Template:Weibull likelihood ratio confidence bounds

Likelihood Ratio Confidence Bounds

As covered in Chapter Confidence Bounds, the likelihood confidence bounds are calculated by finding values for θ₁ and θ₂ that satisfy:

[math]\displaystyle{ -2\cdot \text{ln}\left( \frac{L(\theta _{1},\theta _{2})}{L(\hat{\theta }_{1}, \hat{\theta }_{2})}\right) =\chi _{\alpha ;1}^{2} }[/math]

This equation can be rewritten as:

[math]\displaystyle{ L(\theta _{1},\theta _{2})=L(\hat{\theta }_{1},\hat{\theta } _{2})\cdot e^{\frac{-\chi _{\alpha ;1}^{2}}{2}} }[/math]

For complete data, the likelihood function for the Weibull distribution is given by:

[math]\displaystyle{ L(\beta ,\eta )=\prod_{i=1}^{N}f(x_{i};\beta ,\eta )=\prod_{i=1}^{N}\frac{ \beta }{\eta }\cdot \left( \frac{x_{i}}{\eta }\right) ^{\beta -1}\cdot e^{-\left( \frac{x_{i}}{\eta }\right) ^{\beta }} }[/math]

For a given value of α, values for β and η can be found which represent the maximum and minimum values that satisfy the above equation. These represent the confidence bounds for the parameters at a confidence level δ, where α = δ for two-sided bounds and α = 2δ − 1 for one-sided.

Similarly, the bounds on time and reliability can be found by substituting the Weibull reliability equation into the likelihood function so that it is in terms of β and time or reliability, as discussed in Chapter 6. The likelihood ratio equation used to solve for bounds on time (Type 1) is:

[math]\displaystyle{ L(\beta ,t)=\prod_{i=1}^{N}\frac{\beta }{\left( \frac{t}{(-\text{ln}(R))^{ \frac{1}{\beta }}}\right) }\cdot \left( \frac{x_{i}}{\left( \frac{t}{(-\text{ ln}(R))^{\frac{1}{\beta }}}\right) }\right) ^{\beta -1}\cdot \text{exp}\left[ -\left( \frac{x_{i}}{\left( \frac{t}{(-\text{ln}(R))^{\frac{1}{\beta }}} \right) }\right) ^{\beta }\right] }[/math]

The likelihood ratio equation used to solve for bounds on reliability (Type 2) is:

[math]\displaystyle{ L(\beta ,R)=\prod_{i=1}^{N}\frac{\beta }{\left( \frac{t}{(-\text{ln}(R))^{ \frac{1}{\beta }}}\right) }\cdot \left( \frac{x_{i}}{\left( \frac{t}{(-\text{ ln}(R))^{\frac{1}{\beta }}}\right) }\right) ^{\beta -1}\cdot \text{exp}\left[ -\left( \frac{x_{i}}{\left( \frac{t}{(-\text{ln}(R))^{\frac{1}{\beta }}} \right) }\right) ^{\beta }\right] }[/math]