An asymptotical method to estimate the parameters of a deteriorating system under condition-based maintenance

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Preprint submitted on 11 Jul 2013 (v1), last revised 4 Apr 2017 (v2)

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An asymptotical method to estimate the parameters of a deteriorating system under condition-based maintenance

Philippe Briand, Edwige Idée, Céline Labart

To cite this version:

Philippe Briand, Edwige Idée, Céline Labart. An asymptotical method to estimate the parameters of a deteriorating system under condition-based maintenance. 2013. �hal-00843566v1�

An asymptotical method to estimate the parameters of a deteriorating system under

condition-based maintenance

Philippe Briand¹, Edwige Idée² and Céline Labart³

Laboratoire de Mathématiques, CNRS UMR 5127, Université de Savoie, Campus Scientifique, 73376 Le Bourget du Lac, France.

July 12, 2013

Abstract

In this paper, we develop a new method to estimate the parameters of a deteriorating system under perfect condition-based maintenance. This method is based on the asymptotical behavior of the system, which is studied by using the renewal process theory. We obtain a Central Limit Theorem (CLT in the following) for the parameters. We compare the accuracy and the speed of the method with the maximum likelihood one (ML method in the following) on different examples.

1 Introduction

Many systems suffer from increasing wear with usage and age and are subject to failures result- ing from deteriorations. The deterioration and failures might lead to high costs, then preventive maintenance is necessary. In the past several decades, maintenance, replacement and inspection problems have been widely studied (see the surveys [McC65], [BP96], [PV76], [ON76] and [SS81], [VFF89] among others). If the deterioration of a system can be observed while inspecting, it is more judicious to set up the maintenance policy on the state of the system rather than on its age.

Deterioration systems and their optimal maintenance policy have been studied in the literature (see [MK75], [OKM86], [TvdDS85], [BSK96], [Wan00] and [GBD02]). In this paper, we consider a system subject to random deteriorations, which can lead to failures. As long as the system oper- ates, it is monitored by planned inspections. At these inspections, the system can be in two states : sane or damaged. If the system is found out to be damaged, a preventive perfect repairing (“as good as new”) is performed. If the system fails, an unplanned inspection is performed immediately and a corrective perfect repairing (“as good as new”) is done.

When the deterioration of the system can be continuously measured, the deterioration is usu- ally represented by a stochastic process with stationary and independent increments and the states of the system are fixed by some thresholds on the stochastic process (see [GBD02], [GDBR02] for example). However, some deteriorations can not be easily measured and the state of the system is only known when inspecting. In order to deal with this kind of problem, we assume that the tran- sition time from repairing to damaging and the transition time from damaging to failure are two positive random variables, whose parameters (µ, λ) are unknowns (we consider that failures only ensue from deteriorations). Moreover, we assume that there exists some uncertainty on planned

1philippe.briand@univ-savoie.fr.

2edwige.idee@univ-savoie.fr.

3celine.labart@univ-savoie.fr.

inspection dates (which can be due to physical or financial constraints). The purpose of the present paper is to propose a new method, based on the asymptotic behavior of the system, to estimate the unknown parameters of the transition times. At some timet, we assume that we only know the number of inspections before t, and the state of the system at the inspection dates. Then, we have at hand N_t^r, the number of repairs before time t, N_tⁱ, the number of inspections before timet, andN_t^f, the number of failures before timet. By using the renewal processes theory, we write (µ, λ) as the limit of a function of (^N_t^tr,^N_t^ti,^N

f t

t ). Moreover, we get a CLT for this triplet of variables, which gives us a confidence interval for (µ, λ).

Practically, this method not only applies to long term systems. We can also deal with iden- tical units, repaired at least one time, operating independently and simultaneously in a similar environment and being analogously exploited. Putting repairing times of these units end to end is equivalent to studying a single system on a long period of time. The paper is organized as follows : Section 2 introduces some notations and presents the assumptions on the model, Section 3 recalls standard results on renewal and renewal reward processes. Section 4 states a CLT for our parameters. Sections 5 and 6 present some applications and numerical examples.

2 Model Assumptions

In the following, we represent the time from repairing to damaging by a positive random variable (r.v. in the following)Y^sand the time from damaging to failure by a positive r.v. Y^d. We modelize the uncertainty of inspection dates (which can be due to physical or financial constraints) by a sequence of random variables. The elapsed time between two inspection dates is a random variable C, and ifC_idenotes the time spent between the (i−1)th and theith inspection,D_i:=C₁+· · ·+Ci

is the age of a system at theith inspection (with convention D₀= 0). This enables to define the index of the inspection following the damage

K^r= inf{n≥1 :D_n≥Y^s}= 1 +X

n≥1

1_Dn<Ys.

and the inter-repairing time

X^r= min DK_r, Y^s+Y^d .

This means that we repair the system as soon as a damage is detected (X^r=DK_r) (see Figure 1) or as soon as the system fails (X^r=Y^s+Y^d) (see Figure 2). In the following, we assume that the law ofY^sdepends of a parameterµ, the law ofY^d depends of a parameterλ, and the law of C is known.

0 Ys

Inspection and Repairing Inspection

Nothing to

Inspection Nothing to do do

Yd

C1 C2 C3

Xr=D(Kr)

Figure 1: The system is repaired as soon as the damage is detected We introduce the following notations :

0 Ys Inspection

Nothing to

Inspection Nothing to do do

Yd

C1 C2

Xr=Ys+Yd

Unplanned Inspection Repairing C3

Figure 2: The system is repaired as soon as it fails Definition 2.1.

• Fs (resp. Fd) denotes the cumulative distribution function of Y^s(resp. Y^d)

• R_s(resp. R_d) denotes the survival function of Y^s(resp. Y^d)

• n_µ denotes the law ofY^s

• L denotes the Laplace transform of C : ∀s≥0,L(s) :=E(e^−sC)

• B(t) represents the index of the inspection following t : B(t) := inf{n≥1 :Dn ≥t} = 1 +P

n≥11D_n<t. Then,K^r=B(Y^s)

• B(t)represents the index of the inspection beforet : B(t) := sup{n≥0 :Dn≤t}

• V^s denotes the age of the system when a damage is detected : V^s=D_Kr

• Z^d denotes the time of failure : Z^d=Y^s+Y^d

• Pd denotes the probability that the system fails before the inspection following the damage occurs : Pd=P(V^s≥Z^d) =P(D_B(Ys)−Y^s≥Y^d)

• m_x denotesE(X^r)andm_k denotesE(K^r) With these notations, X^r= min(V^s, Z^d).

Hypothesis 2.2. In the following, we assume that the r.v. Y^s and Y^d are such that the law of Y^s depends on a parameter µ taking values in U and the law of Y^d depends on a parameter λ taking values in V. The r.v. (Ci)i≥1 are independent with common law C, and such that P(C >0) = 1. We also assume that they are independent ofY^s andY^d. Moreover, we assume E((Y^s)²) +E(C²)<∞.

Remark 2.3. Under Hypothesis 2.2, V(X^r) +V(K^r) < ∞. We first prove that E((K^r)²) <

∞. E((K^r)²) = R∞

0 E(B(t)²)nµ(dt) = R∞ 0

E(B(t)²)

t² t²nµ(dt). (B(t))t≥0 is a renewal process with interarrival time C, then ^E(B(t)_t2 ²⁾ → ₍ ¹

E(C))². Since ∃c >0 such that ∀t ≥0 E(B(t)²) ≤ce^t, we get∀ε >0,∃t0 such thatE((K^r)²)≤ce^t0+ ( ¹

E(C)² +ε)E((Y^s)²). ConcerningE((X^r)²), we have E((X^r)²)≤E((V^s)²) andV^s≤Y^s+C_B(Ys). Then,E((V^s)²)≤2(E((Y^s)²) +E(C²

B(Y^s))). Since we have

E(C_B(Y² _s)) =

∞

X

k=1

E(C_k²1_B(Ys)=k) =

∞

X

k=1

E(C_k²1Dk−1<Y^s≤D_k)≤

∞

X

k=1

E(C²)R^s(Dk−1) =E(C²)E(K^r), the result follows (the last equality comes from (1)).

Proposition 2.4. Under Hypothesis 2.2, we have E[K^r] =X

n≥0

E[Rs(Dn)], (1)

1−Pd= Z +∞

0

E h

Rd

D_B(t)−ti

nµ(dt), (2)

E[X^r] =E[Y^s] + Z +∞

0

Z +∞

0

P

D_B(t)−t > y

R^d(y)dy nµ(dt). (3) If in addition we assume thatY^d follows the exponenial law, we get

1−Pd= (1−L(λ))X

k≥1

E

"

e^−λDk Z D_k

0

e^λtnµ(dt)

#

−L(λ)nµ({0}), (4)

E[X^r] =E[Y^s] + 1

λP V^s≥Z^d

=E[Y^s] + 1

λPd. (5)

Proof. By independence, one gets (1) : E[K^r] = 1 +X

n≥1

E[1D_n<Y^s] = 1 +X

n≥1

E[Rs(Dn)] =X

n≥0

E[Rs(Dn)]. Let us prove (2). By independence, we have

Pd=P

D_B(Ys)−Y^s≥Y^d

= Z +∞

0

P

D_B(t)−t≥Y^d

nµ(dt) = Z +∞

0

E h

Fd

D_B(t)−ti nµ(dt), and the result follows. IfY^d follows an exponential law of parameter λ,

1−Pd = Z +∞

0

E

e^{−λ DB(t)−t} nµ(dt) = Z +∞

0

X

k≥1

E h

e^{−λ(Dk−t)}1_B(t)=ki

nµ(dt),

= Z +∞

0

X

k≥1

E h

e^{−λ(Dk−t)}1D_k−1<t≤Dk

i

nµ(dt),

=X

k≥1

E Z +∞

0

e^{−λ(Dk−t)} 1_t≤Dk−1_t≤Dk−1 nµ(dt)

,

=X

k≥1

E

e^−λDk Z +∞

0

e^λt1_t≤Dknµ(dt)−e^−λCke^−λDk−1 Z +∞

0

e^λt1_t≤Dk−1nµ(dt)

.

By independence, 1−P_d =X

k≥1

E

e^−λDk

Z +∞

0

e^λt1_t≤Dkn_µ(dt)

−L(λ)E

e^−λDk−1 Z +∞

0

e^λt1_t≤Dk−1n_µ(dt)

,

and (4) follows.

In order to prove (3), we compute the survival function of X^r. P(X^r> x) =P min V^s, Z^d

> x

=P

D_B(Ys)> x, Y^s+Y^d> x ,

=P(Y^s> x) +P

D_B(Ys)> x, Y^s+Y^d> x, Y^s≤x ,

=Rs(x) +E h

1D

B(Y s)>x1Y^s≤xRd(x−Y^s)i ,

=Rs(x) + Z +∞

0

P

D_B(t)−t > x−t

R^d(x−t)1x−t>0nµ(dt).

Then,

E[X^r] = Z +∞

0

P(X^r> x)dx

=E[Y^s] + Z +∞

0

Z +∞

0

P

D_B(t)−t > x−t

R^d(x−t)1_x−t>0nµ(dt)dx,

=E[Y^s] + Z +∞

0

Z +∞

0

P

D_B(t)−t > x−t

R^d(x−t)1_x−t>0dx nµ(dt),

and the change of variabley=x−tgives the result. IfY^dfollows the exponential law with density fd,Rd(y) = ^fd_λ^(y), and (5) follows.

3 Renewal Theory

3.1 Renewal and Renewal Reward processes

In this section, we recall classical results on renewal processes. We refer to [CT97], [Asm03] [AJ99, Appendix B] among others. Let (Xn)n≥0 be a sequence of nonnegative independent identically distributed (i.i.d.) r.v. with distribution F. We assume thatP(X = 0)<1. We also define the sequence (T_n)_n

T₀= 0, T_n=

n

X

i=1

X_i, n∈N^∗

and

N_t= sup{j:t_j≤t}=

∞

X

i=1

1_Tj≤t

(N_t)_t≥0is a renewal process, and we have

Theorem 3.1 (Almost sure convergence, L¹ convergence and CLT forN_t). We have

t→∞lim Nt

t = 1

E(X)a.s. and inL¹. Assume that V(X)<∞. Then,Nt satisfies the following CLT

t→∞lim

√ t

Nt

t − 1 E(X)

law

= N

0, V(X) E(X)³

.

Let (Xn, Yn)n≥1 denote a sequence of i.i.d. pairs of r.v.. Yi can be interpreted as the reward associated with thej^thinterarrival time Xj (Yj may depend onXj). The processZt defined by

Zt=

N_t

X

i=1

Yi

satisfies the following theorem

Theorem 3.2 (Almost sure convergence, L¹ convergence and CLT forZt). We have

t→∞lim Z_t

t = E(Y)

E(X) almost surely and inL¹.

Assume that V(Y)<∞andV(X)<∞. Then,Z_t satisfies the following CLT

t→∞lim

√ t

Zt

t − E(Y) E(X)

law

= N



0,V(Y − ^E(Y)

E(X)X) E(X)²



.

The second part of Theorem 3.2 ensues from Rényi’s theorem, recalled below, which will be useful in the following.

Theorem 3.3 ([Rén57], Theorem 1). Let (ξn)_n≥1 be a sequence of i.i.d. square integrable r.v.

such that σ² = E[ξ²]. Let (νt)t≥0 be a process taking values in N such that νt/t converges in probability whent→ ∞to a constant c >0. Then

Pν_t i=1ξi

√ν_t −→ N(0, σ²), Pν_t

i=1ξi

√t −→ N(0, c σ²).

3.2 Age and Survival of a renewal process

Let (At, St)_t≥0 denotes the age and the survival of a renewal process : At :=t−TN_t andSt :=

TN_t+1−t. The following result gives the limiting laws of both processes :

Proposition 3.4. Let us consider a renewal process with interarrival timeX such thatE(X)<∞.

Then

t→∞lim P(A_t≤x) = lim

t→∞P(S_t≤x) = 1 E[X]

Z x 0

P(X > u)du, and their densities are given bypA∞(x) =pS∞(x) = 1

E[X]P(X > x).

Lemma 3.5. The process (^B(A√_t^t))t≥0 converges in probability to0 whent tends to infinity.

Proof. Let us first prove that (^√At_t)t≥0converges in probability to 0 whenttends to infinity. Indeed, by using the Markov inequality, we get ∀ε <1,P

√At

t > ε

=P A√t

t∧1> ε

≤ ^E _At

√t∧1

ε . Since the age of a renewal process converges in law whent→+∞(see Proposition 3.4), we get

∀a >0, lim sup

t→∞ E

At

√t ∧1

≤lim sup

t→∞ E

At

√a∧1

=E A∞

√a ∧1

≤E[A∞]

√a .

Then, ∀ε <1,∀a >0, lim sup_t→∞P _A

√t

t > ε

≤ ^E_ε^[A√∞_a^] and the result follows. Let us now prove the Lemma. ∀ε <1,

P

B(At)

√t > ε

=P

B(A_t)> ε√ t

=P





dε√ te

X

i=1

C_i≤A_t



=P



 ε ε√

t

dε√ te

X

i=1

C_i≤ At

√t



.

Letη >0. It remains to split the last probability in two parts by introducing the set{

1 ε√

t

Pdε√ te

i=1 Ci−E(C) >

η} and its complement. The strong law of large numbers and the convergence in probability of (^A√t

t)_t≥0 end the proof.

4 Main Results

4.1 Number of repairs, number of failures and number of inspections

Definition 4.1. Since the system is repaired as good as new at each repairing date, the number of repairs at time t is a renewal process given by

N_t^r=

∞

X

i=1

1_T^r

i≤t,

whereT_i^r=Pi j=1X_j^r.

Definition 4.2. LetX^f be the r.v. representing the time between two consecutive failures : X^{f law}=

τ

X

i=1

Xi, whereτ= inf

i≥1 :V_i^s≥Z_i^d inf∅= +∞.

The number of failures at time t is given by N_t^f =

∞

X

i=1

1_Tf i≤t,

whereT_i^f =Pi

j=1X_j^f. Then,N_t^f is a renewal process with interarrival time X^f. Remark 4.3. N_t^f is also a renewal reward process, sinceN_t^f =PN_t^r

i=11_Vs i≥Z_i^d.

Remark 4.4. The number of damages N^d is a renewal process with interarrival time X^{d law}= Pσ

i=1X_i, where σ= inf

i≥1 :Z_i^d> V_i^s . As forN^f, we can writeN_t^d=P^Nt^r

i=11_Zd i≥V_i^s. Definition 4.5. The number of inspections at timet, denoted N_tⁱ, is given by

N_tⁱ =

N_t^r

X

i=1

K_i^r+B(t−T_N^r

t),

whereK_i^r denotes the number of inspections on the interval]T_i−1^r , T_i^r].

Theorem 4.6(Almost sure and L¹ convergences for ^N_t^tr, ^N

f t

t and ^N_t^ti). The following results hold almost surely and inL¹

t→+∞lim N_t^r

t = 1 mx

, lim

t→+∞

N_t^f t = P_d

mx

, and lim

t→+∞

N_tⁱ t = m_k

mx

. (6)

Proof. The first and second results ensue from Theorem 3.1 and from Wald’s identity, since E(X^f) = E(τ)E(X^r) and τ has a geometric law of parameter P_d. From the definition of N_tⁱ, we getP^N_t^r

i=1Ki≤N_tⁱ≤P^N_t^r+1

i=1 Ki. Then, 1

N_t^r

N_t^r

X

i=1

K_i^r≤ N_tⁱ

N_t^r ≤N_t^r+ 1 N_t^r

1 N_t^r+ 1

N_t^r+1

X

i=1

K_i^r.

Since lim_t→∞N_t^r = ∞ almost surely, we get lim_t→∞^N_N^trⁱ t

= E(K^r). The strong law of large numbers for renewal processes yields the almost sure convergence of ^N_t^ti.

Let us now deal with the L¹-convergence. We have E[

N_t^r

X

i=1

K_i^r]≤E[N_tⁱ]≤E[

N_t^r+1

X

i=1

K_i^r]. (7)

We first deal with the right hand side. N_t^r+ 1 is an F-stopping time ({N_t^r+ 1 =k} ={N_t^r = k−1}={T_k−1^r ≤t < T_k^r}), then

E N_tⁱ

≤E





N_t^r+1

X

k=1

K_i^r



=E[N_t^r+ 1]E[K_i^r]. (8) Concerning the left hand side, we write

E





N_t^r

X

k=1

K_i^r



=E



 X

k≥1

K_i^r1T_k≤t



=X

k≥1

E h

K_i^r1T_k^r≤t

i .

By symetry, we getE hP^Nt^r

k=1K_i^ri

=E h

K₁^rP

k≥11T_k≤t

i

=E[K₁^rN_t^r]. Combining this result with (7) and (8), we obtain

E[K₁^rN_t^r]≤E N_tⁱ

≤E[N_t^r+ 1]E[K^r].

Since lim_t→∞N_t^r/t −→ 1/E[X^r], Fatou’s lemma gives lim inf_t→+∞¹_tE[K₁^rN_t^r] ≥ ^E[Kr]

E[X^r]. The elementary renewal theorem gives lim_t→+∞¹_tE[N_t^r+ 1]E[K^r] = ^E[Kr]

E[X^r] and the last result follows.

Theorem 4.7 (CLT for ^N_t^tr, ^N

f t

t and ^N_t^ti). Let us denoteX :=X^r,I:=1_Vs≥Z^d,K:=K^r. The following result holds

√ t N_t^r

t − 1 mx

,N_t^f t − P_d

mx

,N_tⁱ t −m_k

mx

!

−→ N(0, R), where

R= (mx)⁻³





V[X] Cov[X, PdX−mxI] Cov[X, mkX−mxK]

Cov[X, PdX−mxI] V[PdX−mxI] Cov[PdX−mxI, mkX−mxK]

Cov[X, mkX−mxK] Cov[PdX−mxI, mkX−mxK] V[mkX−mxK]



.

Proof. Let (Xn, Yn, Zn) be a sequence of i.i.d. square integrable r.v. in R³+. One denotes λx = E[X], λy = E[Y] and λz = E[Z]. We denote S_n^x = X1+. . .+Xn, S^y_n = Y1+. . .+Yn, S_n^z=Z1+. . .+Zn andN_t^x= sup{n≥1 :S^x_n≥t}.

We consider the triplet Qn = (S_n^x−nλx, λxS_n^y−λyS_n^x, λxS_n^z−λzS_n^x). CLT gives us that

√1

nQn−→ N(0, V) where V =





V[X] Cov[X, λxY −λyX] Cov[X, λxZ−λzX] Cov[X, λxY −λyX] V[λxY −λyX] Cov[λxY −λyX, λxZ−λzX] Cov[X, λxZ−λzX] Cov[λxY −λyX, λxZ−λzX] V[λxZ−λzX]





By applying Rényi’s theorem (see Theorem 4.6) to the real r.v. u·QN_t^x/p

N_t^x we obtain 1

pN_t^xQN_t^x −→ N(0, V), and 1

√tQN_t^x−→ N(0, λ⁻¹_x V).

LetA^x_t denote the age of the renewal processN^x, i.e. A^x_t :=t−S_N^xx

t. We have QN_t^x =

S_N^xx

t −λxN_t^x, λxS^y_Nx

t −λyS_N^xx

t, λxS_N^zx

t −λzS_N^xx t

=

t−λxN_t^x, λxS_N^yx

t −λyt, λx

S_N^zx

t +B(A^x_t)

−λzt

+ (1, λy, λz)A^x_t −(0,0, λx)B(A^x_t).

Combining Lemma 3.5 and Slutsky’s Lemma yields

√1 t

t−λxN_t^x, λxS_N^yx

t −λyt, λx

S^z_Nx

t +B(A^x_t)

−λzt

−→ N(0, λ⁻¹_x V) Combining this result and the application (x, y, z)7−→(−x, y, z)/λxgives

√1 t

N_t^x− t λ_x, S_N^yx

t −λyt λ_x, S_N^zx

t +B(A^x_t)−λzt λ_x

−→ N(0, R), where

R=λ⁻³_x





V[X] Cov[X, λyX−λxY] Cov[X, λzX−λxZ] Cov[X, λyX−λxY] V[λyX−λxY] Cov[λyX−λxY, λzX−λxZ]

Cov[X, λzX−λxZ] Cov[λyX−λxY, λzX−λxZ] V[λzX−λxZ]



.

To conclude, it remains to chooseX =X^r,Y =1_Vs>Z^d andZ=K^r. We getS_N^yx

t =N_t^f and S_N^zx

t +b^A_c^xtc=N_tⁱwhich gives the result.

4.2 Estimation of the parameters

Theorem 4.6 gives us the almost sure convergence of_Nr t

t ,^N

f t

t ,^N_t^ti

. Combining these limits yields

t→∞lim N_tⁱ

N_t^r =m_k, lim

t→∞

N_t^f N_t^r =P_d.

mk depends only on µ. Then, if mk = f(µ), where f is a continuous and strictly monotone function, we get that µ = f⁻¹(mk). Pd depends on µ and λ. Then, if Pd = g(µ, λ), where λ7−→g(µ, λ) is a continuous and strictly monotone function for allµ, we get thatλ=g_µ⁻¹(µ, Pd) (g⁻¹_µ denotes the inverse ofλ7−→g(µ, λ)).

Lemma 4.8. Assume thatg is aC¹function fromU×V to[0,1], strictly monotone inλfor all µ. Then,g_µ⁻¹, the inverse ofλ7−→g(µ, λ), isC¹ fromU×[0,1]toV.

Proof. For allµandλ, we haveg⁻¹_µ (µ, g(µ, λ)) =λ. By differentiating this equality w.r.t. λ, we get

∂_yg_µ⁻¹(µ, g(µ, λ))∂_yg(µ, λ) = 1. Since for all µ g is C¹ from U ×V to [0,1] and strictly mono- tone in λ, we have ∂_yg_µ⁻¹(µ, p) = ¹

∂_yg(µ,g_µ⁻¹(µ,p)) for all (µ, p) ∈ U ×[0,1]. By differentiating g_µ⁻¹(µ, g(µ, λ)) = λ w.r.t. µ, we get ∂_xg_µ⁻¹(µ, g(µ, λ)) + ∂_yg_µ⁻¹(µ, g(µ, λ))∂_xg(µ, λ) = 0, i.e.

∂_xg⁻¹_µ (µ, p) =−∂yg⁻¹_µ (µ, p)∂_xg(µ, g⁻¹_µ (µ, p)) for all (µ, p)∈U×[0,1].

Let us introduce

µt:=f⁻¹ N_tⁱ

N_t^r

, λt:=g_µ⁻¹ µt,N_t^f N_t^r

!

(9) The following Theorem gives a CLT for√

t(µ_t−µ, λ_t−λ).

Theorem 4.9. Assume thatmk =f(µ),mx=h(µ, λ)andPd=g(µ, λ), wheref is aC¹ strictly monotone function fromU toR+, and g is aC¹function fromU×V to[0,1], strictly monotone inλfor all µ. Let(µt, λt)be defined by (9). It holds

t→∞lim

√

t(µt−µ, λt−λ)^law= N(0,Σ²) whereΣ²=ARA^T,R is given in Theorem 4.7 and

A= h(µ, λ) f⁰(µ)

−f(µ) 1 0

f(µ)^∂_∂^µg(µ,λ)

λg(µ,λ)−g(µ, λ)_∂^f0(µ)

λg(µ,λ) −^∂_∂^µg(µ,λ)

λg(µ,λ)

f⁰(µ) h(µ,λ)∂_λg(µ,λ)

! .

Remark 4.10. If the survival function of Y^s is strictly monotone in µ, f(µ) (defined by (1)) is strictly monotone. By using (2), we get thatg(µ, λ) =R∞

0 E(Fd(D_B(t)−t))nµ(dt). Then, If the survival function ofY^d is strictly monotone inλ,λ7−→g(µ, λ) is strictly monotone.

Proof. Let us first considerµt−µ. A Taylor expansion gives µ_t−µ=f⁻¹

N_tⁱ N_t^r

−f⁻¹(m_k) = N_tⁱ

N_t^r −m_k

(f⁻¹)⁰(ζ_t), (10)