Template:Example: Likelihood Ratio Bounds on Reliability (Type 2)

Likelihood Ratio Bounds on Reliability (Type 2)

For the data given in Example 1, determine the 90% two-sided confidence bounds on the reliability estimate for $$t=45\,\!$$. The ML estimate for the reliability at $$t=45\,\!$$ is 14.816%.

Solution

In this example, we are trying to determine the 90% two-sided confidence bounds on the reliability estimate of 14.816%. As was mentioned, we need to rewrite the likelihood ratio equation so that it is in terms of $$R\,\!$$ and $$\beta .\,\!$$ This is again accomplished by substituting the Weibull reliability equation into the $$\eta \,\!$$ term in the likelihood ratio equation to form a likelihood equation in terms of $$R\,\!$$ and $$\beta \,\!$$:


 * $$\begin{align}

& L(\beta ,R)= & \underset{i=1}{\overset{N}{\mathop \prod }}\,f({{x}_{i}};\beta ,t,R) \\ & &   \end{align}\,\!$$


 * $$=\underset{i=1}{\overset{5}{\mathop \prod }}\,\frac{\beta }{\left( \tfrac{t} \right)}\cdot {{\left( \frac{{{x}_{i}}}{\left( \tfrac{t} \right)} \right)}^{\beta -1}}\cdot \text{exp}\left[ -{{\left( \frac{{{x}_{i}}}{\left( \tfrac{t}{{{(-\text{ln}(R))}^{\tfrac{1}{\beta }}}} \right)} \right)}^{\beta }} \right]\,\!$$

where $${{x}_{i}}\,\!$$ are the original time-to-failure data points. We can now rearrange the likelihood ratio equation to the form:


 * $$L(\beta ,R)-L(\widehat{\beta },\widehat{\eta })\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}=0\,\!$$

Since our specified confidence level, $$\delta \,\!$$, is 90%, we can calculate the value of the chi-squared statistic, $$\chi _{0.9;1}^{2}=2.705543.\,\!$$ We can now substitute this information into the equation:


 * $$\begin{align}

L(\beta ,R)-L(\widehat{\beta },\widehat{\eta })\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}= & 0 \\ \\  L(\beta ,R)-1.714714\times {{10}^{-9}}\cdot {{e}^{\tfrac{-2.705543}{2}}}= & 0 \\ \\  L(\beta ,R)-4.432926\cdot {{10}^{-10}}= & 0 \end{align}\,\!$$

It now remains to find the values of $$\beta \,\!$$ and $$R\,\!$$ that satisfy this equation. This is an iterative process that requires setting the value of $$\beta \,\!$$ and finding the appropriate values of $$R\,\!$$. The following table gives the values of $$R\,\!$$ based on given values of $$\beta \,\!$$.



These points are represented graphically in the following contour plot:



As can be determined from the table, the lowest calculated value for $$R\,\!$$ is 2.38%, while the highest is 44.26%. These represent the 90% two-sided confidence limits on the reliability at $$t=45\,\!$$.