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1 Math-Net.Ru All Russian mathematical portal E. Carlstein, S. Lele, Nonparametric change-point estimation for data from an ergodic sequence, Teor. Veroyatnost. i Primenen., 1993, Volume 38, Issue 4, Use of the all-russian mathematical portal Math-Net.Ru implies that you have read and agreed to these terms of use Download details: IP: July 3, 2018, 15:33:10

2 910 Carlatein E., Lele S r. CARLSTEIN E.*, LELE S." NONPARAMETRIC CHANGE-POINT ESTIMATION FOR DATA FROM AN ERGODIC SEQUENCE*** В рамках схемы серий предполагается, что наблюдаемая последовательность {Л',", 1 < г < п} такова, что X? = ГУ,/(1 ^ : < [в п ]) + V;/([«] + 1 s t < п), где (Ui,V{) стационарная эргодическая последовательность, относительно маргинальных распределений которой предполагается лишь то, что они различны, а в момент изменения вероятностных характеристик, такой, что в (0; 1). Основной результат состоит в доказательстве того, что построенная в работе последовательность (в п ) п^1 непараметрических оценок является состоятельной (в п в). Ключевые слова и фразы: непараметрическое оценивание момента вероятностных характеристик, состоятельность оценок. 1. Introduction. Suppose we observe a sequence of random variables X", X X%, where X" has marginal distribution F for 1 < t < [вп], XJ 1 has marginal distribution G for [вп] + 1 < t < n; here [y] denotes the greatest integer not exceeding y. The unknown parameter в (0,1) is the change-point to be estimated. The purpose of this note is simply to show that в can be consistently estimated in a fully nonparametric scenario: (i) No knowledge of F or G is required. There are no parametric assumptions (e.g., normality) and no regularity conditions (e.g., continuity) oh F or G; (ii) No restrictions are imposed on the strength of serial dependence in {X": 1 < i ^ n}, beyond the minimal condition of ergodicity; there are no assumptions about the dependence mechanism (e.g., autoregression), and there are no "mixing" conditions; (iii) No prior restrictions on в are needed. In fact, a whole class of such fully nonparametric estimators will be presented. A huge amount of work has been done on change-point estimation in general (see [13] for an annotated bibliography and [11] for a recent extensive review); the importance of the nonparametric approach in particular is well established (see [5] for a review). Most of the nonparametric methods assume independence in {X": 1 ^ t ^ n}, and most of them assume prior knowledge about how F and G differ (e.g., in their means, medians, or other measure of level). However, there has recently been a trend towards eliminating prior assumptions on F and G, and towards allowing for serial dependence in {X": 1 ^ ii ^ n} (see [1]-[10] and [12]). Within the nonparametric context, it is certainly desirable to minimize the assumptions on F and G. Since change-point data is inherently time-sequential, it is natural and practical to allow for serial dependence. It is desirable to minimize any restriction on the strength of this serial dependence, because the joint distribution structure of {X?: 1 ^ i < n} is much more obscure than the marginal distribution structure (i.e., F and G), University of North Carolina at Chapel Hill, CB#3260, Phillips Hall, Chapel Hill, N.C , USA. ** Johns Hopkins University, Visiting the University of North Carolina at Chapel "Hill, August *** Research supported by National Science Foundation Grant #DMS

3 Nonparametric change-point estimation for data from an ergodic sequence 911 so it is unrealistic to assume knowledge about the former when the latter is completely unknown. Moreover, the usual "mixing" assumptions are impossible to check for a given set of data. Our main consistency result (in Section 3) shows how far we can relax the restrictions on F, G, and the dependence structure. Let us briefly contrast our fully nonparametric approach (i.e., (i), (ii), (iii) above) with the related works ([1] [10] and [12]) which have recently appeared in the literature. In [4] and [5], it is assumed that the X?s are independent and that F and G are continuous. In [2], [6], [8], and [9], it is assumed that F and G are both discrete with finite support, and that the sequence of X"s satisfies a strong-mixing condition; it is also assumed that 9 is in the known interval [a,fi], where 0 < a < fi < 1. In [7], either one of two possible scenarios is required: F and G are both discrete with finite support, and the X"s are strongmixing; or, F and G are both continuous with support in [0,1] and satisfy a Lipschitz condition, and the A'"s are ф-mixing. In [1], it is assumed that the X"s are independent (or possibly m-dependent) and that F and G are continuous; it is also assumed that в is in the known interval [a, fi], 0 < a < /3 < 1. In [3] and [10], the X"s are again assumed to be independent. In [12], it is assumed that the X"s arise from a Gaussian process, with F and G sharing the same mean. We shall impose none of these assumptions. One final point of comparison between our approach and the works cited above: in each of the references [2], [4]-[10], and [12], a certain "norm" is used to calculate the basic statistic (this notion of a "norm" will be made precise in Section 2). Specifically, a Kolmogorov-Smirnov norm is used in [4], [5], [7], [10], [12]; and a Cramer-von Mises norm is used in [2], [6], [8], [9], [12]. Our method is based on a general "Mean-Dominant norm" (defined in Section 2), which includes as special cases the Kolmogorov-Smirnov norm, the Cramer-von Mises norm, as well as many other norms. Thus, our approach actually provides a whole class of consistent nonparametric estimators; moreover, our general formulation allows us to simultaneously analyze estimators based on all these norms with one unified theoretical argument (see Sections 2 and 4). 2. The estimator. The estimator is constructed from the pre-t empirical c.d.f. (cumulative distribution function): and the post-t empirical c.d.f.: The index t corresponds to possible values of the change-point estimator; we will only need to consider t T n, where and {a n : n ^ 1} is any deterministic sequence satisfying a J. 0 and na n oo as n + oo. For fixed t T, compute the differences between the pre-t and post-t empirical c.d.f.s at the sample observations, i.e., 1 <» < n. These n differences are now combined via a "Mean-Dominant norm yielding the criterion function ibjj'. D n (t) :=«(l-t)5 (d n l,<2,.-..o-

4 912 Carlstein E., Lele S. Our fully nonparametric estimator в is then defined to be: в п T n such that D (0 ) = max D n (t). i T The "Mean-Dominant norm" S (-, ) is any function satisfying the following natural conditions (whenever the arguments d; and d{- are all nonnegative): (1) [Symmetry] S n (-,,..., ) is symmetric in n arguments; (2) [Homogeneity] S (cdi, cd2,ed ) = c5 n (rfi, d 2,...,d n ) whenever с > 0; (3) [Triangle Inequality] S n (di + di,d2 + i <*n + <0 ^ 5 (di,d2i, d n ) + S (di, d' 2,,d' n ); (4) [Identity] S (l,l,...,l) = 1; (5) [Monotonicity] S n (di, d 2,.., d n ) < S n (d[, d' 2,..., d' n ) whenever d; < dj. V t; (6) [Mean-Dominance] S (di, d 2,..., d ) > Ei<i<n d '/ n - Special cases of the "Mean-Dominant norm" include the Kolmogorov-Smirnov norm S* s (d 1,d 2,...,d n ):= sup {di}, l<i<n the Cramer-von Mises norm S (d b d 2 d ):=f -V and the arithmetic-mean norm 5i m (di,d,,...,rf ):= K.-<n n The intuition behind our method is as follows. The c.d.f.s th n (x) and h n (x) are empirical approximations of the unknown distributions th(x) := / {t < в) F(x) + I{t>e] (6F(x) + (*-*) G(z))/t and ftt(s) := I {t < в] ((<? - 1) F(x) + {1-в) G(z))/(1 - i) + / {t > в} G(x), respectively. Therefore the differences dj,,- are empirical approximations of 6 ni := i*(jf") - hi(x?)\ = С (/ {* < в} (1 ~ *)/(! -<) + /{<> «}«/«), 1 < «< n. And, the criterion function D n (t) is an empirical approximation of the corresponding function A n (t) :=t(l-t)s n (S nl,s i n2,...,s t nn). Now, by S n 's homogeneity, we have A (t) := p(t)s n (S nl,s n2,...,6 nn ), where p(t) := I {t < в} t (1 - в) + I {t > в} (1 - t) в.

5 Nonparametric change-point estimation for data from an ergodic sequence 913 Notice that the maximizer of A n (t) over t (0,1) is precisely at t = 9. Thus, the maximizer of the analogous sample-based criterion function D n (t) is a reasonable estimator of в. This logic applies for any "Mean-Dominant norm" Sn - 3. Main result. The data {X": 1 < i < n) are embedded in a stationary ergodic sequence {(Ui,Vi): -oo < i < +00}, where /, has marginal distribution F, and V{ has marginal distribution G. Specifically, the data arise as X? = U { I l < i < [9n]} + VJ {[9n] + 1 < i < n}. There will be no further constraints on the serial dependence structure of {X": 1 <» ^ n}. The only assumption on the unknown marginal distributions is simply that F Ф G. The unknown change-point parameter 9 (0,1) is unrestricted. In this scenario, our fully nonparametric estimator 9 n (defined in Section 2) is consistent, p Theorem. 9 9 as n See Section 4 for a proof of this result. 4. Proof. We begin by presenting three preliminary Lemmas which will be used in proving the Theorem. Lemma 1. Denote / \F(x) - G(x)\dF(x), -OO ' ' ОО. - \F(x)-G(x)\dG(x), p :=9PF + (1-9)UQ- Then и > 0. Proof. By assumption we have Л := jx R: p(x) - G(x) > 0 j ф 0. It suffices to show that either / л df(x) > 0 or / л dg(x) > 0. The case where Л contains a discontinuity point of F or G is trivial, so we will now presume that F and G are continuous at each x Л. Select xo Л with (say) F(x 0 ) > G(x 0 ). Then := {У (-00,xo): F(x) > G(x) Vx (y,x 0 ]} is nonempty, by continuity. Denote yo := inf {y <r}. If yo = 00, then ( 00, xo] С Л and therefore L df(x) > F(x 0) > G(x 0 ) > 0. л If yo > 00, then F(yo) < G(yo) and also (yo,xo] С Л, yielding IA df(x) > F(x 0 ) - F(y 0 ) > F(x 0 ) - G(y 0 ) > G(x 0 ) - G(y 0 ) > 0. Lemma 1 is proved. 8 Теория вероятностей и ее применения, 4

6 914 Carhtein E., Lele S. Lemma 2. Let {Y{i oo < t < } be a stationary ergodic sequence, where Q is the marginal distribution of Y{. Let {a n : я ^ 1} be a deterministic sequence satisfying a n I 00 as n 00, and let {N N : я ^ 1} be integer-valued random variables for which N N > a n Vn. Define for m? ^ mi + 1, ГГ12 1Уд(тх, m 2 ) := sup -oo<y<oo mj mi Q(y) Then WQ(0, N N ) 0 as n -* 00. Proof. Follows directly from the Theorem in [14]. Lemma 3. Denote A n ' T n П {[a n, в - a ] U (в + a n, 1 - a ]}. Let {R n'< rx, ^ 1} be a (possibly random) sequence with R n = r ((Ui, Vj), (J/ 2, V 2 ),(U n, V )) for a deterministic function r : R 3 n -» A n Then I I P Дп(Дп) - Dn(Rn)\ '0 as n-+oo. Proof. Denote е,- :=,Я 4- Я,-, where Яг,.' := \h n R,(Xr) - h Rn (Xr)\, nih := д ft"(x?) - Rnh(X?)\, so that d*" < e nl - + й^/ 1 and therefore: Г := D (R n ) - A (R n ) < S (e i, e n 2,..., e ) < S n (nlh, nih,..., nnh) + S n (H n i, H 2,H n ) by (5) and (3). The same bound holds for Г and hence for Г. Now, [Rn>e}( I{Xf < X*} n тад \j=nr n +1 t] + F(x?)([ Дя Ь»Дп _ 0-Д.Л v ' 'Д и - яд г - Rn) 4- US + i^^ ( ^I{R n > e}w G (nr n,n) + l{r n < б} (lr>(nfl,[0»]) + W' e ([«], n) 4- (2/n)(l - R n )) < W G (/ {Я > 0} я й 4- / {Д < 6} [0п], я) + W> (/{R n < 0} я Л, [вп]) + (2/я) (1 - в) =: Я. (In W/ ( ), f/j plays the role of Yf, in W'G(), V,- plays the role of Y<.) An analogous bound (say, Я can be obtained for, #). Since Я and Я do not depend on t, we have Г < Я + Я, by (5), (2), and (4).

7 Nonparametric change-point estimation for data from an ergodic sequence 915 P We will now show that H n 0; a similar argument applies to Я. We deal explicitly with the second summand in II n; the first summand can be handled analogously to the second, and the third summand goes to zero deterministically. Denote Zi := ({/,-, V',) and *:= (Z k + l,z k + 3,...,Z k + n ). Note that: W F (l{r n (Z n ) < в}пг п (г п ),[вп]^ = W F {l{r n {Z- [B^- 1 ) < в}пг п (2- [,>п] - 1 ) - [вп] -1, -l) = w F (а, [вп] -1 {r n (z- [)n] - 1 ) < в} nr n (z-^- 1 )^, where the last expression uses Yj- = (/_,-. We can now apply Lemma 2, since»- ( ) A n and so: [вп] -/ r (-) < 0}nr n (-) > [0n]-n(0-a ) ^ (na - 1) oo. This establishes W F (I{R n < 0}пЯ,[0п]) 0. The following notation and definitions will be needed to prove the Theorem. The random variable в п is defined as follows: if в A n, then let в п = в п ; if в п A n, then let в п.an satisfy О п (в п ) = тах < е д п D n (t). Note that, in either case, we have в v4 n and В п (в п ) = тах< д т D n (t). Also define the nonrandom entity <n := / jp ([(в - a n ) n]/n) ^ p (([(0 + a ) n] + l)/n) [(0 -a )n] /n + / p ([(«- a n )n]/n) < p (([(0 + a ) n] + l)/«) } ([(в + о») я] + l)/n, which satisfies i, ; A and A (t ) = тах < 6 д п A n (t). P In order to prove the Theorem, it suffices to establish that <? в as n + oo. To see that this is sufficient, write Р{\в п -в\>е}= р{\0 п -в\>е, в п еа п }-гр{\вп-в\>е, вп А п ). The latter probability is zero for n sufficiently large; the other probability is bounded by Р{\в п -в\ >e}. that: To establish в -^-+ в, denote 6 := Ei<i<n *ni/ n а ш * ^ : = min{0,1 в}, and notice * :=An(e)-AnC0n)=(p(e)-p(e n ))Sn(6 nl,6 n2,...,6 e nn) > (/{A < в}(в-в ){1-в) + 1{вп >в}{в п -в)в) s n > в -в\вб п, where the first inequality follows from (6). Therefore Р{\в п -в\ > e} < Р{Ф > евёп, P{*n > ев6 п, S n < v) < Р{* п >ев»}+ P{6 n <»}, where v. a/2 > 0 by Lemma 1. To deal with the latter probability, write A - [H n +(п-[вп])1 п On =,

8 916 Carlstein E., Lele S. where с._ ST* ni 7._ ni Note that { <!/}=>{«/< S - u\ < «+ в -Ai G + 2 fl-[fln]/n }. Since the term in the last modulus goes to zero deterministically, we only need to consider \S n UG\ (the remaining term is handled analogously). Now D * P by the Ergodic Theorem, and S n = 6 n, so that 5 UQ\ 0. p To prove the Theorem, it now suffices to show that Ф * 0. We have Фп < д»(*) - Д»(*») + Д»(*«) - D n (9 n )\ + \D n (fin) - Дп(*»), where the first modulus is deterministically dominated (using (5) and (4)) by: p(9)-p(t ) = /{* < 0}(0-t )(l -0) + /{t > <?}(* -9)9 < or 4-n" 1, and where the second modulus is dominated by z> (* ) - Д (* ) + \D n {9 n ) - Д (в ) because either Дп(«п) > D n (9 n ) > D n (t n ) or D (9 n ) ^ Д (1») > Д (0») holds. Thus, applying Lemma 3 with R n = t n and with R n = 9 n completes proof of the Theorem. СПИСОК ЛИТЕРАТУРЫ 1. Asatryan D., Safaryan I. Nonparametric methods for detecting changes in the properties of random sequences. In: Detection of Changes in Random Processes. Ed. by L. Telksnys. New York: Optimization Software, 1986, p Brodskii В., Darkhovskii B. The a posteriori method of detecting the disruption of a random field. In: Detection of Changes in Random Processes. Ed. by L. Telksnys. New York: Optimization Software, 1986, p Carlstein E. Nonparametric change-point estimation. Ann. Statist., 1988, v. 16, Ws 1, p Csorgo M., Horvdth L. Nonparametric tests for the changepoint problem. J. Statist. Planning Inference, 1987, v. 17, 1, p Csorgo M., Horvdth L. Nonparametric methods for changepoint problems. In: Handbook of Statistics, v. 7. Ed. by P. Krishnaiah and C. R. Rao. The Netherlands: Elsevier, 1988, p

9 Nonparametric change-point estimation for data from an ergodic sequence Darkhovskii B. On two estimation problems for times of change of the probabilistic characteristics of a random sequence. Theory Probab. Appl., 1984, v. 29, 3, p Darkhovskii B. A nonparametric method of estimating intervals of homogeneity for a random sequence. Theory Probab. Appl., 1985, v. 30, 4, p Darkhovskii B. A general method for estimating the instant of change in the probabilistic characteristics of a random sequence. In: Detection of Changes in Random Processes. Ed. by L. Telksnys. New York: Optimization Software, 1986, p Darkhovskii В., Brodskii B. A posteriori detection of the "disorder" time of a random sequence. Theory Probab. Appl., 1980, v. 25, 3, p Deshayes J., Picard D. Convergence de processus a double indice: application aux tests de rupture dans un modele. C. R. Acad. Sci. Paris, 1981, v. 292, p Krishnaiah P., Miao B. Review about estimation of change points. In: Handbook of Statistics, v. 7. Ed. by P. Krishnaiah and C. R. Rao. The Netherlands: Elsevier, 1988, p Picard D. Testing and estimating change-points in time series. Adv. Appl. Probab., 1985, v. 17, N«4, p Shaban S. Change-point problem and two-phase regression: an annotated bibliography. International Statist. Review, 1980, v. 48, p Tucker H. A generalization of the Glivenko-Cantelli theorem. Ann. Math Statist., 1959, v. 30, 3, p Поступила в редакцию