IJQSR International Journal of Quality, Statistics, and Reliability 1687-7152 1687-7144 Hindawi Publishing Corporation 675830 10.1155/2012/675830 675830 Research Article Uniqueness of Maximum Likelihood Estimators for a Backup System in a Condition-Based Maintenance Duan Qihong Wei Ying Chen Xiang Dohi Tadashi 1 Department of Statistics School of Mathematics and Statistics Xi’an Jiaotong University Shaanxi, Xi’an 710049 China xjtu.edu.cn 2012 28 11 2012 2012 14 05 2012 09 11 2012 09 11 2012 2012 Copyright © 2012 Qihong Duan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A parameter estimation problem for a backup system in a condition-based maintenance is considered. We model a backup system by a hidden, three-state continuous time Markov process. Data are obtained through condition monitoring at discrete time points. Maximum likelihood estimates of the model parameters are obtained using the EM algorithm. We establish conditions under which there is no more than one limitation in the parameter space for any sequence derived by the EM algorithm.

1. Introduction

Suppose a backup system is represented by a continuous time homogeneous Markov chain X={Xt:t0} with a state space S={0,1,2}. States 0, 1, and 2 are the healthy state, unhealthy state, and the failure state, respectively. Assume that the system is in a healthy state at time 0, and the transition rate matrix is given by (1)Λ=(-1/θ1,1/θ1,00,-1/θ2,1/θ20,0,0), where θi(α,+) for i=1,2 are unknown. Here α>0 is a known extreme edge of the parameters. Suppose the system is observed at time points 0,Δ,2Δ,, where [kΔ,(k+1)Δ] is a benchmark interval. While the system is failed at an inspection, a new system replaces it. Let two processes Y={Y(k):k=1,,n}, and let R={R(k):k=1,,n} be a record of the system. Here R(k)=1 if the system is failed during (k-1,k), and R(k)=0 otherwise. And Y(k)=XkΔ. As a path of X is a stepped right-continuous function, Y(k)=0 when a replacement occurs at time kΔ. Moreover, we set Y(0)=0 for convenience. The process R represents the replacement of the system, and the process Y represents observable information of the system collected through condition monitoring.

The maximum likelihood estimates (MLE) of the model parameters for such models have been studied by [1, 2]. As the stochastic processes, Y,R are not Markov processes and the sample path of X is not observable, the likelihood function of incomplete data (Y,R) is complex. Hence, it is difficult to obtain directly the MLE θ^=argmaxθΘlnL(θY,R), where θ=(θ1,θ2) and Θ={(x,y):x,y>β}. Here β>0 is a prearranged constant. Both [1, 2] suggest the EM algorithm (see e.g., [3, 4]). Let θ(0) be initial values of the unknown parameters. The EM algorithm works as follows.

The E step. For n>0, compute the following pseudo-likelihood function: (2)Q(θθ(n))=Eθ(n)(lnL(θD)Y,R). Here D is the complete data set of the process X. The forms of the complete data set may be different for different purpose. For example, the forms are different in [1, 2].

The M step. Choose θ(n+1)=M~(θ(n)). Here (3)M~(θ)=argsupθΘQ(θθ). The E and M steps are repeated. According to the theory of EM algorithms (Theorem  1 in ), L(θ(n+1)Y,R)L(θ(n)Y,R) for any given initial value θ(0) and n=0,1,. It is clear that if an MLE θ^ in Θ is one of these fixed-points when it exists.

In this paper, we consider the uniqueness of the MLE θ^. As the likelihood function lnL(θY,R) of incomplete data is complex, we do not follow the classical method by which the uniqueness of a MLE is demonstrated by establishing the global concavity of the log-likelihood function (see e.g., .) Alternatively, we investigate conditions under which the operator M~ is a contraction. The conditions ensure that the MLE is unique if it exists. Moreover, the conditions implies that there is not more than one limitations in Θ for different sequences derived by the EM algorithm. For the complete data set we present in the next section, we have the following main theorem of this paper.

Theorem 1.

There is not more than one fixed-point of the operator M~ in Θ provided that <(2/2)α and the record {Y,R} has at least one replacement.

2. Complete Data Set

To establish the expression of operators Q, we present our construction of the path of X and the complete data set D of the process X. Suppose that random variables Tki, i=1,2, k=0,1, are independent, Tki has an exponential distribution, and ETki=θi/Δ. As every state of X has an exponential duration distribution, we may construct a path of X through the approach introduced by Theorem  5.4 in . The path of X restricted on t[0,Δ) has the following form: (4)Xt10t<Δ=jforΔi=1jT1it<Δ(i=1j+1T1i1),j=0,1, and Xt10tΔ=2 for Δ(T11+T12)t<Δ. If m=limtkΔXt2 for k1, we can construct the following path of X restricted on t[kΔ,(k+1). Consider (5)Xt1kt<k+1={j,Δi=mjTkit-kΔ<Δ(i=mj+1Tki1),j2,2,Δi=m2Tkit-kΔ<Δ. If limtkΔXt=2, that is, the system is failed during (kΔ,(k+1)Δ), a new system replaces it at time (k+1)Δ and a path of X restricted on t[kΔ,(k+1)Δ) is presented by setting m=1 in (5). In this paper, the complete data set D={Tki:i=1,2,k=0,1,,n-1}.

Our choice of the complete data set D ensures a simple form of the operator Q. The log-likelihood function has the following form: (6)lnL(θD~)=2nlnΔ-n(lnθ1+lnθ2)-Δk=0n-1(Tk1θ1+Tk2θ2). And the Markovian property of the process X implies that (7)Eθ(TkiY)=Eθ(TkiY(k-1),Y(k),R(k)), where Y(0)=0.

In Table 1, we denote by ni the number of different values of the triple (Y(k-1),Y(k),R(k)). For example, n1 is the number of triple (Y(k-1),Y(k),R(k))=(0,0,0). It follows from (6) that (8)Q(θθ)=2nlnΔ-n(lnθ1+lnθ2)-Δi=15ni(Mi1θ1+Mi2θ2).

The number for values of (Y(k-1),Y(k),R(k)).

 Values ( 0,0 , 0 ) ( 0,1 , 0 ) ( 1,1 , 0 ) ( 0,0 , 1 ) ( 1,0 , 1 ) Number n 1 n 2 n 3 n 4 n 5

Here the following forms of Mim for m=1,2 follows from (7) and Table 1. Consider (9)M1m=Eθ(T1mT111),(10)M2m=Eθ(T1mT121),(11)M3m=Eθ(T1mT12<1),(12)M4m=Eθ(T1mT11<1,T11+T121),(13)M5m=Eθ(T1mT11+T12<1).

For any given θ, it is obvious that the function Q~(λ1,λ2)=Q(1/λ1,1/λ2θ) is a concave function for 0<λ1,λ2<. Hence there is a unique vector θ'=M(θ) satisfying (14)Q(θθ)θi=0,i=1,2. Moreover, it follows from the definition (3) that if M~(θ)Θ, M(θ)=M~(θ). Therefore, every fixed-point of M~ in Θ is a fixed-point of M and vice versa. In this paper, we will prove Theorem 1 through studying the number of fixed-points of M in Θ. Here we present the from of M(θ)=(M1(θ),M2(θ)) derived by (14). Consider (15)Mm(θ)=Δni=15niMim,m=1,2.

3. Two Lemmas Lemma 2.

Consider the following: (16)|M4mθi|Δθi2,i=1,2,m=1,2.

Proof.

Writing σ={(x1,x2)x1,x20,x1+x21} and (17)V=σΔ2xmθ1θ2e(-Δ/θ1)x1-(Δ/θ2)x2dx1dx2,D=σΔ2θ1θ2e(-Δ2/θ1)x1-(Δ/θ2)x2dx1dx2, we have M4m=V/D. Hence, (18)M4mθi=M4m(V-1Vθi-D-1Dθi).

As 0xm1 on σ, we have that 0VD and 0M4m1. Therefore, (19)|M4mθi||(V-1Vθi-D-1Dθi)|. Moreover, we have (20)Vθi=-Vθi+Δ3θi2σxmxiθ1θ2e-(Δ/θ1)x1-(Δ/θ2)x2dx1dx2. As 0xi1 on the region σ, it follows from the definition to V that there is 0<η1<1 such that (21)Vθi=-Vθi+η1ΔVθi2. Similarly, there is η2(0,1) such that (22)Dθi=-Dθi+η2ΔDθi2. The result follows from (19), (20), and (22).

Lemma 3.

Let a function A:RR be A(0)=1/2 and A(x)=1/(1-ex)+1/x for x0. We have 0A'(x)-1/12 for any xR.

Proof.

By a routine analysis, we may obtain that A'(0)=-1/12 and for x0, (23)A(x)=(e-x/2-ex/2)-2-x-2. Moreover, we may obtain that for x>0, (24)A′′(x)=A2(x)x-3(ex-1)-3, where A2(x)=(ex-1)3-x3ex(1+ex). We have (25)A2(x)=-exA3(x), where A3(x)=3x2(1+ex)+x3(1+2ex)-6(ex-1)2. A routine analysis may confirm that A3(4)(x)=exA4(x), where A4(x)=96+96x+27x2+2x3-96ex. Then it follows from ex1+x+x2/2+x3/6 that A4(x)0. Moreover, as A3(0)=A3(0)=A3(0)=A3(3)(0)=0, we have that for any x>0, there is ξ[0,x] such that (26)A3(x)=124A3(4)(ξ)x4=124A4(ξ)eξx40. Then it follows from A2(0)=0 and (25) that A2(x)0. Hence, it follows from (24) that A′′(x)0 for any x>0. Therefore, for any x>0(27)-112=A(0)A(x)limx+A(x)=0. The result follows from the above formula, and the fact that A'(x) is an even function.

4. Proof of the Main Theorem

We may derive that M(θ) is a contraction by investigating the Jacobian matrix (M1,M2)/(θ1,θ2).

By a routine analysis, it follows from (12) that (28)M21=1+A(Δθ1-Δθ2),M22=1+θ2Δ+A(Δθ1-Δθ2). Hence, it follows from Lemma 3 that (29)|M21θ2|=Δθ22|A'(Δθ1-Δθ2)|Δ12θ22. Similarly, we have (30)|M21θ1|Δ12θ12. As Δ<α<θ2, it follows from (28) and Lemma 3 that (31)|M22θ1|+|M22θ2|=1Δ.

It follows from (11) that M52=A(θ2/Δ), and M51=θ1/Δ. Therefore, (32)M52θ1=0,M51θ1=0,M51θ2=1Δ. From the definition (11), we have that M52=A(Δ/θ2). Then, it follows from Lemma 3 that (33)|M52θ2|Δ12θ22.

It follows from (9) and (10) that M11=1+θ1/Δ, M12=θ2/Δ, and hence for m=1,2, (34)|M1mθ1|+|M1mθ2|1Δ,|M3mθ1|+|M3mθ2|1Δ.

Write S=max{S1,S2}<1, where for i=1,2, (35)Si=supθΘ(|Miθ1|+|Miθ2|). As Mi/θj are continuous on the convex set Θ, it follows from Theorem  5.19 in  that for x=(x1,x2),y=(y1,y2)Θ, there exist zΘ such that (36)M(x)-M(y)=(M1,M2)(θ1,θ2)|z(x-y). Therefore, we have that M(x)-M(y)Sx-y, where x=max{|x1|,|x2|}. Hence, there is not more than one fixed-point of M in Θ when S<1. As every fixed-point of M~ in Θ is a fixed-point of M, we will prove the theorem by indicating S<1.

It follows from (29) to (35), (15), and Lemma 2 that (37)S1n1n+n3n+n5n+n4n2Δ2α2+n2nΔ26α2,(38)S2n1n+n3n+n2n+n4n2Δ2α2+n5nΔ212α2. Then it follows from Δ<2α/2 that S11 and S21.

The record has at least one replacement. That is, there is k1 such that R(k)=1. As a new system replaces the old failed system at time kΔ, we have that Y(k)=0. Now the theorem will be accomplished in two cases.

Case 1.

The case Y(k-1)=0. We have that n40. It follows from (37) and (38) that (39)S1-n4n(1-2Δ2α2)<1.

Case 2.

The case Y(k-1)0. According to the condition-based maintenance policy, a failed system is replaced at an inspection. Hence Y(k-1)=1 follows from Y(k-1)0. As Y(k-1)=1, Y(k)=0, and R(k)=1, we have that n50. Assume that the last replacement occurs at sΔ if any replacement occurs before (k-1)Δ. And we write s=0 when there is no replacement before (k-1)Δ. Now we study the sequence Y(s+1),,Y(k-1), which consists of digits 0 and 1, corresponding healthy and unhealthy states of a system without replacement. It is obvious that there is s+1tk-1 such that Y(t)=1 and Y(t-1)=0. Then, n20 follows from Y(t-1)=0, Y(t)=1 and R(t)=0. It follows from (37) and n20 that (40)S11-n2n(1-Δ26α2)<1. Moreover, it follows from (38) and n50 that (41)S21-n5n(1-Δ212α2)<1.

5. Example and Discussion

We will apply the EM algorithm to a simulation dataset. Based on this example, we will show the efficiency and accuracy of the EM algorithm. Moreover, by this example, we will show some limitations and shortcomings of Theorem 1. In this example, we make ensembles of 103 consecutive inspection of a simulating backup system defined by (1). The true parameters are θ1=10 and θ2=10/3, which is adopted from .

We describe first the process of iterations described by the EM algorithm (2) and (3). For a given couple of initial value θ(0) in a given parameter space Θ, we may derive M(θ(0)) from (15). If M(θ(0))Θ, it is the unique solution to (3), and we have that θ(1)=M(θ(0)). If we are fortunate and can repeat the operation again and again, then we obtain a sequence θ(n)Θ,n=1,2,.

It follows from the expression (8) of Q(θθ) that Q is continuous with respect to both θ' and θ. Similar to the discussion of the Theorem  1 in , we can prove that if the limitation of the sequence θ(n)Θ,n=1,2, exists and is also in Θ, then the limitation is a fixed-point of the operator M~. Theorem 1 shows that the limitation is unique for all such sequences.

In the first experiment, we run the EM algorithm for different initial values. In this experiment, we set β=1 and the parameter space Θ={(x,y):x,y>β}. We run the algorithm for 64 couples of initial values which are chosen randomly from 2 to 12. For each couple of initial value, we run the algorithm for 200 iterations. Figure 1 draws the final estimations of the parameters for initial values. We can see that the algorithm converges to the same result for a great range of initial values. As Theorem 1 points out that the number of fixed points is not more than 1. So we can conclude that there is a unique fixed points, and hence it is the MLE of the model parameters on the parameter space Θ={(x,y):x,y>1}.

Estimated parameters for different initial values.

Sometimes, the above procedure of θ(n),n=1,2, must stop without the output of the estimated parameters. In general, for a θ(n)Θ, if M(θ(n)) derived from (15) is not an element of Θ, then M~(θ(n))M(θ(n)). For this case, the solution M~(θ(n)) to (3) is on the boundary of Θ. As we do not obtain the explicit expression M~(θ(n)) for this case, the procedure is aborted.

In the following second experiment, for the same dataset of the first experiment, we run the EM algorithm for another parameter space Θ={(x,y):x,y>β'} with β'=12. We run the algorithm for 32 couples of initial values which are chosen randomly from 20 to 80. As we predict, the procedure is aborted for every couple of initial values. For these initial values, Figure 2 draws the maximal iteration numbers before the procedure is aborted.

Maximal iteration number for different initial values.

As we know, MLE θ^ in Θ, when it exists, is one of fixed-points of the operator M~. However, there may be other fixed-points of M~, such as stationary points of Q. Theorem 1 provides a sufficient condition under which such fixed-point does not exist. Our first experiment and Figure 1 illustrate this fact. However, in some cases, there is not a fixed point of M~ on the parameter space Θ. Such a phenomenon occurs when we set a wrong parameter space as in the second experiment.

6. Conclusion

A parameter estimation problem for a backup system has been considered. We established an EM algorithm, which can be used to iteratively determine the maximum likelihood estimators given observations of the system at discrete time points. It has been found that for any initial values, the sequence derived by the EM algorithm converges to a unique point when the limitation belongs to the specified parameter space.

Acknowledgment

This research was supported by the National Natural Science Foundation of China Grant Nos. 50977073 and 70971109.

Kim M. J. Makis V. Jiang R. Parameter estimation in a condition-based maintenance model Statistics and Probability Letters 2010 80 21-22 1633 1639 2-s2.0-77955842827 10.1016/j.spl.2010.07.002 Duan Q. Chen X. Zhao D. Zhao Z. Parameter estimation of a multi-state model for an aging piece of equipment under condition-based maintenance Mathematical Problems in Engineering 2012 2012 19 10.1155/2012/347675 347675 Dempster A. P. Laird N. M. Rubin D. B. Maximum likelihood from incomplete data via the EM algorithm Journal of the Royal Statistical Society B 1977 39 1 1 38 Wu C. On the convergence properties of the EM algorithm Annals of Statistics 1982 11 95 103 Akashi K. On uniqueness of the conditional maximum likelihood estimation for a binary panel model Economics Letters 2011 112 2 148 150 2-s2.0-79957492329 10.1016/j.econlet.2011.03.041 Aragón J. Eberly D. Eberly S. Existence and uniqueness of the maximum likelihood estimator for the two-parameter negative binomial distribution Statistics and Probability Letters 1992 15 5 375 379 2-s2.0-38249008072 Seregin A. Uniqueness of the maximum likelihood estimator for k-Monotone densities Proceedings of the American Mathematical Society 2010 138 12 4511 4515 2-s2.0-78649888502 10.1090/S0002-9939-2010-10496-3 Bhattacharya R. Waymire E. Stochastic Processes With Applications 1990 New York, NY, USA John Wiley & Sons Rudin W. Principles of Mathematical Analysis 1976 New York, NY, USA McGraw-Hill