knitr::opts_chunk$set( collapse = TRUE, screenshot.force = FALSE, comment = "#>" ) library(weibulltools)

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 [^note1]. This procedure is called **x on y** rank regression.

[^note1]: Berkson, J.: *Are There Two Regressions?*,
*Journal of the American Statistical Association 45 (250)*,
DOI: 10.2307/2280676, 1950, pp. 164-180

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 [^note2].

[^note2]: ReliaSoft Corporation: *Life Data Analysis Reference Book*,
online: ReliaSoft, accessed 19 December 2020

$$\hat{\eta}=\exp(\hat{a})=\exp(\hat{\mu})$$

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* [^note3] have shown that even in small
samples it is difficult to find an estimator that regularly has better properties
than ML estimators.

[^note3]: 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

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-defectives
(*censored*) observations. The data can be found in *Statistical Methods for Reliability Data* [^note4].

[^note4]: Meeker, W. Q.; Escobar, L. A.: *Statistical Methods for Reliability Data*,
*New York, Wiley series in probability and statistics*, 1998, p. 630

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.

shock_tbl <- reliability_data(data = shock, x = distance, status = status) shock_tbl

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 [^note5].

[^note5]: Meeker, W. Q.; Escobar, L. A.: *Statistical Methods for Reliability Data*,
*New York, Wiley series in probability and statistics*, 1998, p. 131

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

object:

# Data: alloy_tbl <- reliability_data(data = alloy, x = cycles, status = status) alloy_tbl

`weibulltools`

`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: shock_cdf <- estimate_cdf(shock_tbl, methods = "johnson") # Estimating two-parameter Weibull: rr_weibull <- rank_regression(shock_cdf, distribution = "weibull") rr_weibull # Probability plot: weibull_grid <- plot_prob( 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: weibull_plot <- plot_mod( weibull_grid, x = rr_weibull, title_trace = "Rank Regression" ) weibull_plot

# Again estimating Weibull: ml_weibull <- ml_estimation( shock_tbl, distribution = "weibull" ) ml_weibull # Add ML estimation to weibull_grid: weibull_plot2 <- plot_mod( 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_lognormal <- ml_estimation( alloy_tbl, distribution = "lognormal" ) ml_lognormal # Three-parameter Log-normal: ml_lognormal3 <- ml_estimation( alloy_tbl, distribution = "lognormal3" ) ml_lognormal3

# Constructing probability plot: tbl_cdf_john <- estimate_cdf(alloy_tbl, "johnson") lognormal_grid <- plot_prob( 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: lognormal_plot <- plot_mod( lognormal_grid, x = ml_lognormal, title_trace = "Two-parametric log-normal" ) lognormal_plot

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

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