimatio of Chage-Poit Locatio Caria Gersteberger We itroduce a robust estimator of the locatio parameter for the chage-poit i the mea based o the Wilcoxo statistic ad establish its cosistecy for L ear epoch depedet processes. It is show that the cosistecy rate depeds o the magitude of chage. A simulatio study is performed to evaluate fiite sample properties of the Wilcoxo-type estimator i stadard cases, as well as uder heavy-tailed distributios ad disturbaces by outliers, ad to compare it with a CUSUM-type estimator. It shows that the Wilcoxo-type estimator is equivalet to the CUSUM-type estimator i stadard cases, but outperforms the CUSUM-type estimator i presece of heavy tails or outliers i the data. KEYWORDS: Wilcoxo statistic; chage-poit estimator; ear epoch depedece Itroductio I may applicatios it ca ot be assumed that observed data have a costat mea over time. Therefore, extesive research has bee doe i testig for chage-poits i the mea, see e.g. Giraitis et al. 996, Csörgö ad Horváth 997, Lig 2007, ad others. A umber of papers deal with the problem of estimatio of the chage-poit locatio. Bai 994 estimates the uow locatio poit for the brea i the mea of a liear process by the method of least squares. Atoch et al. 995 ad Csörgö ad Horváth 997 established the cosistecy rates for CUSUM-type estimators for idepedet data, while Csörgö ad Horváth 997 cosidered wealy depedet variables. Horváth ad Koosza 997 established cosistecy of CUSUM-type estimators of locatio of chage-poit for strogly depedet variables. Koosza ad Leipus 998, 2000 discussed CUSUM-type estimators for depedet observatios ad ARCH models. I spite of umerous studies o testig for chages ad estimatig for chage-poits, however, just a few procedures are robust agaist outliers i the data. I a recet wor Dehlig et al. 205 address the robustess problem of testig for chage-poits by itroducig a Wilcoxo-type test which is applicable uder short-rage depedece see also Dehlig et al. 203 for the log-rage depedece case. I this paper we suggest a robust Wilcoxo-type estimator for the chage-poit locatio based o the idea of Dehlig et al. 205 ad applicable for L ear epoch depedet Date: Jauary 9, 207. * Faultät für Mathemati, Ruhr-Uiversität Bochum, 44780 Bochum, Germay
processes. The Wilcoxo chage-poit test statistic is defied as W = {Xi X j } /2 i= j=+ ad couts how ofte a observatio of the secod part of the sample, X +,..., X, exceeds a observatio of the first part, X,..., X. Assumig a chage i mea happes at the time, the absolute value of W is expected to be large. Hece, the Wilcoxotype estimator for the locatio of the chage-poit, ˆ = mi { : l< W l } = W, 2 ca be defied as the smallest for which the Wilcoxo test statistic W attais its imum. Sice the Wilcoxo test statistic is a ra-type statistic, outliers i the observed data ca ot affect the test statistic sigificatly. O the cotrary, the CUSUMtype test statistic C = X i X i, i= which compares the differece of the sample mea of the first observatios ad the sample mea over all observatios, ca be sigificatly disturbed by a sigle outlier. The outlie of the paper is as follows. I Sectio 2 we discuss the cosistecy ad the rates of the estimator ˆ i 2. Sectio 3 cotais the simulatio study. Sectio 4 provides useful properties of the Wilcoxo test statistic ad the proof of the mai result. Sectios 5 ad 6 cotai some auxiliary results. 2 Defiitios, assumptios ad mai results i= Assume the radom variables X,..., X follow the chage-poit model { Y i + µ, i X i = Y i + µ +, < i, 3 where the process Y j is a statioary zero mea short-rage depedet process, deotes the locatio of the uow chage-poit ad µ ad µ + are the uow meas. We assume that Y has a cotiuous distributio fuctio F with bouded secod derivative ad that the distributio fuctios of Y Y, satisfy Px Y Y y C y x, 4 for all 0 x y, where C does ot deped o ad x, y. We allow the magitude of the chage vary with the sample size. Assumptio 2.. a The chage-poit = [θ], 0 < θ <, is proportioal to the sample size. 2
b The magitude of chage depeds o the sample size, ad is such that 0, 2,. 5 Next we specify the assumptios o the uderlyig process Y j. The followig defiitio itroduces the cocept of a absolutely regular process which is also ow as β-mixig. Defiitio 2.. A statioary process Z j j Z is called absolutely regular if β = sup E sup P A F + P A 0 A F as, where F b a is the σ-field geerated by radom variables Z a,..., Z b. The coefficiets β are called mixig coefficiets. For further iformatio about mixig coditios see Bradley 2002. The cocept of absolute regularity covers a wide rage of processes. However, importat processes lie liear processes or AR processes might ot be absolutely regular. To overcome this restrictio, i this paper we discuss fuctioals of absolutely regular processes, i.e. istead of focusig o the absolute regular process Z j itself, we cosider process Y j with Y j = fz j, Z j, Z j 2,..., where f : R Z R is a measurable fuctio. The followig ear epoch depedece coditio esures that Y j maily depeds o the ear past of Z j. Defiitio 2.2. We say that statioary process Y j is L ear epoch depedet L NED o a statioary process Z j with approximatio costats a, 0, if coditioal expectatios EY G, where G is the σ-field geerated by Z,..., Z, have property E Y EY G a, = 0,, 2,... ad a 0,. Note that L NED is a special case of more geeral L r ear epoch depedece, where approximatio costats are defied usig L r orm: E Y EY G r a, r. L r NED processes are also called r-approximatig fuctioals. I testig problems cosidered i this paper we allow for heavy-tailed distributios. Hece, we deal with L ear epoch depedece, which assumes existece of oly the first momet E Y. The cocept of ear epoch depedece is applicable e.g. to GARCH, processes, see Hase 99, ad liear processes, see Example 2. below. Borovova et al. 200 provide additioal examples ad iformatio about properties of L r ear epoch depedet process. Example 2.. Let Y j be a liear process, i.e. Y t = j=0 ψ jz t j, where Z j is whiteoise process ad the coefficiets ψ j, j 0, are absolutely summable. Sice Z j is statioary ad Z t j is G measurable for t j, we get E Y t EY t G j=+ ψ j E Z t j EZ t j G 2 j=+ ψ j E Z t j = 2 E Z j=+ ψ j. 3
Thus, the liear process Y j is L NED o Z j with approximatio costats a = 2 E Z j=+ ψ j. We will assume that the process Y j i 3 is L ear epoch depedet o some absolutely regular process Z j. I additio, we impose the followig coditio o the decay of the mixig coefficiets β ad approximatio costats a : 2 β + a <. 6 = The ext theorem states the rates of cosistecy of the Wilcoxo-type chage-poit estimator ˆ give i 2 ad the estimator ˆθ = ˆ/ of the true locatio parameter θ for the chage-poit = [θ]. Theorem 2.. Let X,..., X follow the chage-poit model 3 ad Assumptio 2. be satisfied. Assume that Y j is a statioary zero mea L ear epoch depedet process o some absolutely regular process Z j ad 6 holds. The, ˆ = OP 2, 7 ad ˆθ θ = OP 2. 8 The rate of cosistecy of ˆθ i 8 is give by 2. The assumptio 2 i 5 implies ˆ = o P ad yields cosistecy of the estimator: ˆθ p θ. I particular, for /2+ɛ, ɛ > 0, the rate of cosistecy i 8 is 2ɛ : ˆθ θ = O P 2ɛ. The same cosistecy rate ɛ for the CUSUM-type chage-poit locatio estimator θ C = C /, give by { C = mi : i i X j i j= X j = j= X j j= X j }, 9 was established by Atoch et al. 995 for idepedet data ad by Csörgö ad Horváth 997 for wealy depedet data. 3 Simulatio results I this simulatio study we compare the fiite sample properties of the Wilcoxo-type chage-poit estimator ˆ, give i 2, with the CUSUM-type estimator C, give i 9. We refer to the Wilcoxo-type chage-poit estimator by W ad to the CUSUM-type estimator by C. j= 4
We geerate the sample of radom variables X,..., X usig the model { Y i + µ, i X i = Y i + µ +, < i 0 where Y i = ρy i +ɛ i is a AR process. I our simulatios we cosider ρ = 0.4, which yields a moderate positive autocorrelatio i X i. The iovatios ɛ i are geerated from a stadard ormal distributio ad a Studet s t-distributio with degree of freedom. We cosider the time of chage = [θ], θ = 0.25, 0.5, 0.75, the magitude of chage = 0.5,, 2 ad the sample sizes = 50, 00, 200, 500. All simulatio results are based o 0.000 replicatios. Note that we report estimatio results ot for ˆ ad C, but ˆθ = ˆ/ ad θ C = C /. Figure cotais the histogram based o the sample of 0.000 values of Wilcoxo-type estimator ˆθ ad the CUSUM-type estimator θ C, for the model 0 with =, θ = 0.5, = 50 ad idepedet stadard ormal iovatios ɛ i. Both estimatio methods give very similar histograms. Table reports the sample mea ad the sample stadard deviatio based o 0.000 values of ˆθ ad θ C for other choices of parameters ad θ. It shows that performace of both estimators improves whe the sample size ad the magitude of chage are risig, ad whe the chage happes i the middle of the sample. I geeral, Wilcoxotype estimator performs i all experimets as good as the CUSUM-type estimator. Figure 2 shows the histogram based o 0.000 values of ˆθ ad θ C, for the model 0 with t -distributed heavy-tailed iid iovatios ɛ i, =, θ = 0.5 ad = 500. For heavytailed iovatios ɛ i, both estimators deviate from the true value of the parameter θ more sigificatly tha uder ormal iovatios. Nevertheless, the Wilcoxo-type estimator seems to outperform the CUSUM-type estimator. Figure 3 shows the histogram based o 0.000 values for ˆθ ad θ C whe the data X,..., X is geerated by 0 with =, θ = 0.5, = 200 ad ɛ i NIID0, ad cotais outliers. The outliers are itroduced by multiplyig observatios X [0.2], X [0.3], X [0.6] ad X [0.8] by the costat M = 50. The histogram shows that the Wilcoxo-type estimator is rarely affected by the outliers, whereas the CUSUM-type estimator suffers large distortios. Table 2 reports the sample mea ad the sample stadard deviatio based o 0.000 values of ˆθ ad θ C for = ad θ = 0.5 for sample size = 50, 00, 200, 500 i the case of the ormal, ormal with outliers ad t -distributed iovatios. Figures, 2 ad 3 presets results for = 50, 200, 500. I geeral, we coclude that the Wilcoxo-type chage-poit locatio estimator performs equally well as the CUSUM-type chage-poit estimator i stadard situatios, but outperforms the CUSUM-type estimator i presece of heavy tails ad outliers. 5
Frequecy 0 500 500 2500 3500 Frequecy 0 500 500 2500 3500 0.0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 a CUSUM θ C b Wilcoxo ˆθ Figure : Histogram based o 0.000 values for the Wilcoxo-type estimator ˆθ ad the CUSUM-type estimator θ C. X i follows the model 0 with =, θ = 0.5, = 50 ad ormal iovatios ɛ i NIID0,. =50 =00 =200 =500 θ C W C W C W C W 0.5 0.25 mea 0.46 0.46 0.43 0.44 0.40 0.40 0.34 0.34 sd 0.2 0.2 0.20 0.20 0.8 0.8 0.3 0.3 0.50 mea 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 sd 0.8 0.8 0.6 0.6 0.3 0.3 0.08 0.08 0.75 mea 0.54 0.54 0.57 0.56 0.6 0.6 0.66 0.66 sd 0.20 0.20 0.20 0.20 0.8 0.8 0.3 0.3 0.25 mea 0.39 0.39 0.35 0.35 0.3 0.3 0.28 0.28 sd 0.8 0.8 0.4 0.4 0.0 0.0 0.05 0.06 0.50 mea 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 sd 0.2 0.2 0.09 0.09 0.05 0.05 0.02 0.02 0.75 mea 0.6 0.60 0.65 0.65 0.69 0.69 0.72 0.72 sd 0.7 0.7 0.5 0.5 0.0 0.0 0.05 0.06 2 0.25 mea 0.30 0.3 0.28 0.29 0.27 0.28 0.26 0.26 sd 0.0 0.0 0.06 0.07 0.04 0.04 0.02 0.02 0.50 mea 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 sd 0.05 0.05 0.03 0.03 0.02 0.0 0.0 0.0 0.75 mea 0.69 0.68 0.72 0.7 0.73 0.73 0.74 0.74 sd 0.09 0.0 0.06 0.07 0.04 0.04 0.02 0.02 Table : Sample mea ad the sample stadard deviatio based o 0.000 values of ˆθ ad θ C. X i follows the model 0 with ormal iovatios ɛ i NIID0,. 6
Frequecy 0 200 400 600 Frequecy 0 500 000 500 2000 0.0 0.2 0.4 0.6 0.8.0 0.0 0.2 0.4 0.6 0.8.0 a CUSUM θ C b Wilcoxo ˆθ Figure 2: Histogram of CUSUM-type estimator θ C ad Wilcoxo-type estimator ˆθ based o 0.000 values of θ C ad ˆθ for the model 0 with iid t -distributed iovatios, =, θ = 0.5 ad = 500. Frequecy 0 500 500 2500 Frequecy 0 000 2000 3000 4000 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 a CUSUM θ C b Wilcoxo ˆθ Figure 3: Histogram based o 0.000 values of θ C ad ˆθ for the model 0 with ormal iovatios ɛ i NIID0,, =, θ = 0.5, = 200 ad outliers. =50 =00 =200 =500 Iovatios C W C W C W C W ormal mea 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 sd 0.2 0.2 0.09 0.09 0.05 0.05 0.02 0.02 t mea 0.52 0.50 0.5 0.50 0.5 0.50 0.50 0.50 sd 0.23 0.20 0.24 0.9 0.24 0.7 0.25 0.4 ormal with mea 0.50 0.49 0.50 0.50 0.50 0.50 0.5 0.50 outliers sd 0.7 0.3 0.6 0.09 0.5 0.06 0.09 0.02 Table 2: Sample mea ad the sample stadard deviatio of ˆθ ad θ C based o 0.000 replicatios for the ormal, ormal with outliers ad t -distributed iovatios, = ad θ = 0.5. 7
4 Useful properties of the Wilcoxo test statistic ad proof of Theorem 2. This sectio presets some useful properties of the Wilcoxo test statistic ad the proof of Theorem 2.. Throughout the paper without loss of geerality, we assume that µ = 0 ad > 0. We let C deote a geeric o-egative costat, which may vary from time to time. The otatio a b meas that two sequeces a ad b of real umbers have property a /b c, as, where c 0 is a costat. g = sup x gx stads for the supremum orm of fuctio g. By d we deote the covergece i distributio, by p the covergece i probability ad by d = we deote equality i distributio. 4. U-statistics ad Hoeffdig decompositio The Wilcoxo test statistic W i uder the chage-poit model 3 ca be decomposed ito two terms W = = = where { i= i= j=+ {Xi X j } /2 j=+ {Y i Y j } /2 + i= j= + {Y j <Y i Y j + }, i= j=+ {Y i Y j } /2 + i= j=+ {Y j <Y i Y j + }, <, { U + U,, U + U,, <, U = U, = U, = i= j=+ i= j= + i= j=+ {Yi Y j } /2,, 2 {Yj <Y i Y j + },, 3 {Yj <Y i Y j + }, <. 4 The first term U depeds oly o the uderlyig process Y j, while the terms U, ad U, deped i additio o the chage-poit time ad the magitude of the chage i the mea. The term U ca be writte as a secod order U-statistic U = h Y i, Y j Θ,, i= j=+ 8
with the erel fuctio h x, y = {x y} ad the costat Θ = E h Y, Y 2 = /2, where Y ad Y 2 are idepedet copies of Y. We apply to U Hoeffdig s decompositio of U-statistics established by Hoeffdig 948. It allows to write the erel fuctio as the sum h x, y = Θ + h x + h 2 y + g x, y, 5 where h x = E h x, Y 2 Θ = /2 F x, h2 y = E h Y, y Θ = F y /2, g x, y = h x, y h x h 2 y Θ. By defiitio of h ad h 2, E h Y = 0 ad E h 2 Y = 0. Hece, E gx, Y = E gy, y = 0, i.e. gx, y is a degeerate erel. The term U, i 3 ad U, i 4 ca be writte as a U-statistic U, = h Y i, Y j,, i= j= + with the erel h x, y = hx, y + hx, y = {y<x y+ }. The Hoeffdig decompositio allows to write the erel as with Θ = E {Y 2 Y Y 2 + }, h x, y = Θ + h, x + h 2, y + g x, y, 6 h, x = E h x, Y 2 Θ = F x F x Θ, h 2, y = E h Y, y Θ = F y + F y Θ, g x, y = h x, y h, x h 2, y Θ. By assumptio the distributio fuctio F of Y has bouded probability desity f ad bouded secod derivative. This allows to specify the asymptotic behaviour of Θ, as, Θ = E {Y 2 <Y Y 2 + } = P Y 2 < Y Y 2 + = F y + F y df y = f 2 y dy + o. 7 R Note that E h, Y = 0 ad E h 2, Y = 0. Therefore, g x, y is a degeerate erel, i.e. E g x, Y = E g Y, y = 0. Furthermore, h, 0, as, sice h, x Fx Fx Θ C + Θ C, 8 where C > 0 is a costat ad 0, as. R 9
4.2 -cotiuity property of erel fuctios h ad h Asymptotic properties of ear epoch depedet processes Y j itroduced i Sectio 2 are well ivestigated i the literature, see e.g. Borovova et al. 200. I the cotext of chage-poit estimatio we are iterested i asymptotic properties of the variables hy i, Y j, where hx, y = {x y} is the Wilcoxo erel, ad also i properties of the terms h Y j ad h, Y j of the Hoeffdig decompositio of the erels i 5 ad 6. We will eed to show that the variables hy i, Y j, h Y j ad h, Y j retai some properties of Y j. To derive them, we will use the fact that the erels h i 5 ad h i 6 satisfy the -cotiuity coditio itroduced by Borovova et al. 200. Defiitio 4.. We say that the erel h x, y is -cotiuous with respect to a distributio of a statioary process Y j if there exists a fuctio φɛ 0, ɛ 0 such that φ ɛ 0, ɛ 0, ad for all ɛ > 0 ad h Y, Y h Y E E, Y { Y Y ɛ} h Y, Y h Y, Y { Y Y ɛ} φ ɛ, 9 φ ɛ, ad h E Y, Y 2 h Y, Y 2 { Y Y ɛ} φ ɛ, 20 h E Y 2, Y h Y 2, Y { Y Y ɛ} φ ɛ, where Y 2 is a idepedet copy of Y ad Y is ay radom variable that has the same distributio as Y. For a uivariate fuctio gx we defie the -cotiuity property as follows. Defiitio 4.2. The fuctio g x is -cotiuous with respect to a distributio of a statioary process Y j if there exists a fuctio φɛ 0, ɛ 0 such that φ ɛ 0, ɛ 0, ad for all ɛ > 0 g E Y g Y { Y Y ɛ} φ ɛ, 2 where Y is ay radom variable that has the same distributio as Y. Corollary 4. below establishes the -cotiuity of fuctios hx, y = {x y} ad h x, y = {y<x y+ },. For h, we assume that 9 ad 20 hold with the same φɛ for all. We start the proof by showig the -cotiuity of the more geeral erel fuctio hx, y; t = {x y t}. Lemma 4.. Let Y j be a statioary process, Y have distributio fuctio F which has bouded first ad secod derivative ad Y Y, satisfy 4. The the fuctio hx, y; t = {x y t} is -cotiuous with respect to the distributio fuctio of Y j. 0
Proof. The proof is similar to the proof of -cotiuity of the erel fuctio hx, y; t = { x y t} give i Example 2.2 of Borovova et al. 200. Note that {Y Y t} {Y Y t} = 0 if Y Y t ad Y Y t; or Y Y > t ad Y Y > t. The differece is ot zero if Y Y t ad Y Y > t; or Y Y > t ad Y Y t. Let Y Y ɛ, where ɛ > 0. The Y Y < t ɛ implies Y Y < t, ad Y Y > t + ɛ implies Y Y > t. Hece, {Y Y t} {Y Y t} { Y Y ɛ} {t ɛ Y Y t+ɛ}. Therefore, {Y E Y t} {Y { Y Y P t ɛ Y Y t + ɛ C ɛ, 22 Y t} ɛ} because of assumptio 4. Similar argumet yields {Y E Y t} {Y Y t} { Y Y ɛ} P t ɛ Y Y t + ɛ C ɛ, {Y E Y 2 t} {Y Y 2 t} { Y Y Pt ɛ Y Y 2 t + ɛ C 2 ɛ, ɛ} E {Y 2 Y t} {Y 2 Y t} { Y Y ɛ} Pt ɛ Y Y 2 t + ɛ C 2 ɛ, where Y 2 is a idepedet copy of Y, otig that by the mea value theorem ad d Fy/dy C, Pt ɛ Y Y 2 t + ɛ = F y + t + ɛ F y + t ɛ df y R Cɛ fydy = C 2 ɛ. R These bouds imply 9 ad 20 with φɛ = Cɛ, where C does ot deped o t. This completes the proof. Corollary 4.. Assume that assumptios of Lemma 4. are satisfied. The, i Fuctio hx, y = {x y} is -cotiuous with respect to the distributio fuctio of Y j. ii Fuctio h x, y = {y<x y+ } is -cotiuous with respect to the distributio fuctio of Y j. Proof. i follows from Lemma 4., otig that {x y} = hx, y; 0. ii We eed to verify 9 ad 20. Write h x, y = hx, y hx, y + = {x y} {x y+ }. The by 22, E h Y, Y h Y, Y { Y Y E ɛ} {Y Y } {Y Y } { Y Y ɛ} + E {Y Y + } {Y Y + } { Y Y Cɛ. ɛ}
Similar argumet yields E h Y, Y h Y, Y { Y Y Cɛ, ɛ} E h Y, Y 2 h Y, Y 2 { Y Y Cɛ, ɛ} E h Y 2, Y h Y 2, Y { Y Y ɛ} Hece, 9 ad 20 hold with φɛ = Cɛ. Cɛ. Note that coditio 4 is satisfied if variables Y, Y,, have joit probability desities that are bouded by the same costat C for all. If the joit desity does ot exist, for examples of verificatio of coditio 4 see pages 435, 436 of Borovova et al. 200. Lemma 2.5 of Borovova et al. 200 yields that if a geeral fuctio hx, y is - cotiuous, i.e. satisfies 9 ad 20 with fuctio φɛ the E h x, Y 2, where Y 2 is a idepedet copy of Y, is also -cotiuous ad satisfies the coditio i 2 with the same fuctio φɛ. Hece, h i x ad h i, x, i =, 2 are -cotiuous ad satisfy the coditio i 2 with φɛ = Cɛ. Next we tur to -cotiuity property of gx, y. By Hoeffdig decompositio 5, gx, y = hx, y Θ h x h 2 y. Sice hx, y, h x ad h 2 x i 5 are - cotiuous ad satisfy 9, 20 ad 2 with the same fuctio φɛ = Cɛ, the gx, y is also -cotiuous with fuctio φɛ = Cɛ. Ideed, E gy, Y gy, Y { Y Y ɛ} E hy, Y hy, Y { Y Y ɛ} +E h Y h Y { Y Y ɛ} 2φɛ ad similarly, E gy, Y gy, Y { Y Y 2φɛ. ɛ} Usig the same argumet, it follows that the fuctio g x, y = h x, y Θ h, x h 2, x i the Hoeffdig decompositio 6 is also -cotiuous ad satisfies 9, 20 with φɛ = Cɛ. 4.3 NED property of h Y j ad h, Y j I Propositio 2. of Borovova et al. 200 it is show that if Y j is L NED o a statioary absolutely regular process Z j with approximatio costats a ad gx is -cotiuous with fuctio φ, the gy j is also L NED o Z j with approximatio costats φ 2a + 2 2a g. Thus, the processes h Y j ad h 2 Y j i 5 ad h, Y j ad h 2, Y j i 6 are L NED processes with approximatio costats a = C a. Corollary 3.2 of Wooldridge ad White 988 provides a fuctioal cetral limit theorem for partial sum process i= Ỹi,, where Ỹj is L 2 NED o a strogly mixig 2
process Z j. To apply this result to h Y j which is L NED o Z j with approximatio costats a, we eed to show that h Y j is also L 2 NED process. Note that the variables η := h Y Eh Y G have property E η 2 = E η 2 { } η a 2 + E η 2 { } η >a 2 a 2 E η + a E η 4 a + a C =: a. The last iequality holds, because by Defiitio 2.2 of L ear epoch depedece, E h Y Eh Y G a ad because h Y /2. Therefore the process h Y j is L 2 NED o Z j with approximatio costat a. Sice absolute regular process Z j is strogly mixig process, from Corollary 3.2 of Wooldridge ad White 988, we obtai [t] d /2 h Y i σw t 0 t, 0 t i= where W t is a Browia motio ad σ 2 = = Covh Y 0, h Y. Sice h 2 x = h x, all properties of h Y j remai valid also for h 2 Y j. 4.4 Proof of Theorem 2. First we show cosistecy property ˆ = o P of the estimate ˆ = arg W. To prove it, we verify that for ay ɛ > 0, lim P ˆ ɛ =. 23 This meas that the estimated value ˆ with probability tedig to is i a eighbourhood of the true value : P ˆ [ ɛ, + ɛ]. We will show that as, P W < W. 24 : ɛ Sice W : ɛ W, this proves 23. By, W = { U + U,, U + U,, <. Theorem 6. implies U = O P 3/2 ad Propositio 5. below yields U, Θ = o P 3/2, U, Θ = o P 3/2. 3
Hece, Thus, W = Θ + U, Θ + U = Θ + O P 3/2, W ɛ Θ + O P 3/2, ɛ W + ɛ Θ + O P 3/2. +ɛ W : ɛ W ɛδ + O P 3/2, where δ = mi, Θ. By defiitio = [θ] θ, ad by 7 ad 5, Θ c. Hece, δ = o 3/2 ad ɛδ + O P 3/2 = ɛδ + O P 3/2 δ = ɛδ + o P which proves 24. Next we establish the rate of covergece i 7, = O P / 2. Set a = M. 2 The for fixed M > 0, a, as. We will verify that lim P ˆ a, as M, which implies 7. As i 24, we prove this by showig lim < W, : a as M. 25 Defie V := W 2 W 2. If W attais its imum at, it is easy to see that V attais its imum at the same. Hece, ˆ = mi{ : l W l = W } = mi{ : l V l = V }. Thus, istead of 25 it remais to show that lim P V : < 0, M. 26 a Defie := mi{ : ɛ ; V = α l β V l }. Sice by 23 ˆ is a cosistet estimator of, it holds lim Pˆ = =. So, i the proof of 26 it suffices to cosider over, such that ɛ, a, which correspods to ɛ a ad + a < + ɛ. Let us start with ɛ a. Sice > 0, relatio 26 holds for such, if lim P ɛ a V 2 < 0, M. 27 4
Note that V 2 = W 2 W 2 2 W W = By, W = U + U,. The, where W W 2 + 2 W W = Θ + δ, + δ 2,, δ, = U U, δ 2, = U, U, W. 28 Θ. Observe that by 7, Θ c, c > 0, ad / θ. Therefore, /Θ c 0, where c 0 = θc. Moreover, a δ i, = o P, i =, 2, by 47 ad 48 of Lemma 5.4. Hece, W W I tur, = c 0 + o P, W W W = U, + U 2 = c 2 0 2 + o P. ad U, = Θ + δ 3,, where δ 3, = U, Θ. By Propositio 5., δ 3, / = o P /2. Sice Θ c 0 θc 0 ad = o, this implies U, = c 0 + o P. Next, by Theorem 6. below, U = O P 3/2, ad hece, U / = O P /2. Therefore, W / = c 0 + o P. Hece, for ɛ a, 29 W = c 0 + o P ɛ c 0 + o P. 30 Usig 29 ad 30 i 28, it follows V 2 2 ɛ c2 0 2 + o P c 2 0 2 + o P 2 ɛ c 0 2 + o P > 0. 5
This proves 27. Similar argumet yields lim P V +a < 0, M, +ɛ which completes the proof of 26 ad the theorem. 5 Auxiliary results This sectio cotais auxiliary results used i the proof of Theorem 2.. We establish asymptotic properties of the quatities U, U, ad U, defied i 2-4 ad appearig i the decompositio of W. The followig lemma derives a Háje-Réyi type iequality for L NED radom variables. Lemma 5.. Let Y j be a statioary L ear epoch depedet process o some absolutely regular process Z j, satisfyig 6. Assume that E Y j = 0 ad Y j K a.s. for some K 0. The, for all fixed ɛ > 0, for all m, P m where C > 0 does ot deped o m,, ɛ. i= Y i > ɛ C m ɛ 2, 3 Proof. To prove 3, we use the Háje-Réyi type iequality of Theorem 6.3 established i Koosza ad Leipus 2000, ɛ 2 P m i= Y i > ɛ + 2 =m m m 2 E 2 Y i + i= + 2 E Y + + 2 2 E 2 Y i =m Y j + j= =m i= + 2 E Y 2 +. 32 First we boud E i= Y i 2. Uder assumptios of this lemma, by Lemma 6. below, for i, j 0 Cov Y i, Y i+j = E Y i Y i+j 4Ka j 3 + 2K2 β j 3 Ca j 3 + β j. 33 3 By statioarity of Y j, E Y i Y j = Cov Y i, Y j = Cov Y 0, Y i j. 6
Hece, 2 E Y i = i= E Y i Y j i,j= C i,j= i,j= by 33 ad 6. Sice Y j K, the E Y + j= Cov Y 0, Y i j a i j + β i j C a 3 3 3 + β C, 3 Y j K E Y j K E j= Usig these bouds i 32 together with we obtai 3: ɛ 2 P i= =0 i= Y i 2 /2 C. + 2 2, + 2 + 2 2 + 2 2 4 3, m i= [ Y i > ɛ C m + =m 2 + =m ] 3/2 C m. The ext lemma establishes asymptotic bouds of the sums S = i= h, Y i, S 2 = h 2, Y j. 34 Lemma 5.2. Assume that Y j is a statioary zero mea L ear epoch depedet process o some absolutely regular process Z j ad 6 holds. Furthermore, let Assumptio 2. be satisfied ad S i, i =, 2, be as i 34. The S /2 i = o P, i =, 2. 35 Proof. To show 35 for i =, we will use the iequality give i Theorem 6.2. Defie S = i= /2 h, Y i,, ad set S 0 = 0. We eed to evaluate ES l S 4 for < l. Note that E S l S 4 = 2 E l i=+ h, Y i 4 j= l = 2 4 E h, Y i, i= 7
where the last equality holds because h, Y j is a statioary process. Sice h, Y j is L NED o a absolutely regular process, see Sectio 4.3, E h, Y 0 = 0 ad h, x C by 8, the by Lemma 6. ad the commet below l E h, Y i i= where C does ot deped o l, or. Thus, 4 Cl 2 2, P S l S λ λ 4 E S l S 4 Cl 2 2 λ 4 2 = l 2 λ 4 u,i, i=+ where u,i = C /2. Hece, S j satisfies assumptio 53 of Theorem 6.2 with β = 4, α = 2. Therefore, by 54, for ay fixed ɛ > 0, as, S P /2 ɛ K ɛ 4 i= u,i 2 = KC 2 ɛ 4 0, sice 0. The proof of 35 for i = 2 follows usig a similar argumet as i the proof for i =. Propositio 5.. Assume that Y j is L ear epoch depedet process o some absolutely regular process Z j ad 6 holds. Furthermore, let Assumptio 2. be satisfied. The U, Θ = o P 36 ad 3/2 3/2 where Θ is the same as i 7. Proof. By the Hoeffdig decompositio 6, Hece, U, Θ = U, Θ = o P, 37 h x, y Θ = h, x + h 2, y + g x, y. = i= j= + h, Y i + i= h, Y i + h 2, Y j + g Y i, Y j j= + h 2, Y j + i= j= + g Y i, Y j. 8
Deote U g, = i= j= + g Y i, Y j, U g, = i= j=+ g Y i, Y j. 38 Sice, ad i= + h 2,Y j = S 2 S 2, the U, S Θ U, Θ S where S i, i =, 2 are defied i 34. Therefore, + S 2 + S 2 + + S 2 + S 2 +,,,, U g U g U 3/2, Θ /2 S + S 2 + S 2 U + 3/2 g,. 39 The degeerate erel g is bouded ad -cotiuous, see Subsectios 4. ad 4.2. Thus, by Propositio 6. below, 3/2 3/2 U g, g Y i, Y j + i= j=+ 3/2 i= j=+ g Y i, Y j = o P. 40 Similar argumet implies < 3/2 U g, = o P. Usig i 39 the bouds 40 ad 35 of Lemma 5.2 we obtai U 3/2, Θ = op which proves 36. The proof of 37 follows usig similar argumet. Deote Ũ g = i= j=+ gy i, Y j. 4 Lemma 5.3. Assume that Y j is L ear epoch depedet process o some absolutely regular process Z j ad 6 holds. Furthermore, let Assumptio 2. be satisfied ad let 9
a = M/ 2, M > 0, ad Ũ g, U g, ad U g, are defied as i 4, 38. The there exists C > 0 such that for ay ɛ > 0, P P P Ũg : a a +a > ɛ Cɛ 2 2 a +, 42 U g, > ɛ Cɛ 2 2 a +, 43 U g, where C does ot deped o ɛ, ad a. > ɛ Cɛ 2 2 a +, Proof. Recall { : a} = { a} { + a}. We cosider oly the case a sice the proof for +a is similar. Proof of 42. Defie R = Ũ g Ũ g,, Ũ g 0 = 0 ad R 0 = 0. The = i= R i. Iequality 55 of Theorem 6.3, applied to the radom variables R i with c = / yields Ũ g ρ := ɛ 2 P = a 2 E R2 + i= a = a + 2 2 E R + R i > ɛ 2 2 2 E R i i= j= a R j + 2 E R2 +. 44 I Subsectios 4. ad 4.2, we showed that erel fuctio gx, y is bouded ad - cotiuous. Therefore, by Lemma 6.2 below [ E Ũ g Lemma 6.2 also yields E R 2 + = E Ũ g The, E R + 2] = E i= 2 + Ũ g 3 C R j j= = R i 2 C, =,...,. 45 /2 E R+ 2 E + 2 = C, =,...,. 46 2 /2 R j C. j= 20
From 44, 45 ad 46, usig 2 2 2 3, we obtai [ ρ C 2 + a = { 3 + + 2 Notig that, 3 2, it follows a ρ C + 2 2 C + 2. a = Proof of 43. It follows a similar lie to the proof of 42. Deote R = U g,,. We verified i Subsectios 4. ad 4.2 that fuctio g x, y is bouded U g ad -cotiuous. Therefore, by Lemma 6.2 below, ad [ E E R 2 + = E U g U g, 2] = E i= +, U g, = E j= + Combiig both bouds, we obtai E R+ R j j= R i 2 C, 2 }]. =,..., g Y +, Y j 2 C, =,...,. /2 E R 2 /2 + 2 E R j C. j= Usig the same argumet as i the proof of 42, we obtai P a U g, > ɛ Cɛ 2 + 2. a This completes proof of 43 ad the lemma. Lemma 5.4. Assume that Y j is a statioary zero mea L ear epoch depedet process o some absolutely regular process Z j ad 6 holds. Furthermore, let Assumptio 2. be satisfied ad let a = M/ 2, M > 0. The, as, M, i For ay ɛ > 0, P : a U U > ɛ 0. 47 2
ii For ay ɛ > 0, P ad P a +a where Θ is the same as i 7. U, U, U, U, Θ > ɛ 0, 48 Θ > ɛ 0, Proof. Notice that { : a} = { a} { + a}. We will prove relatios 47 ad 48 for a. The proof for +a is similar. Notice, that a = M 2 = om sice 2 by assumptio 5. Therefore, for a fixed M, a = o ad a > as. i Deote S = h Y i. i= By Hoeffdig s decompositio 5, for, ad usig h x = h 2 x, it follows U = = i= j=+ h Y i + h 2 Y j + gy i, Y j h Y i i= where Ũ g U U j=+ h Y j + i= j=+ is defied i 4. Hece, = S S S + Ũ g Therefore, for a, U U S S + S + =: ρ + ρ 2 + ρ 3 + ρ 4. gy i, Y j = S S + Ũ g, Ũ g Ũ g. a + Ũ g It suffices to show that for ay ɛ > 0, as, for l =,..., 4, P > ɛ 0, M, 49 a ρl which proves 47 for a. 22
For l =, statioarity of the process h Y j yields { S S = i=+ h Y i {, d= } a} S, a. Therefore, a ρ d = : a S d = a j Sj j. Sice h Y j is L NED o a absolutely regular process Z j, E h Y = 0 ad h x /2, the by Lemma 5., a S = O P. 50 a Thus, a ρ = O P a = O P M = o P, as M, which proves 49 for l =. For l = 2, by 50, S / = O P /2. Thus, ρ 2 = S = O P = o P, sice by 5, which proves 49 for l = 2. To show 49 for l = 3, recall that gx, y 3/2 is -cotiuous, see Subsectio 4.3. Therefore, by Lemma 6.2, g Ũ 2 E C, which implies that Ũ g = O P. Thus, ρ 3 Ũ g = a = O P a = O P = O P = O P = o P, as M, a M M which proves 49 for l = 3. 23
Fially, for l = 4, by Lemma 5.3, P a ρ4 > ɛ = P a C 2 ɛ 2 a + Ũ g > ɛ which proves 49 for l = 4 ad completes the proof of i. = C ɛ 2 M + 3 2 0, as M, ii Let S, S2 ad U g, be defied as i 34 ad 38. By Hoeffdig s decompositio 6, for, U, Θ = Hece, = i= j= + h, Y i + i= h, Y i + h 2, Y j + g Y i, Y j j= + U, U, Θ h 2, Y j + = S S + S 2 S 2 Therefore, for a, U, U, Θ S S + S 2 S 2 + =: ν + ν 2 + ν 3 + ν 4. i= j= + g Y i, Y j = S + S 2 S 2 + U g,. + U g U g, a, U g,. + U g, It suffices to show that for ay ɛ > 0, as, for l =,..., 4, P a νl > ɛ 0, M, 5 which proves 48 for a. The process h, Y j is statioary ad L NED o a absolutely regular process, see Sectio 4.3. Furthermore, it has zero mea ad h, C by 8. Hece, by the same argumet as for ρ, usig Lemma 5., it follows a ν d = S j a j j = O P a = O P M = o P, 24
as M. Lemma 5.2 yields /2 S 2 = o P. Therefore, a ν2 /2 S 2 + /2 S 2 = o P, sice. We showed i Subsectios 4. ad 4.2 that the fuctio g x, y is bouded ad - cotiuous. Hece, by Lemma 6.2, g U, 2 E C. Therefore, the claim a ν 3 = o P follows usig the same argumet as i the proof of 49 for l = 3. By Lemma 5.3, P a ν4 > ɛ = P a C 2 ɛ 2 a + U g, > ɛ = C ɛ 2 M + 3 2 0, as M, which proves 5 for l = 4. This completes the proof of 48 ad the lemma. 6 Auxiliary results from the literature This sectio cotais results from the literature used i the proofs of this paper. Lemma 6. states a correlatio ad a momet iequality for L NED radom variables, established by Borovova et al. 200. Lemma 6.. Lemma 2.8 ad 2.24, Borovova et al. 200 Let Y j be L ear epoch depedet o a absolutely regular, statioary process with mixig coefficiets β ad approximatio costats a, ad such that Y 0 K a.s. The, for all i, 0, CovY i, Y i+ 4Ka 3 + 2K2 β 3. I additio, if =0 2 a + β <, the there exists C > 0 such that for all 4 E Yi E Y i C 2. 52 i= The proof of Lemma 2.24 i Borovova et al. 200 shows that 52 holds with C = C 0 K 2, where C 0 > 0 does ot deped o K ad. I Theorem 3 of Dehlig et al. 205 the asymptotic distributio of the Wilcoxo test statistic for L NED radom process is obtaied. We use this result to show the cosistecy of the Wilcoxo-type estimator ˆ. 25
Theorem 6.. Theorem 3, Dehlig et al. 205 Assume that Y j is statioary ad L ear epoch depedet process o some absolutely regular process Z j ad 6 holds. The, 3/2 < i= j=+ {Yi Y j } /2 d σ sup B τ, 0 τ where B τ 0 τ is the stadard Browia bridge process, σ 2 = = ad F deotes the distributio fuctio of Y j. Cov F Y, F Y 0, We use the followig results from Dehlig et al. 205 to hadle the degeerate part gx, y of the Hoeffdig decompositio 5. Propositio 6.. Propositio, Dehlig et al. 205 Let Y j be statioary ad L ear epoch depedet o a absolutely regular process with mixig coefficiets β ad approximatio costats a satisfyig β + a + φa <, = with φɛ as i Defiitio 4.. If gx, y is a -cotiuous bouded degeerate erel, the, as, 3/2 g Y i, Y j p 0. i= j=+ Lemma 6.2. Lemma ad 2, Dehlig et al. 205 Uder assumptios of Propositio 6. there exists C > 0 such that for all m, 2, E 3 E i= j=+ i= j=+ gy i, Y j gy i, Y j 2 C, m i= j=m+ 2 gy i, Y j C m 2. I our proofs we use the imal iequality of Billigsley 999, which is valid for statioary/o-statioary ad idepedet/depedet radom variables ξ i. Theorem 6.2. Theorem 0.2, Billigsley 999 Let ξ,..., ξ be radom variables ad S = i= ξ,, S 0 = 0 deotes the partial sum. Suppose that there exist α >, β > 0 ad o-egative umbers u,,..., u, such that P S j S i λ j α λ β u,l, 53 l=i+ 26
for λ > 0, 0 i j. The for all λ > 0, 2, where K > 0 depeds oly o α ad β. By the Marov iequality, 53 is satisfied if P S λ K α λ β u,l, 54 l= j α E S j S i β u,l. l=i+ I the proof of Lemma 5. we use a Háje-Réyi type iequality established by Koosza ad Leipus 2000. Theorem 6.3. Theorem 4., Koosza ad Leipus 2000 Let X,..., X be ay radom variables with fiite secod momets ad c,..., c be ay o-egative costats. The ɛ 2 P m c i= Acowledgemet X i > ɛ c 2 m + 2 m E X i X j + i,j= =m =m c 2 + E X + c 2 + c2 X j + j= E X i X j i,j= =m c 2 + E X2 +. The author would lie to tha Herold Dehlig, Liudas Giraitis ad Isabel Garcia for valuable discussios. The research was supported by the Collaborative Research Cetre 823 Statistical modellig of oliear dyamic processes ad the Korad-Adeauer- Stiftug. 55 Refereces Atoch, J., Hušová, M. ad Veraverbee, N. 995. Chage-poit problem ad bootstrap. J. Noparametr. Stat. 5, 23-44. Bai, J. 994. Least squares estimatio of a shift i liear processes. J. Time Series Aal. 5, 453-472. Billigsley, P. 999. Covergece of Probability Measures, 2d ed. Wiley, New Yor. Borovova, S., Burto, R. ad Dehlig, H. 200. Limit theorems for fuctioals of mixig processes with applicatios to U-statistics ad dimesio estimatio. Tras. Amer. Math. Soc. 353, 426-438. Bradley, R.C. 2002. Itroductio to Strog Mixig Coditios. Kedric Press, Heber City. 27
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