Template:Parameter estimation fielded rga

Parameter Estimation
Suppose that the number of systems under study is $$K$$  and the  $${{q}^{th}}$$  system is observed continuously from time  $${{S}_{q}}$$  to time  $${{T}_{q}}$$,  $$q=1,2,\ldots ,K$$. During the period $$[{{S}_{q}},{{T}_{q}}]$$, let  $${{N}_{q}}$$  be the number of failures experienced by the  $${{q}^{th}}$$  system and let  $${{X}_{i,q}}$$  be the age of this system at the  $${{i}^{th}}$$  occurrence of failure,  $$i=1,2,\ldots ,{{N}_{q}}$$. It is also possible that the times $${{S}_{q}}$$  and  $${{T}_{q}}$$  may be observed failure times for the  $${{q}^{th}}$$  system. If $${{X}_{{{N}_{q}},q}}={{T}_{q}}$$  then the data on the  $${{q}^{th}}$$  system is said to be failure terminated and  $${{T}_{q}}$$  is a random variable with  $${{N}_{q}}$$  fixed. If $${{X}_{{{N}_{q}},q}}<{{T}_{q}}$$  then the data on the  $${{q}^{th}}$$  system is said to be time terminated with  $${{N}_{q}}$$  a random variable. The maximum likelihood estimates of $$\lambda $$  and  $$\beta $$  are values satisfying the Eqns. (lambdaPowerLaw) and (BetaPowerLaw).


 * $$\begin{align}

& \widehat{\lambda }= & \frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\left( T_{q}^{\widehat{\beta }}-S_{q}^{\widehat{\beta }} \right)} \\ & \widehat{\beta }= & \frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\widehat{\lambda }\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\left[ T_{q}^{\widehat{\beta }}\ln ({{T}_{q}})-S_{q}^{\widehat{\beta }}\ln ({{S}_{q}}) \right]-\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{\mathop{\sum }}}\,\ln ({{X}_{i,q}})} \end{align}$$

where $$0\ln 0$$  is defined to be 0. In general, these equations cannot be solved explicitly for $$\widehat{\lambda }$$  and  $$\widehat{\beta },$$  but must be solved by iterative procedures. Once $$\widehat{\lambda }$$  and  $$\widehat{\beta }$$  have been estimated, the maximum likelihood estimate of the intensity function is given by:


 * $$\widehat{u}(t)=\widehat{\lambda }\widehat{\beta }{{t}^{\widehat{\beta }-1}}$$

If $${{S}_{1}}={{S}_{2}}=\ldots ={{S}_{q}}=0$$  and  $${{T}_{1}}={{T}_{2}}=\ldots ={{T}_{q}}$$   $$\,(q=1,2,\ldots ,K)$$  then the maximum likelihood estimates  $$\widehat{\lambda }$$  and  $$\widehat{\beta }$$  are in closed form.


 * $$\begin{align}

& \widehat{\lambda }= & \frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{K{{T}^{\beta }}} \\ & \widehat{\beta }= & \frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{\mathop{\sum }}}\,\ln (\tfrac{T})} \end{align}$$

The following examples illustrate these estimation procedures.

Example 1
For the data in Table 13.1, the starting time for each system is equal to $$0$$  and the ending time for each system is 2000 hours. Calculate the maximum likelihood estimates $$\widehat{\lambda }$$  and  $$\widehat{\beta }$$.

Solution Since the starting time for each system is equal to zero and each system has an equivalent ending time, the general Eqns. (lambdaPowerLaw) and (BetaPowerLaw) reduce to the closed form Eqns. (sample1) and (sample2). The maximum likelihood estimates of $$\hat{\beta }$$  and  $$\hat{\lambda }$$  are then calculated as follows:


 * $$\begin{align}

& \widehat{\beta }= & \frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{\mathop{\sum }}}\,\ln (\tfrac{T})} \\ & = & 0.45300 \end{align}$$


 * $$\begin{align}

& \widehat{\lambda }= & \frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{K{{T}^{\beta }}} \\ & = & 0.36224 \end{align}$$



The system failure intensity function is then estimated by:


 * $$\widehat{u}(t)=\widehat{\lambda }\widehat{\beta }{{t}^{\widehat{\beta }-1}},\text{ }t>0$$

Figure wpp intensity is a plot of $$\widehat{u}(t)$$  over the period (0, 3000). Clearly, the estimated failure intensity function is most representative over the range of the data and any extrapolation should be viewed with the usual caution.