VARIANCE ESTIMATION AND CHANGE POINT DETECTION FOR THE HYBRIDS OF EMPIRICAL AND PARTIAL-SUM PROCESSES

DETECTION FOR THE HYBRIDS OF EMPIRICAL AND PARTIAL-SUM PROCESSES

S. ALVAREZ-ANDRADE

We deal with a variance estimator based on invariance principle for the so-called hybrids of empirical and partial sums processes defined as in (1), as well as in the study on a possible change in the variance in an At Most One Change (AMOC) spirit, and a study of the domain of attraction for this process.

AMS 2010 Subject Classification: 60F15, 60F17, 60K99, 62E20.

Key words: empirical process, Gaussian process, partial sum, strong approxima- tion.

1. INTRODUCTION

We consider the hybrids of empirical and partial-sum processes given by (1) A(n, t) =¯ X

1≤i≤n

H(Xi)1{X_i≤t}i, −∞< t <∞,

where 1A denotes the indicator function of an event A. This process is de- fined the assuming regularity conditions (A), (B), (C), (D), (E) or (F), on the sequences of random variables {X_i}_i≥1 and {_i}_i≥1 given below.

Some relevant related works are, for instance, Diebolt [10] considering the process (1) in order to study a family of nonparametric tests for the regression function in the nonlinear regression model (see also [7]), Diebolt et al. [11]

introducing methods for testing the goodness-of fit of linear or nonlinear para- metric autoregression models of order one, Diebolt and Zuber [12] studying the goodness-of-fit tests for parametric possibly nonlinear heteroscedastic re- gression models. WhenH = 1, Maumy [19] considered the related compound empirical process, Haeusler and Mason [14] defined the “randomly weighted empirical process” associated withA(n, t), Horv´athet al. [16] showed how to detect a possible change in the distribution of independent observations based on approximations for the particular bootstrapped empirical process based on

REV. ROUMAINE MATH. PURES APPL.,55(2010),2, 79–91

independent weights and, finally, the author in [1], [2] and [3] established re- sults like the almost sure central limit theorem, the asymptotic behavior of the tails and approximations for the local times for the hybrid process.

Our purpose here is to establish a variance estimator based on an invari- ance principle (IP) as well as some result concerning a possible change in the variance of the distribution of the i’s in an At Most One Change (AMOC) spirit. Finally, we study the domain of attraction for some normalized incre- ments for this process. In particular, as a first result we establish that a variance estimator obtained by using an IP is given by

σb²_m(t) = 1 m

m

X

i=1

(A(ni, t)−A(ni−1, t))²

h_N ,

see Proposition 2.1 below.

The regularity conditions related to the process (1) considered here are:

(A) the sequences of real random variables {X_i}i≥1 and {_i}i≥1 are in- dependent;

(B) {X_i,1 ≤ i < ∞} are independent, identically distributed (i.i.d.) random variables with common distribution functionF;

(C){_i,0≤i <∞}are i.i.d. random variables with E[_i] =µ,E[²_i] = 1 and, without any loss of generality, we consider that µ= 1, whenµ6= 0;

(D) the functionHis positive and is of bounded variation on the real line;

(E)₁ has a finite moment generating function in a neighborhood of 0;

(F) E[|₁|^r]<∞ forr >2.

This paper is organized as follows. In Section 2 we recall the change point approach given in [16] and state our results. Section 3 is devoted to proofs of the results.

Remark 1.1. Without loss of generality, there are i.i.d. random vari- ables {Y_i,1 ≤ i < ∞} uniform on [0,1] such that Xi = Q(Yi), with Q(y) = inf{x:F(x)≥y}, i.e., the quantile function of F (see [15], p. 5). Then we can consider

(2) A(n, t) = X

1≤i≤n

V(Yi)1_{Yi≤t}i, 0≤t≤1, in the place of ¯A(n, t), because

(3) A(n, t) =¯ X

1≤i≤n

H(Q(Y_i))1_{Q(Yi)≤t}i = X

1≤i≤n

V(Y_i)1_{Yi≤F(t)}i,

holds withV =H◦Q. Therefore, by (D), we can assume without any loss of generality that

sup

t∈[0,1]

|V(t)| ≤1.

Without any loss of generality, we will assume throughout this paper that all random variables and stochastic processes we need are defined on the same probability space. By [10] this space is well defined.

Let us recall some important results concerning approximations of the process ¯A(n, t). Diebolt [10], showed that A^∗(n, t) =n^−1/2A(n, t) (with µ= 0 in condition (C) converges weakly to a time transformed Wiener process (Brownian motion) and obtained upper bounds for the rate of convergence.

The time transformation for the limiting Wiener process is given by

(4) G¯n(t) =

Z ∞ 0

H²(s)dFn(s), whereF_n(t) = ¹_nP

1≤i≤n1_{Xi≤t},−∞< t <∞, i.e., the empirical distribution function of X1, X2, . . . .

Later, Horv´ath (2000) showed that the random time change “ ¯G_n(t)” can be replaced by a non-random time change

(5) G(t) =¯

Z t

−∞

H²(s)dF(s),

whereF is the common distribution function of theXi’s, without reducing the rate of the approximation given in [10]. He also gave an almost sure approxi- mation of the two-parameter process {A(n, t),−∞< t <∞,1≤n <∞} by a two-parameter Wiener process (cf. Theorem 1 below).

Remark 1.2. As a consequence of Remark 1.1, A^∗(n, t) is of the form A^∗(n, t) =A(n, t)/√

n, where we replace ¯A(n, t) byA(n, t) without any loss of generality, and we can replace the times changes (corresponding to ¯Gn(t) and G(t)) by¯

G_n(t) = Z t

0

V²(s)dE_n(s) and G(t) = Z t

0

V²(s)ds, t∈[0,1], where

(6) E_n(t) = 1

n

n

X

i=1

1_{Yi≤t}, t∈[0,1]

denotes the uniform empirical distribution function of the i.i.d. sequence {Y_i, 1≤i <∞}of uniform distribution on [0,1].

Theorem 1 ([15]). Under conditions (A), (B), (C, with µ = 0), (D) and (F), we can define a two-time parameter Wiener process{Γ(x, y), 0≤x, y <∞}such that, with probability one,

sup

−∞<t<∞

A(n, t)¯

√n −Γ( ¯G(t), n)

=O

n^1/2−δ(r)(logn)^1/2

,

with δ(r) = (r−2)/4(r+ 1) and r≥5.

Recall that the two-parameter Wiener process {Γ(x, y), 0≤ x, y <∞}

is a Gaussian process (also called Brownian sheet) with E[Γ(x, y)] = 0 and E[Γ(x, y)Γ(x⁰, y⁰)] = min(x, x⁰) min(y, y⁰) (see [6]).

It was also established that for a Lipschitz functionalgof order one and W a Wiener process, such that g(W(·)) has a bounded density, under the assumptions of Theorem 1 we have

sup

−1<x<1

P

g n^−1/2A¯n(·)

≤x

−P g W( ¯G(·))

≤x =o

n−(p−2)/(2(p+1)) .

Remark 1.3. An example of of Lipschitz functional of order one is g(φ) =kφk_[0,1]= sup_t∈[0,1]|φ(t)|. In this case the distribution functionψ(s) = P

kWk_[0,1]≤s , s ≥ 0, has a well-known analytical expression ([20], e.g.

(7), p. 34). Another example is g(φ) = R₁

0 φ²(s)ds1/2

. Once again, the distribution of

R1

0 W²(s)ds1/2

has a known analytical expression.

2. CHANGE POINT DETECTION AND RESULTS

In this section before stating our results, we recalling applications given in [16] to change point detection.

Let Y₁, Y₂, . . . , Y_n be independent random variables with distribution functions H₍₁₎(t), H₍₂₎(t), . . . , H_(n)(t). We wish to test the null hypothesis

H0: H₍₁₎(t) =H₍₂₎(t) =· · ·=H_(n)(t) for all t against the alternative

H_a: there is k, 1≤k < n, such thatH₍₁₎(t) =· · ·=H_(k)(t), H_(k+1)(t) =· · ·=H_(n)(t) for allt

and

H_(k)(t₀)6=H_(k+1)(t₀) with somet₀.

We divide the data into two subsets, before and after thekth observation, and compute the corresponding empirical distribution functions

H_k(t) = 1 k

k

X

i=1

1_{Yi≤t} and H_k^∗(t) = 1 n−k

n

X

i=k+1

1_{Yi≤t}, 1≤k≤n.

We reject H0 if Rn = max

1≤k<n k(n−k)

n^3/2 sup

−∞<t<∞

|H_k(t)−H_k^∗(t)| is large. Let H denote the common distribution function under H₀. IfH₀ holds, then

R_n−→^D sup

0≤x<1

sup

−∞<t<∞

Γ_H(t, x),

where ΓH(t, x) is a Gaussian process. The distribution function of the limiting random variable is unknown. If H_a holds and k= [nθ] with some 0 < θ <1, then

R_n−→ ∞.^P

In our approach, we assume that we have observed {A(n, t), 0 ≤ t ≤ 1, 1 ≤ n ≤N} atni = ihN for i = 0, . . . ,[N/hN], so that nm =n_[N/hN] ≈ N, where [x] is the greatest integer satisfying x−1 <[x]≤x and hN =O(N^α) with α ∈ (1−2δ(r),1) and δ(r) has been defined in Theorem 1. We put log(t) = log(t∨e) and let log₂(t) denote the iterated logarithm.

In order to obtain a variance estimator based on an IP, our first result provides the rate of convergence.

Proposition 1. Under the assumptions of Theorem 1, an estimator of σ²(t) =σ²G(t) is given by

σb²_m(t) = 1 m

m

X

i=1

(A(n_i, t)−A(ni−1, t))² hN

.

Moreover, we have

(7) sup

0≤t≤1

σb²_m(t)−σ²(t)

=OP sup N^1−2δ(r)log(N)

h_N ,

hN

N

δ!!

for some 0< δ <1/2.

Now, our purpose is to test for a change in the variance of the sequence of observations {_i}_i≥1, i.e., condition (C) withµ= 0 and var(₁) =σ². One of the key tools in change point analysis is to make use of an IP. Let us introduce the context. Consider the stochastic process{A(n, t), 1≤n≤N, 0≤t≤1}.

We would like to test the null hypothesis

H₀ : k^∗=N (no change in the variance over{1, . . . , N}) against the alternative

Ha: 1≤k^∗ < N and σ² 6=σ^2∗(change in the variance) after k^∗. Remark that a change in the variance when the meanµis known, corres- ponds to a case considered in Section 2.8.7 of [5], namely, it is indicated that testing H₀ against H₁ means that we are looking for a change in the mean of

V(Yi)i1{Y_i≤t}

2

.

Let us recall thatA(n, t)=^d P[N(t)]

i=1 i, where N(t) =Pn

i=1V(Yi)1{Y_i≤t}, t∈[0,1] and= denotes equality in distribution or law.^D

In the same way as in [16], let us define the process Z_n(t) =

( A(n, t), 1≤n≤k^∗, A(k^∗, t) +A^∗(n, t), k^∗ < n≤N, with Z0(t) = 0.

Theorem 1 withσ² not necessarily equal to 1, yields approximations for Zn(t), 0≤n≤k^∗, and Zn(t)−A(k^∗, t) = A^∗(n, t), n > k^∗, where A^∗(n, t)=^D P[N(t)]

i=k^∗+1^∗_i and var(^∗_i) =σ^∗2. Let

(8) T(t, ni, ni−1) =A(ni, t)−A(ni−1, t), i∈ {1, . . . , m}, t∈[0,1]. By [15] the random variables T(t, ni, ni−1) are independent and let

Z_k(t) =

k

X

i=1

T²(t, ni, ni−1)

h_N − k

m

m

X

i=1

T²(t, ni, ni−1)

h_N , k∈ {1, . . . , m}. Finally, let

Mfm = sup

0≤t≤1 1≤k≤mmax

Zk(t) pσ²(t)mh_N.

Proposition2. Assume conditions(A), (B), (C,withµ= 0 andvar(_i)

=σ²),(D), (F)hold andN^1/2−δ(r) =o(hN). Moreover, suppose that there is a two-parameter Gaussian processesΓas in Theorem1. Then under H₀ we have

Mf_m−→^D sup

0≤t≤1

|B(t)| as m→ ∞, where B(t) is a standard Brownian bridge.

Proposition 3. Assume that conditions (A), (B), (C, with µ = 0 and var(_i) =σ²), (D), (F) hold and N^1/2−δ(r)=o(h_N). Then under H_a we have

Mf_m −→ ∞^P as m→ ∞,

(i.e., the first observed time-change in variance is atk∗ ∈ {2, . . . , m}).

The two propositions below refer to the domains of attraction for some normalized increments of process (1).

Proposition 4. Assume that conditions (A), (B), (C, with µ = 0 and var(i) = σ²), (D) and (F, with r ≥ 5) hold. Let {h_n:n= 1,2, . . .} be a sequence of integers satisfying 1 ≤ h_n ≤ n, h_n ↑ ∞, h_n/n → 0 and n^1−2δ(r)log₂(n/h_n)/h_n→0 as n→ ∞. Consider

A⁺(n, t) = max

A(k, t)−A(k−hn, t)

√h_n :h_n≤k≤n

.

Under H₀, with a_n= (2 log₂(n/h_n))^1/2 and b_n= 2 log₂(n/h_n) + log₃(n/h_n)− log(4π), for each t∈[0,1] and n≥3 we have

anA⁺(n, t)−bn

−→D Z,

where the random variable Z satisfiesP(Z ≤x) = exp (−e^−x).

The next result is stated without proof because it can be obtained in the same way as Proposition 4.

Proposition 5. Assume that conditions (A), (B), (C, with µ = 0 and var(_i) = σ²), (D) and (F, with r ≥ 5) hold. Let {h_n:n= 1,2, . . .} be a sequence of integers satisfying 1 ≤ hn ≤ n, hn ↑ ∞, hn/n → 0 and n^1−2δ(r)log₂(n/h_n)/h_n→0 as n→ ∞. Consider

B(n, t) = max

A(k, t)

√

k :hn≤k≤n

.

Under H₀, with a_n and b_n as in Proposition 4, for each t ∈ [0,1] and n ≥3 we have

anB(n, t)−bn d

−→Z,

where the random variable Z satisfiesP(Z ≤x) = exp (−e^−x).

3. PROOFS

This subsection is devoted to the proofs of our results.

Proof of Proposition 1. Roughly speaking, the scheme of the proof will be based on the evaluation of the distance

bσ_m²(t)−σ²(t) =

σb²_m(t)−σe²_m(t) +σe²_m(t)−σ²(t)

from the estimator to the variance, where σe²_m(t) (will be defined later) is obtained from the IP given in Theorem 1. Before establishing our proof we outline the following facts.

Fact1: by using an IP given by Theorem 1, we have sup

0≤t≤1

i=1,...,mmax |T(t, ni, ni−1)−σ{Γ(n_i, G(t))−Γ(ni−1, G(t))}|= (9)

=Oa.s.

N^1/2−δ(r)(log(N))^1/2

.

Fact2: by (2.1) of [15] and (1.11) of [6], forn_i< k≤n_i+1 we can define Sk−n_i(G(t)) =

k−ni

X

j=1

Wj+ni(G(t)), t∈[0,1],

where the {W_l(t)}_l≥1 are independent standard Brownian motions such that (10) Γ(k, G(t))−Γ(n_i, G(t)) =Sk−n_i(G(t)).

Fact3: by Fact 1 and Fact 2 we have the equations

bσ_m²(t)−eσ_m²(t) =

1 m

m

X

i=1

T²(t, ni, ni−1) hN

− 1 mσ²

m

X

i=1

S_n²_i−ni−1(G(t)) hN

=

=

1 m

m

X

i=1

T²(t, n_i, ni−1)−σ²S_n²_i−ni−1(G(t)) h_N

=

1 m

m

X

i=1

1

h_NI1(i, t)I2(i, t) ,

where

I₁(i, t) =T(t, n_i, ni−1)−σS_ni−ni−1(G(t)) and

I₂(i, t) =T(t, n_i, ni−1) +σS_ni−ni−1(G(t)).

Now, by using (9), we have

(11) sup

0≤t≤1

1≤i≤mmax |I₁(i, t)|=Oa.s.

N^1/2−δ(r)(log(N))^1/2

and sup

0≤t≤1

1≤i≤mmax |I₂(i, t)| ≤ max

1≤i≤m|I₁(i, t)|+ 2 max

1≤i≤mσ

S_ni−ni−1(G(t)) . Then the law of the iterated logarithm (LIL) yields

sup

0≤t≤1

1≤i≤mmax |I₂(i, t)|=Oa.s.

N^1/2−δ(r)(log(N))^1/2

+Oa.s.

phNlog (N/hN)

and comparing the rates we see that

(12) sup

0≤t≤1

1≤i≤mmax |I₂(i, t)|=O_a.s.

N^α/2(log(N))^1/2 ,

and from (11), (12) in conjunction with Fact 3 we deduce that

(13) sup

0≤t≤1

σb²_m(t)−σe²_m(t) =

1 m

m

X

i=1

1 hN

I1(i, t)I2(i, t)

=Oa.s.

Nα(1/2−δ(r))log(N) h_N

! .

Fact5: we now deduce that

(14) sup

0≤t≤1

σe²_m(t)−σ²(t) =oP

1 m^δ

, 0< δ <1/2,

because the term on the right hand side is equal in law to σ²(t)χ²_m−1/m, where χ²_m−1 is a chi-squared distributed random variable with m−1 degrees of freedom.

So, by the LIL, we have σ²(t)χ²_m−1

m −σ²(t) =o_P 1

m^δ

, 0≤δ ≤1/2,

that yields (14). Result (7) is now a direct consequence of (13) and (14).

Proof of Proposition2. Similarly to (9), letni =ihN,i= 1, . . . ,[N/hN], such that n_m =n_[N/hN]≈N. Under our conditions we can take for instance h_N = [N^α], 1−2δ(r)< α <1. Consider for eacht∈[0,1] random variables

S_ni−ni−1(G(t)) pG(t)hN

, 0≤i≤m,

independent and of common law N(0,1). By Theorem 1 we have T²(t, n_i, ni−1) =n

σS_ni−ni−1(G(t)) +O_a.s.

N^1/2−δ(r)(log(N))^1/2o2

. Then

sup

1≤t≤1

sup

0≤k≤m

k

X

i=1

T²(t, ni, ni−1)−σ²G(t)hN

−

−σ²

k

X

i=1

S_n²_i−ni−1(G(t))−hNG(t)

=

=O_P

N^1/2−δ(r)(log(N))^1/2p

mh_Nσ²(t)) log₂(m)

+O_P mN^1−2δ(r)log(N) , where we have used the LIL. Now, this last rate can be written as

(15) O_P

N^1−δ(r)log(N) (log₂(N/h_N))^1/2

+O_P N^2−2δ(r)log(N) h_N

! .

Moreover,

(16) sup

1≤t≤1

sup

0≤k≤m

σ²

k

X

i=1

S_n²_i−ni−1(G(t))−hNG(t) hN

−√

2cW(kG(t))

=Oa.s.(log(N)),

where we have used the strong approximation result of [17] applied to the random variables

S_n²_i−ni−1(G(t))

h_N −G(t)

satisfying condition (E), and Wc denotes a standard Brownian motion.

From (15) and (16) we obtain (17) sup

1≤t≤1

sup

0≤k≤m

σ²

k

X

i=1

T²(t, n_i, ni−1)(G(t)

h_N −σ²(t)−√

2cW(kG(t))

=

=OP





N^1−δ(r) h_N

!2

log(N) (log₂(N/hN))^1/2





and

k m

m

X

i=1

T²(t, n_i, ni−1)(G(t) hN

−√ 2k

mcW(mG(t)) =

=OP





N^1−δ(r) h_N

!2

log(N) (log₂(N/hN))^1/2



. Finally,

Mf=

√2

√m

Wc(kG(t))− k

mWc(kG(t))

+ +O

N^1−2δ(r) h²_N

!2

log(N) (log₂(N/h_N))^1/2

!

and

√1 m

cW(kG(t))− k

mWc(kG(t))

=B_m(G(t)),

where Bm(·) is a Brownian bridge for each m. By continuity ofBm(G(t)) we obtain sup

0≤s≤1

|B(s)|, where we took s= _m^k. This last argument completes the proof of Proposition 2.

Proof of Proposition 3. Using Theorem 1 for T(t, n_i, ni−1) we consider two cases: the approximation is given by Γ if i ≤ k^∗ and by Γ⁰ for i > k^∗. Here, Γ and Γ⁰ are independent two-parameter Gaussian processes.

Let Z_k^∗ =

k^∗

X

i=1

T²(t, ni, ni−1)−k^∗ m

m

X

i=1

T²(t, ni, ni−1) =

k^∗

X

i=1

T²(t, ni, ni−1)−

−k^∗ m

k^∗

X

i=1

T²(t, n_i, ni−1) +

m

X

i=1

T²(t, n_i, ni−1)−

k^∗

X

i=1

T²(t, n_i, ni−1)

! .

Now, putting

I =

k^∗

X

i=1

T²(t, n_i, ni−1) G(t)hN

,

II =

m

X

i=1

T²(t, n_i, ni−1) G(t)h_N −

k^∗

X

i=1

T²(t, n_i, ni−1) G(t)h_N =

m

X

k^∗+1

T^2∗(t, n_i, ni−1) G(t)h_N , where T^2∗(t, n_i, ni−1) is defined as in (8) but replacing A(n_i, t) by A^∗(n_i, t).

By the IP, we have

I =σ²

k^∗

X

i=1

S_n²_i−ni−1(G(t))

G(t)h_N +OP(1).

Also, k^∗

mII = k^∗σ^2∗

m

m

X

i=k^∗+1

S_n^2∗_i−ni−1(G(t))

G(t)h_N +O_P(1) = k^∗σ^2∗

m

m

X

i=k^∗+1

χ²_i

and

k^∗

mII = k^∗σ² m

k^∗

X

i=1

S_n^2∗_i−ni−1(G(t)) G(t)hN

+O_P(1) = k^∗σ² m

k^∗

X

i=1

χ².

Then

k^∗

mII−k^∗ mI

=

k^∗σ^2∗

m

m

X

i=1

χ²− k^∗(σ^2∗−σ²) m

k^∗

X

i=1

χ² .

By the strong law of large numbers we have Mf_m=O_P

sup

0≤t≤1

√1

mI + k^∗ m^3/2

σ^2∗−σ²

(m−k^∗)

,

so that the second part of the rate goes to infinity if k^∗

m^3/2(m−k^∗) = h^3/2_N

√

N(1−k^∗h²_N/N)→ ∞.

This completes the proof of Proposition 3.

Proof of Proposition4. Define Sn(t) =

Pn i=1Zi(t)

√

nG(t) , where Zi(t) =iV(Yi) I{Y_i≤t} and I{·} is the set indicator function.

We have EZi(t) = 0 and Var(Zi(t)) = Rt

0V²(s)ds for t ∈ [0,1]. More- over, by our assumptions, we can consider a sequence of Wiener processes {W_n^∗(t), t∈[0,1]}, satisfying

W_n^∗(t) = W^∗(nt)

√n and W_n^∗ Z t

0

V⁴(s)ds d

= Z t

0

V²(s)dW n(s), where = denoted equality in law (or distribution).^d

By Theorem 1 of [15] we have sup

0≤t≤1

A(n, t)

√n −W_n^∗(G(t))

=O 1

√n

a.s.

and, as in (5.30) of [4], we deduce that

a_n max

(log(n))³≤i≤n

A(i, t)

√

i − sup

(log(n))³≤i≤n

W_i^∗(G(t))

!

→0 a.s.

Then

a_nA⁺(n, t)−b_n=a_n max

hn≤k≤n

W(kG(t))−W((k−h_n)G(t))

√h_n

−b_n+ +O_a.s.

rlog(n/h_n)/h_n n

! , or

hnmax≤k≤n

W(kG(t))−W((k−hn)G(t))

√hn

=

=D max

1≤s≤n/hn

n

Wf(sG(t))−Wf((1−s)G(t))o . Then by Theorem 12.3.5 of [18] the proof is complete.

Proof of Proposition5. Similar to the proof of Theorem 2 in [13].

REFERENCES

[1] S. Alvarez-Andrade, On almost sure central limit theorem for hybrids empirical and partial sum-processes.Int. J. Pure Appl. Math.34(2007), 31–41.

[2] S. Alvarez-Andrade,Invariance for the tail of hybrids of empirical and partial-sum pro- cesses.Rev. Roumaine Math. Pures Appl.52(2007), 305–313.

[3] S. Alvarez-Andrade,Some asymptotic properties of the hybrids of empirical and partial- sum processes.Rev. Mat. Iberoamericana24(2008), 31–41.

[4] I. Berkes and E. Cs´aki,A universal result in almost sure central limit theory.Stochastic Process. Appl.94(2001), 105–134.

[5] M. Cs¨org˝o and L. Horv´ath,Weighted Approximations in Probability and Statistics. Wi- ley, Chichester, 1993.

[6] M. Cs¨org˝o and P. R´ev´esz,Strong Approximations in Probability and Statistics. Academic Press, New York, 1981.

[7] J. Diebolt, Deux tests non param´etriques pour un mod`ele autoregressif non lin´eaire.

C. R. Acad. Sci. Paris Ser. I Math.301(1985), 649–652.

[8] J. Diebolt, Testing the functions defining a nonlinear autoregressive time series. Sto- chastic Processes. Appl.36(1990), 85–106.

[9] J. Diebolt and N. La¨ıb, Un principe d’invariance faible pour l’´etude d’un test non param´etrique relatif `a un mod`ele de regr´ession non lin´eaire. C. R. Acad. Sci. Paris Ser. I Math.312(1991), 887–891.

[10] J. Diebolt, A non parametric test for the regression function: asymptotic theory. J.

Statist. Plann. Inference44(1995), 1–17.

[11] J. Diebolt, N. La¨ıb and J. Gatchou-Wandji,Limiting distributions of weighted processes of residuals for parametric autoregressive models of time series.Rapport Technique 1, IMAG, 1997.

[12] J. Diebolt and J. Zuber,Goodness -of fit test for non linear heteroscedastic regressions models.Statist. Probab. Lett.42(1999), 53–60.

[13] A. Gut and J. Steinebach,Truncated sequential change-point detection based on renewal counting processes.Scand. J. Statist.29(2002), 693–719.

[14] E. Haeusler and D. Mason, Weighted approximations to continuous time martingales with applications.Scand. J. Statist.26(1999), 281–295.

[15] L. Horv´ath, Approximations for hybrids of empirical and partial sums processes. J.

Statist. Plann. Inference88(2000), 1–18.

[16] L. Horv´ath, P. Kokoszka and J. Steinebach,Approximations for weighted bootstrap pro- cesses with an application.Statist. Probab. Lett.48(2000), 59–70.

[17] J. Koml´os, P. Major and G. Tusn´ady,An approximation of partial sums of independent R.V.’s and the sample D.F.I.Z. Wahrsch. Verw. Gebiete.32(1975), 111–131.

[18] M.R. Leadbetter, G. Lindgren and H. Rootz´en, Extremes and Related Properties of Random Sequences and Processes.Springer-Verlag, 1983.

[19] M. Maumy,Etude du processus empirique compos´e.PhD Thesis, University of Paris VI, 2002.

[20] G. Shorack and J. Wellner, Empirical Processes with Applications to Statistics.Wiley, New York, 1986.

Received 1 July 2009 Universit´e de Technologie de Compi`egne LMAC

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