This document introduces two methods for the parameter estimation of
lifetime distributions. Whereas *Rank Regression (RR)* fits a
straight line through transformed plotting positions (transformation is
described precisely in
`vignette(topic = "Life_Data_Analysis_Part_I", package = "weibulltools")`

),
*Maximum likelihood (ML)* strives to maximize a function of the
parameters given the sample data. If the parameters are obtained, a
cumulative distribution function *(CDF)* can be computed and
added to a probability plot.

In the theoretical part of this vignette the focus is on the
two-parameter Weibull distribution. The second part is about the
application of the provided estimation methods in {weibulltools}. All
implemented models can be found in the help pages of
`rank_regression()`

and `ml_estimation()`

.

The Weibull distribution is a continuous probability distribution,
which is specified by the location parameter \(\mu\) and the scale parameter \(\sigma\). Its *CDF* and *PDF
(probability density function)* are given by the following
formulas:

\[F(t)=\Phi_{SEV}\left(\frac{\log(t) - \mu}{\sigma}\right)\]

\[f(t)=\frac{1}{\sigma t}\;\phi_{SEV}\left(\frac{\log(t) - \mu}{\sigma}\right)\] The practical benefit of the Weibull in the field of lifetime analysis is that the common profiles of failure rates, which are observed over the lifetime of a large number of technical products, can be described using this statistical distribution.

In the following, the estimation of the specific parameters \(\mu\) and \(\sigma\) is explained.

In *RR* the *CDF* is linearized such that the true,
unknown population is estimated by a straight line which is analytically
placed among the plotting pairs. The lifetime characteristic, entered on
the x-axis, is displayed on a logarithmic scale. A double-logarithmic
representation of the estimated failure probabilities is used for the
y-axis. Ordinary Least Squares *(OLS)* determines a best-fit line
in order that the sum of squared deviations between this fitted
regression line and the plotted points is minimized.

In reliability analysis, it became prevalent that the line is placed
in the probability plot in the way that the horizontal distances between
the best-fit line and the points are minimized ^{1}. This procedure is
called **x on y** rank regression.

The formulas for estimating the slope and the intercept of the regression line according to the described method are given below.

Slope:

\[\hat{b}=\frac{\sum_{i=1}^{n}(x_i-\bar{x})\cdot(y_i-\bar{y})}{\sum_{i=1}^{n}(y_i-\bar{y})^2}\]

Intercept:

\[\hat{a}=\bar{x}-\hat{b}\cdot\bar{y}\]

With

\[x_i=\log(t_i)\;;\; \bar{x}=\frac{1}{n}\cdot\sum_{i=1}^{n}\log(t_i)\;;\]

as well as

\[y_i=\Phi^{-1}_{SEV}\left[F(t)\right]=\log\left\{-\log\left[1-F(t_i)\right]\right\}\;and \; \bar{y}=\frac{1}{n}\cdot\sum_{i=1}^{n}\log\left\{-\log\left[1-F(t_i)\right]\right\}.\]

The estimates of the intercept and slope are equal to the Weibull parameters \(\mu\) and \(\sigma\), i.e.

\[\hat{\mu}=\hat{a}\]

and

\[\hat{\sigma}=\hat{b}.\]

In order to obtain the parameters of the shape-scale parameterization
the intercept and the slope need to be transformed ^{2}.

\[\hat{\eta}=\exp(\hat{a})=\exp(\hat{\mu})\]

and

\[\hat{\beta}=\frac{1}{\hat{b}}=\frac{1}{\hat{\sigma}}.\]

The *ML* method of Ronald A. Fisher estimates the parameters
by maximizing the likelihood function. Assuming a theoretical
distribution, the idea of *ML* is that the specific parameters
are chosen in such a way that the plausibility of obtaining the present
sample is maximized. The likelihood and log-likelihood are given by the
following equations:

\[L = \prod_{i=1}^n\left\{\frac{1}{\sigma t_i}\;\phi_{SEV}\left(\frac{\log(t_i) - \mu}{\sigma}\right)\right\}\]

and

\[\log L = \sum_{i=1}^n\log\left\{\frac{1}{\sigma t_i}\;\phi_{SEV}\left(\frac{\log(t_i) - \mu}{\sigma}\right)\right\}\]

Deriving and nullifying the log-likelihood function according to parameters results in two formulas that have to be solved numerically in order to obtain the estimates.

In large samples, ML estimators have optimality properties. In
addition, the simulation studies by *Genschel and Meeker* ^{3} have shown
that even in small samples it is difficult to find an estimator that
regularly has better properties than ML estimators.

To apply the introduced parameter estimation methods the
`shock`

and `alloy`

datasets are 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.

```
<- 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
```

The dataset `alloy`

in which the cycles until a fatigue
failure of a special alloy occurs are inspected. The data is also taken
from Meeker and Escobar ^{5}.

Again, the data have to be formatted as a
`wt_reliability_data`

object:

```
# Data:
<- reliability_data(data = alloy, x = cycles, status = status)
alloy_tbl
alloy_tbl#> Reliability Data with characteristic x: 'cycles':
#> # A tibble: 72 × 3
#> x status id
#> <dbl> <dbl> <chr>
#> 1 300 0 ID1
#> 2 300 0 ID2
#> 3 300 0 ID3
#> 4 300 0 ID4
#> 5 300 0 ID5
#> 6 291 1 ID6
#> 7 274 1 ID7
#> 8 271 1 ID8
#> 9 269 1 ID9
#> 10 257 1 ID10
#> # ℹ 62 more rows
```

`rank_regression()`

and `ml_estimation()`

can
be applied to complete data as well as failure and (multiple)
right-censored data. Both methods can also deal with models that have a
threshold parameter \(\gamma\).

In the following both methods are applied to the dataset
`shock`

.

```
# rank_regression needs estimated failure probabilities:
<- estimate_cdf(shock_tbl, methods = "johnson")
shock_cdf
# Estimating two-parameter Weibull:
<- rank_regression(shock_cdf, distribution = "weibull")
rr_weibull
rr_weibull #> Rank Regression
#> Coefficients:
#> mu sigma
#> 10.2596 0.3632
# 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"
)
# Add regression line:
<- plot_mod(
weibull_plot
weibull_grid,x = rr_weibull,
title_trace = "Rank Regression"
)
weibull_plot
```

```
# Again estimating Weibull:
<- ml_estimation(
ml_weibull
shock_tbl, distribution = "weibull"
)
ml_weibull #> Maximum Likelihood Estimation
#> Coefficients:
#> mu sigma
#> 10.2299 0.3164
# Add ML estimation to weibull_grid:
<- plot_mod(
weibull_plot2
weibull_grid, x = ml_weibull,
title_trace = "Maximum Likelihood"
)
weibull_plot2
```

Finally, two- and three-parametric log-normal distributions are
fitted to the `alloy`

data using maximum likelihood.

```
# Two-parameter log-normal:
<- ml_estimation(
ml_lognormal
alloy_tbl,distribution = "lognormal"
)
ml_lognormal#> Maximum Likelihood Estimation
#> Coefficients:
#> mu sigma
#> 5.1278 0.3276
# Three-parameter Log-normal:
<- ml_estimation(
ml_lognormal3
alloy_tbl,distribution = "lognormal3"
)
ml_lognormal3#> Maximum Likelihood Estimation
#> Coefficients:
#> mu sigma gamma
#> 4.5015 0.6132 72.0727
```

```
# Constructing probability plot:
<- estimate_cdf(alloy_tbl, "johnson")
tbl_cdf_john
<- plot_prob(
lognormal_grid
tbl_cdf_john,distribution = "lognormal",
title_main = "Log-normal Probability Plot",
title_x = "Cycles",
title_y = "Probability of Failure in %",
title_trace = "Failed units",
plot_method = "ggplot2"
)
# Add two-parametric model to grid:
<- plot_mod(
lognormal_plot
lognormal_grid,x = ml_lognormal,
title_trace = "Two-parametric log-normal"
)
lognormal_plot
```

```
# Add three-parametric model to lognormal_plot:
<- plot_mod(
lognormal3_plot
lognormal_grid, x = ml_lognormal3,
title_trace = "Three-parametric log-normal"
)
lognormal3_plot
```

Berkson, J.:

*Are There Two Regressions?*,*Journal of the American Statistical Association 45 (250)*, DOI: 10.2307/2280676, 1950, pp. 164-180↩︎ReliaSoft Corporation:

*Life Data Analysis Reference Book*, online: ReliaSoft, accessed 19 December 2020↩︎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. 630↩︎Meeker, W. Q.; Escobar, L. A.:

*Statistical Methods for Reliability Data*,*New York, Wiley series in probability and statistics*, 1998, p. 131↩︎