4.6.3 Beyond Normality

Consider a family of probability distributions involving two parameters, ${\theta }_1$ and ${\theta }_2$ , and let $F\left(x;{\theta }_1,{\theta }_2\right)$ denote the corresponding cdf’s. The family of normal distributions is one such family, with ${\theta }_1=\mu ,{\theta }_2=\sigma$ , and $F\left(x;\mu ,\sigma \right)=\Phi \left[\left(x-\mu \right)/\sigma \right]$ . Another example is the Weibull family, with ${\theta }_1=\alpha ,{\theta }_2=\beta$ , and

$$F\left(x;\alpha ,\beta \right)=1-e^{-{\left(x/\beta \right)}^{\alpha }}$$

Still another family of this type is the gamma family, for which the cdf is an integral involving the incomplete gamma function that cannot be expressed in any simpler form.

The parameters ${\theta }_1$ and ${\theta }_2$ are said to be location and scale parameters, respectively, if $F\left(x;{\theta }_1,{\theta }_2\right)$ is a function of $\left(x-{\theta }_1\right)/{\theta }_2$ . The parameters $\mu$ and $\sigma$ of the normal family are location and scale parameters, respectively. In general, changing ${\theta }_1$ shifts the location of the corresponding density curve to the right or left, and changing ${\theta }_2$ amounts to stretching or compressing the horizontal measurement scale. Another example is given by the cdf

$$F\left(x;{\theta }_1,{\theta }_2\right)=1-e^{-e^{\left(x-{\theta }_1\right)/{\theta }_2}}-\infty <x<\infty$$

A random variable with this cdf is said to have an extreme value distribution. It is used in applications involving component lifetime and material strength.

Although the form of the extreme value cdf might at first glance suggest that ${\theta }_1$ is the point of symmetry for the density function, and therefore the mean and median, this is not the case. Instead, $P\left(X\le {\theta }_1\right)=F\left({\theta }_1;{\theta }_1,{\theta }_2\right)=1-e^{-1}=.632$ , and the density function $f\left(x;{\theta }_1,{\theta }_2\right)=F^{\prime }\left(x;{\theta }_1,{\theta }_2\right)$ is negatively skewed (a long lower tail). Similarly, the scale parameter ${\theta }_2$ is not the standard deviation $(\mu ={\theta }_1-.5772{\theta }_2$ and $\sigma =1.283{\theta }_2$ ). However, changing the value of ${\theta }_1$ does rigidly shift the density curve to the left or right, whereas a change in ${\theta }_2$ rescales the measurement axis.

The parameter $\beta$ of the Weibull distribution is a scale parameter, but $\alpha$ is not a location parameter. A similar comment applies to the parameters $\alpha$ and $\beta$ of the gamma distribution. And for the lognormal distribution, $\mu$ is not a location parameter, nor is $\sigma$ a scale parameter. In the usual form, the density function for any member of these families is positive for $x>0$ and 0 otherwise. Examples and exercises in the two previous sections introduced a third location (i.e., threshold) parameter $\gamma$ for these three distributions; this shifts the density function so that it is positive if $x>\gamma$ and zero otherwise.

When the family under consideration has only location and scale parameters, the issue of whether any member of the family is a plausible population distribution can be addressed via a single, easily constructed probability plot. One first obtains the percentiles of the standard distribution, the one with ${\theta }_1=0$ and ${\theta }_2=1$ , for percentages $100\left(i-.5\right)/n\left(i=1,\dots ,n\right)$ . The $n$ (standardized percentile, observation) pairs give the points in the plot. This is exactly what we did to obtain an omnibus normal probability plot. Somewhat surprisingly, this methodology can be applied to yield an omnibus Weibull probability plot. The key result is that if $X$ has a Weibull distribution with shape parameter $\alpha$ and scale parameter $\beta$ , then the transformed variable $\ln \left(X\right)$ has an extreme value distribution with location parameter ${\theta }_1=\ln \left(\beta \right)$ and scale parameter $1/\alpha$ . Thus a plot of the (extreme value standardized percentile, $\ln \left(x\right)$ ) pairs showing a strong linear pattern provides support for choosing the Weibull distribution as a population model.

EX 4.31

The gamma distribution is an example of a family involving a shape parameter for which there is no transformation $h\left(\cdot \right)$ such that $h\left(X\right)$ has a distribution that depends only on location and scale parameters. Construction of a probability plot necessitates first estimating the shape parameter from sample data (some methods for doing this are described in Chapter 6). Sometimes an investigator wishes to know whether the transformed variable $X^{\theta }$ has a normal distribution for some value of $\theta$ (by convention, $\theta =0$ is identified with the logarithmic transformation, in which case $X$ has a lognormal distribution). The book Graphical Methods for Data Analysis, listed in the Chapter 1 bibliography, discusses this type of problem as well as other refinements of probability plotting. Fortunately, the wide availability of various probability plots with statistical software packages means that the user can often sidestep technical details.