Template:Beta Binomial Confidence Bounds

Beta Binomial Confidence Bounds

Another less mathematically intensive method of calculating confidence bounds involves a procedure similar to that used in calculating median ranks (see Chapter Parameter Estimation). This is a non-parametric approach to confidence interval calculations that involves the use of rank tables and is commonly known as beta-binomial bounds (BB). By non-parametric, we mean that no underlying distribution is assumed. (Parametric implies that an underlying distribution, with parameters, is assumed.) In other words, this method can be used for any distribution, without having to make adjustments in the underlying equations based on the assumed distribution. Recall from the discussion on the median ranks that we used the binomial equation to compute the ranks at the 50% confidence level (or median ranks) by solving the cumulative binomial distribution for [math]\displaystyle{ Z }[/math] (rank for the [math]\displaystyle{ {{j}^{th}} }[/math] failure):

[math]\displaystyle{ P=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix} N \\ k \\ \end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}} }[/math]

where [math]\displaystyle{ N }[/math] is the sample size and [math]\displaystyle{ j }[/math] is the order number.
The median rank was obtained by solving the following equation for [math]\displaystyle{ Z }[/math]:

[math]\displaystyle{ 0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix} N \\ k \\ \end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}} }[/math]

The same methodology can then be repeated by changing [math]\displaystyle{ P }[/math] for [math]\displaystyle{ 0.50 }[/math] [math]\displaystyle{ (50%) }[/math] to our desired confidence level. For [math]\displaystyle{ P=90% }[/math] one would formulate the equation as

[math]\displaystyle{ 0.90=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix} N \\ k \\ \end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}} }[/math]

Keep in mind that one must be careful to select the appropriate values for [math]\displaystyle{ P }[/math] based on the type of confidence bounds desired. For example, if two-sided 80% confidence bounds are to be calculated, one must solve the equation twice (once with [math]\displaystyle{ P=0.1 }[/math] and once with [math]\displaystyle{ P=0.9 }[/math]) in order to place the bounds around 80% of the population.

Using this methodology, the appropriate ranks are obtained and plotted based on the desired confidence level. These points are then joined by a smooth curve to obtain the corresponding confidence bound.

This non-parametric methodology is only used by Weibull++ when plotting bounds on the mixed Weibull distribution. Full details on this methodology can be found in Kececioglu [20]. These binomial equations can again be transformed using the beta and F distributions, thus the name beta binomial confidence bounds.