Subsections

• 12.4.1 Intensity Function
• 12.4.2 Multiple Counting Processes
• 12.4.3 Power Law Model
• 12.4.4 Models Suitable for Conditional Estimation

12.4 Multiple Failures and Counting Processes

The standard methods of survival analysis can be generalized to include multiple failures simply defined as a series of well-defined event occurrences. For example, in software reliability, engineers are often interested in detecting software bugs. Inference from a single counting process has been studied in detail ([8,23]), with multiple independent processes being considered as a means to estimate a common cumulative mean function from a nonparametric or semi-parametric viewpoint ([20,26]). [16] discussed problems associated with parametric conditional inference in models with a common trend parameter or possibly different base-line intensity parameters.

12.4.1 Intensity Function

For multiple failures, intensity functions correspond to hazard functions in that the intensity function is defined as discussed next.

In time interval $[t_0,t]$ we define the number of occurrences of events or failures as $N(t)$. The Poisson counting process $\{N(t): t\ge t_0\}$ is given such that it satisfies the following three conditions for $t\ge t_0$.

1. $\Pr\{N(t_0)=0\}=1$
2. The increment $N_{s,t}=N(t)-N(s) \ (t_0\ge s<t)$ has a Poisson distribution with the mean parameter $\Lambda_t - \Lambda_s$, for some positive and increasing function in $t$.
3. $\{N_t: t\ge t_0\}$ is a process of independent increments. That is, for any ($t_0<)$ $t_1<t_2<\cdots <t_n$, $n$ increments, $N(t_1)-N(t_0),\ldots, N(t_n)-N(t_{n-1})$ are mutually independent.

For this counting process $\{N(t): t\ge t_0\}$ we can define the intensity function as

$$\lambda (t) = \lim_{\rm\Delta \rightarrow 0} \frac{1}{\rm\Delta }\Pr\{N(t+\rm\Delta t) - N(t) =1 \vert H(t)\}\,,$$ (12.26)

where $H(t)$is the history of the process up to $t$:

$$\notag H(t)=\{N(u):\ t_0 \le u \le t\}\,.$$

Note that

$$\notag \Lambda (t) =\int_{t_0}^t \lambda (t) {\text{d}}t\,.$$

Expectation of $E[N_{s,t}]$ becomes

$$E[N_{s,t}]=\sum_{n=0}^\infty n \Pr \{N_{n,s}=n\}=\Lambda_t - \Lambda_s\,,$$ (12.27)

and

$$\lambda (t) = \frac{{\text{d}}}{{\text{d}}t}\Lambda_t = \frac{{\text{d}}}{{\text{d}}t}E[N(t)]\,.$$ (12.28)

The nonparametric estimate of the intensity function is easy to determine and is quite useful for observing the trend of a series of events. If a data set of failure times $\{t_1,t_2, \ldots, t_n\}$ is available, assuming constant intensity in $(t_{k_1}, t_k]$, then

$$\notag \lambda(t) = \lambda_k \ \ (t_{k-1}<t\le t_k)\,,$$

and the nonparametric ML estimates becomes

$$\lambda_k = \frac{1}{t_k - t_{k-1}} \ \ \ (k= 1, \ldots, n)\,,$$ (12.29)

where $t_0=0$.

12.4.2 Multiple Counting Processes

We assume several independent counting processes $\{N_k(t_k)$, i.e., $0<t_k\le \tau_k, k=1,\ldots, K\}$. The cumulative mean function for $N_k(t)$ is expressed by

$$M_k(t) = E\left\{N_k(t)\right\}\,.$$ (12.30)

[26] described a method for estimating the cumulative mean function of an identically distributed process without assuming any Poisson process structure, while [20] developed robust variance estimates based on the Poisson process. All these methods are basically concerned with nonparametric estimation. Here, parametric models for effectively acquiring information on the trend of an event occurrence are dealt with. [16] considered generalized versions of two primal parametric models to multiple independent counting processes under the framework of a nonhomogeneous Poisson process.

Cox and Lewis ([8]) considered a log-linear model for trend testing a singe counting process, i.e.,

$$\lambda(t) =\exp(\alpha + \beta t)\,,$$ (12.31)

where $\lambda(t)$ is the intensity function corresponding to the derivative of the mean function in the continuous case. Note that for a single case the subscript $k$ is omitted. They assumed the above nonhomogeneous Poisson process and gave a simple test statistic for $H_0:\beta =0$ against $H_A:\beta \neq 0$, i.e.,

$$U= \frac{\sum_{i=1}^n t_i-\frac{1}{2}\tau_0}{\tau_0\sqrt{\frac{n}{12}}}\,.$$ (12.32)

The distribution of this statistic steeply converges to the standard normal distribution when $n\rightarrow \infty$. This statistic is sometimes called the $U$ statistic and is frequently applied to trend testing in reliability engineering.

[16] generalized this log-linear model to the multiple case, with the log-linear model for $k$-th individual being

$$\lambda_k(t) = \exp\left(\alpha _k + \beta t \right)\,.$$ (12.33)

In this modeling we assume the common trend parameter $\beta$ and are mainly interested in estimating and testing this parameter. The full likelihood for the model becomes

$$L(\beta, \alpha _1, \alpha _2 , \ldots, \alpha _K) = \prod _{k=1}^K \left[\left\{\prod _{i=1}^{n_k}\lambda _k(t_{ki})\right\}\exp\left\{ -\int _0^{\tau _k}\lambda _k (u) {\text{d}}u\right\}\right]$$ (12.34)

$$=\exp\left\{ \sum _{k=1}^K n_k \alpha _k + \beta \sum _{k=1}^K \s... ...1}{\beta} \sum _{k=1}^K e^{\alpha _k}\left(e^{\beta\tau _k}-1\right)\right\}\,.$$

If $K$ is large, it is difficult to compute all parameter estimates based on such full likelihood.

Given $N_k(\tau _k)=n_k, k=1,2, \ldots, K$, conditional likelihood is considered as

$$CL(\beta \vert N_k(\tau _k)=n_k, i=1, \ldots, K)= \frac{\prod _{k... ...ta\sum \sum t_{ki}}} {\prod _{k=1}^K \left(e^{\beta \tau _k}-1\right)^{n_k}}\,.$$ (12.35)

Note that the nuisance parameter $\alpha_k$'s do not appear. Fisher information is calculated as

$$I(\beta) = E\left[-\frac{ \partial^2 \log CL } {\partial \beta ^2 } \right]$$

$$= \left\{ \begin{array}{ll} \sum _{k=1}^K n_k \left\{\frac{1}{\be... ...\\ \frac{1}{12} \sum _{k=1}^K n_k \tau_k^2 & (\beta =0) \end{array} \right.\, .$$ (12.36)

The test statistic obtained from the above calculations becomes

$$U_k = \frac{\left. \log CL \right\vert _{\beta =0}}{\sqrt{I(0)}}$$

$$= \frac{\sum _{k=1}^K \sum _{i=1}^{n_k}t_{ki}-\frac{1}{2}\sum _{k=1}^K n_k\tau_k} {\sqrt{\frac{1}{12}\sum _{k=1}^K n_k \tau _k^2}}\,.$$ (12.37)

To obtain the conditional estimate, numerical calculations are required such as Newton-Raphson method. However, the log conditional likelihood and its derivatives are not computable at the origin of the parameter $\beta$. In such a case, Taylor series expansions of the log conditional likelihood are used around the origin ([16]).

12.4.3 Power Law Model

[9] considered the power law model, sometimes called the Weibull process model. This model was generalized to the multiple case using the following intensity for the $k$-th individual ([16]):

$$\lambda _k(t) = \theta _k m t^{m-1}\,.$$ (12.38)

In this case it is easy to calculate the MLE. Direct calculation of the likelihood gives rise to the MLE $\widehat{m}$ and $\widehat{\theta _k}$ i.e.,

$$\widehat{m} = \frac{\sum _{k=1}^K n_k}{\sum _{k=1}^K \sum _{i=1}^{n_k}\log \left(\frac{\tau_k}{t_{ki}}\right)},$$ (12.39)

$$\widehat{\theta _k} = \frac{n_k}{\tau _k^{\widehat{m}}}\,.$$ (12.40)

Putting

$$Z = \frac{2m\sum_{k=1}^K n_k}{\widehat{m}}\,,$$ (12.41)

the distribution of $Z$ becomes a chi-square with $2 \sum _{k=1}^K n_k$ degrees of freedom. Based on this result we can make an inference of the common parameter $m$.

12.4.4 Models Suitable for Conditional Estimation

Estimation based on conditional likelihood allows effectively eliminating the nuisance parameter and obtaining information on the structure parameter. Let us now consider the class of nonhomogeneous Poisson process models which are specified by the intensity parameterized by two parameters. The first parameter $\alpha$ is concerned with the base line occurrences for the individual, while the second parameter $\beta$ is concerned with the trend of intensity. For simplicity, the property of the intensity for $K=1$ is examined. Using conditional likelihood is convenient because the nuisance parameter $\alpha$ need not be known. This is of great importance in multiple intensity modeling, i.e.,

Theorem 4 ([16])  
Conditional likelihood does not include the nuisance parameter $\alpha$ iff the intensity is factorized as two factors, a function of $\alpha$ and a function of $\beta$ and the time $t$, in the class of nonhomogeneous Poisson process models. That is, the intensity is expressed as

$$\lambda(t;\alpha,\beta) = h(\alpha)g(\beta;t)\,,\ \ \ a.s.$$ (12.42)

Several intensity models for software reliability are described in [23]: the log-linear model, geometric model, inverse linear model, inverse polynomial model, and power law model, all of which are included in this class satisfying the condition of the theorem.

Acknowledgements.     This work has been partially supported financially by Chuo University as one of the 2003 Research Projects for Promotion of Advanced Research at Graduate School.