The Lognormal Distribution

The lognormal distribution is commonly used to model the lives of units whose failure modes are of a fatigue-stress nature. Since this includes most, if not all, mechanical systems, the lognormal distribution can have widespread application. Consequently, the lognormal distribution is a good companion to the Weibull distribution when attempting to model these types of units. As may be surmised by the name, the lognormal distribution has certain similarities to the normal distribution. A random variable is lognormally distributed if the logarithm of the random variable is normally distributed. Because of this, there are many mathematical similarities between the two distributions. For example, the mathematical reasoning for the construction of the probability plotting scales and the bias of parameter estimators is very similar for these two distributions.

Lognormal Probability Density Function
The lognormal distribution is a 2-parameter distribution with parameters $${\mu }'$$  and  $$\sigma'$$. The $$pdf$$  for this distribution is given by:


 * $$f({t}')=\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{t}^{\prime }}-{\mu }'}{} \right)}^{2}}}}$$

where,


 * $${t}'=\ln (t)$$. $$t$$ values are the times-to-failure, and


 * $$\mu'=\text{mean of the natural logarithms of the times-to-failure,}$$


 * $$\sigma'=\text{standard deviation of the natural logarithms of the times-to-failure}$$

The lognormal $$pdf$$  can be obtained, realizing that for equal probabilities under the normal and lognormal  $$pdf$$ s, incremental areas should also be equal, or:


 * $$f(t)dt=f({t}')d{t}'$$

Taking the derivative yields:


 * $$d{t}'=\frac{dt}{t}$$

Substitution yields:


 * $$\begin{align}

f(t)= & \frac{f({t}')}{t} \\ f(t)= & \frac{1}{t\cdot \sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-{\mu }'}{} \right)}^{2}}}} \end{align}$$

where:


 * $$f(t)\ge 0,t>0,-\infty <{\mu }'<\infty ,{{\sigma' }}>0$$

Probability Plotting
As described before, probability plotting involves plotting the failure times and associated unreliability estimates on specially constructed probability plotting paper. The form of this paper is based on a linearization of the $$cdf$$  of the specific distribution. For the lognormal distribution, the cumulative density function can be written as:


 * $$F({t}')=\Phi \left( \frac{{t}'-{\mu }'} \right)$$

or:


 * $${{\Phi }^{-1}}\left[ F({t}') \right]=-\frac+\frac{1}\cdot {t}'$$

where:


 * $$\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{2}}}dt$$

Now, let:


 * $$y={{\Phi }^{-1}}\left[ F({t}') \right]$$


 * $$a=-\frac$$

and:


 * $$b=\frac{1}$$

which results in the linear equation of:


 * $$y=a+b{t}'$$

The normal probability paper resulting from this linearized $$cdf$$  function is shown next.

The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see The Normal Distribution Chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.

The process of lognormal probability plotting is illustrated in the following example.

Example 1:

Rank Regression on Y
Performing a rank regression on Y requires that a straight line be fitted to a set of data points such that the sum of the squares of the vertical deviations from the points to the line is minimized.

The least squares parameter estimation method, or regression analysis, was discussed in Parameter Estimation Chapter and the following equations for regression on Y were derived, and are again applicable:


 * $$\hat{a}=\bar{y}-\hat{b}\bar{x}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}-\hat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}}{N}$$

and:


 * $$\hat{b}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}{{y}_{i}}-\tfrac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}}{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,x_{i}^{2}-\tfrac{N}}$$

In our case the equations for $${{y}_{i}}$$  and $$x_{i}$$ are:


 * $${{y}_{i}}={{\Phi }^{-1}}\left[ F(t_{i}^{\prime }) \right]$$

and:


 * $${{x}_{i}}=t_{i}^{\prime }$$

where the $$F(t_{i}^{\prime })$$  is estimated from the median ranks. Once $$\widehat{a}$$  and  $$\widehat{b}$$  are obtained, then  $$\widehat{\sigma }$$  and  $$\widehat{\mu }$$  can easily be obtained from the above equations.

Example 2:

Rank Regression on X
Performing a rank regression on X requires that a straight line be fitted to a set of data points such that the sum of the squares of the horizontal deviations from the points to the line is minimized.

Again, the first task is to bring our $$cdf$$  function into a linear form. This step is exactly the same as in regression on Y analysis and all the equations apply in this case too. The deviation from the previous analysis begins on the least squares fit part, where in this case we treat $$x$$  as the dependent variable and  $$y$$  as the independent variable. The best-fitting straight line to the data, for regression on X (see Chapter Parameter Estimation), is the straight line:


 * $$x=\widehat{a}+\widehat{b}y$$

The corresponding equations for   and  $$\widehat{b}$$  are:


 * $$\hat{a}=\overline{x}-\hat{b}\overline{y}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}}{N}-\hat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}$$

and:


 * $$\hat{b}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}{{y}_{i}}-\tfrac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}}{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,y_{i}^{2}-\tfrac{N}}$$

where:


 * $${{y}_{i}}={{\Phi }^{-1}}\left[ F(t_{i}^{\prime }) \right]$$

and:


 * $${{x}_{i}}=t_{i}^{\prime }$$

and the $$F(t_{i}^{\prime })$$  is estimated from the median ranks. Once $$\widehat{a}$$  and  $$\widehat{b}$$  are obtained, solve the linear equation for the unknown  $$y$$, which corresponds to:


 * $$y=-\frac{\widehat{a}}{\widehat{b}}+\frac{1}{\widehat{b}}x$$

Solving for the parameters we get:


 * $$a=-\frac{\widehat{a}}{\widehat{b}}=-\frac{\sigma'}$$

and:


 * $$b=\frac{1}{\widehat{b}}=\frac{1}{\sigma'}$$

The correlation coefficient is evaluated as before using equation in the previous section.

Example 3:

Maximum Likelihood Estimation
As it was outlined in Chapter Parameter Estimation, maximum likelihood estimation works by developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function. This can be achieved by using iterative methods to determine the parameter estimate values that maximize the likelihood function. However, this can be rather difficult and time-consuming, particularly when dealing with the three-parameter distribution. Another method of finding the parameter estimates involves taking the partial derivatives of the likelihood equation with respect to the parameters, setting the resulting equations equal to zero, and solving simultaneously to determine the values of the parameter estimates. The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the lognormal distribution are covered in Appendix D .

Note About Bias

See the discussion regarding bias with the normal distribution for information regarding parameter bias in the lognormal distribution.

Confidence Bounds
The method used by the application in estimating the different types of confidence bounds for lognormally distributed data is presented in this section. Note that there are closed-form solutions for both the normal and lognormal reliability that can be obtained without the use of the Fisher information matrix. However, these closed-form solutions only apply to complete data. To achieve consistent application across all possible data types, Weibull++ always uses the Fisher matrix in computing confidence intervals. The complete derivations were presented in detail for a general function in Chapter Confidence Bounds. For a discussion on exact confidence bounds for the normal and lognormal, see Chapter The Normal Distribution.

Bounds on the Parameters
The lower and upper bounds on the mean, $${\mu }'$$, are estimated from:


 * $$\begin{align}

& \mu _{U}^{\prime }= & {{\widehat{\mu }}^{\prime }}+{{K}_{\alpha }}\sqrt{Var({{\widehat{\mu }}^{\prime }})}\text{ (upper bound),} \\ & \mu _{L}^{\prime }= & {{\widehat{\mu }}^{\prime }}-{{K}_{\alpha }}\sqrt{Var({{\widehat{\mu }}^{\prime }})}\text{ (lower bound)}\text{.} \end{align}$$

For the standard deviation, $${\widehat{\sigma}'}$$,  $$\ln $$  is treated as normally distributed, and the bounds are estimated from:


 * $$\begin{align}

& {{\sigma}_{U}}= & \cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var}}}}\text{ (upper bound),} \\ & {{\sigma }_{L}}= & \frac\text{ (lower bound),} \end{align}$$

where $${{K}_{\alpha }}$$  is defined by:


 * $$\alpha =\frac{1}{\sqrt{2\pi }}\int_^{\infty }{{e}^{-\tfrac{2}}}dt=1-\Phi ({{K}_{\alpha }})$$

If $$\delta $$  is the confidence level, then  $$\alpha =\tfrac{1-\delta }{2}$$  for the two-sided bounds and  $$\alpha =1-\delta $$  for the one-sided bounds.

The variances and covariances of $${{\widehat{\mu }}^{\prime }}$$  and  $$$$  are estimated as follows:


 * $$\left( \begin{matrix}

\widehat{Var}\left( {{\widehat{\mu }}^{\prime }} \right) & \widehat{Cov}\left( {{\widehat{\mu }}^{\prime }}, \right) \\ \widehat{Cov}\left( {{\widehat{\mu }}^{\prime }}, \right) & \widehat{Var}\left( \right)  \\ \end{matrix} \right)=\left( \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{({\mu }')}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {\mu }'\partial } \\ {} & {} \\   -\tfrac{{{\partial }^{2}}\Lambda }{\partial {\mu }'\partial } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma'^{2}}  \\ \end{matrix} \right)_{{\mu }'={{\widehat{\mu }}^{\prime }},=}^{-1}$$

where $$\Lambda $$  is the log-likelihood function of the lognormal distribution.

Bounds on Time(Type 1)
The bounds around time for a given lognormal percentile, or unreliability, are estimated by first solving the reliability equation with respect to time, as follows:


 * $${t}'({{\widehat{\mu }}^{\prime }},)={{\widehat{\mu }}^{\prime }}+z\cdot $$

where:


 * $$z={{\Phi }^{-1}}\left[ F({t}') \right]$$

and:


 * $$\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({t}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz$$

The next step is to calculate the variance of $${T}'({{\widehat{\mu }}^{\prime }},):$$


 * $$\begin{align}

& Var({{{\hat{t}}}^{\prime }})= & {{\left( \frac{\partial {t}'}{\partial {\mu }'} \right)}^{2}}Var({{\widehat{\mu }}^{\prime }})+{{\left( \frac{\partial {t}'}{\partial } \right)}^{2}}Var \\ & & +2\left( \frac{\partial {t}'}{\partial {\mu }'} \right)\left( \frac{\partial {t}'}{\partial } \right)Cov\left( {{\widehat{\mu }}^{\prime }}, \right) \\ & &  \\  & Var({{{\hat{t}}}^{\prime }})= & Var({{\widehat{\mu }}^{\prime }})+{{\widehat{z}}^{2}}Var+2\cdot \widehat{z}\cdot Cov\left( {{\widehat{\mu }}^{\prime }}, \right) \end{align}$$

The upper and lower bounds are then found by:


 * $$\begin{align}

& t_{U}^{\prime }= & \ln {{t}_{U}}={{{\hat{t}}}^{\prime }}+{{K}_{\alpha }}\sqrt{Var({{{\hat{t}}}^{\prime }})} \\ & t_{L}^{\prime }= & \ln {{t}_{L}}={{{\hat{t}}}^{\prime }}-{{K}_{\alpha }}\sqrt{Var({{{\hat{t}}}^{\prime }})} \end{align}$$

Solving for $${{t}_{U}}$$  and  $${{t}_{L}}$$  we get:


 * $$\begin{align}

& {{t}_{U}}= & {{e}^{t_{U}^{\prime }}}\text{ (upper bound),} \\ & {{t}_{L}}= & {{e}^{t_{L}^{\prime }}}\text{ (lower bound)}\text{.} \end{align}$$

Bounds on Reliability (Type 2)
The reliability of the lognormal distribution is:


 * $$\hat{R}(t;{{\hat{\mu }}^{'}},{{\hat{\sigma }}^{'}})=\int_{t'}^{\infty }{\frac{1}{{^{'}}\sqrt{2\pi }}}{{e}^{-\frac{1}{2}{{\left( \frac{x-{{{\hat{\mu }}}^{'}}}{{{{\hat{\sigma }}}^{'}}} \right)}^{2}}}}dx$$

where $$t'=\ln (t)$$. Let $$\hat{z}(x)=\frac{x-{{{\hat{\mu }}}^{'}}}$$, the above equation then becomes:


 * $$\hat{R}\left( \hat{z}(t') \right)=\int_{\hat{z}(t')}^{\infty }{\frac{1}{\sqrt{2\pi }}}{{e}^{-\frac{1}{2}{{z}^{2}}}}dz$$

The bounds on $$z$$  are estimated from:


 * $$\begin{align}

& {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align}$$

where:


 * $$\begin{align}

& Var(\hat{z})=\left( \frac{\partial {z}}{\partial \mu '} \right)_{\hat{\mu }'}^{2}Var\left( \hat{\mu }' \right)+\left( \frac{\partial {z}}{\partial \sigma '} \right)_{\hat{\sigma }'}^{2}Var\left( \hat{\sigma }' \right) \\ & +2\left( \frac{\partial{z}}{\partial \mu '} \right)_{\hat{\mu }'}^ – \left( \frac{\partial {z}}{\partial \sigma '} \right)_{\hat{\sigma }'}^ – Cov\left( \hat{\mu }',\hat{\sigma }' \right) \end{align}$$ or:


 * $$Var(\hat{z})=\frac{1}{{{{\hat{\sigma }}}^{'2}}}\left[ Var\left( \hat{\mu }' \right)+{{{\hat{z}}}^{2}}Var\left( \sigma ' \right)+2\cdot \hat{z}\cdot Cov\left( \hat{\mu }',\hat{\sigma }' \right) \right]$$

The upper and lower bounds on reliability are:


 * $$\begin{align}

& {{R}_{U}}= & \int_^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\ & {{R}_{L}}= & \int_^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)} \end{align}$$

Example 4:

Bounds on Parameters
As covered in Chapter Parameter Estimation, the likelihood confidence bounds are calculated by finding values for $${{\theta }_{1}}$$  and  $${{\theta }_{2}}$$  that satisfy:


 * $$-2\cdot \text{ln}\left( \frac{L({{\theta }_{1}},{{\theta }_{2}})}{L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})} \right)=\chi _{\alpha ;1}^{2}$$

This equation can be rewritten as:


 * $$L({{\theta }_{1}},{{\theta }_{2}})=L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}$$

For complete data, the likelihood formula for the normal distribution is given by:


 * $$L({\mu }',)=\underset{i=1}{\overset{N}{\mathop \prod }}\,f({{x}_{i}};{\mu }',)=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot \cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-{\mu }'}{{{\sigma'}}} \right)}^{2}}}}$$

where the $${{x}_{i}}$$  values represent the original time-to-failure data. For a given value of $$\alpha $$, values for  $${\mu }'$$  and  $$$$  can be found which represent the maximum and minimum values that satisfy likelihood ratio equation. These represent the confidence bounds for the parameters at a confidence level $$\delta ,$$  where  $$\alpha =\delta $$  for two-sided bounds and  $$\alpha =2\delta -1$$  for one-sided.

Example 5:

Bounds on Time and Reliability
In order to calculate the bounds on a time estimate for a given reliability, or on a reliability estimate for a given time, the likelihood function needs to be rewritten in terms of one parameter and time/reliability, so that the maximum and minimum values of the time can be observed as the parameter is varied. This can be accomplished by substituting a form of the normal reliability equation into the likelihood function. The normal reliability equation can be written as:


 * $$R=1-\Phi \left( \frac{\text{ln}(t)-{\mu }'} \right)$$

This can be rearranged to the form:


 * $${\mu }'=\text{ln}(t)-\cdot {{\Phi }^{-1}}(1-R)$$

where $${{\Phi }^{-1}}$$  is the inverse standard normal. This equation can now be substituted into likelihood function to produce a likelihood equation in terms of $$,$$   $$t$$  and  $$R$$:


 * $$L(,t/R)=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot \cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-\left( \text{ln}(t)-\cdot {{\Phi }^{-1}}(1-R) \right)} \right)}^{2}}}}$$

The unknown variable $$t/R$$  depends on what type of bounds are being determined. If one is trying to determine the bounds on time for a given reliability, then $$R$$  is a known constant and  $$t$$  is the unknown variable. Conversely, if one is trying to determine the bounds on reliability for a given time, then $$t$$  is a known constant and  $$R$$  is the unknown variable. Either way, the above equation can be used to solve the likelihood ratio equation for the values of interest.

Example 6:

Example 7:

Bounds on Parameters
From Chapter Parameter Estimation, we know that the marginal distribution of parameter $${\mu }'$$  is:


 * $$\begin{align}

f({\mu }'|Data)= & \int_{0}^{\infty }f({\mu }',|Data)d \\ = & \frac{\int_{0}^{\infty }L(Data|{\mu }',)\varphi ({\mu }')\varphi d}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data|{\mu }',)\varphi ({\mu }')\varphi d{\mu }'d} \end{align}$$

where:
 * $$\varphi $$ is  $$\tfrac{1}$$, non-informative prior of  $$$$.

$$\varphi ({\mu }')$$ is an uniform distribution from - $$\infty $$  to + $$\infty $$, non-informative prior of  $${\mu }'$$. With the above prior distributions, $$f({\mu }'|Data)$$  can be rewritten as:


 * $$f({\mu }'|Data)=\frac{\int_{0}^{\infty }L(Data|{\mu }',)\tfrac{1}d}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data|{\mu }',)\tfrac{1}d{\mu }'d}$$

The one-sided upper bound of  $${\mu }'$$  is:


 * $$CL=P({\mu }'\le \mu _{U}^{\prime })=\int_{-\infty }^{\mu _{U}^{\prime }}f({\mu }'|Data)d{\mu }'$$

The one-sided lower bound of $${\mu }'$$  is:


 * $$1-CL=P({\mu }'\le \mu _{L}^{\prime })=\int_{-\infty }^{\mu _{L}^{\prime }}f({\mu }'|Data)d{\mu }'$$

The two-sided bounds of $${\mu }'$$  is:


 * $$CL=P(\mu _{L}^{\prime }\le {\mu }'\le \mu _{U}^{\prime })=\int_{\mu _{L}^{\prime }}^{\mu _{U}^{\prime }}f({\mu }'|Data)d{\mu }'$$

The same method can be used to obtained the bounds of $$$$.

Bounds on Time (Type 1)
The reliable life of the lognormal distribution is:


 * $$\ln T={\mu }'+{{\Phi }^{-1}}(1-R)$$

The one-sided upper on time bound is given by:


 * $$CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\ln t\le \ln {{t}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'+{{\Phi }^{-1}}(1-R)\le \ln {{t}_{U}})$$

The above equation can be rewritten in terms of $${\mu }'$$  as:


 * $$CL=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln {{t}_{U}}-{{\Phi }^{-1}}(1-R)$$

From the posterior distribution of $${\mu }'$$  get:


 * $$CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln {{t}_{U}}-{{\Phi }^{-1}}(1-R)}L(,{\mu }')\tfrac{1}d{\mu }'d}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(,{\mu }')\tfrac{1}d{\mu }'d}$$

The above equation is solved w.r.t. $${{t}_{U}}.$$  The same method can be applied for one-sided lower bounds and two-sided bounds on Time.

Bounds on Reliability (Type 2)
The one-sided upper bound on reliability is given by:


 * $$CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln t-{{\Phi }^{-1}}(1-{{R}_{U}}))$$

From the posterior distribution of $${\mu }'$$  is:


 * $$CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln t-{{\Phi }^{-1}}(1-{{R}_{U}})}L(,{\mu }')\tfrac{1}d{\mu }'d}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(,{\mu }')\tfrac{1}d{\mu }'d}$$

The above equation is solved w.r.t. $${{R}_{U}}.$$  The same method is used to calculate the one-sided lower bounds and two-sided bounds on Reliability.

Example 8: