Permutation principles for the change analysis of stochastic processes under strong invariance

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1 Permutatio priciples for the chage aalysis of stochastic processes uder strog ivariace Claudia Kirch, Josef Steiebach September 2, 2004 Abstract Approximatios of the critical values for chage-poit tests are obtaied through permutatio methods. Both, abrupt ad gradual chages are studied i models of possibly depedet observatios satisfyig a strog ivariace priciple, as well as gradual chages i a i.i.d. model. he theoretical results show that the origial test statistics ad their correspodig permutatio couterparts follow the same distributioal asymptotics. Some simulatio studies illustrate that the permutatio tests behave better tha the origial tests if performace is measured by the α- ad β-error, respectively. Itroductio A series of papers has bee published o the use of permutatio priciples for obtaiig reasoable approximatios to the critical values of chage-poit tests. his approach was first suggested by Atoch ad Hušková [] ad later pursued by other authors cf. Hušková [7] for a recet survey. But, so far, it has mostly bee dealt with abrupt chages ad idepedet observatios. I may practical applicatios, however, smooth gradual chages are more realistic, so are depedet observatios. I this paper we shall discuss the use of permutatio priciples i the followig three models: Gradual chage i the mea of idepedet, idetically distributed i.i.d. observatios Hušková ad Steiebach [8] ivestigated the followig model: γ i m X i = µ + d + e i, i =,...,,. + where x + = 0, x; µ, d = d, ad m = m are ukow parameters, ad e,..., e are i.i.d. radom variables with Ee i = 0, 0 < var e i = σ 2 <, E e i 2+δ < for some δ > 0..2 he parameter γ is supposed to be kow. ote that i cotrast to abrupt chages the biggest differece i the mea here is ot d, but d m γ, ad thus depeds o, m ad γ. Oe is iterested i testig the hypotheses H 0 : m = vs. H : m <, d 0. he followig test statistic, which is based o the likelihood ratio approach i case of ormal errors {e i }, has bee used: = i kγ +X i X ˆσ k< k i2γ k iγ 2 /2, where ˆσ deotes a suitable estimator of σ. Asymptotic critical values for the correspodig test ca be chose accordig to the followig ull asymptotics cf. Hušková ad Steiebach [8]:

2 heorem.. Let X, X 2,... be i.i.d. r.v. s with var X = σ 2 > 0, ad E X 2+δ < for some δ > 0. he, for all x R, as, P α β x exp 2e x, where α = 2 log log ad β = β γ is as follows: i for γ > 2 : β = 2 log log + log 4π /2 2γ + ; 2γ ii for γ = 2 : β = 2 log log + log log log log log4π; 2 iii for 0 < γ < 2 : β = 2 log log + 2γ Cγ /2γ+ H 2γ+ log log log + log, 22γ + π2 2γ/2γ+ with H γ as i Remark of Leadbetter et al. [0] e.g. H =, H 2 = / π, ad C γ = 2γ + 0 x γ x + γ x γ γx γ dx. Moreover, ˆσ is assumed to be a estimator of σ satisfyig ˆσ σ = o P log log as. 2 Abrupt chage i the mea or variace of a stochastic process uder strog ivariace his model has bee cosidered by Horváth ad Steiebach i [6]. Suppose oe observes a stochastic process {Zt : 0 t < } havig the followig structure: { at + by t, 0 t, Zt = Z + a t + b Y t,.3 < t, where a, b, a, b are ukow parameters, ad {Y t : 0 t < } resp. {Y t : 0 t < } are uobserved stochastic processes satisfyig the followig strog ivariace priciples: For every > 0, there exist two idepedet Wieer processes {W t : 0 t } ad {W t : 0 t }, ad some δ > 0, such that, for, sup Y t W t = O /2+δ a.s..4 0 t ad sup Y t W t = O /2+δ a.s..5 0 t Moreover, we assume Y 0 = 0 ad Y 0 = 0. It should be oted that oly weak ivariace has bee assumed i [6], istead of the strog rates of.4 ad.5, which are required for later use here. Moreover, the processes {Zt}, {Y t}, ad {Y t} could be replaced by a family of processes {Z t}, {Y t}, ad {Y t}, > 0, sice the asymptotic aalysis is merely based o the approximatig family of Wieer processes {W t} ad {W t}, respectively. Oe is iterested i testig the hypothesis of o chage, i.e. H 0 : =, agaist the alterative of a chage i the mea at 0,, i.e. H : 0 < < ad a a, resp. a chage i the variace at 0,, i.e. H 2 : 0 < < ad b b, but a = a. 2

3 Basic examples satisfyig coditios.3-.5 are partial sums of i.i.d. radom variables ad reewal processes based o i.i.d. waitig times, but also sums of depedet observatios for details we refer to Horváth ad Steiebach [6]. It is assumed, that the process {Zt : t 0} has bee observed at discrete time poits t i = t i, = i, i =. Let Z i, = Zt i Zt i ad Z i, = Zt i Zt i Z. he followig statistics will be used: M = k where Z = Z i,, ad { b k } Zi, Z,.6 resp. b2 = M = k { 2 Zi, Z, ĉ k } Z 2 i, Z 2,.7 where Z 2 = Z 2 i,, ad ĉ 2 := Z i, Z Zl, Z. l= Remark.. he statistic M uses a slightly differet variace estimator ĉ 2 tha the oe give i Horváth ad Steiebach [6]. It possesses, however, the same asymptotic behavior, sice the ratio of the two ormalizatios coverges i probability to uder the ull hypothesis, ad to some positive costat uder the alterative cf. heorem i Kirch [9]. his modificatio is ecessary for applyig the permutatio method, sice, uder the alterative, the permutatio statistic correspodig to the statistic used i [6] does ot coverge to sup 0 t Bt, but to c sup 0 t Bt, c > 0, c i geeral, where c is the asymptotic ratio of the two variace estimators. Here {Bt : 0 t } deotes a Browia bridge. he followig ull asymptotics hold uder the above coditios cf. Horváth ad Steiebach [6]: heorem.2. If = ad = o 2/2+δ as, the, uder H 0, M D sup Bt, 0 t where {Bt : 0 t } is a Browia bridge. heorem.3. If = ad = o /2 /2+δ as, the, uder H 0, M D sup Bt, 0 t where {Bt : 0 t } is a Browia bridge. 3 Gradual chage i the mea of a stochastic process uder strog ivariace his model has bee cosidered by Steiebach i [3]. Suppose oe observes a stochastic process {St : 0 t < } havig the followig structure: { at + by t, 0 t, St := S + at + dt +γ + b Y t,.8 < t, where a, b, b ad {Y t}, {Y t} are as i model 2 above, d = d is ukow, γ > 0 is kow. Agai, the biggest differece i the mea here depeds o, ad γ, similarly as i the first model. ote that, istead of.4, Steiebach [3] assumed the followig weak ivariace priciple for the process 3

4 {Y t : 0 t < }, amely that, for every > 0, there is a Wieer process {W t : 0 t } such that sup Y Y t W t /t /2+δ = O P..9 t he reaso is that small approximatio rates were required ear the chage-poit, but oly i a weak sese, whereas we eed strog approximatios for our permutatio priciples below. Here, too, the processes {Zt}, {Y t}, ad {Y t} could be replaced by a family of processes {Z t}, {Y t}, ad {Y t}, > 0. Oe is ow iterested i testig the ull hypothesis of o chage i the drift, i.e. H 0 : = agaist the alterative of a smooth gradual chage i the drift, i.e. H : 0 < <, d 0. Basic examples fulfillig the coditios above are agai partial sums of i.i.d. radom variables ad reewal processes based o i.i.d. waitig times cf. Steiebach [3] for more details. As i model 2, we assume that we have observed {St : t 0} at discrete time poits t i = i, ad set S i, = St i St i. he followig test statistic is used: 2 = b 2 k< i kγ + S i, S k i2γ where S = S i,, ad b 2 = Si, S 2. k iγ 2 /2,.0 Steiebach assumed i [3] a slightly differet weight, which is asymptotically equivalet to the oe used above. However, it turs out, that the above weight gives much better results for the permutatio statistic, which is due to the fact, that it is the imum-likelihood statistic uder Gaussia errors. he results obtaied i [3] remai valid. Remark.2. he magitude of d is completely differet from that of d i the first model. However, d := d + γ +γ is comparable to d, which ca easily be see via the mea value theorem. Similar to heorem., the followig ull asymptotic applies cf. Steiebach [3]: heorem.4. If.9 holds, = ad = O as, the, uder H 0, for all x R : P α 2 β x exp 2e x, where α = 2 log log ad β = β γ is as i heorem. with replacig. 2 Rak ad permutatio statistics i case of a gradual chage uder i.i.d. errors I order to derive distributioal asymptotics for the permutatio statistics, we shall make use of the followig theorem for the correspodig rak statistics. I the case γ = was prove by Slabý i [2]. heorem 2.. Let R = R,..., R be a radom permutatio of,...,, ad a,..., a be scores satisfyig a i a 2 D, 2. ad a i a 2+δ D 2, 2.2 4

5 where D, D 2 ad δ are some positive costats, ad a = a i. he, for fixed γ > 0 ad all x R, as P α a β x exp 2e x, where a = σ a k< i kγ +a R i a k i2γ k iγ 2 /2 ; Here σ 2 a = a i a 2, the variace of a R, α = 2 log log ad β = β γ is as i heorem.. I the proof of this theorem we apply the followig weak embeddig: heorem 2.2. Let a,..., a be scores satisfyig 2. ad 2.2. he, o a rich eough probability space, there is a sequece of stochastic processes { Π k : k } =, 2,... with { } D= { Π k : k σ 2 a } k a π i a : k, where π,..., π is a radom permutatio of, 2,...,, σa 2 := a i a 2, a = a i, ad there is a fixed Browia bridge {Bt : 0 t } such that, for 0 ν < mi, δ 22+δ, 4 k< k k ν Π k Bk/ k k = O P. he proof goes alog the lies of heorem of Eimahl ad Maso [4], by replacig the Hájek-Réyi iequality cf. [4], p. 0 resp. Lemma 3 there with the followig lemmas: Lemma 2.. Let M0 = 0, M,..., Mm, m, be a mea 0, square-itegrable martigale, ad a... am 0 be costats. he, for < s 2 ad λ > 0, P a i Mi > λ 2 s i m λ s m a s i E Mi Mi s. Proof. Cofer Lemma i Häusler ad Maso [5], or Lemma 5..2 i Kirch [9] together with Eimahl [3]. Lemma 2.2. Let a,..., a be scores with a i = 0, ad π,..., π be a radom permutatio as i heorem 2.2. he, for i ad s 2, E i a π j s 2 mi i, i j= a j s. j= Proof. Cofer Lemma i Kirch [9] ad Maso []. ow we have the tools to prove heorem 2.: 5

6 Proof of heorem 2.. First ote that k k 2 k i 2γ i γ = k i γ k k k k 0 j= j γ x 2γ dx = k 2γ + k2γ+. 2 k + k i 2γ 2.3 ow, from heorem 2.2 with ν = 0, uiformly i k [, 2 ] : σ a k a R i+ a = k B = k B k k + O P + O P k. Sice { B } k D= { : k = 0,..., W k k W : k = 0,..., }, where {W t : t 0} is a stadard Wieer process, we coclude from the law of the iterated logarithm i kγ +a R i a σ a log <k< k = σ a = O P + O P <k<log <k<log <k<log i2γ k k = o P log log. k iγ 2 /2 l= lγ l γ k l+ a R i+ a k k iγ 2 /2 i2γ l= lγ l γ W k l + k l+ W k k iγ 2 /2 i2γ k l= lγ l γ k l + k iγ 2 /2 k i2γ Hece it suffices to ivestigate the imum over k [, log ]. Let := i kγ +X i l= X l k log k k 2 /2 iγ i2γ resp. := k log i kγ + Π i Π i k i2γ k iγ 2 /2 be the correspodig test statistics based o i.i.d. 0, radom variables X i resp. o the distributioally equivalet versios of a R i. We choose X i such that B k k = X i k X i, with {Bt} deotig the Browia bridge of heorem 2.2. By the same applicatio of the law of the iterated logarithm as above, i kγ +X i X i log k k = o k iγ 2 /2 P log log. Sice α β = i2γ α β + α, ad sice heorem. implies that α β has a limitig Gumbel distributio, it suffices to show that α = o P. We set Y i := Π i 6

7 Π i X i X, where X = X i, ad S l := l Y i. he, k log k k 2 i k γ +Y i i2γ iγ where 0 < ν < mi k log k< k k k k 2 i2γ iγ l= ν Π k B k k k S l + k l γ l γ k k l + k l k + /2 ν k log ν l= k k l γ l γ, 2 i2γ iγ δ 22+δ, 4 k< as i heorem 2.2. his theorem also implies k k which meas, that it suffices to show k k log ν l= ν k k Π k B l + k l k + /2 ν k k l 2 i2γ iγ k = O P, γ l γ = o log log /2. he latter rate ca be obtaied through a straightforward calculatio, takig 2.3 ito accout together with the followig estimate: i 2γ k k 2 i γ c γ k 2γ+ for all γ, 2.4 where c γ > 0 ad γ depeds oly o γ. his completes the proof. For details we refer to Kirch [9], Corollary We are ow ready to study the followig permutatio statistic: R = i kγ +X Ri X ˆσ k< k, k iγ 2 /2 i2γ where R = R,..., R is a radom permutatio of,...,. We cosider ow the coditioal distributio of R give the origial observatios X,..., X, i.e. the radomess is oly geerated by the radom permutatio R = R,..., R. he followig theorem proves that this statistic coditioally o the give observatios has a.s. the same asymptotic behavior - both uder the ull hypothesis ad uder the alterative - as that of uder the ull hypothesis cf. heorem.. heorem 2.3. Let X,..., X be observatios satisfyig. ad.2. Moreover, let d = d D. he, for all x R, as, P α R β x X,..., X exp 2e x a.s., where α, β = β γ are as i heorem.. Proof. It is sufficiet to verify the assumptios of heorem 2. with a i = X i, i =,...,. First we have X = µ + e + d γ l= l m γ +. 7

8 Hece X i X 2 e i e 2 + 2d γ m i m γ +e i 2d γ l γ l= e i. It is eough to show that the secod term coverges to 0 a.s., because the, by the strog law of large umbers, lim if X i X 2 var e a.s. ow, by partial summatio, i m γ +e i = S m γ + S i i + m γ + i m γ +, 2.5 where S i := i j= e j, ad, from the law of the iterated logarithm, γ+ S i i + m γ + i m γ + = O γ+ = o a.s., i 3/4 i + m γ + i m γ + where the last estimate follows via the mea value theorem. Usig 2.5 together with the strog law of large umbers, we get ideed, as, d γ+ i m γ +e i = d m γ + γ 0 a.s. S d γ+ O the other had, for suitable costats c ad C, ad 0, Xi X 2+δ = c C e i e + d γ i m γ + S i i + m γ + i m γ + l m γ 2+δ + l= m e i 2+δ + c e 2+δ + c d 2+δ 2γ δγ + c d 2+δ a.s. m 2γ δγ 2 δ A applicatio of heorem 2. ow completes the proof. l= l γ 2+δ i 2γ+γδ 3 Permutatio statistics for chages of stochastic processes uder strog ivariace ext we study models 2 ad 3 of Sectio. For model 2, we first eed to ivestigate the asymptotic behavior of the correspodig rak statistic: 8

9 heorem 3.. Let R,..., R be a radom permutatio of,...,, ad a,..., a be scores satisfyig the followig coditios: ad he, as, a i = 0, k a 2 i, 3. i a2 i k a R i D sup Bt, where {Bt : 0 t } deotes a Browia bridge. Proof. It follows from heorem 24.2 i Billigsley [2]. 0 t Lemma 3.. a Let X,..., X be idepedet r.v. s with E Xi 4 D < for all i,. he X i E X i 0 a.s.. b Let {W t: t 0},, be Wieer processes ad f be a fuctio of, the W f = O f log a.s.. Proof. a It follows immediately from Markov s iequality. b Cf. Kirch [9], heorem I the sequel we assume that there is a --correspodece betwee ad, which is ecessary to get a coutable triagular array i, ad, i tur, allows us to use the precedig lemma. Moreover, we assume = θ, 0 < θ, ad = o 2/2+δ. Let = = θ + o ad b Y i Y i, i, Y i = by Y + b Y +, i = +, b Y i Y 3.3 i, i + 2. Lemma 3.2. a It holds, as, b i For s = 2, 3, 4, as, Y = log Y i = O a.s. s 2/2 s/2 where W has a stadard ormal distributio. ii For ν > 0, as, Y i s E W s θb s + θb s a.s., ν 2/2 ν/2 Yi Y ν = O a.s. c For ν > 0, as, ν 2/2 ν/2 i Yi Y ν = o a.s. 9

10 Proof. he proof makes use of.3.5 i combiatio with Lemma 3. for details cofer Kirch [9], heorem We are ow prepared to ivestigate the followig permutatio statistics ad { k } M R = ZRi, Z, k b { k 2 M R = Z R k ĉ i, Z 2 }. Here agai, R = R,..., R deotes a radom permutatio of,...,. heorem 3.2. Let {Zt : t 0} be a process accordig to model.3. Let = θ, 0 < θ, = o 2/2+δ, ad i b also a = a. he, for all x R, as, a b P M R x Zt, 0 t P sup Bt x 0 t P M R x Zt, 0 t P sup Bt x 0 t a.s. a.s., where {Bt : 0 t } is a Browia bridge. Proof. First ote that, for the icremets of {Zt}, we have a + Y i, i, Z i, = a + a + + Y +, i = +, a + Y i, i + 2, with Y i as i 3.3. ow, for the proof of a, cosider the scores a i = Z i, Z i,, i =,...,. Obviously, a i = 0 ad a2 i =, which meas that it is sufficiet to verify assumptio 3.2 of heorem 3.. I the sequel, c ad C deote suitable costats which may be differet i differet places. We first cosider the case θ < ad a a. Here, for sufficietly large, ˆb b2 = = Zi, 2 Z 2 a 2 i a2 + Y i 2 Y 2 2 a + a Y + 2ab Y + 2a b Y Y a + a + Y c a.s., 0

11 where a, i, a i = a + a +, i = +, a, i + 2, ad a = a i = a + a. he last iequality i 3.4 follows from the fact that the first terms are the domiatig oes. Ideed, sice θ <, a a, for sufficietly large, a 2 i a2 a a 2 a 2 2 a 2 2 2aa = + o a 2 θ θ + a 2 θ θ 2aa θ θ = + o θ θa a 2 a 2 2 a c a.s. ext we prove that the other terms are of smaller order ad hece are egligible. Lemma 3. b gives 2ab Y + 2a b Y Y + = 2ab W + 2a b log = O a.s. W W + + O /2+δ Sice ad +, we also get 2 a + a + Y + 2 a + a b W W + a + a b W + /2+δ + O log = O a.s. Lemma 3.2 further implies Y i 2 Y 2 2 a + a Y = O + log + log a.s., which proves 3.4. ote that a a, i, a i a = a a ϑ, i = +, a a, i + 2, for some 0 ϑ, hece a i a 2 = i a a 2, /2, a a 2, > /

12 O combiig 3.4, Lemma 3.2 a ad Lemma 3. b i we fially get 3.2, sice i a2 i 2 b 2 2 c 0 a i a i b 2 Y i 2 Y 2 a a c a.s. Y i Y 2 i 3.8 O the other had, if θ = or a = a, we obtai from Lemma 3.2, b2 = Z i, Z 2 = θb 2 + θb 2 c > 0 Y i 2 Y 2 a.s., for sufficietly large. Usig Lemma 3.2 c, we arrive at 3.2, i.e. which completes the proof of a. i a2 i = b2 0 a.s., Y i Y 2 i For the proof of b, cosider a i = ĉ Yi Y 2 l= Yl Y 2. It suffices agai to verify the assumptios of heorem 3.. Sice a = a, we get a2 i =. Similarly as above, Lemma 3.2 gives ad b 2 = 2 Yi Y 4 3 θb 4 + θb 4 a.s., 2 Y i 2 Y 2 θb 2 + θb 2 2 From Jese s iequality we coclude lim ĉ2 = lim 2 So, a applicatio of Lemma 3.2 results i k a2 k = ĉ 2 C 2 0 which completes the proof of b. Yi Y 4 b2 2 = 3θb 4 + θb 4 θb 2 + θb 2 2 2θb 4 + θb 4 > 0 k Yk Y 2 k Y k Y 4 + a.s., a.s. Yi Y Y i Y 2 a.s. 3. Fially we tur to model 3 of Sectio ad ivestigate the permutatio aalogue of.0, i.e. the statistic 2 R = i kγ + S Ri, S b 2 k< k i2γ k iγ 2 /2. he followig asymptotic applies: 2

13 heorem 3.3. Let {St : t 0} be a process accordig to model.8. Assume = θ, 0 < θ, ad log = omi 2/2+δ, /2+γ. he, for all x R, as, P α 2 R β x St, 0 t exp 2e x a.s., where α, β = β γ are as i heorem. with replacig. Proof. First ote that, for the icremets of {St}, we have Y i, i, S i, = Y + + d +γ +, i = +, Yi + d i +γ +γ i, i + 2. I case of the ull hypothesis, i.e. for θ =, we ca immediately verify the assumptios of heorem 2. for a i := S i, by usig Lemma 3.2. O the other had, i case of θ <, we use a i := S +γ i,. First, via the mea value theorem, +δ +γ2+δ i +γ i +γ 2+δ + + +γ2+δ = O i= +2 +δ +γ2+δ +γ2+δ 2+δ = O, which, together with Lemma 3.2, gives +γ S i, +γ S 2+δ = O a.s. I order to verify the secod assumptio of heorem 2., we first realize, by usig partial summatio, the mea value theorem ad Lemmas 3. resp. 3.2, that i +γ +γ i 2+2γ Y i i= +2 +γ + + Y + = 2 Y +γ 2+2γ 2+2γ k= +2 +γ by + b Y k k + +γ 2 k +γ + k +γ by 2+2γ + b Y γ 2 + +γ = o + O by + b Y k +γ k= + log = o + O = o a.s. /2+γ + O /2+δ +γ 3.2 3

14 ext we have 2+2γ i= +2 + γ2 i +γ i +γ 2 + i= + 2γ i θ x 2γ + γ2 θ dx + γ2 = + o 2γ + θ2γ+, which shows that d 2 2+2γ d γ i= +2 i +γ i +γ 2 + +γ θ2γ+ + o 2γ + 2 γ 2 + θ2γ +. O combiig 3.2, 3.3 ad Lemma 3.2, we get ideed, for large, 2+2γ 2 Si, S cθ with some cθ > 0, which completes the proof. + 2+γ + 2+γ Simulatios So far, we have oly prove that the permutatio priciple is asymptotically applicable for processes satisfyig models to 3. ow we wat to describe the results of some simulatio studies to get a idea, how good the permutatio method is i compariso to the origial method. 4. Gradual chages i the mea of a i.i.d. sequece I a first study, we geerated data accordig to model. usig the followig parameters: = 00, 200, ormally distributed errors Laplace distributed errors γ 90% 95% 97.5% 99% 90% 95% 97.5% 99% able 4..: Simulated critical values uder the ull hypothesis 4

15 ormally distributed errors Laplace distributed errors γ d m 90% 95% 97.5% 99% 90% 95% 97.5% 99% ormally distributed errors Laplace distributed errors m 90% 95% 97.5% 99% 90% 95% 97.5% 99% able 4..2: Simulated critical values usig the permutatio method 5

16 γ 90%-Quatile 95%-Quatile 97.5%-Quatile 99%-Quatile able 4..3: Asymptotic critical values m = 4, 2 ad 3 4, d = 0, 4, 2,, 2, 4, γ = 4, 2, ad 2, ormally ad Laplace distributed errors each stadardized. his gives 256 combiatios of the above parameters ote that we are i the case of the ull hypothesis for d = 0. First we compare the exact critical values with the asymptotic oes by geeratig 0, 000 series X,..., X accordig to the ull hypothesis. he critical values we got from these series ca be foud i able 4... For compariso, able 4..3 shows the asymptotic oes γ = 4 is missig, sice H is ukow cf. also 4 Remark i Leadbetter et al. [0]. First ote that the asymptotic critical values are rather too small especially for γ = 2. 90%-Quatile 95%-Quatile 97.5%-Quatile 99%-Quatile γ d simul. asym. simul. asym. simul. asym. simul. asym able 4..4: Simulated α- resp. β-errors i %, 000 repetitios, m = 2 6

17 90% 95% 97.5% 99% 90% 95% 97.5% 99% Partial sums Poisso Process able 4.2.: Simulated critical values uder the ull hypothesis ext we were iterested i the critical values obtaied via permutatio, which we simulated usig the followig algorithm:. Geerate a series X,..., X accordig to the give parameters. 2. Geerate a radom permutatio R = R,..., R of,..., ad calculate R. 3. Repeat step 2 0, 000 times. 4. Calculate the empirical quatiles of these 0, 000 values. he result ca be foud i able It is importat that, for the differet combiatios of parameters, we always used the same seed i step, which meas that we also always used the same permutatios for the calculatio of the empirical quatiles as log as the series had the same legth. We realize, that the quatiles are quite good, but decrease the more obvious the chage which is quite surprisig cosiderig that the test statistic icreases if there is a chage cf. Hušková ad Steiebach [8], Sectio 4. I additio, we approximated the α- resp. β-errors usig the followig simulatio:. Geerate a series X,..., X accordig to the give parameters. 2. Calculate the critical values usig the permutatio priciple compare steps 2-4 above. 3. Calculate the value of the statistic ad see, if we had rejected the ull hypothesis usig the quatiles from step 2 resp. the asymptotic oes. 4. Repeat steps -3, 000 times. 5. Calculate the empirical α- resp. β-errors from the, 000 simulatios above. able 4..4 cotais the results for ormally distributed errors ad a chage at 2. We realize that both methods give good results for γ > 2. For γ = 2 the α-error is far too high for the asymptotic method, especially with the 90%- ad 95%-quatile, which is due to the fact, that the asymptotic critical value is too small compare ables 4.. ad Moreover, we were iterested i the stadard deviatio of the critical values obtaied by the permutatio method. Uder the ull hypothesis = 00, γ = 2, ormally distributed errors we got a stadard deviatio of 0.82 for the 90%-quatile ad of for the 99%-quatile. he result is similar for differet parameters. Here we used, 000 trials of step to 4 of the first simulatio described above. For our simulatios we used the software package R, Versio.2.3. O a Celero with 466 MHz ad 384 MB RAM the calculatio of the permutatio quatiles takes approximately 0 secods i the case of 00 observatios, ad 30 secods i the case of 200 usig 0, 000 permutatios. his meas that the method is ideed applicable. 4.2 Chage i the mea of a stochastic process uder strog ivariace he followig simulatios are based o partial sums of ormally distributed radom variables with variace cf. Horváth ad Steiebach [6], Example., ad o a Poisso process cf. Horváth ad Steiebach [6], Example.2. More specifically, we simulated the icremets of the partial sums as i.i.d. r.v. s, ad the icremets of the Poisso process were take at times, 2,... istead of i, i =,...,, sice this meas oly a scalig of the uderlyig r.v. s. Other tha that, we used the followig parameters: 7

18 90% 95% 99% able 4.2.2: Asymptotic quatiles = 00, 200 = 4, 2, 3 4 a a = 0,, 2, 3, 4 Here is the chage poit, ad we are i the case of the ull hypothesis for a a = 0. Oce agai we geerated 0, 000 series of icremets Z,..., Z for the differet parameters uder the ull hypothesis. he resultig quatiles ca be foud i able he asymptotic critical values are give i able for compariso. First ote that the asymptotic quatiles are slightly too large. Moreover, we realize that the exact oes are a little larger for the partial sums tha for the Poisso process. o study the critical values obtaied from the permutatio method, we used the same algorithm as i Sectio 4.. he results ca be foud i able We realize that these critical values give better estimates tha the asymptotic oes. It also does ot seem to be importat where exactly the chage poit is located. ext we simulated the α- resp. β-errors, as we did i Sectio 4., with a chage at = 3 4. he results ca be foud i able Expectedly, the α-errors are smaller for the asymptotic method, but the a a 90% 95% 97.5% 99% 90% 95% 97.5% 99% Partial sums Poisso Process able 4.2.3: Simulated critical values usig the permutatio method 8

19 90%-quatile 95%-quatile 99%-quatile a a simul. asym. simul. asym. simul. asym. Partial sums Poisso Process able 4.2.4: Simulated α- resp. β-errors i %, 000 trials, = 3 4 β-errors are larger. Particularly, this is sigificat for smaller samples ad i case of the Poisso process. I the latter cases we actually do get better results usig the permutatio method. Moreover, we were iterested i the stadard deviatio of the critical values obtaied by the permutatio method. Uder the ull hypothesis Poisso process, =00, we got a stadard deviatio of 0.0 for the 90%-quatile ad of 0.09 for the 99%-quatile, for the partial sums the stadard deviatio was eve smaller. he results are comparable for differet parameters. As before, we used, 000 repetitios of step to 4 of the first simulatio above. Agai, computig time is ot a problem here. For example, the calculatio of the permutatio quatiles for a series of legth 00 takes approximately 3 secods, ad for legth 200 approximately 5 secods, usig a Celero with 466 MHz ad 384 MB RAM ad the software package R, Versio Gradual chage i the mea of a stochastic process uder strog ivariace he followig simulatios are based o partial sums of ormally distributed r.v. with variace cf. Steiebach [3], Example. ad o a Poisso process cf. Steiebach [3], Example.2. More precisely, we simulated the icremets of the partial sums as i.i.d. r.v. s, ad the icremets of the Poisso process were take at times, 2,... istead of i, i =,...,, as above. he followig parameters were chose: = 00, 200 = 4, 2, 3 4 d = 0, 4, 2,, 2, 4 Here is the chage poit, ad the ull hypothesis is give for d = 0. d is as i Remark.2 i order to be able to compare the results with those of Sectio 4.. More precisely, the icremets of the chage 9

20 Partial sums Poisso process γ d 90% 95% 97.5% 99% 90% 95% 97.5% 99% Partial sums Poisso process 90% 95% 97.5% 99% 90% 95% 97.5% 99% able 4.3.: Simulated critical values usig the permutatio method 20

21 Partial sums Poisso Process γ 90% 95% 97.5% 99% 90% 95% 97.5% 99% able 4.3.2: Simulated critical values uder the ull hypothesis were chose as d +γ i +γ γ + i +γ +. ote that the latter expressio does ot deped o, but oly o. Oce agai, we geerated 0, 000 series of icremets S,..., S uder the ull hypothesis for the various choices of parameters. he resultig quatiles ca be foud i able he asymptotic critical values are the same as i Sectio 4. ad are give i able First we realize that the asymptotic quatiles are agai too small this time eve more sigificatly. For compariso, we simulated the critical values obtaied through the permutatio method as before. he results ca be foud i able As i Sectio 4., the critical values are quite good, but declie as the chage becomes more obvious. ote that here the cosistecy of the test is ot guarateed, sice the estimator for b is ubouded uder the alterative which violates coditio 2.4 of Steiebach [3]. ext we simulated the α- resp. β-errors, as we did i Sectio 4., with a chage at = 2. he results ca be foud i able he α-errors are far too high for the asymptotic method for γ = 0.5, which is due to the fact, that the asymptotic critical values are too small compare ables ad he permutatio method, however, gives good results. For γ = both methods give comparable results. Whe we used d, istead of d, ad = which chages d slightly, the critical values decreased sigificatly. evertheless, this did ot seem to affect the permutatio method at all apparetly the permutatio quatiles were still smaller tha the value of the test statistic for the upermuted observatios. With the asymptotic method, however, we oly obtaied good β-errors for smaller d s, but observed a sudde jump i the β-errors up to 00% as soo as d got larger. his jump e.g. occurred at d = 2 for the 90%-quatile with γ = 0.5, = 00, 200. Agai, we were also iterested i the stadard deviatio of the critical values obtaied by the permutatio method. Uder the ull hypothesis Poisso process, = 00, γ =, we got a stadard deviatio of 0.28 for the 90%-quatile ad of 0.96 for the 99%-quatile, for the partial sums the stadard deviatio was eve smaller. As before, we used, 000 repetitios of step to 4 from the first simulatio. For our simulatios we used agai the software package R, Versio.2.3. O a Celero with 466 MHz ad 384 MB RAM the calculatio of the permutatio quatiles takes approximately 0 secods i the case of 00 observatios, ad 30 secods i the case of 200 usig 0, 000 permutatios. 2

22 90%-Quatile 95%-Quatile 97.5%-Quatile 99%-Quatile γ d simul. asym. simul. asym. simul. asym. simul. asym. Partial sums Poisso process able 4.3.3: Simulated α- resp. β-errors i %, 000 repetitios, = 2 22

23 Refereces [] Atoch, J. ad Hušková, M. Permutatio tests for chage poit aalysis. Statist. Probab. Lett., 53:37 46, 200. [2] Billigsley, P. Covergece of Probability Measures. Wiley, ew York, 968. [3] Eimahl, U. Persoal commuicatio, [4] Eimahl, U. ad Maso, D.M. Approximatios to permutatio ad exchageable processes. J. heor. Probab., 5:0 26, 992. [5] Häusler, E. ad Maso, D.M. Weighted approximatio to cotiuous time martigales with applicatios. Scad. J. Statist., 26:28 295, 999. [6] Horváth, L. ad Steiebach, J. estig for chages i the mea or variace of a stochastic process uder weak ivariace. J. Statist. Pla. Ifer., 9: , [7] Hušková, M. Permutatio priciple ad bootstrap i chage poit aalysis. Fields Ist. Comm., o appear. [8] Hušková, M. ad Steiebach, J. Limit theorems for a class of tests of gradual chages. J. Statist. Pla. Ifer., 89:57 77, [9] Kirch, C. Permutatiosprizipie i der Chagepoit-Aalyse. Diploma thesis, Philipps-Uiversität Marburg, [0] Leadbetter, M.R., Lidgre, L., ad Rootzé, H. Extremes ad Related Properties of Radom Sequeces ad Processes. Spriger, ew York, 983. [] Maso, D.M. Persoal commuicatio, [2] Slabý, A. Robust Chage-Poit Detectio i the Locatio Model. PhD thesis, Karls-Uiversität, Prag, [3] Steiebach, J. Some remarks o the testig of smooth chages i the liear drift of a stochastic process. heory Probab. Math. Statist., 6:64 75,