The UNIVARIATE Procedure

Formulas for Fitted Continuous Distributions

Subsections:

The following sections provide information about the families of parametric distributions that you can fit with the HISTOGRAM statement. Properties of these distributions are discussed by Johnson, Kotz, and Balakrishnan (1994, 1995).

Beta Distribution

The fitted density function is

\[ p(x) = \left\{ \begin{array}{ll} hv \frac{(x-\theta )^{\alpha -1}(\sigma +\theta -x)^{\beta -1}}{ B(\alpha ,\beta )\sigma ^{(\alpha +\beta -1)}} & \mbox{for }\theta < x < \theta + \sigma \\ 0 & \mbox{for }x \leq \theta \text { or } x \geq \theta + \sigma \end{array} \right. \]

where $B(\alpha ,\beta )=\frac{\Gamma (\alpha )\Gamma (\beta )}{\Gamma (\alpha +\beta )}$ and

  • $\theta =$ lower threshold parameter (lower endpoint parameter)

  • $\sigma =$ scale parameter $(\sigma >0)$

  • $\alpha =$ shape parameter $(\alpha >0)$

  • $\beta =$ shape parameter $(\beta >0)$

  • $h =$ width of histogram interval

  • $v =$ vertical scaling factor

and

\[ v = \left\{ \begin{array}{ll} n & \mbox{the sample size, for VSCALE=COUNT} \\ 100 & \mbox{for VSCALE=PERCENT} \\ 1 & \mbox{for VSCALE=PROPORTION} \end{array} \right. \]

Note: This notation is consistent with that of other distributions that you can fit with the HISTOGRAM statement. However, many texts, including Johnson, Kotz, and Balakrishnan (1995), write the beta density function as

\[ p(x) = \left\{ \begin{array}{ll} \frac{(x - a)^{p - 1} (b - x)^{q - 1} }{B(p ,q)(b - a )^{p + q - 1} } & \mbox{for }a < x < b \\ 0 & \mbox{for }x \leq a \text { or } x \geq b \end{array} \right. \]

The two parameterizations are related as follows:

  • $\sigma = b - a$

  • $\theta = a$

  • $\alpha = p$

  • $\beta = q$

The range of the beta distribution is bounded below by a threshold parameter $\theta = a$ and above by $\theta + \sigma = b$. If you specify a fitted beta curve by using the BETA option, $\theta $ must be less than the minimum data value and $\theta + \sigma $ must be greater than the maximum data value. You can specify $\theta $ and $\sigma $ with the THETA= and SIGMA= beta-options in parentheses after the keyword BETA. By default, $\sigma =1$ and $\theta =0$. If you specify THETA=EST and SIGMA=EST, maximum likelihood estimates are computed for $\theta $ and $\sigma $. However, three- and four-parameter maximum likelihood estimation does not always converge.

In addition, you can specify $\alpha $ and $\beta $ with the ALPHA= and BETA= beta-options, respectively. By default, the procedure calculates maximum likelihood estimates for $\alpha $ and $\beta $. For example, to fit a beta density curve to a set of data bounded below by 32 and above by 212 with maximum likelihood estimates for $\alpha $ and $\beta $, use the following statement: 

histogram Length / beta(theta=32 sigma=180);

The beta distributions are also referred to as Pearson Type I or II distributions. These include the power function distribution ($\beta =1$), the arc sine distribution ($\alpha =\beta =\frac{1}{2}$), and the generalized arc sine distributions ($\alpha +\beta =1$, $\beta \neq \frac{1}{2}$).

You can use the DATA step function QUANTILE to compute beta quantiles and the DATA step function CDF to compute beta probabilities.

Exponential Distribution

The fitted density function is

\[ p(x) = \left\{ \begin{array}{ll} \frac{hv}{\sigma } \exp (-(\frac{x - \theta }{\sigma })) & \mbox{for }x \geq \theta \\ 0 & \mbox{for }x < \theta \end{array} \right. \]

where

  • $\theta =$ threshold parameter

  • $\sigma =$ scale parameter $(\sigma >0)$

  • $h =$ width of histogram interval

  • $v =$ vertical scaling factor

and

\[ v = \left\{ \begin{array}{ll} n & \mbox{the sample size, for VSCALE=COUNT} \\ 100 & \mbox{for VSCALE=PERCENT} \\ 1 & \mbox{for VSCALE=PROPORTION} \end{array} \right. \]

The threshold parameter $\theta $ must be less than or equal to the minimum data value. You can specify $\theta $ with the THRESHOLD= exponential-option. By default, $\theta =0$. If you specify THETA=EST, a maximum likelihood estimate is computed for $\theta $. In addition, you can specify $\sigma $ with the SCALE= exponential-option. By default, the procedure calculates a maximum likelihood estimate for $\sigma $. Note that some authors define the scale parameter as $\frac{1}{\sigma }$.

The exponential distribution is a special case of both the gamma distribution (with $\alpha =1$) and the Weibull distribution (with $c=1$). A related distribution is the extreme value distribution. If $Y=\exp (-X)$ has an exponential distribution, then X has an extreme value distribution.

You can use the DATA step function QUANTILE to compute exponential quantiles and the DATA step function CDF to compute exponential probabilities.

Gamma Distribution

The fitted density function is

\[ p(x) = \left\{ \begin{array}{ll} \frac{hv}{\Gamma (\alpha )\sigma } (\frac{x - \theta }{\sigma })^{\alpha - 1} \exp (-(\frac{x - \theta }{\sigma })) & \mbox{for }x > \theta \\ 0 & \mbox{for }x \leq \theta \end{array} \right. \]

where

  • $\theta =$ threshold parameter

  • $\sigma =$ scale parameter $(\sigma >0)$

  • $\alpha =$ shape parameter $(\alpha >0)$

  • $h =$ width of histogram interval

  • $v =$ vertical scaling factor

and

\[ v = \left\{ \begin{array}{ll} n & \mbox{the sample size, for VSCALE=COUNT} \\ 100 & \mbox{for VSCALE=PERCENT} \\ 1 & \mbox{for VSCALE=PROPORTION} \end{array} \right. \]

The threshold parameter $\theta $ must be less than the minimum data value. You can specify $\theta $ with the THRESHOLD= gamma-option. By default, $\theta =0$. If you specify THETA=EST, a maximum likelihood estimate is computed for $\theta $. In addition, you can specify $\sigma $ and $\alpha $ with the SCALE= and ALPHA= gamma-options. By default, the procedure calculates maximum likelihood estimates for $\sigma $ and $\alpha $.

The gamma distributions are also referred to as Pearson Type III distributions, and they include the chi-square, exponential, and Erlang distributions. The probability density function for the chi-square distribution is

\[ p(x) = \left\{ \begin{array}{ll} \frac{1}{2\Gamma (\frac{\nu }{2})} \left( \frac{x}{2} \right)^{\frac{\nu }{2} - 1} \exp (-\frac{x}{2}) & \mbox{for }x > 0 \\ 0 & \mbox{for }x \leq 0 \end{array} \right. \]

Notice that this is a gamma distribution with $\alpha = \frac{\nu }{2}$, $\sigma =2$, and $\theta =0$. The exponential distribution is a gamma distribution with $\alpha =1$, and the Erlang distribution is a gamma distribution with $\alpha $ being a positive integer. A related distribution is the Rayleigh distribution. If $R=\frac{\max (X_1,\ldots ,X_ n)}{\min (X_1,\ldots ,X_ n)}$ where the $X_{i}$’s are independent $\chi ^{2}_{\nu }$ variables, then $\log R$ is distributed with a $\chi _{\nu }$ distribution having a probability density function of

\[ p(x) = \left\{ \begin{array}{ll} \left[2^{\frac{\nu }{2}-1}\Gamma (\frac{\nu }{2})\right] ^{-1}x^{\nu -1} \exp (-\frac{x^2}{2}) & \mbox{for }x > 0 \\ 0 & \mbox{for }x \leq 0 \end{array} \right. \]

If $\nu =2$, the preceding distribution is referred to as the Rayleigh distribution.

You can use the DATA step function QUANTILE to compute gamma quantiles and the DATA step function CDF to compute gamma probabilities.

Gumbel Distribution

The fitted density function is

\[ p(x) = \frac{hv}{\sigma }e^{-(x-\mu )/\sigma } \exp \left( -e^{-(x-\mu )/\sigma }\right) \]

where

  • $\mu =$ location parameter

  • $\sigma =$ scale parameter $(\sigma >0)$

  • $h =$ width of histogram interval

  • $v =$ vertical scaling factor

and

\[ v = \left\{ \begin{array}{ll} n & \mbox{the sample size, for VSCALE=COUNT} \\ 100 & \mbox{for VSCALE=PERCENT} \\ 1 & \mbox{for VSCALE=PROPORTION} \end{array} \right. \]

You can specify $\mu $ and $\sigma $ with the MU= and SIGMA= Gumbel-options, respectively. By default, the procedure calculates maximum likelihood estimates for these parameters.

Note: The Gumbel distribution is also referred to as Type 1 extreme value distribution.

Note: The random variable X has Gumbel (Type 1 extreme value) distribution if and only if $e^ X$ has Weibull distribution and $\exp ((X-\mu )/\sigma )$ has standard exponential distribution.

Inverse Gaussian Distribution

The fitted density function is

\[ p(x) = \left\{ \begin{array}{ll} hv \left(\frac{\lambda }{2\pi x^3}\right)^{1/2} \exp (-\frac{\lambda }{2\mu ^2 x}(x-\mu )^2) & \mbox{for }x > 0 \\ 0 & \mbox{for }x \leq 0 \end{array} \right. \]

where

  • $\mu =$ location parameter $(\mu >0)$

  • $\lambda =$ shape parameter $(\lambda >0)$

  • $h =$ width of histogram interval

  • $v =$ vertical scaling factor

and

\[ v = \left\{ \begin{array}{ll} n & \mbox{the sample size, for VSCALE=COUNT} \\ 100 & \mbox{for VSCALE=PERCENT} \\ 1 & \mbox{for VSCALE=PROPORTION} \end{array} \right. \]

The location parameter $\mu $ has to be greater then zero. You can specify $\mu $ with the MU= iGauss-option. In addition, you can specify shape parameter $\lambda $ with LAMBDA= iGauss-option. By default, the procedure calculates maximum likelihood estimates for $\mu $ and $\lambda $.

Note: The special case where $\mu =1$ and $\lambda =\phi $ corresponds to the Wald distribution.

You can use the DATA step function QUANTILE to compute inverse Gaussian quantiles and the DATA step function CDF to compute inverse Gaussian probabilities.

Lognormal Distribution

The fitted density function is

\[ p(x) = \left\{ \begin{array}{ll} \frac{hv}{\sigma \sqrt {2\pi }(x - \theta )} \exp \left(-\frac{(\log (x-\theta )-\zeta )^{2}}{2\sigma ^{2}}\right) & \mbox{for }x > \theta \\ 0 & \mbox{for }x \leq \theta \end{array} \right. \]

where

  • $\theta =$ threshold parameter

  • $\zeta =$ scale parameter $(-\infty < \zeta < \infty )$

  • $\sigma =$ shape parameter $(\sigma >0)$

  • $h =$ width of histogram interval

  • $v =$ vertical scaling factor

and

\[ v = \left\{ \begin{array}{ll} n & \mbox{the sample size, for VSCALE=COUNT} \\ 100 & \mbox{for VSCALE=PERCENT} \\ 1 & \mbox{for VSCALE=PROPORTION} \end{array} \right. \]

The threshold parameter $\theta $ must be less than the minimum data value. You can specify $\theta $ with the THRESHOLD= lognormal-option. By default, $\theta =0$. If you specify THETA=EST, a maximum likelihood estimate is computed for $\theta $. You can specify $\zeta $ and $\sigma $ with the SCALE= and SHAPE= lognormal-options, respectively. By default, the procedure calculates maximum likelihood estimates for these parameters.

Note: The lognormal distribution is also referred to as the $S_ L$ distribution in the Johnson system of distributions.

Note: This book uses $\sigma $ to denote the shape parameter of the lognormal distribution, whereas $\sigma $ is used to denote the scale parameter of the other distributions. The use of $\sigma $ to denote the lognormal shape parameter is based on the fact that $\frac{1}{\sigma }(\log (X-\theta )-\zeta )$ has a standard normal distribution if X is lognormally distributed. Based on this relationship, you can use the DATA step function PROBIT to compute lognormal quantiles and the DATA step function PROBNORM to compute probabilities.

Normal Distribution

The fitted density function is

\[ p(x) = \begin{array}{ll} \frac{hv}{\sigma \sqrt {2\pi }} \exp \left(-\frac{1}{2} (\frac{x - \mu }{\sigma })^{2}\right) & \mbox{for }-\infty < x < \infty \end{array} \]

where

  • $\mu =$ mean

  • $\sigma =$ standard deviation $(\sigma >0)$

  • $h =$ width of histogram interval

  • $v =$ vertical scaling factor

and

\[ v = \left\{ \begin{array}{ll} n & \mbox{the sample size, for VSCALE=COUNT} \\ 100 & \mbox{for VSCALE=PERCENT} \\ 1 & \mbox{for VSCALE=PROPORTION} \end{array} \right. \]

You can specify $\mu $ and $\sigma $ with the MU= and SIGMA= normal-options, respectively. By default, the procedure estimates $\mu $ with the sample mean and $\sigma $ with the sample standard deviation.

You can use the DATA step function QUANTILE to compute beta quantiles and the DATA step function CDF to compute normal probabilities.

Note: The normal distribution is also referred to as the $S_ N$ distribution in the Johnson system of distributions.

Generalized Pareto Distribution

The fitted density function is

\[ p(x) = \left\{ \begin{array}{ll} \frac{hv}{\sigma }(1 - \alpha (x-\theta )/\sigma )^{1/\alpha -1} & \mbox{if } \alpha \neq 0 \\ \frac{hv}{\sigma } \exp (-x/\sigma ) & \mbox{if }\alpha = 0 \end{array} \right. \]

where

  • $\theta =$ threshold parameter

  • $\alpha =$ shape parameter

  • $\sigma =$ shape parameter $(\sigma >0)$

  • $h =$ width of histogram interval

  • $v =$ vertical scaling factor

and

\[ v = \left\{ \begin{array}{ll} n & \mbox{the sample size, for VSCALE=COUNT} \\ 100 & \mbox{for VSCALE=PERCENT} \\ 1 & \mbox{for VSCALE=PROPORTION} \end{array} \right. \]

The support of the distribution is $x>0$ for $\alpha \leq 0 $ and $0<x<\sigma /\alpha $ for $\alpha >0$.

Note: Special cases of Pareto distribution with $\alpha =0$ and $\alpha =1$ correspond respectively to the exponential distribution with mean $\sigma $ and uniform distribution on the interval $(0,\sigma )$.

The threshold parameter $\theta $ must be less than the minimum data value. You can specify $\theta $ with the THETA= Pareto-option. By default, $\theta =0$. You can also specify $\alpha $ and $\sigma $ with the ALPHA= and SIGMA= Pareto-options,respectively. By default, the procedure calculates maximum likelihood estimates for these parameters.

Note: Maximum likelihood estimation of the parameters works well if $\alpha <\frac{1}{2}$, but not otherwise. In this case the estimators are asymptotically normal and asymptotically efficient. The asymptotic normal distribution of the maximum likelihood estimates has mean $(\alpha ,\sigma )$ and variance-covariance matrix

\[ \frac{1}{n}\left( \begin{array}{ccc} (1-\alpha )^2 & \sigma (1-\alpha )\\ \sigma (1-\alpha ) & 2\sigma ^2(1-\alpha ) \end{array}\right). \]

Note: If no local minimum found in the space

\[ \{ \alpha <0,\sigma >0\} \cup \{ 0<\alpha \leq 1,\sigma /\alpha >\max (X_ i)\} , \]

there is no maximum likelihood estimator. More details on how to find maximum likelihood estimators and suggested algorithm can be found in Grimshaw(1993).

Power Function Distribution

The fitted density function is

\[ p(x) = \left\{ \begin{array}{ll} hv \frac{\alpha }{\sigma }\left(\frac{x-\theta }{\sigma }\right)^{\alpha -1} & \mbox{for }\theta < x < \theta + \sigma \\ 0 & \mbox{for }x \leq \theta \text { or }x \geq \theta + \sigma \end{array} \right. \]

where

  • $\theta =$ lower threshold parameter (lower endpoint parameter)

  • $\sigma =$ scale parameter $(\sigma >0)$

  • $\alpha =$ shape parameter $(\alpha >0)$

  • $h =$ width of histogram interval

  • $v =$ vertical scaling factor

and

\[ v = \left\{ \begin{array}{ll} n & \mbox{the sample size, for VSCALE=COUNT} \\ 100 & \mbox{for VSCALE=PERCENT} \\ 1 & \mbox{for VSCALE=PROPORTION} \end{array} \right. \]

Note: This notation is consistent with that of other distributions that you can fit with the HISTOGRAM statement. However, many texts, including Johnson, Kotz, and Balakrishnan (1995), write the density function of power function distribution as

\[ p(x) = \left\{ \begin{array}{ll} \frac{p}{b-a}\left(\frac{x-a}{b-a} \right)^{p-1} & \mbox{for }a < x < b \\ 0 & \mbox{for }x \leq a \text { or }x \geq b \end{array} \right. \]

The two parameterizations are related as follows:

  • $\sigma = b - a$

  • $\theta = a$

  • $\alpha = p$

Note: The family of power function distributions is subclass of beta distribution with density function

\[ p(x) = \left\{ \begin{array}{ll} hv \frac{(x-\theta )^{\alpha -1}(\sigma +\theta -x)^{\beta -1}}{ B(\alpha ,\beta )\sigma ^{(\alpha +\beta -1)}} & \mbox{for }\theta < x < \theta + \sigma \\ 0 & \mbox{for }x \leq \theta \text { or }x \geq \theta + \sigma \end{array} \right. \]

where $B(\alpha ,\beta )=\frac{\Gamma (\alpha )\Gamma (\beta )}{\Gamma (\alpha +\beta )}$ with parameter $\beta =1$. Therefore, all properties and estimation procedures of beta distribution apply.

The range of the power function distribution is bounded below by a threshold parameter $\theta = a$ and above by $\theta + \sigma = b$. If you specify a fitted power function curve by using the POWER option, $\theta $ must be less than the minimum data value and $\theta + \sigma $ must be greater than the maximum data value. You can specify $\theta $ and $\sigma $ with the THETA= and SIGMA= power-options in parentheses after the keyword POWER. By default, $\sigma =1$ and $\theta =0$. If you specify THETA=EST and SIGMA=EST, maximum likelihood estimates are computed for $\theta $ and $\sigma $. However, three-parameter maximum likelihood estimation does not always converge.

In addition, you can specify $\alpha $ with the ALPHA= power-option. By default, the procedure calculates maximum likelihood estimate for $\alpha $. For example, to fit a power function density curve to a set of data bounded below by 32 and above by 212 with maximum likelihood estimate for $\alpha $, use the following statement: 

histogram Length / power(theta=32 sigma=180);

Rayleigh Distribution

The fitted density function is

\[ p(x) = \left\{ \begin{array}{ll} hv \frac{x-\theta }{\sigma ^2}e^{-(x-\theta )^2/(2\sigma ^2)} & \mbox{for }x \geq \theta \\ 0 & \mbox{for }x <\theta \end{array} \right. \]

where

  • $\theta =$ lower threshold parameter (lower endpoint parameter)

  • $\sigma =$ scale parameter $(\sigma >0)$

  • $h =$ width of histogram interval

  • $v =$ vertical scaling factor

and

\[ v = \left\{ \begin{array}{ll} n & \mbox{the sample size, for VSCALE=COUNT} \\ 100 & \mbox{for VSCALE=PERCENT} \\ 1 & \mbox{for VSCALE=PROPORTION} \end{array} \right. \]

Note: The Rayleigh distribution is Weibull distribution with density function

\[ p(x) = \left\{ \begin{array}{ll} hv \frac{k}{\lambda } \left(\frac{x-\theta }{\lambda }\right)^{k-1} \exp (-(\frac{x-\theta }{\lambda })^ k) & \mbox{for }x \geq \theta \\ 0 & \mbox{for }x <\theta \end{array} \right. \]

and with shape parameter $k=2$ and scale parameter $\lambda = \sqrt {2}\sigma $.

The threshold parameter $\theta $ must be less than the minimum data value. You can specify $\theta $ with the THETA= Rayleigh-option. By default, $\theta =0$. In addition you can specify $\sigma $ with the SIGMA= Rayleigh-option. By default, the procedure calculates maximum likelihood estimate for $\sigma $.

For example, to fit a Rayleigh density curve to a set of data bounded below by 32 with maximum likelihood estimate for $\sigma $, use the following statement: 

histogram Length / rayleigh(theta=32);

Johnson $S_ B$ Distribution

The fitted density function is

\[ p(x) = \left\{ \begin{array}{ll} \frac{\delta hv}{\sigma \sqrt {2\pi } } \left[ \left( \frac{x - \theta }{\sigma } \right) \left( 1 - \frac{x - \theta }{\sigma } \right) \right]^{-1} \times & \\ \exp \left[ -\frac{1}{2} \left( \gamma + \delta \log ( \frac{x - \theta }{\theta + \sigma -x} ) \right)^2 \right] & \mbox{for } \theta < x < \theta + \sigma \\ 0 & \mbox{for } x \leq \theta \text { or } x \geq \theta + \sigma \end{array} \right. \]

where

  • $\theta =$ threshold parameter $(-\infty < \theta < \infty )$

  • $\sigma =$ scale parameter $(\sigma > 0)$

  • $\delta =$ shape parameter $(\delta >0)$

  • $\gamma =$ shape parameter $(-\infty < \gamma < \infty )$

  • $h =$ width of histogram interval

  • $v =$ vertical scaling factor

and

\[ v = \left\{ \begin{array}{ll} n & \mbox{the sample size, for VSCALE=COUNT} \\ 100 & \mbox{for VSCALE=PERCENT} \\ 1 & \mbox{for VSCALE=PROPORTION} \end{array} \right. \]

The $S_ B$ distribution is bounded below by the parameter $\theta $ and above by the value $\theta + \sigma $. The parameter $\theta $ must be less than the minimum data value. You can specify $\theta $ with the THETA= $S_ B$-option, or you can request that $\theta $ be estimated with the THETA = EST $S_ B$-option. The default value for $\theta $ is zero. The sum $\theta + \sigma $ must be greater than the maximum data value. The default value for $\sigma $ is one. You can specify $\sigma $ with the SIGMA= $S_ B$-option, or you can request that $\sigma $ be estimated with the SIGMA = EST $S_ B$-option.

By default, the method of percentiles given by Slifker and Shapiro (1980) is used to estimate the parameters. This method is based on four data percentiles, denoted by $x_{-3z}$, $x_{-z}$, $x_{z}$, and $x_{3z}$, which correspond to the four equally spaced percentiles of a standard normal distribution, denoted by $-3z$, $-z$, z, and $3z$, under the transformation

\[ z = \gamma + \delta \log \left( \frac{x - \theta }{\theta + \sigma - x} \right) \]

The default value of z is 0.524. The results of the fit are dependent on the choice of z, and you can specify other values with the FITINTERVAL= option (specified in parentheses after the SB option). If you use the method of percentiles, you should select a value of z that corresponds to percentiles which are critical to your application.

The following values are computed from the data percentiles:

\[ \begin{array}{lcl} m & = & x_{3z} - x_{z} \\ n & = & x_{-z} - x_{-3z} \\ p & = & x_{z} - x_{-z} \\ \end{array} \]

It was demonstrated by Slifker and Shapiro (1980) that

\[ \begin{array}{ll} \frac{mn}{p^2} > 1 & \mbox{for any }S_ U\text { distribution} \\ \frac{mn}{p^2} < 1 & \mbox{for any }S_ B\text { distribution} \\ \frac{mn}{p^2} = 1 & \mbox{for any }S_ L\text { (lognormal) distribution} \\ \end{array} \]

A tolerance interval around one is used to discriminate among the three families with this ratio criterion. You can specify the tolerance with the FITTOLERANCE= option (specified in parentheses after the SB option). The default tolerance is 0.01. Assuming that the criterion satisfies the inequality

\[ \frac{mn}{p^2} < 1 - \mbox{tolerance} \]

the parameters of the $S_ B$ distribution are computed using the explicit formulas derived by Slifker and Shapiro (1980).

If you specify FITMETHOD = MOMENTS (in parentheses after the SB option), the method of moments is used to estimate the parameters. If you specify FITMETHOD = MLE (in parentheses after the SB option), the method of maximum likelihood is used to estimate the parameters. Note that maximum likelihood estimates may not always exist. Refer to Bowman and Shenton (1983) for discussion of methods for fitting Johnson distributions.

Johnson $S_ U$ Distribution

The fitted density function is

\[ p(x) = \left\{ \begin{array}{ll} \frac{ \delta hv}{\sigma \sqrt {2\pi } } \frac{ 1 }{ \sqrt { 1 + \left( (x - \theta ) / \sigma \right)^2 } } \times & \\ \exp \left[ -\frac{1}{2} \left( \gamma + \delta \sinh ^{-1} \left( \frac{x - \theta }{\sigma } \right) \right)^2 \right] & \mbox{for } x > \theta \\ 0 & \mbox{for } x \leq \theta \end{array} \right. \]

where

  • $\theta =$ location parameter $(-\infty < \theta < \infty )$

  • $\sigma =$ scale parameter $(\sigma > 0)$

  • $\delta =$ shape parameter $(\delta >0)$

  • $\gamma =$ shape parameter $(-\infty < \gamma < \infty )$

  • $h =$ width of histogram interval

  • $v =$ vertical scaling factor

and

\[ v = \left\{ \begin{array}{ll} n & \mbox{the sample size, for VSCALE=COUNT} \\ 100 & \mbox{for VSCALE=PERCENT} \\ 1 & \mbox{for VSCALE=PROPORTION} \end{array} \right. \]

You can specify the parameters with the THETA=, SIGMA=, DELTA=, and GAMMA= $S_ U$-options, which are enclosed in parentheses after the SU option. If you do not specify these parameters, they are estimated.

By default, the method of percentiles given by Slifker and Shapiro (1980) is used to estimate the parameters. This method is based on four data percentiles, denoted by $x_{-3z}$, $x_{-z}$, $x_{z}$, and $x_{3z}$, which correspond to the four equally spaced percentiles of a standard normal distribution, denoted by $-3z$, $-z$, z, and $3z$, under the transformation

\[ z = \gamma + \delta \sinh ^{-1} \left( \frac{x - \theta }{\sigma } \right) \]

The default value of z is 0.524. The results of the fit are dependent on the choice of z, and you can specify other values with the FITINTERVAL= option (specified in parentheses after the SU option). If you use the method of percentiles, you should select a value of z that corresponds to percentiles that are critical to your application.

The following values are computed from the data percentiles:

\[ \begin{array}{lcl} m & = & x_{3z} - x_{z} \\ n & = & x_{-z} - x_{-3z} \\ p & = & x_{z} - x_{-z} \\ \end{array} \]

It was demonstrated by Slifker and Shapiro (1980) that

\[ \begin{array}{ll} \frac{mn}{p^2} > 1 & \mbox{for any }S_ U\text { distribution} \\ \frac{mn}{p^2} < 1 & \mbox{for any }S_ B\text { distribution} \\ \frac{mn}{p^2} = 1 & \mbox{for any }S_ L\text { (lognormal) distribution} \\ \end{array} \]

A tolerance interval around one is used to discriminate among the three families with this ratio criterion. You can specify the tolerance with the FITTOLERANCE= option (specified in parentheses after the SU option). The default tolerance is 0.01. Assuming that the criterion satisfies the inequality

\[ \frac{mn}{p^2} > 1 + \mbox{tolerance} \]

the parameters of the $S_ U$ distribution are computed using the explicit formulas derived by Slifker and Shapiro (1980).

If you specify FITMETHOD = MOMENTS (in parentheses after the SU option), the method of moments is used to estimate the parameters. If you specify FITMETHOD = MLE (in parentheses after the SU option), the method of maximum likelihood is used to estimate the parameters. Note that maximum likelihood estimates do not always exist. Refer to Bowman and Shenton (1983) for discussion of methods for fitting Johnson distributions.

Weibull Distribution

The fitted density function is

\[ p(x) = \left\{ \begin{array}{ll} hv\frac{c}{\sigma } (\frac{x - \theta }{\sigma })^{c-1} \exp (-(\frac{x- \theta }{\sigma })^ c) & \mbox{for } x > \theta \\ 0 & \mbox{for }x \leq \theta \end{array} \right. \]

where

  • $\theta =$ threshold parameter

  • $\sigma =$ scale parameter $(\sigma >0)$

  • $c =$ shape parameter $(\mi{c} >0)$

  • $h =$ width of histogram interval

  • $v =$ vertical scaling factor

and

\[ v = \left\{ \begin{array}{ll} n & \mbox{the sample size, for VSCALE=COUNT} \\ 100 & \mbox{for VSCALE=PERCENT} \\ 1 & \mbox{for VSCALE=PROPORTION} \end{array} \right. \]

The threshold parameter $\theta $ must be less than the minimum data value. You can specify $\theta $ with the THRESHOLD= Weibull-option. By default, $\theta =0$. If you specify THETA=EST, a maximum likelihood estimate is computed for $\theta $. You can specify $\sigma $ and c with the SCALE= and SHAPE= Weibull-options, respectively. By default, the procedure calculates maximum likelihood estimates for $\sigma $ and c.

The exponential distribution is a special case of the Weibull distribution where $c=1$.

You can use the DATA step function QUANTILE to compute Weibull quantiles and the DATA step function CDF to compute Weibull probabilities.