Lognormal Distribution - Parameter Estimation

Techniques

Cumulative Distribution Function

Cumulative Hazard Function

Weibull Distribution

Normal Distribution

Exponential Distribution

Exam

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The lognormal probability density function is

where μ is the location parameter or log mean, and σ is the scale parameter or log standard deviation. The location parameter is the mean of the data set after transformation by taking the logarithm, and the scale parameter is the standard deviation of the data set after transformation.

If x is a lognormally distributed random variable, then y = ln(x) is a normally distributed random variable. The location parameter is equal to the mean of the logarithm of the data points, and the shape parameter is equal to the standard deviation of the logarithm of the data points. Thus, the lognormal distribution does not have to be dealt with as a separate distribution. By taking the logarithm of the data points, the techniques developed for the normal distribution can be used to estimate the parameters of the lognormal distribution.

Three popular methods for parameter estimation for the lognormal distribution when censored data are encountered are

• maximum likelihood estimation,
• hazard plotting, and
• probability plotting.

After distribution parameters have been estimated, reliability estimations and predictions are used for evaluations.  The maximum likelihood estimation section explains how this can be done manually, but because of the complexity of the calculations, manual methods are not recommended.  The predicting module explains how to estimate reliability using The Reliability and Maintenance Analyst software package.