The objective of this section is to construct empirical estimators for two failure rate functions of a discrete time semi-Markov system.

Firstly, we give a method for computing the reliability of a discrete time semi-Markov system and we propose an empirical estimator.

Let E be partitioned into two subsets U and D, respectively for the up states and for the down states, where E = U U D and U n D = %. Without loss of generality, we can suppose that

U = {1,..., .s} and D = {s1 + 1, ...,s}, with 0 < si < s.

We will partition all vectors and matrix-valued functions according to this partition. For instance, the transition matrix P of the semi-Markov process and the initial distribution vector a can be written as follows:

For m,n G N* such that m > n, let 1m_n denote the m-column vector whose first n elements are 1 and the last m - n elements are 0; for m G N*, let 1m denote the m-column vector whose all elements are 1.

Let Td denotes the first passage time in the subset D, i.e., Td := inf{n G N; Zn G D}. The reliability at time k G N is given by

We define a new semi-Markov process Y = (Yn; n G N) with state space Ey = U U {A}, where A is an absorbing state

_ i Zn if n <Td Yn = < n ., ^ , n G N. A if n > t d

The semi-Markov kernel of the process Y is qY (k) = \qu(k) qi2(k) h-r.

, k e N, where 0im is an m-dimensional row vector whose all elements are 0. Let Py denote the transition function of Y. Thus, for all k e N, the reliability is given by

R(k) = P(Yfc e U ) = EE P(Yfc = j | Yo = i) P(Yo = i)

jeuieu

= [ai, 0] Py (k) 1m+i,m = ai • Pii(k) • 1Sl = ai ^ii * (I - Hi)(k) 1Si = ai ^ii * (I - diag(Q • 1)ii)1Si.

We propose the following estimator for the system reliability:

= ai [¿a(•, M) * (i - diag(Q(;M) • 1)i^] (k)1Si, (22)

where the estimators ip and Q are defined in (18).

All the results which follow are proved under Assumptions (1), (2) and

Theorem 1. ( see [BL04a]) The estimator of the reliability of a discrete time semi-Markov system is strongly consistent, i.e., max | R(k,M) - R(k) |—^ 0.

Let us denote by T the discrete random variable describing the lifetime of the system. We consider two different definitions of the failure rate function. • BMP-failure rate function A(k)

It is the usual failure rate, defined by [BMP63] as the conditional probability that the failure of the system occurs at time k, given that the system has worked until time k - 1 (see also [BP75] and [SL02] in the case of Markov chains).

It is worth noticing that the failure rate in discrete case is a probability function and not a general positive function as in continuous time case. For any k > 1

A(k) := P(T = k | T > k) ^ WW), R(k - ^ 1 - R—), R(k - l)={)

I 0, otherwise. I 0, otherwise and

A new definition of the discrete failure rate function is proposed in [RG92] for solving some of the problems raised by the use of the usual failure rate function A(k) in discrete time. A detailed argument for the introduction of the new definition of the failure rate function is given in [Bra01, BGX01].

( in R(k-i) k> i r(k) :=l ln R(k) , k > 1 . V ' ln R(0), k = 0

The two failure rate functions are related by r(k) = - ln(1 - A(k)).

We propose the following estimators for the failure rates:

I 0, otherwise

The following results concern the uniform strong consistence and the asymptotic normality of the empirical estimators of the failure rates.

Theorem 2. The estimators of the failure rates of a discrete time semi-Markov system are strongly consistent, in the sense that max I X(k,M) - X(k) ^ 0

Notation: For a matrix function A e ME(N), we denote by A+ e Me(N) the matrix function defined by A+(k) := A(k + 1), k e N.

Theorem 3. For any fixed k e N we have the following convergence in distribution

ai (k) = R2(k)J2 D - l{iEU} ^ a(t)Vti] * qij (k - 1)

+R2(k - 1)J2 D - 1{ieu} E a(t)*ti] * qij (k) - T2(k)

+2R(k - 1)R(k)Y, [l{ieu }DjY, a(tWi + 1{ieu }DUj )+Y. a(t)*t>

j=i teu teu

- Dj)+Du - 1{ieu}(£ aWti) (E a(t)*+)] * qij(k - 1)

teu teu where s

Ti(k) :=Y, [R(k)Du * qij (k - 1) - R(k - 1)Du * qij(k) j=1

- R(k)l{ieu a(t)^ti * Qij (k - 1) + R(k - 1)l{ie_u } ^&(t)^ti * Qij(k)

teu teu

Dij a(n)^ni * ^jr * (j - diag(Q • l)) , neureu is the mean recurrence time of state i for the embedded Markov chain ((Jn); n e N) and pii is the mean recurrence time of the state i for the DTMRP (J,S).

Corollary 1. For any fixed k e N, \/M[r(k,M) - r(k)] converges in distribution to a zero mean normal random variable with variance ar(k) = (1 - A(k))2 al (k) = R2(k -1)R2(k) al(k), where a2(k) is given in Equation (23).

Asymptotic Confidence Intervals for Failure Rates

Let us now give the asymptotic confidence intervals for failure rates, using the above asymptotic results.

For k G N,k < M, replacing q(k),Q(k),^(k),&(k) respectevely by q(k,M), Q(k,M), $(k,M), <P(k,M) in Equation (23), we obtain an estimator a2(k) of the variance a2(k). From the strong consistency of the estimators q(k,M), Q(k, M), $(k,M) and $(k,M) (see [BL04b]), we obtain that a2x(k) converges almost surely to a^ (k), as M tends to infinity.

For k G N,k < M, the estimated asymptotic confidence interval of BMP-failure rate function X(k) at level 100(1 — 7)%, 7 G (0,1), is given by

X(k,M) — u-/ < X(k) < X(k,M) + ui-Y/2 , (24)

where uY is the 7 — quantile of an N(0,1)- distributed variable. In the same way, we obtain the asymptotic confidence interval of RG-failure rate function.

4 Proofs

In order to prove the above results, we need the following lemmas.

Lemma 2. Let A G Me (N) be a matrix function and let q be the semi-Markov kernel of the DTMRP (J,S). For any fixed k G N, i,j, l and r G E, we have

1 Aij (k — x), x < k j 1{x= })(k) = ^ otherwise.

Lemma 3. Let A,B G Me(N) be two matrix functions. For any fixed k,y G N, i,j, l, r, u and v G E, we have

Z.^jAuv * 1{x< }(k)qij(x) = A uv * Qij (k>) .

* 1{x=-}(k)Bir * 1{x<}(k)qij (x) = A uv\ )£ Bir (t) * qij(k).

Lemma 4. Let A,B G Me (N) be two matrix functions. For any fixed k G N, i, j, l, r, u and v G E, we have

1. E (Auv * 1{x=})(k)(Bir * 1{x=.})(k - l)qij (x) = (A+v Bir) * qij (k - 1), x=0 tt

3. E (A uv * 1{x=})(k — 1)(Bir * 1{x<})(k)qij (x)

In the sequel, we give the proofs of Theorem 2, 3 and Corollary 1. All along this section, we will use the notation Aqij(k,M) := qj (k,M) -qij(k), APij(k,M) := Pij(k,M) - Pij(k), etc. We will also omit the censoring time M as an argument of the estimators; for instance, we write qij(k) instead of qj (k, M). Proof of Theorem 2. We have max I q(k, M) - X(k) |

0<k<M R(k - 1, M) 0<k<M R(k - 1) R(k - 1, M)

>From the uniform strong consistency of the reliability estimator we conclude that the right-hand side term converges almost surely to zero, when M tends to infinity.

Using the relation between the BMP-failure rate and the RG-failure rate r(k) = — ln(1 — A(k)), we infer the consistency of the RG-failure rate. □

Proof of Theorem 3.

vM [\(k,M) — A(k)] _ VM[R(k)(R(k — 1, M) — R(k — 1)) — (R(k, M) — R(k))R(k — 1)]

>From the consistency of the reliability estimator (see Theorem 1), we have R(k — 1, M) : R(k — 1), so, in order to obtain the asymptotic normality

for the BMP-failure rate, we need only to prove that

V^\R(k)(R(k—1, M)—R(k—1)] — (R(k, M)—R(k)] R(k—1)} V : N(0, a\(k)).

We obtain that

VM R(k)(R(k — 1, M) — R(k — 1)^ — (li(k,M) — R(k)j R(k — 1) has the same limit in distribution as s s

VMY,R(k)DUi * Aqn(k — 1) — VM^Y<R(k)(Y< a(n)^nr) * AQ rl(k — ^ l,r = 1 rEU l = 1 nEU

—VmY, R(k — 1)DUl * Aqrl(k) + VM^Y,R(k — 1)^2 a(n)^nr) * AQrl(k)

^ WW) [R(k)DUl * [1{Jn-l=r,Jn = l,Xn = } — qrl(-)l{Jn-1=r^ (k — 1)

— R(k)1{rEU a(t)^t^ * (1{Jn-i=r,Jn = l,Xn< } — Qrl (')1 {J(-i=r^) (k — ^

— R(k — 1)DUl * (1{.Jn-i=r,.Jn=l,X(= } — qrl(-)1{Jn-i=r}) (k)

— R(k — 1)1{rEUa(t)'^ * (1{J(-i=r,J(=l,X(<■} — Qrl(')1{J(-i=r^j (k)

tr tEU

Vm where we have defined the function f : E x E x N ^ R by f (i,j,x)

We will obtain the desired result from the Central limit theorem for discrete time Markov renewal processes (see [PS64] and [MP68]). Using Lemmas 2, 3 and 4, we obtain:

+R(k - 1)1{ieu} (y^ a(t)^ti) * qij(k) - Ti(k)pij teu

>From Lemmas 3 and 4 we get

(nm) 2{ R2 (k)(DUj )2 * qij (k -1) + R2(k - 1)(DU )2 * qij (k)

1=0 teU

+R2(k - 1)1{ieu}(£ Y,a(t)Uti(1)) 2 * qij (k) + T2(k)pij

1=0 teu

1=0 teu

+2R(k - 1)R(k) [DUj(■)1{ieu}(E E a(t)Uti(l)) + * qij(k - 1)

+2R(k - 1)R(k) (DUj)+(^)1{ieu}(E $>(t)V«(0) * qij(k - 1)

1=0 teU

-2R2(k - 1) [DUj(■)1{ieU}(E Y^a(t)Uti(l))\ * qij(k)

1=0 teU

-2R(k - 1)R(k)1{ieU} [(E E a(t)Uti(l)) (E E a(t)Uti(l)) + * qij(k - 1)

-2Ti(k) [R(k)DUj * qij(k - 1) - R(k - 1)DU * qij(k)

+R2(k - 1)J2 [DUj - 1{ieu} E °<t)*ti] * qij (k) - Ti (k)

+2R(k - 1)R(k)J2 [l{ieu}DjJ2 a(t)*+ + 1{ieu} (DUa(t)Wti j=i teu teu

teu teu

Since Ni(M)/M a's'—► 1/pii (see, e.g., [LOOl] ), applying the central limit

theorem, we obtain the desired result. □

Proof of Corollary 1. The relation between the BMP-failure rate and the RG-failure rate can be written in the form r(k) = ^(\(k)), where ^ is the function defined by $(x) := - ln(1 - x). Using delta method and the asymptotic normality of the BMP-failure rate (see Theorem 3), we obtain that the RG-failure rate converges in distribution to a zero mean normal random variable with the variance ^(k), given by

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