AND RELATED PROCESSES

AND RELATED PROCESSES

S. ALVAREZ-ANDRADE and L. BORDES

We deal with asymptotic properties of the [αm]th order statistic of a type-II progressively censored sample of sizem(quantile process) and with the counting processm⁻¹Pm

i=11(Xi:m:n ≤ t), where the Xi:m:n are the observed m-sample obtained by progressively censoring from the sampleX1, . . . , Xn.

AMS 2000 Subject Classication: Primary 60F17, 60G50; Secondary 62G20.

Key words: Progressive censoring, quantile process, counting process.

1. INTRODUCTION

Progressive censoring schemes are very useful in life-test experiments and in clinical studies. Montanari and Cacciari [13] reported results of progressively censored data aging tests on XLPE-insulated cable models under combined thermal-electrical stresses. Bhattacharya [5] gives an example, in a clinical trial study. The monograph of Balakrishnan and Aggarwala [2] (see also [5]) gives an interesting review of the background and of developments in this eld.

Progressive censoring is a particular model based on order statistics and record values, see for instance [12].

Let X₁, . . . , X_n be independent and identically distributed (i.i.d.) ran- dom lifetimes of n items. A progressive type-II right censored sample may be obtained in the following way: at the time of the rst failure, noted X_1:m:n, r1 surviving items are removed at random from then−1remaining surviving items, at the time of the next failure, noted X2:m:n,r2 surviving items are re- moved at random from then−r₁−2 remaining items, and so on. At the time of the mth failure, all the remainingrm =n−m−r1− · · · −rm−1 surviving items are censored.

Our rst result concerns the asymptotic behaviour of the [αm]th order statistic of a type-II progressively censored sample of sizem(quantile process), dened by (X_[αm]:m:n)α∈[0,1], where [x]is the greatest integer satisfying [x]≤ x <[x] + 1. Our aim is to establish a result of the type

√m X_[αm]:m:n−u_α

=C_αW⁰(α) +O δ_m⁰

REV. ROUMAINE MATH. PURES APPL., 53 (2008), 4, 267275

in probability. HereW⁰is a standard Brownian motion,uαis theα-quantile of the distribution function, C_α is an expression depending onα while δ⁰_m gives the rate of convergence (see our Theorem 2). In order to establish our rate of convergence result (Theorem 2) in Section 2, we begin by establishing in Theorem 1 a result about the associated triangular array resulting from the almost sure (a.s.) representation (see (1) below). Theorem 3 is related to the counting processm⁻¹Pm

i=11(X_i:m:n≤t), for which a Poisson approximation is given. Finally, in Theorem 4, we give a result on induced order statistics under progressively type-II censoring scheme applied to the X sample. In Section 2, we make precise the regularity conditions and state our results. Section 3 is devoted to the proof of our results.

2. ASSUMPTIONS AND RESULTS

From now on, we assume that theX_ihave a common distribution function F with density f. We denote by λ the hazard rate function i.e. λ(t) = f(t)/R(t), whereR(·) = 1−F(·)is the survival function (reliability function) andΛ(t) =Rt

0f(x)/R(x)dxthe cumulative hazard rate function. Denelog₂n as the two-iterated logarithm, withlogn= log(n∨e).

Let us rst introduce the assumptions under which these asymptotic prop- erties are obtained. All asymptotic results are given for m→+∞.

A1. α∈[0, a]and 0< a <1.

A2. (ri)i≥1 is a bounded sequence of nonnegative integers,ri≤K for all i≥1, such thatr¯=m⁻¹Pm

i=1r_i =r+O

qlog₂m m

, wherer is a nonnegative number.

A3. τ1 is a real number such that F(τ)<1.

A4. Let G= 1−R^r+1. There exists a real number ε∈[0, a) such that G⁻¹([ε, a])⊂(c, b)⊂R⁺ and λis continuous and strictly positive on(c, b).

Now, we introduce some useful notation. Let(α^m_j )1≤j≤m be a triangular array of non-negative integers dened byα^m₁ =m,α^m_j =rj+· · ·+r_m+m−j+1 for 2 ≤ j ≤ m−1, and α_m^m = rm+ 1. Denote by uα the α-quantile of the distribution function G,u_α =G⁻¹(α), where G⁻¹ is taken in the generalized inverse sense (G⁻¹(x) = inf{y:G(y)> x}) whenG is not invertible.

Finally, we recall the almost sure representation given in [2]: there ex- ists a triangular array Z_j^(m)

1≤j≤m of i.i.d. exponentially distributed random variables with mean 1 such that

Λ(X_i:m:n) =

i

X

j=1

Z_j^(m) α^m_j a.s.

(1)

Theorem 1. Under assumptions A1A4 and in a probability space (Ω₀,F₀, P₀) rich enough containing all random variables and processes that we need, we have

(2)

[αm]

X

j=1

Z˜_j^(m)−1 α^m_j −

W⁰

_α(1−a)

a(1−α)

θm(a)

=O δ_m

θm(a) + 1 m

in probability, where W⁰ is a standard Brownian motion, θ_m(a) = q

1/(P[am]

k=1(α_k^m)⁻²) and δ_m is a non-increasing positive sequence such that δm=o(√

m) as m→ ∞.

Remark 2.1. On account of the a.s. representation (1), we have to deal with the triangular array Z_j^(m)

1≤j≤m of i.i.d. exponentially distributed ran- dom variables with mean1. In Theorem 1, we writeZ˜_j^(m) in place ofZ_j^(m) be- cause the probability space (Ω0,F₀, P0) is a common probability space where Z˜_j^(m) and W⁰ are well dened. In the proof of Theorem 1 (see Fact 4), we justify the existence of this space by standard arguments.

Theorem 2. Under the assumptions of Theorem 1, we have

(3) √

m X_[αm]:m:n−uα

=

√a (r+ 1)λ(uα)√

1−aW⁰(α) +O

δm+ 1 m

in probability, where {W⁰(t), 0≤t≤1} is a standard Brownian motion.

In the next theorem, we are concerned with the behaviour of the counting process

Nm(t) = 1 m

m

X

i=1

1(Xi:m:n≤t), t≥0.

(4)

Our result is based on a Poisson approximation for the empirical distribution function. Roughly speaking, our result can be seen as a corollary of Lemma 3.1 of [11].

Theorem 3. If the assumptions of Theorem1 are satised, then we can dene Poisson processes Mm(x) with E[Mm(x)] =x such that

sup

0≤t<∞

N_m(t)− M_m(t) Mm(m)

=O

rlog₂m m

! (5)

in probability.

Let (X_i, Y_i), i = 1,2. . ., be independent copies of (X, Y) ∈ R⁺×R⁺. Suppose a progressively censored scheme is adopted for the X-sample and let

Y1:m:n, . . . , Ym:m:nthe corresponding values ofY. The random variablesYi:m:n, i= 1, . . . , m, are called induced order statistics (cf. [4] and [8] for instance).

Set m(x) = E[Y|X = x], σ²(x) = Var(Y|X = x), β(x) = E[(Y − m(x))⁴|X=x], and dene

A_m(t) =m^−1/2

m

X

i=1

(Y_i:m:n−m(X_i:m:n))1(X_i:m:n≤t), t≥0.

Theorem 4. Under the assumptions of Theorem1, letσ²(x)be of bounded variation andβ(x)bounded. Then we can dene Poisson processesM_m(x)with E[Mm(x)] =x and Brownian motions Wm such that

sup

t≥0

|A_m(t)−W_m(G_m(t))| →0 (6)

in probability, where

Gm(t) = (Mm(m))^−1/2

Z 1−R^r+1(t) 0

σ²(s)dMm(ms).

3. PROOFS

Proof of Theorem 1. Keeping in mind approximations results for tri- angular arrays given in [10], set α^m_j ξ˜^(m)_j = θ_m(a) ˜Z_j^(m) −1

with θ_m(a) = q

1/(P[am]

k=1(α_k^m)⁻²). Dene the polygonal processS˜_(m)(t) by S_(m)(t) =

r

X

k=1

ξ˜^(m)_k + t−t_m,r tm,r+1−tm,r

ξ˜_r+1^(m), (7)

where 0 < r <[αm]and t_m,r = Pr

k=1σ²_m,k for 1≤r ≤[αm], with t_m,[αm]=

α(1−a)

a(1−α) for α∈[0, a]andm large enough (see Lemma 1 of [1]).

Under our assumptions, we have the following facts:

Fact 1. Fromm−k+ 1≤α^m_k ≤(K+ 1)(m−k+ 1), fork≤αmwe have α^m_k ≥(1−α)m which yields

αm (1−α)²m² ≥

[αm]

X

j=1

1 (α^m_k)². Fact 2. From Fact 1, for

B_m,j =α^m_j v u u t

[am]

X

k=1

1 (α^m_k)²

we have

c√

m≤Bm,j≤ (K+ 1)√ am (1−α) , (8)

wherec is a constant >0.

Fact 3. For a random variableZ with exponential law of mean 1, we have P

Z−1 Bm,j

≤x

= 1−e^−(1+xBm,j)

1 (1 +xBm,j ≥0), (9)

where1(·)stands for the indicator function.

Fact 4. The result Lm(δ) =

km

X

j=1

E h

ξ˜_j^(m)

21

ξ˜^(m)_j > δ

i

→0 asm→ ∞ (10)

corresponds to Condition (C) in Corollary 4 of [10].

By using Fact 3, the moment part of (10) can be evaluated as E

Z−1 B_m,j

2

1 (|Z−1|> δB_m,j)

=

= Z

R

B_m,jx²e^−(1+xBm,j)1 (1 +xB_m,j ≥0) 1 (|x|> δ) dx.

Then we have to consider the following two cases:

I = Z

R

Bm,jx²

e^(1+xBm,j)1 (1 +xBm,j ≥0) 1 (x > δ) dx= e⁻¹ Z +∞

δBm,j

t²e^−t (B_m,j)²dt, II =

Z

R

B_m,jx²

e^(1+xBm,j)1 (1 +xBm,j ≥0) 1 (x <−δ) dx≤

≤

Z 1−δB_m,j 0

u−1 Bm,j

2

e^−udu.

Using (8), we have1−δB_m,j ≤1− min

1≤j≤[αm]B_m,jδ≤1−cδ√

m, henceII →0.

Let us now evaluate I. The sum term is bounded byδ²P[αm]

j=1 e^−δc

√m, which goes to 0 by the integral test, so (10) is obtained by (8).

By (7) and Corollary 4 of [10], we haveP S_(m)|B(C[0,1])→^w W|B(C[0,1]).

On account of this last argument, the probability space(Ω₀,F₀, P₀)exists and is well dened (see Theorem 11.7.1 of [9]). In that space, we can dene S˜m

and a standard Wiener process W⁰ such that P Sm =P0S˜m and W⁰ = W in law, withd( ˜S_(m), W⁰)→0 both in probability and a.s.

Fact 5. By [9], there exist positive numbers δm → 0 such that ρ(P S_(m), W) ≤δ_m, i.e., where ρ(·,·) stands for the Prohorov distance which associates a metric with weak convergence.

This last inequality yields

[αm]

X

j=1

Z˜_j^(m)−1 α^m_j −

W⁰_α(1−a)

a(1−α)

θm(a)

=O δ_m

θm(a) + 1 m

(11) ,

in probability, where the rate of convergence is obtained as in the proof of Theorem 11.7.1 of [9], which completes the proof.

Proof of Theorem 2. By (1) and Assumption A4,Λ⁻¹admits a rst order Taylor expansion. Then we have

X_[αm]:m:n^a.s.= Λ⁻¹

Λ(uα) +

[αm]

X

j=1

Z˜_j^(m)

α^m_j −Λ(uα) (12)

= Λ⁻¹(Λ(uα)) + β_[αm]

λ(Λ⁻¹(Λ(φm))), whereβ_[αm]=P[αm]

j=1 Z_j^(m)/α^m_j −Λ(uα) andφm is such that Λ(φm) belongs to the segment with extremities Λ(uα) and Λ(uα) +β_[αm].

It is sucient to study the asymptotic behaviour of β_[αm]. We have

√mβ_[αm]=I^(m)+II^(m)+III^(m)

with

I^(m) =√

mP[αm]

j=1

Z˜^(m)_j −1 α^m_j ,

II^(m)=√ m

[αm]

X

j=1

1

α^m_j − 1

(r+ 1)(m−j+ 1)

!

=o(1)

and

III^(m) =√ m





[αm]

X

j=1

1

(r+ 1)(m−j+ 1)−Λ(u_α)



=O 1

m

,

where theo( )andO( )terms are obtained by Corollary 1 of [1] under Assump- tions A1A4. Moreover, I^(m)/√

m is given by (2). Then we can write

√mβ_[αm]=

√mW⁰_α(1−a)

a(1−α)

θ_m(a) +O

δ_m+ 1 m

in probability and, by (12),

√m X_[αm]:m:n−u_α

=

√mW⁰_α(1−a)

a(1−α)

θm(a)λ(uα) +O

δ_m+ 1 m

in probability, with √ m θm(a) →

r a

(1 +r)²(1−a), which completes the proof.

Proof of Theorem 3. As in [6], let us dene the counting process Nbm(t) = 1

m

m

X

i=1

1(Ybi:m:n≤Λ(t)), t≥0, (13)

where

Yb_i:m:n=

i

X

j=1

Z_j^(m)

(r+ 1)(m−j+ 1), 1≤i≤m.

Remark 3.1. In [6] it was stated that theYb_i:m:n have the same distribu- tion as an order statistic of an m-sample of exponential random variables of mean 1/(r+ 1).

In order to prove our result we establish the following facts.

Fact 6. Having in mind Remark 3.1, we can rewrite (13) as Nb_m(τ) = 1

m

m

X

i=1

1(U_i:m:n≤τ), (14)

whereτ = 1−exp{(r+ 1)Λ(t)}, t≥0, and theUi:m:nare the order statistics of an m-sample of uniformly distributed on [0,1] random variables (see [2]

about monotone transformations in this context).

On account of the above transformation, we can also rewrite (4) as N_m(τ) = 1

m

m

X

i=1

1(V_i:m:n≤τ), (15)

where1−exp{(r+ 1)Y_i:m:n}=V_i:m:n withY_i:m:n= Λ(X_i:m:n) i= 1, . . . , m. Fact 7. By Lemma 3.1 of [11] and (14), we can dene Poisson processes M_m(τ)with E[M_m(τ)] =τ such that

(16) P

sup

0≤τ≤1

Nbm(τ)− Mm(τ) M_m(m)

> 1

m(x+clogm)

≤exp (−cx) for all x >0.

Fact 8. On account of Assumption A2, we can mimic the proof of Propo- sition 3 of [6], but we use the law of the iterated logarithm instead of the strong law of large numbers, to get

sup

0≤t≤τ1

Nbm(t)−Nm(t) =O

rlog₂m m

!

in probability.

Facts 6, 7 and 8 suce to establish (5).

Proof of Theorem 4. Consider the monotone transformations dened in (14) and for 0≤t≤1 consider

Sm([mt]) =m^−1/2

[mt]

X

i=1

(Yi:m:n−m(Ui:m:n)), k= 1, . . . , m.

Sums of conditional independent random variables can be represented by Brow- nian motion at stopping times. This is proved in [3] by applying the Skorokhod embedding theorem. Then we obtain the representation of S_m([mt])byW_m(t) in some common probability space. Further, we have

sup

0≤t≤1

W_m Z t

0

σ²(s)dN_m(s)

−M_m^−1/2W_m Z t

0

σ²(s)dM_m(s)

≤I+II,

where

I = sup

0≤t≤1

W_m Z t

0

σ²(s)dN_m(s)

−W_m Z t

0

σ²(s)d ˜N_m(s)

and

II= sup

0≤t≤1

W_m Z t

0

σ²(s)d ˜N_m(s)

−M_m^−1/2W_m Z t

0

σ²(s)dM_m(s)

, where without loss of generality we have considered t=τ (see (14)).

Remark by using (16) that sup

0≤t≤1

Z _t

0

σ²(s)dN_m(s)− Z _t

0

σ²(s)d ˜N_m(s)

≤C sup

0≤t≤1

N_m(t)−N˜_m(t) ≤1, whereC is a constant.

Since σ² is of bounded variation, we apply Lemma 1.1.1 of [7] to get I → 0. In the same way as in Lemma 3.2 of [11], we obtain that II → 0 in probability.

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Received 26 October 2007 Université de Technologie de Compiègne

Revised 30 January 2008 L.M.A.C.

B.P. 529, 60205 Compiègne Cedex, France and

Université de Pau L.M.A.

B.P. 1155, 64013 Pau Cedex, France