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1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.6. Probability Distributions
1.3.6.6. Gallery of Distributions

1.3.6.6.9. Lognormal Distribution

Probability Density Function A variable X is lognormally distributed if \(Y = \ln(X)\) is normally distributed with "LN" denoting the natural logarithm. The general formula for the probability density function of the lognormal distribution is

\( f(x) = \frac{e^{-((\ln((x-\theta)/m))^{2}/(2\sigma^{2}))}} {(x-\theta)\sigma\sqrt{2\pi}} \hspace{.2in} x > \theta; m, \sigma > 0 \)

where σ is the shape parameter (and is the standard deviation of the log of the distribution), θ is the location parameter and m is the scale parameter (and is also the median of the distribution). If x = θ, then f(x) = 0. The case where θ = 0 and m = 1 is called the standard lognormal distribution. The case where θ equals zero is called the 2-parameter lognormal distribution.

The equation for the standard lognormal distribution is

\( f(x) = \frac{e^{-((\ln x)^{2}/2\sigma^{2})}} {x\sigma\sqrt{2\pi}} \hspace{.2in} x > 0; \sigma > 0 \)

Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.

Note that the lognormal distribution is commonly parameterized with

\( \mu = \log(m) \)

The μ parameter is the mean of the log of the distribution. If the μ parameterization is used, the lognormal pdf is

\( f(x) = \frac{e^{-(\ln(x - \theta) - \mu)^2/(2\sigma^2)}} {(x - \theta)\sigma\sqrt{2\pi}} \hspace{.2in} x > 0; \sigma > 0 \)

We prefer to use the m parameterization since m is an explicit scale parameter.

The following is the plot of the lognormal probability density function for four values of σ.

plot of the lognormal probability density function for
 four values of sigma

There are several common parameterizations of the lognormal distribution. The form given here is from Evans, Hastings, and Peacock.

Cumulative Distribution Function The formula for the cumulative distribution function of the lognormal distribution is

\( F(x) = \Phi(\frac{\ln(x)} {\sigma}) \hspace{.2in} x \ge 0; \sigma > 0 \)

where \(\Phi\) is the cumulative distribution function of the normal distribution.

The following is the plot of the lognormal cumulative distribution function with the same values of σ as the pdf plots above.

plot of the lognormal cumulative distribution function

Percent Point Function The formula for the percent point function of the lognormal distribution is

\( G(p) = \exp(\sigma\Phi^{-1}(p)) \hspace{.2in} 0 \le p < 1; \sigma > 0 \)

where \(\Phi^{-1}\) is the percent point function of the normal distribution.

The following is the plot of the lognormal percent point function with the same values of σ as the pdf plots above.

plot of the lognormal percent point function

Hazard Function The formula for the hazard function of the lognormal distribution is

\( h(x,\sigma) = \frac{(\frac{1} {x\sigma})\phi(\frac{\ln x} {\sigma})} {\Phi(\frac{-\ln x} {\sigma})} \hspace{.2in} x > 0; \sigma > 0 \)

where \(\phi\) is the probability density function of the normal distribution and \(\Phi\) is the cumulative distribution function of the normal distribution.

The following is the plot of the lognormal hazard function with the same values of σ as the pdf plots above.

plot of the lognormal hazard function

Cumulative Hazard Function The formula for the cumulative hazard function of the lognormal distribution is

\( H(x) = -\ln(1 - \Phi(\frac{\ln(x)} {\sigma})) \hspace{.2in} x \ge 0; \sigma > 0 \)

where \(\Phi\) is the cumulative distribution function of the normal distribution.

The following is the plot of the lognormal cumulative hazard function with the same values of σ as the pdf plots above.

plot of the lognormal cumulative hazard function

Survival Function The formula for the survival function of the lognormal distribution is

\( S(x) = 1 - \Phi(\frac{\ln(x)} {\sigma}) \hspace{.2in} x \ge 0; \sigma > 0 \)

where \(\Phi\) is the cumulative distribution function of the normal distribution.

The following is the plot of the lognormal survival function with the same values of σ as the pdf plots above.

plot of the lognormal survival function

Inverse Survival Function The formula for the inverse survival function of the lognormal distribution is

\( Z(p) = \exp(\sigma\Phi^{-1}(1-p)) \hspace{.2in} 0 \le p < 1; \sigma > 0 \)

where \(\Phi^{-1}\) is the percent point function of the normal distribution.

The following is the plot of the lognormal inverse survival function with the same values of σ as the pdf plots above.

plot of the lognormal inverse survival function

Common Statistics The formulas below are with the location parameter equal to zero and the scale parameter equal to one.

Mean \( e^{0.5\sigma^{2}} \)
Median Scale parameter m (= 1 if scale parameter not specified).
Mode \( \frac{1} {e^{\sigma^{2}}} \)
Range 0 to \(\infty\)
Standard Deviation \( \sqrt{e^{\sigma^{2}} (e^{\sigma^{2}} - 1)} \)
Skewness \( (e^{\sigma^{2}}+2) \sqrt{e^{\sigma^{2}} - 1} \)
Kurtosis \( (e^{\sigma^{2}})^{4} + 2(e^{\sigma^{2}})^{3} + 3(e^{\sigma^{2}})^{2} - 3 \)
Coefficient of Variation \( \sqrt{e^{\sigma^{2}} - 1} \)

Parameter Estimation The maximum likelihood estimates for the scale parameter, m, and the shape parameter, σ, are
    \( \hat{m} = \exp{\hat{\mu}} \)
and
    \( \hat{\sigma} = \sqrt{\frac{\sum_{i=1}^{N}{(\ln{(X_{i})} - \hat{\mu})^{2}}} {N}} \)
where
    \( \hat{\mu}= \frac{\sum_{i=1}^{N}{\ln{X_i}}} {N} \)
If the location parameter is known, it can be subtracted from the original data points before computing the maximum likelihood estimates of the shape and scale parameters.
Comments The lognormal distribution is used extensively in reliability applications to model failure times. The lognormal and Weibull distributions are probably the most commonly used distributions in reliability applications.
Software Most general purpose statistical software programs support at least some of the probability functions for the lognormal distribution.

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