Lognormal Distribution Examples

These examples also appear in the Life Data Analysis Reference book.

Lognormal Distribution General Example Complete Data
Determine the lognormal parameter estimates for the data given in the following table.

Solution

Using Weibull++, the computed parameters for maximum likelihood are:


 * $$\begin{align}

& {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & {\hat{\sigma '}}= & 1.10 \end{align}$$

For rank regression on $$X$$


 * $$\begin{align}

& {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & = & 1.24 \end{align}$$

For rank regression on $$Y:$$


 * $$\begin{align}

& {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & = & 1.36 \end{align}$$

Lognormal Distribution General Example Complete Data RRX
From Kececioglu [20, p. 347]. Fifteen identical units were tested to failure and following is a table of their times-to-failure:

Times-to-Failure Data

$$\begin{matrix} \text{Data Point Index} & \text{Time-to-Failure, hr} \\ \text{1} & \text{62}\text{.5} \\ \text{2} & \text{91}\text{.9} \\ \text{3} & \text{100}\text{.3} \\ \text{4} & \text{117}\text{.4} \\ \text{5} & \text{141}\text{.1} \\ \text{6} & \text{146}\text{.8} \\ \text{7} & \text{172}\text{.7} \\ \text{8} & \text{192}\text{.5} \\ \text{9} & \text{201}\text{.6} \\ \text{10} & \text{235}\text{.8} \\ \text{11} & \text{249}\text{.2} \\ \text{12} & \text{297}\text{.5} \\ \text{13} & \text{318}\text{.3} \\ \text{14} & \text{410}\text{.6} \\ \text{15} & \text{550}\text{.5} \\ \end{matrix}$$

Solution

Published results (using probability plotting):


 * $$\begin{matrix}

{{\widehat{\mu }}^{\prime }}=5.22575 \\ =0.62048. \\ \end{matrix}$$

Weibull++ computed parameters for rank regression on X are:


 * $$\begin{matrix}

{{\widehat{\mu }}^{\prime }}=5.2303 \\ =0.6283. \\ \end{matrix}$$

The small differences are due to the precision errors when fitting a line manually, whereas in Weibull++ the line was fitted mathematically.

Lognormal Distribution General Example Complete Data Unbiased MLE
From Kececioglu [19, p. 406]. Nine identical units are tested continuously to failure and their times-to-failure were recorded at 30.4, 36.7, 53.3, 58.5, 74.0, 99.3, 114.3, 140.1, and 257.9 hours.

Solution

The results published were obtained by using the unbiased model. Published Results (using MLE):


 * $$\begin{matrix}

{{\widehat{\mu }}^{\prime }}=4.3553 \\ =0.67677 \\ \end{matrix}$$

This same data set can be entered into Weibull++ by creating a data sheet capable of handling non-grouped time-to-failure data. Since the results shown above are unbiased, the Use Unbiased Std on Normal Data option in the User Setup must be selected in order to duplicate these results. Weibull++ computed parameters for maximum likelihood are:


 * $$\begin{matrix}

{{\widehat{\mu }}^{\prime }}=4.3553 \\ =0.6768 \\ \end{matrix}$$

Lognormal Distribution General Example Interval Data
Determine the lognormal parameter estimates for the data given in the table below.

Solution

This is a sequence of interval times-to-failure where the intervals vary substantially in length. Using Weibull++, the computed parameters for maximum likelihood are calculated to be:


 * $$\begin{align}

& {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & = & 0.18 \end{align}$$

For rank regression on $$X\ \ :$$


 * $$\begin{align}

& {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & = & 0.17 \end{align}$$

For rank regression on $$Y\ \ :$$


 * $$\begin{align}

& {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & = & 0.21 \end{align}$$

Lognormal Distribution General Example Suspension Data
From Nelson [30, p. 324]. Ninety-six locomotive controls were tested, 37 failed and 59 were suspended after running for 135,000 miles. Table 9.6 below shows their times-to-failure.

Solution

The distribution used in the publication was the base-10 lognormal. Published results (using MLE):


 * $$\begin{matrix}

{{\widehat{\mu }}^{\prime }}=2.2223 \\ =0.3064 \\ \end{matrix}$$

Published 95% confidence limits on the parameters:


 * $$\begin{matrix}

{{\widehat{\mu }}^{\prime }}=\left\{ 2.1336,2.3109 \right\} \\ =\left\{ 0.2365,0.3970 \right\} \\ \end{matrix}$$

Published variance/covariance matrix:


 * $$\left[ \begin{matrix}

\widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0020 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},)=0.001 \\ {} & {} & {} \\   \widehat{Cov}({{{\hat{\mu }}}^{\prime }},)=0.001 & {} & \widehat{Var}\left(  \right)=0.0016  \\ \end{matrix} \right]$$

To replicate the published results (since Weibull++ uses a lognormal to the base $$e$$ ), take the base-10 logarithm of the data and estimate the parameters using the Normal distribution and MLE.


 * Weibull++ computed parameters for maximum likelihood are:


 * $$\begin{matrix}

{{\widehat{\mu }}^{\prime }}=2.2223 \\ =0.3064 \\ \end{matrix}$$


 * Weibull++ computed 95% confidence limits on the parameters:


 * $$\begin{matrix}

{{\widehat{\mu }}^{\prime }}=\left\{ 2.1364,2.3081 \right\} \\ =\left\{ 0.2395,0.3920 \right\} \\ \end{matrix}$$


 * Weibull++ computed/variance covariance matrix:


 * $$\left[ \begin{matrix}

\widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0019 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},)=0.0009 \\ {} & {} & {} \\   \widehat{Cov}({\mu }',)=0.0009 & {} & \widehat{Var}\left(  \right)=0.0015  \\ \end{matrix} \right]$$