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

## Power Lognormal Distribution

Probability Density Function The formula for the probability density function of the standard form of the power lognormal distribution is

$$f(x;p,\sigma) = (\frac{p} {x\sigma})\phi(\frac{\log x} {\sigma}) (\Phi(\frac{-\log x} {\sigma}))^{p-1} \hspace{.2in} x, p, \sigma > 0$$

where p (also referred to as the power parameter) and σ are the shape parameters, $$\Phi$$ is the cumulative distribution function of the standard normal distribution, and $$\phi$$ is the probability density function of the standard normal distribution.

As with other probability distributions, the power lognormal distribution can be transformed with a location parameter, μ, and a scale parameter, B. We omit the equation for the general form of the power lognormal distribution. 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.

The following is the plot of the power lognormal probability density function with four values of p and σ set to 1.

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

$$F(x;p,\sigma) = 1 - (\Phi(\frac{-\log x} {\sigma}))^{p} \hspace{.2in} x, p, \sigma > 0$$

where $$\Phi$$ is the cumulative distribution function of the standard normal distribution.

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

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

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

where $$\phi^{-1}$$ is the percent point function of the standard normal distribution.

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

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

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

where $$\Phi$$ is the cumulative distribution function of the standard normal distribution, and $$\phi$$ is the probability density function of the standard normal distribution.

Note that this is simply a multiple (p) of the lognormal hazard function.

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

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

$$H(x;p,\sigma) = -\log{((\Phi(\frac{-\log x} {\sigma}))^{p})} \hspace{.2in} x, p, \sigma > 0$$

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

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

$$S(x;p,\sigma) = (\Phi(\frac{-\log x} {\sigma}))^{p} \hspace{.2in} x, p, \sigma > 0$$

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

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

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

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

Common Statistics The statistics for the power lognormal distribution are complicated and require tables. Nelson discusses the mean, median, mode, and standard deviation of the power lognormal distribution and provides references to the appropriate tables.
Parameter Estimation Nelson discusses maximum likelihood estimation for the power lognormal distribution. These estimates need to be performed with computer software. Software for maximum likelihood estimation of the parameters of the power lognormal distribution is not as readily available as for other reliability distributions such as the exponential, Weibull, and lognormal.
Software Most general purpose statistical software programs do not support the probability functions for the power lognormal distribution.