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''This article also appears in the [[Confidence Bounds|Life Data Analysis Reference]] and [[Appendix D: Confidence Bounds|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>

== What Are Confidence Bounds?==

One of the most confusing concepts to a novice reliability engineer is estimating the precision of an estimate. This is an important concept in the field of reliability engineering, leading to the use of confidence intervals (or bounds). In this section, we will try to briefly present the concept in relatively simple terms but based on solid common sense. 


=== The Black and White Marbles ===
To illustrate, consider the case where there are millions of perfectly mixed black and white marbles in a rather large swimming pool and our job is to estimate the percentage of black marbles. The only way to be absolutely certain about the exact percentage of marbles in the pool is to accurately count every last marble and calculate the percentage. However, this is too time- and resource-intensive to be a viable option, so we need to come up with a way of estimating the percentage of black marbles in the pool. In order to do this, we would take a relatively small sample of marbles from the pool and then count how many black marbles are in the sample. 

'''Taking a Small Sample of Marbles'''

First, pick out a small sample of marbles and count the black ones. Say you picked out ten marbles and counted four black marbles. Based on this, your estimate would be that 40% of the marbles are black. 

[[Image:estimation.png|center|200px]] 

If you put the ten marbles back in the pool and repeat this example again, you might get six black marbles, changing your estimate to 60% black marbles. Which of the two is correct? Both estimates are correct! As you repeat this experiment over and over again, you might find out that this estimate is usually between <math>{{X}_{1}}%</math> and <math>{{X}_{2}}%</math>, and you can assign a percentage to the number of times your estimate falls between these limits. For example, you notice that 90% of the time this estimate is between <math>{{X}_{1}}%</math> and <math>{{X}_{2}}%.</math>

''' Taking a Larger Sample of Marbles '''

If you now repeat the experiment and pick out 1,000 marbles, you might get results for the number of black marbles such as 545, 570, 530, etc., for each trial. The range of the estimates in this case will be much narrower than before. For example, you observe that 90% of the time, the number of black marbles will now be from <math>{{Y}_{1}}%</math> to <math>{{Y}_{2}}%</math>, where <math>{{X}_{1}}%<{{Y}_{1}}%</math> and <math>{{X}_{2}}%>{{Y}_{2}}%</math>, thus giving you a more narrow estimate interval. The same principle is true for confidence intervals; the larger the sample size, the more narrow the confidence intervals.

'''Back to Reliability'''

We will now look at how this phenomenon relates to reliability. Overall, the reliability engineer's task is to determine the probability of failure, or reliability, of the population of units in question. However, one will never know the exact reliability value of the population unless one is able to obtain and analyze the failure data for every single unit in the population. Since this usually is not a realistic situation, the task then is to estimate the reliability based on a sample, much like estimating the number of black marbles in the pool. If we perform ten different reliability tests for our units, and analyze the results, we will obtain slightly different parameters for the distribution each time, and thus slightly different reliability results. However, by employing confidence bounds, we obtain a range within which these reliability values are likely to occur a certain percentage of the time. This helps us gauge the utility of the data and the accuracy of the resulting estimates. Plus, it is always useful to remember that each parameter is an estimate of the true parameter, one that is unknown to us. This range of plausible values is called a confidence interval.

=== One-Sided and Two-Sided Confidence Bounds  ===
Confidence bounds are generally described as being one-sided or two-sided. 

'''Two-Sided Bounds''' 

[[Image:two sided bounds.png|center|250px]] 

When we use two-sided confidence bounds (or intervals), we are looking at a closed interval where a certain percentage of the population is likely to lie. That is, we determine the values, or bounds, between which lies a specified percentage of the population. For example, when dealing with 90% two-sided confidence bounds of <span class="texhtml">(''X'',''Y'')</span>, we are saying that 90% of the population lies between <span class="texhtml">''X''</span> and <span class="texhtml">''Y''</span> with 5% less than <span class="texhtml">''X''</span> and 5% greater than <span class="texhtml">''Y''</span>. 

'''One-Sided Bounds''' 

One-sided confidence bounds are essentially an open-ended version of two-sided bounds. A one-sided bound defines the point where a certain percentage of the population is either higher or lower than the defined point. This means that there are two types of one-sided bounds: upper and lower. An upper one-sided bound defines a point that a certain percentage of the population is less than. Conversely, a lower one-sided bound defines a point that a specified percentage of the population is greater than. 

[[Image:one sided bounds.png|center|250px]] 

For example, if <span class="texhtml">''X''</span> is a 95% upper one-sided bound, this would imply that 95% of the population is less than <span class="texhtml">''X''</span>. If <span class="texhtml">''X''</span> is a 95% lower one-sided bound, this would indicate that 95% of the population is greater than <span class="texhtml">''X''</span>. Care must be taken to differentiate between one- and two-sided confidence bounds, as these bounds can take on identical values at different percentage levels. For example, in the figures above, we see bounds on a hypothetical distribution. Assuming that this is the same distribution in all of the figures, we see that <span class="texhtml">''X''</span> marks the spot below which 5% of the distribution's population lies. Similarly, <span class="texhtml">''Y''</span> represents the point above which 5% of the population lies. Therefore, <span class="texhtml">''X''</span> and <span class="texhtml">''Y''</span> represent the 90% two-sided bounds, since 90% of the population lies between the two points. However, <span class="texhtml">''X''</span> also represents the lower one-sided 95% confidence bound, since 95% of the population lies above that point; and <span class="texhtml">''Y''</span> represents the upper one-sided 95% confidence bound, since 95% of the population is below <span class="texhtml">''Y''</span>. It is important to be sure of the type of bounds you are dealing with, particularly as both one-sided bounds can be displayed simultaneously in Weibull++. In Weibull++, we use upper to represent the higher limit and lower to represent the lower limit, regardless of their position, but based on the value of the results. So if&nbsp;obtaining the confidence bounds on the reliability, we would identify the lower value of reliability as the lower limit and the higher value of reliability as the higher limit. If obtaining the confidence bounds on probability of failure we will again identify the lower numeric value for the probability of failure as the lower limit and the higher value as the higher limit.

== Fisher Matrix Confidence Bounds  ==
This section presents an overview of the theory on obtaining approximate confidence bounds on suspended (multiple censored) data. The methodology used is the so-called Fisher matrix bounds (FM), described in [[Appendix D: Weibull References|Nelson [30]]]&nbsp;and [[Appendix D: Weibull References|Lloyd and Lipow [24]]]. These bounds are employed in most other commercial statistical applications. In general, these bounds tend to be more optimistic than the non-parametric rank based bounds. This may be a concern, particularly when dealing with small sample sizes. Some statisticians feel that the Fisher matrix bounds are too optimistic when dealing with small sample sizes and prefer to use other techniques for calculating confidence bounds, such as the likelihood ratio bounds. <br>

=== Approximate Estimates of the Mean and Variance of a Function  ===
In utilizing FM bounds for functions, one must first determine the mean and variance of the function in question (i.e., reliability function, failure rate function, etc.). An example of the methodology and assumptions for an arbitrary function <span class="texhtml">''G''</span> is presented next. 

'''Single Parameter Case''' 

For simplicity, consider a one-parameter distribution represented by a general function <span class="texhtml">''G'',</span> which is a function of one parameter estimator, say <math>G(\widehat{\theta }).</math> For example, the mean of the exponential distribution is a function of the parameter <span class="texhtml">λ</span>: <span class="texhtml">''G''(λ) = 1 / λ = μ</span>. Then, in general, the expected value of <math>G\left( \widehat{\theta } \right)</math> can be found by:

::<math>E\left( G\left( \widehat{\theta } \right) \right)=G(\theta )+O\left( \frac{1}{n} \right)</math>

where <span class="texhtml">''G''(θ)</span> is some function of <span class="texhtml">θ</span>, such as the reliability function, and <span class="texhtml">θ</span> is the population parameter where <math>E\left( \widehat{\theta } \right)=\theta </math> as <math>n\to \infty </math> . The term <math>O\left( \tfrac{1}{n} \right)</math> is a function of <span class="texhtml">''n''</span>, the sample size, and tends to zero, as fast as <math>\tfrac{1}{n},</math> as <math>n\to \infty .</math> For example, in the case of <math>\widehat{\theta }=1/\overline{x}</math> and <span class="texhtml">''G''(''x'') = 1 / ''x''</span>, then <math>E(G(\widehat{\theta }))=\overline{x}+O\left( \tfrac{1}{n} \right)</math> where <math>O\left( \tfrac{1}{n} \right)=\tfrac{{{\sigma }^{2}}}{n}</math>. Thus as <math>n\to \infty </math>, <math>E(G(\widehat{\theta }))=\mu </math> where <span class="texhtml">μ</span> and <span class="texhtml">σ</span> are the mean and standard deviation, respectively. Using the same one-parameter distribution, the variance of the function <math>G\left( \widehat{\theta } \right)</math> can then be estimated by:

::<math>Var\left( G\left( \widehat{\theta } \right) \right)=\left( \frac{\partial G}{\partial \widehat{\theta }} \right)_{\widehat{\theta }=\theta }^{2}Var\left( \widehat{\theta } \right)+O\left( \frac{1}{{{n}^{\tfrac{3}{2}}}} \right)</math>

'''Two-Parameter Case''' 

Consider a Weibull distribution with two parameters <span class="texhtml">β</span> and <span class="texhtml">η</span>. For a given value of <span class="texhtml">''t''</span>, <math>R(t)=G(\beta ,\eta )={{e}^{-{{\left( \tfrac{t}{\eta } \right)}^{\beta }}}}</math>. Repeating the previous method for the case of a two-parameter distribution, it is generally true that for a function <span class="texhtml">''G''</span>, which is a function of two parameter estimators, say <math>G\left( {{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}} \right)</math>, that: 

::<math>E\left( G\left( {{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}} \right) \right)=G\left( {{\theta }_{1}},{{\theta }_{2}} \right)+O\left( \frac{1}{n} \right)</math>

and: 

::<math>\begin{align}
Var( G( {{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}}))= &{(\frac{\partial G}{\partial {{\widehat{\theta }}_{1}}})^2}_{{\widehat{\theta_{1}}}={\theta_{1}}}Var(\widehat{\theta_{1}})+{(\frac{\partial G}{\partial {{\widehat{\theta }}_{2}}})^2}_{{\widehat{\theta_{2}}}={\theta_{2}}}Var(\widehat{\theta_{2}})\\
& +2{(\frac{\partial G}{\partial {{\widehat{\theta }}_{1}}})^2}_{{\widehat{\theta_{1}}}={\theta_{1}}}{(\frac{\partial G}{\partial {{\widehat{\theta }}_{2}}})^2}_{{\widehat{\theta_{2}}}={\theta_{2}}}Cov(\widehat{\theta_{1}},\widehat{\theta_{2}}) \\
& +O(\frac{1}{n^{\tfrac{3}{2}}})
\end{align}

</math>

<br>Note that the derivatives of the above equation are evaluated at <math>{{\widehat{\theta }}_{1}}={{\theta }_{1}}</math> and <math>{{\widehat{\theta }}_{2}}={{\theta }_{2}},</math> where E <math>\left( {{\widehat{\theta }}_{1}} \right)\simeq {{\theta }_{1}}</math> and E <math>\left( {{\widehat{\theta }}_{2}} \right)\simeq {{\theta }_{2}}.</math> <br><br>

'''Parameter Variance and Covariance Determination''' 

The determination of the variance and covariance of the parameters is accomplished via the use of the Fisher information matrix. For a two-parameter distribution, and using maximum likelihood estimates (MLE), the log-likelihood function for censored data is given by: 

::<math>\begin{align}
   \ln [L]= & \Lambda =\underset{i=1}{\overset{R}{\mathop \sum }}\,\ln [f({{T}_{i}};{{\theta }_{1}},{{\theta }_{2}})] \\ 
   & \text{ }+\underset{j=1}{\overset{M}{\mathop \sum }}\,\ln [1-F({{S}_{j}};{{\theta }_{1}},{{\theta }_{2}})] \\ 
   & \text{ }+\underset{l=1}{\overset{P}{\mathop \sum }}\,\ln \left\{ F({{I}_{{{l}_{U}}}};{{\theta }_{1}},{{\theta }_{2}})-F({{I}_{{{l}_{L}}}};{{\theta }_{1}},{{\theta }_{2}}) \right\}  
\end{align}</math>

In the equation above, the first summation is for ''complete data'', the second summation is for ''right censored data'' and the third summation is for ''interval or left censored data''.

Then the Fisher information matrix is given by: 

::<math>{{F}_{0}}=\left[ \begin{matrix}
   {{E}_{0}}{{\left[ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \theta _{1}^{2}} \right]}_{0}} & {} & {{E}_{0}}{{\left[ -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\theta }_{1}}\partial {{\theta }_{2}}} \right]}_{0}}  \\
   {} & {} & {}  \\
   {{E}_{0}}{{\left[ -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\theta }_{2}}\partial {{\theta }_{1}}} \right]}_{0}} & {} & {{E}_{0}}{{\left[ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \theta _{2}^{2}} \right]}_{0}}  \\
\end{matrix} \right]</math>

The subscript <span class="texhtml">0</span> indicates that the quantity is evaluated at <math>{{\theta }_{1}}={{\theta }_{{{1}_{0}}}}</math> and <math>{{\theta }_{2}}={{\theta }_{{{2}_{0}}}},</math> the true values of the parameters. <br>So for a sample of <span class="texhtml">''N''</span> units where <span class="texhtml">''R''</span> units have failed, <span class="texhtml">''S''</span> have been suspended, and <span class="texhtml">''P''</span> have failed within a time interval, and <span class="texhtml">''N'' = ''R'' + ''M'' + ''P'',</span> one could obtain the sample local information matrix by: 

::<math>F={{\left[ \begin{matrix}
   -\tfrac{{{\partial }^{2}}\Lambda }{\partial \theta _{1}^{2}} & {} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\theta }_{1}}\partial {{\theta }_{2}}}  \\
   {} & {} & {}  \\
   -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\theta }_{2}}\partial {{\theta }_{1}}} & {} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \theta _{2}^{2}}  \\
\end{matrix} \right]}^{}}</math>

Substituting&nbsp;the values of the estimated parameters, in this case <math>{{\widehat{\theta }}_{1}}</math> and <math>{{\widehat{\theta }}_{2}}</math>, and then inverting the matrix, one can then obtain the local estimate of the covariance matrix or: 

::<math>\left[ \begin{matrix}
   \widehat{Var}\left( {{\widehat{\theta }}_{1}} \right) & {} & \widehat{Cov}\left( {{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}} \right)  \\
   {} & {} & {}  \\
   \widehat{Cov}\left( {{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}} \right) & {} & \widehat{Var}\left( {{\widehat{\theta }}_{2}} \right)  \\
\end{matrix} \right]={{\left[ \begin{matrix}
   -\tfrac{{{\partial }^{2}}\Lambda }{\partial \theta _{1}^{2}} & {} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\theta }_{1}}\partial {{\theta }_{2}}}  \\
   {} & {} & {}  \\
   -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\theta }_{2}}\partial {{\theta }_{1}}} & {} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \theta _{2}^{2}}  \\
\end{matrix} \right]}^{-1}}</math>

Then the variance of a function (<span class="texhtml">''V''''a''''r''(''G'')</span>) can be estimated using equation for the variance. Values for the variance and covariance of the parameters are obtained from Fisher Matrix equation. Once they have been obtained, the approximate confidence bounds on the function are given as: 

::<math>C{{B}_{R}}=E(G)\pm {{z}_{\alpha }}\sqrt{Var(G)}</math>

which is the estimated value plus or minus a certain number of standard deviations. We address finding <span class="texhtml">''z''<sub>α</sub></span> next.

=== Approximate Confidence Intervals on the Parameters  ===
In general, MLE estimates of the parameters are asymptotically normal, meaning that for large sample sizes,&nbsp;a distribution of parameter estimates from the same population would be very close to the normal distribution. Thus if <math>\widehat{\theta }</math> is the MLE estimator for <span class="texhtml">θ</span>, in the case of a single parameter distribution estimated from a large sample of <span class="texhtml">''n''</span> units, then: 

::<math>z\equiv \frac{\widehat{\theta }-\theta }{\sqrt{Var\left( \widehat{\theta } \right)}}</math>

follows an approximating normal distribution. That is 

::<math>P\left( x\le z \right)\to \Phi \left( z \right)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>

for large <span class="texhtml">''n''</span>. We now place confidence bounds on <span class="texhtml">θ,</span> at some confidence level <span class="texhtml">δ</span>, bounded by the two end points <span class="texhtml">''C''<sub>1</sub></span> and <span class="texhtml">''C''<sub>2</sub></span> where: 

::<math>P\left( {{C}_{1}}<\theta <{{C}_{2}} \right)=\delta </math>

From the above equation: 

::<math>P\left( -{{K}_{\tfrac{1-\delta }{2}}}<\frac{\widehat{\theta }-\theta }{\sqrt{Var\left( \widehat{\theta } \right)}}<{{K}_{\tfrac{1-\delta }{2}}} \right)\simeq \delta </math>

where <span class="texhtml">''K''<sub>α</sub></span> is defined by: 

::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi \left( {{K}_{\alpha }} \right)</math>

Now by simplifying the equation for the confidence level, one can obtain the approximate two-sided confidence bounds on the parameter <span class="texhtml">θ,</span> at a confidence level <span class="texhtml">δ,</span> or: 

::<math>\left( \widehat{\theta }-{{K}_{\tfrac{1-\delta }{2}}}\cdot \sqrt{Var\left( \widehat{\theta } \right)}<\theta <\widehat{\theta }+{{K}_{\tfrac{1-\delta }{2}}}\cdot \sqrt{Var\left( \widehat{\theta } \right)} \right)</math>

The upper one-sided bounds are given by: 

::<math>\theta <\widehat{\theta }+{{K}_{1-\delta }}\sqrt{Var(\widehat{\theta })}</math>

while the lower one-sided bounds are given by: 

::<math>\theta >\widehat{\theta }-{{K}_{1-\delta }}\sqrt{Var(\widehat{\theta })}</math>

If <math>\widehat{\theta }</math> must be positive, then <math>\ln \widehat{\theta }</math> is treated as normally distributed. The two-sided approximate confidence bounds on the parameter <span class="texhtml">θ</span>, at confidence level <span class="texhtml">δ</span>, then become: 

::<math>\begin{align}
  & {{\theta }_{U}}= & \widehat{\theta }\cdot {{e}^{\tfrac{{{K}_{\tfrac{1-\delta }{2}}}\sqrt{Var\left( \widehat{\theta } \right)}}{\widehat{\theta }}}}\text{ (Two-sided upper)} \\ 
 & {{\theta }_{L}}= & \frac{\widehat{\theta }}{{{e}^{\tfrac{{{K}_{\tfrac{1-\delta }{2}}}\sqrt{Var\left( \widehat{\theta } \right)}}{\widehat{\theta }}}}}\text{     (Two-sided lower)}  
\end{align}</math>

The one-sided approximate confidence bounds on the parameter <span class="texhtml">θ</span>, at confidence level <span class="texhtml">δ,</span> can be found from: 

::<math>\begin{align}
  & {{\theta }_{U}}= & \widehat{\theta }\cdot {{e}^{\tfrac{{{K}_{1-\delta }}\sqrt{Var\left( \widehat{\theta } \right)}}{\widehat{\theta }}}}\text{ (One-sided upper)} \\ 
 & {{\theta }_{L}}= & \frac{\widehat{\theta }}{{{e}^{\tfrac{{{K}_{1-\delta }}\sqrt{Var\left( \widehat{\theta } \right)}}{\widehat{\theta }}}}}\text{     (One-sided lower)}  
\end{align}</math>

The same procedure can be extended for the case of a two or more parameter distribution. Lloyd and Lipow [[Appendix: Weibull References|[24]]] further elaborate on this procedure.

=== Confidence Bounds on Time (Type 1)  ===
Type 1 confidence bounds are confidence bounds around time for a given reliability. For example, when using the one-parameter exponential distribution, the corresponding time for a given exponential percentile (i.e., y-ordinate or unreliability, <span class="texhtml">''Q'' = 1 − ''R'')</span> is determined by solving the unreliability function for the time, <span class="texhtml">''T''</span>, or: 

::<math>\begin{align}\widehat{T}(Q)= &-\frac{1}{\widehat{\lambda }}
                    \ln (1-Q)= & -\frac{1}{\widehat{\lambda }}\ln (R)
          \end{align}</math>

Bounds on time (Type 1) return the confidence bounds around this time value by determining the confidence intervals around <math>\widehat{\lambda }</math> and substituting these values into the above equation. The bounds on <math>\widehat{\lambda }</math> are determined using the method for the bounds on parameters, with its variance obtained from the Fisher Matrix. Note that the procedure is slightly more complicated for distributions with more than one parameter.

=== Confidence Bounds on Reliability (Type 2)  ===
Type 2 confidence bounds are confidence bounds around reliability. For example, when using the two-parameter exponential distribution, the reliability function is: 

::<math>\widehat{R}(T)={{e}^{-\widehat{\lambda }\cdot T}}</math>

Reliability bounds (Type 2) return the confidence bounds by determining the confidence intervals around <math>\widehat{\lambda }</math> and substituting these values into the above equation. The bounds on <math>\widehat{\lambda }</math> are determined using the method for the bounds on parameters, with its variance obtained from the Fisher Matrix. Once again, the procedure is more complicated for distributions with more than one parameter.

== 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 [[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 <span class="texhtml">''Z''</span> (rank for the <span class="texhtml">''j''<sup>''t''''h'''</sup></span> failure):

::<math>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 <span class="texhtml">''N''</span> is the sample size and <span class="texhtml">''j''</span> is the order number. <br>The median rank was obtained by solving the following equation for <span class="texhtml">''Z''</span>: 

::<math>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 <span class="texhtml">''P''</span> for <span class="texhtml">0.50</span> <span class="texhtml">(50%)</span> to our desired confidence level. For <span class="texhtml">''P'' = 90%</span> one would formulate the equation as 

::<math>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 <span class="texhtml">''P''</span> 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 <span class="texhtml">''P'' = 0.1</span> and once with <span class="texhtml">''P'' = 0.9</span>) in order to place the bounds around 80% of the population. <br><br>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. 

In Weibull++, this non-parametric methodology is&nbsp;used only&nbsp;when plotting bounds on the mixed Weibull distribution. Full details on this methodology can be found in [[Appendix D: Weibull References|Kececioglu [20]]]. These binomial equations can again be transformed using the beta and F distributions, thus the name beta binomial confidence bounds.

== Likelihood Ratio Confidence Bounds  ==
Another method for calculating confidence bounds is the likelihood ratio bounds (LRB) method. Conceptually, this method is a great deal simpler than that of the Fisher matrix, although that does not mean that the results are of any less value. In fact, the LRB method is often preferred over the FM method in situations where there are smaller sample sizes. 

Likelihood ratio confidence bounds are based on the following likelihood ratio equation: 

::<math>-2\cdot \text{ln}\left( \frac{L(\theta )}{L(\widehat{\theta })} \right)\ge \chi _{\alpha ;k}^{2}</math>

where: 

:*<span class="texhtml">''L''(θ)</span> is the likelihood function for the unknown parameter vector <span class="texhtml">θ</span> 
:*<math>L(\widehat{\theta })</math> is the likelihood function calculated at the estimated vector <math>\widehat{\theta }</math> 
:*<math>\chi _{\alpha ;k}^{2}</math> is the chi-squared statistic with probability <span class="texhtml">α</span> and <span class="texhtml">''k''</span> degrees of freedom, where <span class="texhtml">''k''</span> is the number of quantities jointly estimated

If <span class="texhtml">δ</span> is the confidence level, then <span class="texhtml">α = δ</span> for two-sided bounds and <span class="texhtml">α = (2δ − 1)</span> for one-sided. Recall from the [[Brief Statistical Background]] chapter that if <span class="texhtml">''x''</span> is a continuous random variable with ''pdf''</span>: 

::<span class="texhtml">''f''(''x'';θ<sub>1</sub>,θ<sub>2</sub>,...,θ<sub>''k''</sub>)</span>,

where <span class="texhtml">θ<sub>1</sub>,θ<sub>2</sub>,...,θ<sub>''k''</sub></span> are <span class="texhtml">''k''</span> unknown constant parameters that need to be estimated, one can conduct an experiment and obtain <span class="texhtml">''R''</span> independent observations, <span class="texhtml">''x''<sub>1</sub>,</span> <span class="texhtml">''x''<sub>2</sub>,</span><span class="texhtml">...,''x''<sub>''R''</sub></span>, which correspond in the case of life data analysis to failure times. The likelihood function is given by: 

::<math>L({{x}_{1}},{{x}_{2}},...,{{x}_{R}}|{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}})=L=\underset{i=1}{\overset{R}{\mathop \prod }}\,f({{x}_{i}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}})</math>

::<span class="texhtml">''i'' = 1,2,...,''R''</span>

The maximum likelihood estimators (MLE) of <span class="texhtml">θ<sub>1</sub>,θ<sub>2</sub>,...,θ<sub>''k''</sub>,</span> are obtained by maximizing <span class="texhtml">''L''.</span> These are represented by the <math>L(\widehat{\theta })</math> term in the denominator of the ratio in the likelihood ratio equation. Since the values of the data points are known, and the values of the parameter estimates <math>\widehat{\theta }</math> have been calculated using MLE methods, the only unknown term in the likelihood ratio equation is the <span class="texhtml">''L''(θ)</span> term in the numerator of the ratio. It remains to find the values of the unknown parameter vector <span class="texhtml">θ</span> that satisfy the likelihood ratio equation. For distributions that have two parameters, the values of these two parameters can be varied in order to satisfy the likelihood ratio equation. The values of the parameters that satisfy this equation will change based on the desired confidence level <span class="texhtml">δ;</span> but at a given value of <span class="texhtml">δ</span> there is only a certain region of values for <span class="texhtml">θ<sub>1</sub></span> and <span class="texhtml">θ<sub>2</sub></span> for which the likelihood ratio equation holds true. This region can be represented graphically as a contour plot, an example of which is given in the following graphic. 

[[Image:Examplecontourplot.png|center|350px]] 

The region of the contour plot essentially represents a cross-section of the likelihood function surface that satisfies the conditions of the likelihood ratio equation. 


'''Note on Contour Plots in Weibull++ and ALTA''' 

Contour plots can be used for comparing data sets. Consider two data sets, one for an&nbsp;old product design and another for a new design.&nbsp;The engineer would like to determine if the two designs are significantly different and at what confidence. By plotting the contour plots of each data set in an overlay&nbsp;plot (the same distribution must be fitted to each data set), one can determine the confidence at which the two sets are significantly different. If, for example, there is no overlap (i.e., the two plots do not intersect) between the two 90% contours, then the two data sets are significantly different with a 90% confidence. If&nbsp;the two 95% contours overlap, then the two designs are NOT significantly different at the 95% confidence level. An example of non-intersecting contours is shown next. For details on comparing data sets, see&nbsp;[[Comparing Life Data Sets]]. 

<br>[[Image:Contourplot.png|center|350px]] 


=== Confidence Bounds on the Parameters  ===
The bounds on the parameters are calculated by finding the extreme values of the contour plot on each axis for a given confidence level. Since each axis represents the possible values of a given parameter, the boundaries of the contour plot represent the extreme values of the parameters that satisfy the following: 

::<math>-2\cdot \text{ln}\left( \frac{L({{\theta }_{1}},{{\theta }_{2}})}{L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})} \right)=\chi _{\alpha ;1}^{2}</math>

This equation can be rewritten as: 

::<math>L({{\theta }_{1}},{{\theta }_{2}})=L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}</math>

The task now is to&nbsp;find the values of the parameters <span class="texhtml">θ<sub>1</sub></span> and <span class="texhtml">θ<sub>2</sub></span> so that the equality in&nbsp;the likelihood ratio equation shown above is satisfied. Unfortunately, there is no closed-form solution; therefore, these values must be arrived at numerically. One way to&nbsp;do this is to hold one parameter constant and iterate on the other until an acceptable solution is reached. This can prove to be rather tricky, since there will be two solutions for one parameter if the other is held constant. In situations such as these, it is best to begin the iterative calculations with values close to those of the MLE values, so as to ensure that one is not attempting to perform calculations outside of the region of the contour plot where no solution exists. 

<br>'''Example 1:''' {{Example: Likelihood Ratio Bounds on Parameters }} 


=== Confidence Bounds on Time (Type 1)  ===
The manner in which the bounds on the time estimate for a given reliability are calculated is much the same as the manner in which the bounds on the parameters are calculated. The difference lies in the form of the likelihood functions that comprise the likelihood ratio. In the preceding section, we used the standard form of the likelihood function, which was in terms of the parameters <span class="texhtml">θ<sub>1</sub></span> and <span class="texhtml">θ<sub>2</sub></span>. In order to calculate the bounds on a time estimate, the likelihood function needs to be rewritten in terms of one parameter and time, so that the maximum and minimum values of the time can be observed as the parameter is varied. This process is best illustrated with an example. 

<br>'''Example 2:''' {{Example: Likelihood Ratio Bounds on Time (Type I)}} 


=== Confidence Bounds on Reliability (Type 2)  ===
The likelihood ratio bounds on a reliability estimate for a given time value are calculated in the same manner as were the bounds on time. The only difference is that the likelihood function must now be considered in terms of <span class="texhtml">β</span> and <span class="texhtml">''R''</span>. The likelihood function is once again altered in the same way as before, only now <span class="texhtml">''R''</span> is considered to be a parameter instead of <span class="texhtml">''t''</span>, since the value of <span class="texhtml">''t''</span> must be specified in advance. Once again, this process is best illustrated with an example. 

<br>'''Example 3:''' {{Example: Likelihood Ratio Bounds on Reliability (Type 2)}}

== Bayesian Confidence Bounds  ==
A fourth method of estimating confidence bounds is based on the Bayes theorem. This type of confidence bounds relies on a different school of thought in statistical analysis, where prior information is combined with sample data in order to make inferences on model parameters and their functions. An introduction to Bayesian methods is given in the [[Parameter Estimation]] chapter. Bayesian confidence bounds are derived from Bayes's rule, which states that: 

::<math>f(\theta |Data)=\frac{L(Data|\theta )\varphi (\theta )}{\underset{\varsigma }{\int{\mathop{}_{}^{}}}\,L(Data|\theta )\varphi (\theta )d\theta }</math>

where: 
:*<math>f (</math><span class="texhtml">θ |</span> ''Data'') is the ''posterior pdf'' of <span class="texhtml">θ</span>
:*<span class="texhtml">θ</span> is the parameter vector of the chosen distribution (i.e., Weibull, lognormal, etc.) 
:*<math>L(\bullet )</math> is the likelihood function 
:*<math>\varphi (\theta )</math> is the ''prior pdf'' of the parameter vector <span class="texhtml">θ</span> 
:*<math>\varsigma </math> is the range of <span class="texhtml">θ</span>.

In other words, the prior knowledge is provided in the form of the prior ''pdf'' of the parameters, which in turn is combined with the sample data in order to obtain the posterior ''pdf''. Different forms of prior information exist, such as past data, expert opinion or non-informative (refer to [[Parameter Estimation]]). It can be seen from the above Bayes's rule formula that we are now dealing with distributions of parameters rather than single value parameters. For example, consider a one-parameter distribution with a positive parameter <span class="texhtml">θ<sub>1</sub></span>. Given a set of sample data, and a prior distribution for <span class="texhtml">θ<sub>1</sub>,</span> <math>\varphi ({{\theta }_{1}}),</math> the above Bayes's rule formula can be written as: 

::<math>f({{\theta }_{1}}|Data)=\frac{L(Data|{{\theta }_{1}})\varphi ({{\theta }_{1}})}{\int_{0}^{\infty }L(Data|{{\theta }_{1}})\varphi ({{\theta }_{1}})d{{\theta }_{1}}}</math>

In other words, we now have the distribution of <span class="texhtml">θ<sub>1</sub></span> and we can now make statistical inferences on this parameter, such as calculating probabilities. Specifically, the probability that <span class="texhtml">θ<sub>1</sub></span> is less than or equal to a value <span class="texhtml">''x'',</span> <math>P({{\theta }_{1}}\le x)</math> can be obtained by integrating the posterior probability density function (''pdf''), or: 

::<math>P({{\theta }_{1}}\le x)=\int_{0}^{x}f({{\theta }_{1}}|Data)d{{\theta }_{1}}</math>

The above equation is the posterior ''cdf'', which essentially calculates a confidence bound on the parameter, where <math>P({{\theta }_{1}}\le x)</math> is the confidence level and <span class="texhtml">''x''</span> is the confidence bound. Substituting the posterior ''pdf'' into the above posterior ''cdf'' yields: 

::<math>CL=\frac{\int_{0}^{x}L(Data|{{\theta }_{1}})\varphi ({{\theta }_{1}})d{{\theta }_{1}}}{\int_{0}^{\infty }L(Data|{{\theta }_{1}})\varphi ({{\theta }_{1}})d{{\theta }_{1}}}</math>

The only question at this point is, what do we use as a prior distribution of <span class="texhtml">θ<sub>1</sub>?</span> For the confidence bounds calculation application, non-informative prior distributions are utilized. Non-informative prior distributions are distributions that have no population basis and play a minimal role in the posterior distribution. The idea behind the use of non-informative prior distributions is to make inferences that are not affected by external information, or when external information is not available. In the general case of calculating confidence bounds using Bayesian methods, the method should be independent of external information and it should only rely on the current data. Therefore, non-informative priors are used. Specifically, the uniform distribution is used as a prior distribution for the different parameters of the selected fitted distribution. For example, if the Weibull distribution is fitted to the data, the prior distributions for beta and eta are assumed to be uniform. The above equation can be generalized for any distribution having a vector of parameters <span class="texhtml">θ,</span> yielding the general equation for calculating Bayesian confidence bounds: 

::<math>CL=\frac{\underset{\xi }{\int{\mathop{}_{}^{}}}\,L(Data|\theta )\varphi (\theta )d\theta }{\underset{\varsigma }{\int{\mathop{}_{}^{}}}\,L(Data|\theta )\varphi (\theta )d\theta }</math>

where: 

:*''CL'' is the confidence level
:*<span class="texhtml">θ</span> is the parameter vector 
:*<math>L(\bullet )</math> is the likelihood function 
:*<math>\varphi (\theta )</math> is the prior ''pdf'' of the parameter vector <span class="texhtml">θ</span> 
:*<math>\varsigma </math> is the range of <span class="texhtml">θ</span> 
:*<span class="texhtml">ξ</span> is the range in which <span class="texhtml">θ</span> changes from <span class="texhtml">Ψ(''T'',''R'')</span> till <span class="texhtml">''s''</span> maximum value, or from <span class="texhtml">θ</span> minimum value till <span class="texhtml">Ψ(''T'',''R'')</span> 
:*<span class="texhtml">Ψ(''T'',''R'')</span> is a function such that if <span class="texhtml">''T''</span> is given, then the bounds are calculated for <span class="texhtml">''R''</span> .&nbsp;If <span class="texhtml">''R''</span> is given, then the bounds are calculated for <span class="texhtml">''T''</span>.

If ''T'' is given, then from the above equation and <span class="texhtml">Ψ</span> and for a given ''CL'', the bounds on ''R'' are calculated. If ''R'' is given, then from the above equation and <span class="texhtml">Ψ</span> and for a given ''C'' the bounds on ''T'' are calculated. 

=== Confidence Bounds on Time (Type 1)  ===
For a given failure time distribution and a given reliability <span class="texhtml">''R''</span>, <span class="texhtml">''T''(''R'')</span> is a function of <span class="texhtml">''R''</span> and the distribution parameters. To illustrate the procedure for obtaining confidence bounds, the two-parameter Weibull distribution is used as an example. The bounds in&nbsp;other types of distributions can be obtained in similar fashion. For the two-parameter Weibull distribution: 

::<math>T(R)=\eta \exp (\frac{\ln (-\ln R)}{\beta })</math>

For a given reliability, the Bayesian one-sided upper bound estimate for <span class="texhtml">''T''(''R'')</span> is: 

::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(T\le {{T}_{U}})=\int_{0}^{{{T}_{U}}(R)}f(T|Data,R)dT</math>

where <span class="texhtml">''f''(''T'' | ''Data'',''R'') is the posterior distribution of Time <span class="texhtml">''T''.</span> Using the above equation, we have the following:

::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(T\le {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\eta \exp (\frac{\ln (-\ln R)}{\beta })\le {{T}_{U}})</math>

The above equation can be rewritten in terms of <span class="texhtml">η</span> as: 

::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\eta \le {{T}_{U}}\exp (-\frac{\ln (-\ln R)}{\beta }))</math>

Applying the Bayes's rule by assuming that the priors of <span class="texhtml">β</span> and <span class="texhtml">η</span> are independent, we then obtain the following relationship: 

::<math>CL=\frac{\int_{0}^{\infty }\int_{0}^{{{T}_{U}}\exp (-\frac{\ln (-\ln R)}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int_{0}^{\infty }\int_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }</math>

The above equation can be solved for <span class="texhtml">''T''<sub>''U''</sub>(''R'')</span>, where: 

:*''CL'' is the confidence level,
:*<math>\varphi (\beta )</math> is the prior ''pdf'' of the parameter <span class="texhtml">β</span>. For non-informative prior distribution, <math>\varphi (\beta )=\tfrac{1}{\beta }.</math> 
:*<math>\varphi (\eta )</math> is the prior ''pdf'' of the parameter <span class="texhtml">η.</span> For non-informative prior distribution, <math>\varphi (\eta )=\tfrac{1}{\eta }.</math> 
:*<math>L(\bullet )</math> is the likelihood function.

The same method can be used to get the one-sided lower bound of <span class="texhtml">''T''(''R'')</span> from: 

::<math>CL=\frac{\int_{0}^{\infty }\int_{{{T}_{L}}\exp (\frac{-\ln (-\ln R)}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int_{0}^{\infty }\int_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }</math>

The above equation can be solved to get <span class="texhtml">''T''<sub>''L''</sub>(''R'')</span>. <br>The Bayesian two-sided bounds estimate for <span class="texhtml">''T''(''R'')</span> is: 

::<math>CL=\int_{{{T}_{L}}(R)}^{{{T}_{U}}(R)}f(T|Data,R)dT</math>

which is equivalent to:

::<math>(1+CL)/2=\int_{0}^{{{T}_{U}}(R)}f(T|Data,R)dT</math>

and:

::<math>(1-CL)/2=\int_{0}^{{{T}_{L}}(R)}f(T|Data,R)dT</math>

Using the same method for the one-sided bounds, <span class="texhtml">''T''<sub>''U''</sub>(''R'')</span> and <span class="texhtml">''T''<sub>''L''</sub>(''R'')</span> can be solved. 

=== Confidence Bounds on Reliability (Type 2)  ===
For a given failure time distribution and a given time <span class="texhtml">''T''</span>, <span class="texhtml">''R''(''T'')</span> is a function of <span class="texhtml">''T''</span> and the distribution parameters. To illustrate the procedure for obtaining confidence bounds, the two-parameter Weibull distribution is used as an example. The bounds in other types of distributions can be obtained in similar fashion. For example, for two parameter Weibull distribution: 

::<math>R=\exp (-{{(\frac{T}{\eta })}^{\beta }})</math>

The Bayesian one-sided upper bound estimate for <span class="texhtml">''R''(''T'')</span> is: 

::<math>CL=\int_{0}^{{{R}_{U}}(T)}f(R|Data,T)dR</math>

Similar to the bounds on Time, the following is obtained: 

::<math>CL=\frac{\int_{0}^{\infty }\int_{0}^{T\exp (-\frac{\ln (-\ln {{R}_{U}})}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int_{0}^{\infty }\int_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }</math>

The above equation can be solved to get <span class="texhtml">''R''<sub>''U''</sub>(''T'')</span>. 

The Bayesian one-sided lower bound estimate for R(T) is: 

::<math>1-CL=\int_{0}^{{{R}_{L}}(T)}f(R|Data,T)dR</math>

Using the posterior distribution, the following is obtained: 

::<math>CL=\frac{\int_{0}^{\infty }\int_{T\exp (-\frac{\ln (-\ln {{R}_{L}})}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int_{0}^{\infty }\int_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }</math>

The above equation can be solved to get <span class="texhtml">''R''<sub>''L''</sub>(''T'')</span>. <br>The Bayesian two-sided bounds estimate for <span class="texhtml">''R''(''T'')</span> is: 

::<math>CL=\int_{{{R}_{L}}(T)}^{{{R}_{U}}(T)}f(R|Data,T)dR</math>

which is equivalent to: 

::<math>\int_{0}^{{{R}_{U}}(T)}f(R|Data,T)dR=(1+CL)/2</math>

and

::<math>\int_{0}^{{{R}_{L}}(T)}f(R|Data,T)dR=(1-CL)/2</math>

Using the same method for one-sided bounds, <span class="texhtml">''R''<sub>''U''</sub>(''T'')</span> and <span class="texhtml">''R''<sub>''L''</sub>(''T'')</span> can be solved.

== Simulation Based Bounds ==
The SimuMatic tool in Weibull++ can be used to perform a large number of reliability analyses on data sets that have been created using Monte Carlo simulation. This utility can assist the analyst to a) better understand life data analysis concepts, b) experiment with the influences of sample sizes and censoring schemes on analysis methods, c) construct simulation-based confidence intervals, d) better understand the concepts behind confidence intervals and e) design reliability tests. This section describes how to use simulation for estimating confidence bounds. <br> 

SimuMatic generates confidence bounds and assists in visualizing and understanding them. In addition, it allows one to determine the adequacy of certain parameter estimation methods (such as rank regression on X, rank regression on Y and maximum likelihood estimation) and to visualize the effects of different data censoring schemes on the confidence bounds. 


'''Example:'''
{{:Simulation_Based_Bounds_Example}}