In contrast to point estimation procedures, interval estimation methods, e.g. the computation of confidence intervals, express the uncertainty which is associated with the use of a statistical estimator.

In reliability analysis it is common practice to provide confidence regions for the parameters of a lifetime distribution as well as for quantities that depend on these model parameters.

In this vignette, the determination of confidence intervals for model
parameters, quantiles (lifetime characteristic) and failure probabilites
(*CDF*) is presented.

Confidence intervals can be calculated for every model parameter. For
this, the (approximated) sampling distribution as well as an estimate of
its standard deviation must be given. In the following, the formulas
which strongly depend on the estimation methods, are provided for
*Rank Regression* and *Maximum Likelihood Estimation*.

In *Rank Regression* a linear relationship between the
lifetime characteristic and the failure probability is determined. The
parameters of a simple linear regression model are the intercept and the
slope, which are the location parameter \(\mu\) and the scale parameter \(\sigma\) for the majority of lifetime
distributions.

An approximated two-sided interval for the true parameters on a \(1 - \alpha\) confidence level can be obtained with the formulas

\[\bigg[\hat{\mu}_{\text{lower}} \, ; \hat{\mu}_{\text{upper}} \bigg] = \bigg[\hat{\mu}_{\text{RR}} \pm t_{1 - \frac{\alpha}{2}} \cdot \hat{se}^{\text{HC}}_{\hat{\mu}}\bigg] \qquad and \qquad \bigg[\hat{\sigma}_{\text{lower}} \, ;\hat{\sigma}_{\text{upper}} \bigg] = \bigg[\hat{\sigma}_{\text{RR}} \pm t_{1 - \frac{\alpha}{2}} \cdot \hat{se}^{\text{HC}}_{\hat{\sigma}}\bigg]. \]

For a given sample, \(\hat{\mu}_{\text{RR}}\) and \(\hat{\sigma}_{\text{RR}}\) are the least squares estimates and \(\hat{se}^{\text{HC}}_{\hat{\mu}}\) and \(\hat{se}^{\text{HC}}_{\hat{\sigma}}\) are the respective estimates of the standard deviations. The uncertainty that arises due to the unknown standard deviations must be taken into account by using the quantiles of Student’s t-distribution.

When using *RR* in the context of reliability analysis, the
assumption of homoscedastic error terms is often violated. Therefore,
the computation of the standard errors is based on a
heteroscedasticity-consistent (**HC**) variance-covariance
matrix. Other assumptions of the classical linear regression model like
the need of no serial correlation are questionable as well and have
already been discussed in the literature ^{1}.

Hence, the *Maximum Likelihood Estimation* procedure is
recommended, which is described in the next section.

*ML* estimators are subject to a variety of restrictions but
in return have many useful properties in contrast to other estimation
techniques. One is that *ML* estimators converge in distribution
to a normal distribution and for this reason, normal approximation
confidence intervals for the model parameters can be calculated by
theory.

Using the parameterization introduced above, a two-sided normal approximation confidence interval for the location parameter \(\mu\) can be computed with the equation

\[\bigg[\hat{\mu}_{\text{lower}} \, ; \, \hat{\mu}_{\text{upper}} \bigg] = \bigg[\hat{\mu}_{\text{MLE}} \pm z_{1 - \frac{\alpha}{2}} \cdot \hat{se}_{\hat{\mu}}\bigg].\]

By definition, the scale parameter \(\sigma\) is always positive and thus an
alternative confidence interval is used ^{2}:

\[\bigg[\hat{\sigma}_{\text{lower}} \, ; \, \hat{\sigma}_{\text{upper}} \bigg] = \bigg[\frac{\hat{\sigma}_{\text{MLE}}}{w} \, ; \hat{\sigma}_{\text{MLE}} \cdot w \bigg]\] with

\[w = \exp\left[z_{1 - \frac{\alpha}{2}} \cdot \frac{\hat{se}_{\hat{\sigma}}}{\hat{\sigma}_{\text{MLE}}}\right].\]

In addition to the confidence regions for the distribution-specific parameters, intervals for the regression line are provided as well. These can be aligned according to the probability \(F(t)\) or to the lifetime characteristic \(t\).

Whereas the Beta-Binomial confidence bounds are often used in
combination with *RR*, Fisher’s normal approximation confidence
intervals are only applicable in the case of *MLE*.

To obtain a two-sided non-parametric confidence interval for the
failure probabilities at a given \(1-\alpha\) level, a procedure similar to
*Median Ranks (MR)* is used.

Instead of finding the probability \(p_{\text{MR}}\) for the *j-th* rank
at the \(50\%\) level

\[0.5 = \sum^n_{k = j} \binom{n}{k} \cdot p_{\text{MR}}^k \cdot \left(1-p_{\text{MR}}\right)^{n-k}, \]

the probability \(p_{\text{lower}}\) must be determined for equation

\[\frac{\alpha}{2} = \sum^n_{k = j} \binom{n}{k} \cdot p_{\text{lower}}^k \cdot \left(1-p_{\text{lower}}\right)^{n-k}\]

and \(p_{\text{upper}}\) for the expression

\[1 - \frac{\alpha}{2} = \sum^n_{k = j} \binom{n}{k} \cdot p_{\text{upper}}^k \cdot \left(1-p_{\text{upper}}\right)^{n-k}.\]

The resulting interval \(\left[\hat{F}_{j,
\, {\text{lower}}} \, ; \, \hat{F}_{j, \, {\text{upper}}}\right] =
\left[\hat{p}_{{\text{lower}}} \, ; \,
\hat{p}_{{\text{upper}}}\right]\) is the estimated confidence
region for the true failure probability with respect to the
*j-th* rank.

Once the intervals of the failure probabilities are calculated, a two-sided confidence interval for the lifetime characteristic can be found with the quantile function of the underlying lifetime distribution.

For the Weibull, the quantile function is given by the formula

\[t_{p} = F^{-1}(p) = \exp\left[\mu + \Phi^{-1}_{\text{SEV}}(p) \cdot \sigma\right],\] where \(\Phi^{-1}_{\text{SEV}}\) is the quantile function of the standard smallest extreme value distribution.

The confidence interval for \(t\)
with respect to the estimated *RR* parameters as well as the
lower and upper probability of the *j-th* rank is then computed
by

\[\hat{t}_{j \, ; \, \text{lower}} = \exp\left[\hat{\mu}_{\text{RR}} + \Phi^{-1}_{\text{SEV}}(\hat{F}_{j, \, {\text{lower}}}) \cdot \hat{\sigma}_{\text{RR}}\right]\] and

\[\hat{t}_{j \, ; \, \text{upper}} = \exp\left[\hat{\mu}_{\text{RR}} + \Phi^{-1}_{\text{SEV}}(\hat{F}_{j, \, {\text{upper}}}) \cdot \hat{\sigma}_{\text{RR}}\right].\]

For a particular quantile \(t\) and the vector of parameters \(\hat{\theta}_{MLE}\), a normal approximation confidence interval for the failure probability \(F(t)\) can be obtained by

\[\bigg[\hat{F}_{\text{lower}}(t) \, ; \, \hat{F}_{\text{upper}}(t)\bigg] = \bigg[\hat{F}_{\text{MLE}}(t) \pm z_{1 - \frac{\alpha}{2}} \cdot \hat{se}_{\hat{F}(t)}\bigg].\]

In order to guarantee that the realized confidence interval of \(F(t)\) always is between 0 and 1, the so
called *z-procedure* can be applied ^{3}. Using this technique,
statistical inference is first done for the standardized quantile \(z\) and afterwards entered in \(F(t)\) to obtain the desired interval.

For the Weibull, the *ML* estimator of the standardized
quantile function \(z\) is

\[\hat{z}_{\text{MLE}} = \frac{log(t) - \hat{\mu}_{\text{MLE}}}{\hat{\sigma}_{\text{MLE}}}. \]

First, an approximate confidence interval for \(z\) is determined with the following formula:

\[\bigg[\hat{z}_{\text{lower}} \, ; \, \hat{z}_{\text{upper}}\bigg] = \bigg[\hat{z}_{\text{MLE}} \pm z_{1 - \frac{\alpha}{2}} \cdot \hat{se}_{\hat{z}}\bigg].\]

An approximate formula for the standard error of the estimator \(\hat{z}\) can be derived with the *delta
method*:

\[\hat{se}_{\hat{z}} = \sqrt{\hat{Var}_{\hat{z}}} = \sqrt{\bigg(\frac{\partial{\hat{z}}}{\partial{\hat{\theta}_{\text{MLE}}}}\bigg)^{T}\; \hat{Var}(\hat{\theta}_{\text{MLE}})\; \frac{\partial{\hat{z}}}{\partial{\hat{\theta}_{\text{MLE}}}}} \; .\]

Finally, the estimated bounds of \(z\) are then plugged into the
distribution-specific standard *CDF* to obtain the interval for
\(F(t)\), which is

\[\bigg[\hat{F}_{\text{lower}}(t) \, ; \, \hat{F}_{\text{upper}}(t)\bigg] = \bigg[\Phi_{\text{SEV}}(\hat{z}_{\text{lower}}) \, ; \, \Phi_{\text{SEV}}(\hat{z}_{\text{upper}}) \bigg].\]

In reliability analysis the lifetime characteristic often is defined as a strictly positive quantity and hence, a normal approximation confidence interval for the quantile \(t\) with respect to a particular probability \(p\) and the vector of parameters \(\hat{\theta}_{MLE}\), can be calculated by

\[\bigg[\hat{t}_{\text{lower}}(p) \, ; \, \hat{t}_{\text{upper}}(p)\bigg] = \bigg[\frac{\hat{t}_{\text{MLE}}(p)}{w} \, ; \hat{t}_{\text{MLE}}(p) \cdot w \bigg], \] where \(w\) is

\[w = \exp\left[z_{1 - \frac{\alpha}{2}} \cdot \frac{\hat{se}_{\hat{t}(p)}}{\hat{t}_{\text{MLE}}(p)}\right].\]

For the Weibull, the *ML* equation for the quantile \(t(p)\) is

\[\hat{t}_{\text{MLE}}(p) = \exp\left[\hat{\mu}_{\text{MLE}} + \Phi^{-1}_{\text{SEV}}(p) \cdot \hat{\sigma}_{\text{MLE}}\right]\]

and again, through the use of the *delta method*, a formula
for the standard error of \(\hat{t}_p\)
can be provided, which is

\[\hat{se}_{\hat{t}(p)} = \sqrt{\hat{Var}_{\hat{t}(p)}} = \sqrt{\bigg(\frac{\partial{\hat{t}(p)}}{\partial{\hat{\theta}_{\text{MLE}}}}\bigg)^{T}\; \hat{Var}(\hat{\theta}_{\text{MLE}})\; \frac{\partial{\hat{t}(p)}}{\partial{\hat{\theta}_{\text{MLE}}}}} \; .\]

For the computation of the presented confidence intervals the
`shock`

dataset is used. In this dataset kilometer-dependent
problems that have occurred on shock absorbers are reported. In addition
to failed items the dataset also contains non-defective
(*censored*) observations. The data can be found in
*Statistical Methods for Reliability Data* ^{4}.

For consistent handling of the data, {weibulltools} introduces the
function `reliability_data()`

that converts the original
dataset into a `wt_reliability_data`

object. This formatted
object allows to easily apply the presented methods.

```
# Data:
<- reliability_data(data = shock, x = distance, status = status)
shock_tbl
shock_tbl#> Reliability Data with characteristic x: 'distance':
#> # A tibble: 38 × 3
#> x status id
#> <int> <dbl> <chr>
#> 1 6700 1 ID1
#> 2 6950 0 ID2
#> 3 7820 0 ID3
#> 4 8790 0 ID4
#> 5 9120 1 ID5
#> 6 9660 0 ID6
#> 7 9820 0 ID7
#> 8 11310 0 ID8
#> 9 11690 0 ID9
#> 10 11850 0 ID10
#> # ℹ 28 more rows
```

Before calculating confidence intervals with {weibulltools} one has to conduct the basic steps of the Weibull analysis which are described in the previous vignettes.

```
# Estimation of failure probabilities:
<- estimate_cdf(shock_tbl, methods = "johnson")
shock_cdf
# Rank Regression:
<- rank_regression(shock_cdf, distribution = "weibull")
rr_weibull
# Maximum Likelihood Estimation:
<- ml_estimation(shock_tbl, distribution = "weibull") ml_weibull
```

The confidence intervals for the distribution parameters are included
in the model output of `rank_regression()`

and
`ml_estimation()`

, respectively.

```
# Confidence intervals based on Rank Regression:
$confint
rr_weibull#> 2.5 % 97.5 %
#> mu 10.2079027 10.3112576
#> sigma 0.3428984 0.3835118
# Confidence intervals based on Maximum Likelihood Estimation:
$confint
ml_weibull#> 2.5 % 97.5 %
#> mu 10.0144824 10.4452437
#> sigma 0.2011036 0.4978267
```

The `confint`

element of the model output is a matrix with
the parameter names as row names and the confidence level as column
names. Different levels can be specified using the argument
`conf_level`

.

```
# Confidence intervals based on another confidence level:
<- ml_estimation(shock_tbl, distribution = "weibull", conf_level = 0.99)
ml_weibull_99 $confint
ml_weibull_99#> 0.5 % 99.5 %
#> mu 9.9468049 10.5129212
#> sigma 0.1744101 0.5740192
```

Confidence bounds for failure probabilities can be either determined
with `confint_betabinom()`

or `confint_fisher()`

.
As explained in the theoretical part of this vignette the Beta-Binomial
confidence bounds should be applied to the output of
`rank_regression()`

whereas Fisher’s normal approximation
confidence intervals are only applicable if the parameters and the
variance-covariance matrix were estimated with
`ml_estimation()`

.

```
# Beta-Binomial confidence bounds:
<- confint_betabinom(
conf_bb x = rr_weibull,
b_lives = c(0.01, 0.1, 0.5),
bounds = "two_sided",
conf_level = 0.95,
direction = "y"
)
conf_bb#> # A tibble: 102 × 6
#> x rank prob lower_bound upper_bound cdf_estimation_method
#> * <dbl> <dbl> <dbl> <dbl> <dbl> <chr>
#> 1 6700. 1.00 0.0183 0.000674 0.0927 johnson
#> 2 6910 1.06 0.0199 0.000859 0.0958 johnson
#> 3 7120. 1.13 0.0216 0.00108 0.0990 johnson
#> 4 7330. 1.20 0.0234 0.00134 0.102 johnson
#> 5 7540 1.27 0.0252 0.00165 0.106 johnson
#> 6 7750 1.34 0.0272 0.00201 0.109 johnson
#> 7 7960. 1.42 0.0293 0.00242 0.113 johnson
#> 8 8170 1.51 0.0314 0.00289 0.117 johnson
#> 9 8380 1.59 0.0336 0.00341 0.121 johnson
#> 10 8590 1.68 0.0360 0.00401 0.124 johnson
#> # ℹ 92 more rows
# Fisher's normal approximation confidence intervals:
<- confint_fisher(x = ml_weibull)
conf_fisher
conf_fisher#> # A tibble: 102 × 6
#> x prob std_err lower_bound upper_bound cdf_estimation_method
#> * <dbl> <dbl> <dbl> <dbl> <dbl> <chr>
#> 1 6700. 0.0112 0.916 0.00186 0.0655 <NA>
#> 2 6910 0.0123 0.895 0.00214 0.0691 <NA>
#> 3 7120. 0.0135 0.875 0.00245 0.0729 <NA>
#> 4 7330. 0.0148 0.855 0.00279 0.0767 <NA>
#> 5 7540 0.0162 0.835 0.00317 0.0805 <NA>
#> 6 7750 0.0177 0.817 0.00359 0.0845 <NA>
#> 7 7960. 0.0192 0.799 0.00404 0.0886 <NA>
#> 8 8170 0.0208 0.781 0.00454 0.0927 <NA>
#> 9 8380 0.0225 0.764 0.00509 0.0969 <NA>
#> 10 8590 0.0244 0.747 0.00568 0.101 <NA>
#> # ℹ 92 more rows
```

The outputs of both functions contain the calculated bounds for the failure probabilities ranging from the minimum to the maximum observed failure. Between the observed range of failures an interpolation of quantiles is made for which the intervals of the probabilities are provided as well (supporting points).

In the function call of `confint_betabinom()`

the default
arguments of both functions are listed. With the argument
`b_lives`

, confidence regions for selected probabilities are
included, but only if they are in the range of the estimated failure
probabilities.

The argument `bounds`

is used for the specification of the
bound(s) to be computed. It could be one of
`c("two_sided", "lower", "upper")`

.

If `direction = "y"`

, confidence intervals for the
probabilities are provided.

The visualization of the computed intervals is done with
`plot_conf()`

.

```
# Probability plot
<- plot_prob(
weibull_grid
shock_cdf,distribution = "weibull",
title_main = "Weibull Probability Plot",
title_x = "Mileage in km",
title_y = "Probability of Failure in %",
title_trace = "Defectives",
plot_method = "ggplot2"
)
```

```
# Beta-Binomial confidence intervals:
<- plot_conf(
weibull_conf_bb
weibull_grid,
conf_bb, title_trace_mod = "Rank Regression",
title_trace_conf = "Beta-Binomial Bounds"
) weibull_conf_bb
```

```
# Fisher's normal approximation confidence intervals:
<- plot_conf(
weibull_conf_fisher
weibull_grid,
conf_fisher, title_trace_mod = "Maximum Likelihood",
title_trace_conf = "Fisher's Confidence Intervals"
) weibull_conf_fisher
```

As one can see, `plot_conf()`

not only adds the confidence
limits to an existing probability plot, but also includes the estimated
linearized CDF. There is no need for an additional call of
`plot_mod()`

. In fact, the same routines used by
`plot_mod()`

are called under the hood, which ensures that
confidence bounds are not drawn without the regression line.

The computation and visualization of confidence for the lifetime
characteristic is pretty similar to the presented procedure with regard
to the probabilities. The only difference is that one has to change the
value of the argument `direction`

to `"x"`

.

```
# Computation of confidence intervals for quantiles:
## Beta-Binomial confidence intervals:
<- confint_betabinom(
conf_bb_x x = rr_weibull,
bounds = "upper",
conf_level = 0.95,
direction = "x"
)
conf_bb_x#> # A tibble: 102 × 5
#> x rank prob upper_bound cdf_estimation_method
#> * <dbl> <dbl> <dbl> <dbl> <chr>
#> 1 6700. 1.00 0.0183 11357. johnson
#> 2 6910 1.06 0.0199 11521. johnson
#> 3 7120. 1.13 0.0216 11687. johnson
#> 4 7330. 1.20 0.0234 11855. johnson
#> 5 7540 1.27 0.0252 12024. johnson
#> 6 7750 1.34 0.0272 12195. johnson
#> 7 7960. 1.42 0.0293 12367. johnson
#> 8 8170 1.51 0.0314 12541. johnson
#> 9 8380 1.59 0.0336 12715. johnson
#> 10 8590 1.68 0.0360 12891. johnson
#> # ℹ 92 more rows
## Fisher's normal approximation confidence intervals:
<- confint_fisher(x = ml_weibull, bounds = "lower", direction = "x")
conf_fisher_x
conf_fisher_x#> # A tibble: 102 × 5
#> x prob std_err lower_bound cdf_estimation_method
#> * <dbl> <dbl> <dbl> <dbl> <chr>
#> 1 6700. 0.0112 1943. 4159. <NA>
#> 2 6910 0.0123 1957. 4337. <NA>
#> 3 7120. 0.0135 1970. 4517. <NA>
#> 4 7330. 0.0148 1982. 4698. <NA>
#> 5 7540 0.0162 1993. 4881. <NA>
#> 6 7750 0.0177 2003. 5067. <NA>
#> 7 7960. 0.0192 2011. 5253. <NA>
#> 8 8170 0.0208 2019. 5441. <NA>
#> 9 8380 0.0225 2025. 5631. <NA>
#> 10 8590 0.0244 2031. 5822. <NA>
#> # ℹ 92 more rows
```

```
# Visualization:
## Beta-Binomial confidence intervals:
<- plot_conf(
weibull_conf_bb_x
weibull_grid,
conf_bb_x, title_trace_mod = "Rank Regression",
title_trace_conf = "Beta-Binomial Bounds"
) weibull_conf_bb_x
```

```
## Fisher's normal approximation confidence intervals:
<- plot_conf(
weibull_conf_fisher_x
weibull_grid,
conf_fisher_x, title_trace_mod = "Maximum Likelihood",
title_trace_conf = "Fisher's Confidence Intervals"
) weibull_conf_fisher_x
```

Genschel, U.; Meeker, W. Q.:

*A Comparison of Maximum Likelihood and Median-Rank Regression for Weibull Estimation*, in:*Quality Engineering 22 (4)*, DOI: 10.1080/08982112.2010.503447, 2010, pp. 236-255↩︎Meeker, W. Q.; Escobar, L. A.:

*Statistical Methods for Reliability Data*,*New York, Wiley series in probability and statistics*, 1998, p. 188↩︎Hoang, Y.; Meeker, W. Q.; Escobar, L. A.:

*The Relationship Between Confidence Intervals for Failure Probabilities and Life Time Quantiles*, in: _IEEE Transactions on Reliability 57, 2008, pp. 260-266↩︎Meeker, W. Q.; Escobar, L. A.:

*Statistical Methods for Reliability Data*,*New York, Wiley series in probability and statistics*, 1998, p. 630↩︎