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 [math]\displaystyle{ t=45\,\! }[/math]. The ML estimate for the reliability at [math]\displaystyle{ t=45\,\! }[/math] 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 [math]\displaystyle{ R\,\! }[/math] and [math]\displaystyle{ \beta .\,\! }[/math] This is again accomplished by substituting the Weibull reliability equation into the [math]\displaystyle{ \eta \,\! }[/math] term in the likelihood ratio equation to form a likelihood equation in terms of [math]\displaystyle{ R\,\! }[/math] and [math]\displaystyle{ \beta \,\! }[/math]:

[math]\displaystyle{ \begin{align} & L(\beta ,R)= & \underset{i=1}{\overset{N}{\mathop \prod }}\,f({{x}_{i}};\beta ,t,R) \\ & & \end{align}\,\! }[/math]

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

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

[math]\displaystyle{ L(\beta ,R)-L(\widehat{\beta },\widehat{\eta })\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}=0\,\! }[/math]

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

[math]\displaystyle{ \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}\,\! }[/math]

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

These points are represented graphically in the following contour plot:

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