Estimation of parameters of component lifetime distribution in a coherent system

Abstract

We deal with estimation of parameters of lifetime distribution of components which form a coherent system. It is assumed that the system lifetime and the number of failed components up to the moment of the system failure, including the component which fails at the system failure epoch, are observed for each system.We apply the theory of optimal estimating functions and obtain optimal estimators of the parameters assuming that component lifetime has exponential and Weibull distribution. We apply Fisher’s conditionality principle and obtain properties of estimators of the system based on the conditional distribution of the system life-time, given the number of failed components in the system. Exact conditional distributions of the estimators as well as conditional consistency and asymptotic normality have been discussed for both the models. We also compare the proposed estimators with some of the existing estimators.

Appendix

Here, we state Theorem 2.1 of Godambe and Thompson (1989). Let $\underline{\theta } = (\theta _1,\theta _2, \ldots , \theta _p)'$, a vector of unknown parameters. We assume that the parameter space $\varTheta$ is an open rectangle in $R^p$. Let $\underline{h(\underline{\theta })} = (h_1(\underline{\theta }),h_2(\underline{\theta }), \ldots , h_p(\underline{\theta }))$ be a vector valued function of observations $\underline{X}$ and parameter $\underline{\theta }$. Let F denote the distribution function of $\underline{X}$. It is assumed that $F \in \mathcal {F}$, an appropriate class of distribution functions. We write $\underline{h(\underline{\theta })} = \underline{h}$ for the sake of convenience. Definition (Godambe and Thompson 1989) An estimating function $\underline{h}$ is said to be regular, if, for all $r =1,2,\ldots p$ and $s =1,2,\ldots p$, the followings hold for all $F \in \mathcal {F}$.

• 1. $E[h_r(\underline{\theta } ) ] = 0.$

• 2. $E[h_r^2(\underline{\theta } ) ] < \infty$.

• 3. $\frac{\partial }{\partial \theta _s} E[h_r(\underline{\theta } )]= E[\frac{ \partial }{\partial \theta _s} h_r(\underline{\theta } )].$

• 4. Let $V(\underline{h})$ be the variance-covariance matrix of $\underline{h}$. Then, $V(\underline{h})$ is positive definite.

Now, let $\mathcal {G}$ be an appropriately chosen class of regular estimating functions. Let the $p \times p$ matrix $A(\underline{h}) = (( A(\underline{h})_{rs} ))$ be defined by

$$\begin{aligned} A(\underline{h})_{rs} = E\left[ \frac{\partial h_r}{\partial \theta _s}\right] . \end{aligned}$$

Let $V = V(\underline{h})$. Let $\underline{g^*}$$\in$$\mathcal {G}$ be an estimating function. Let $V^*= V(\underline{g^*})$ denote the variance-covariance matrix of $\underline{g^*}$ and let the matrix $A^* = A(\underline{g^*})$ be defined by $A(g^*)_{rs}=E[\frac{\partial g^*_r}{\partial \theta _s}]$.

Definition. (Godambe and Thompson 1989) An estimating function $\underline{h}^*$$\in$$\mathcal {G}$ is said to be optimal in $\mathcal {G}$, if $A^*$ is non-singular and if for all $F \in \mathcal {F}$,

$$\begin{aligned} V - A [A^*]^{-1}V^* [A^{*'}]^{-1} A' \end{aligned}$$

is positive semi-definite for all $\underline{h} \in \mathcal {G}.$

Now, let $(\tau _1,\tau _2,\ldots ,\tau _m)$ be a set of real valued regular estimating functions. Consider the class $\mathcal {G}$ of estimating functions which are obtained by linear combinations of $\tau _1,\tau _2,\ldots ,\tau _m$. An estimating function $\underline{h}= ( h_1,h_2,\ldots , h_p)$ in such a class is defined by

$$\begin{aligned} h_r = \sum _{j=1}^m \tau _j q(j,r),~~~~ r = 1,2,\ldots p \end{aligned}$$

where q(j, r) is a constant for all r and j. The estimating functions $\tau _i$’s are known as elementary estimating functions. Consider the estimating function $\underline{g^*}=(g_1^*, g_2^*, \ldots ,g_p^*)$ defined by

$$\begin{aligned} g_r^* = \sum _{j=1}^m \tau _j q^*(j,r), \quad r=1,2,\ldots , p, \end{aligned}$$

where

$$\begin{aligned} q^*(j,r) = \frac{E(\partial \tau _j/\partial \theta _r)}{E[\tau _j^2]}. \end{aligned}$$

We note that $g_r^*$ can also be written as $g_r^* = \sum _{j=1}^m g_{jr}^*$, where $g_{jr}^* = \tau _j q^*(j,r).$

Definition. (Godambe and Thompson 1989, Definition 2.2) The estimating functions $\tau _j,j=1,2,\ldots ,m$ are said to be orthogonal if

$$\begin{aligned} E [g_{jr}^* g_{j'r'}^*] =0 , \end{aligned}$$

for all $F \in \mathcal {F}$, for $j \ne j',\;\; j=1,2,\ldots p,\;\; j'=1,2,\ldots p$.

The Theorem 2.1 of Godambe and Thompson (1989) proves that if $\tau _j,j=1,2,\ldots ,m$ are orthogonal, the estimating function $\underline{g^*}$ is optimal in $\mathcal {G}$, as per the definition of optimality given above.

In applications of the above result to our set-up, we have $p=1$ in the case of exponential distribution and $p=2$ in the case of Weibull distribution. Further, m refers to the number of systems observed. The estimating functions $g_j$ as defined in Sects. 2 and 3, equal $\tau _j,~j = 1,2, \ldots m$, the set of elementary estimating functions. The above theory is then applied in the conditional setup, i.e., given $W_1=w_1,W_2=w_2,\ldots , W_m=w_m$. The elementary estimating functions are regular. Further, in view of the fact that the random vectors $(X_1,W_1), (X_2,W_2), \ldots , (X_m,W_m)$ are independent, we have

$$\begin{aligned} E\left[ \frac{\partial g_j}{\partial \theta _r}|W_1=w_1,W_2=w_2,\ldots , W_m=w_m \right] = E\left[ \frac{\partial g_j}{\partial \theta _r}|W_j=w_j\right] \end{aligned}$$

and $Var\left[ g_j|W_1=w_1,W_2=w_2,\ldots , W_m=w_m \right] =Var\left[ g_j|W_j\right]$. Further, the class of estimating functions $\mathcal {G}$ is obtained by considering q(j, r) to be functions of $w_1,w_2,\ldots ,w_m$. The (conditional) orthogonality follows from independence of $(X_1,W_1),(X_2,W_2), \ldots , (X_m,W_m)$.