The RELIABILITY Procedure

Probability Distributions

Subsections:

This section describes the probability distributions available in the RELIABILITY procedure for probability plotting and parameter estimation.

PROBPLOT and RELATIONPLOT Statements

Probability plots can be constructed for each of the probability distributions in Table 16.57.

For all distributions other than the three-parameter Weibull, estimates of two distribution parameters (location and scale or scale and shape) are computed by maximum likelihood or by least squares fitted to points on the probability plot. If one of the parameters is specified as fixed, the other is estimated. In addition, you can specify a fixed threshold, or shift, parameter for distributions for which a threshold parameter is indicated in Table 16.57. If you do not specify a threshold parameter, the threshold is set to 0.

For the three-parameter Weibull distribution described in Table 16.57, the scale, shape, and threshold parameters are estimated by maximum likelihood.

You should not interpret the parameters $\mu $ and $\sigma $ as representing the means and standard deviations for all of the distributions in Table 16.57. The normal is the only distribution in Table 16.57 for which this is the case.

Table 16.57: Distributions and Parameters for PROBPLOT and RELATIONPLOT Statements

   

Parameters

Distribution

Density Function

Location

Scale

Shape

Threshold

Normal

$\frac{1}{\sqrt {2\pi }\sigma }\exp \left(-\frac{(x-\mu )^{2}}{2\sigma ^{2}}\right)$

$\mu $

$\sigma $

   

Lognormal

$\frac{1}{\sqrt {2\pi }\sigma (x-\theta )}\exp \left(-\frac{(\log (x-\theta )-\mu )^{2}}{2\sigma ^{2}}\right)$

$\mu $

$\sigma $

 

$\theta $

Lognormal

$\frac{\log (10)}{\sqrt {2\pi }\sigma (x-\theta )}\exp \left(-\frac{(\log _{10}(x-\theta )-\mu )^{2}}{2\sigma ^{2}}\right)$

$\mu $

$\sigma $

 

$\theta $

(base 10)

         

Extreme value

$\frac{1}{\sigma }\exp \left(\frac{x-\mu }{\sigma }\right) \exp \left(-\exp \left(\frac{x-\mu }{\sigma }\right)\right)$

$\mu $

$\sigma $

   

Weibull

$\frac{\beta }{\alpha ^{\beta }}(x-\theta )^{\beta -1} \exp \left(-\left(\frac{x-\theta }{\alpha }\right)^{\beta }\right)$

  

$\alpha $

$\beta $

$\theta $

Exponential

$\frac{1}{\alpha }\exp \left(-\left(\frac{x-\theta }{\alpha }\right)\right)$

 

$\alpha $

 

$\theta $

Logistic

$\frac{\exp \left(\frac{x-\mu }{\sigma }\right)}{\sigma \left[1+\exp \left(\frac{x-\mu }{\sigma }\right)\right]^{2}}$

$\mu $

$\sigma $

   

Log-logistic

$\frac{\exp \left(\frac{\log (x-\theta )-\mu }{\sigma }\right)}{(x-\theta )\sigma \left[1+\exp \left(\frac{\log (x-\theta )-\mu }{\sigma }\right)\right]^{2}}$

$\mu $

$\sigma $

 

$\theta $

Three-parameter Weibull

$\frac{\beta }{\alpha ^{\beta }}(x-\theta )^{\beta -1} \exp \left(-\left(\frac{x-\theta }{\alpha }\right)^{\beta }\right)$

  

$\alpha $

$\beta $

$\theta $

Gompertz

$\frac{\beta }{\alpha } \exp \left\{ \frac{t}{\alpha } + \beta -\beta \exp \left(\frac{t}{\alpha }\right)\right\} $

  

$\alpha $

$\beta $

 

Three-parameter Gompertz

$\left\{ \frac{\beta }{\alpha } \exp (\frac{t}{\alpha }) + \frac{\theta }{\alpha } \right\} $

  

$\alpha $

$\beta $

$\theta $

 

$\qquad \times \exp \left\{ -\frac{\theta t}{\alpha } +\beta - \beta \exp (\frac{t}{\alpha })\right\} $

        


The threshold parameter for the three-parameter Gompertz distribution is known as the Makeham mortality component and is labeled as such in the output. This component represents an additive risk that is constant over time. The exponential distribution shown in Table 16.57 is a special case of the Weibull distribution with $\beta =1$. The remaining distributions in Table 16.57 are related to one another as shown in Table 16.58. The threshold parameter, $\theta $, is assumed to be 0 in Table 16.58.

Table 16.58: Relationship among Life Distributions

Distribution of T

Parameters

Distribution of Y=$\log $T

Parameters

Lognormal

$\mu $

$\sigma $

Normal

$\mu $

$\sigma $

Weibull

$\alpha $

$\beta $

Extreme value

$\mu = \log \alpha $

$\sigma = \frac{1}{\beta }$

Log-logistic

$\mu $

$\sigma $

Logistic

$\mu $

$\sigma $


MODEL Statement

All the distributions in Table 16.57 except the three-parameter Weibull and the Gompertz distributions are available for regression model estimation by using the MODEL statement. In addition, you can fit regression models with the generalized gamma distribution with the following probability density function $f(t)$:

\[ f(t) = \frac{|\lambda |}{t\sigma \Gamma (\lambda ^{-2})}(\lambda ^{-2})^{(\lambda ^{-2})} \exp \left[\lambda ^{-2}\left(\lambda \left(\frac{\log (t)-\mu }{\sigma }\right)- \exp \left(\lambda \left(\frac{\log (t)-\mu }{\sigma }\right)\right)\right)\right] \]

If a lifetime T has the generalized gamma distribution, then the logarithm of the lifetime $X=\log (T)$ has the generalized log-gamma distribution, with the following probability density function $g(x)$. When the gamma distribution is specified, the logarithms of the lifetimes are used as responses, and the generalized log-gamma distribution is used to estimate the parameters by maximum likelihood.

\[ g(x) = \frac{|\lambda |}{\sigma \Gamma (\lambda ^{-2})}(\lambda ^{-2})^{(\lambda ^{-2})} \exp \left[\lambda ^{-2}\left(\lambda \left(\frac{x-\mu }{\sigma }\right)- \exp \left(\lambda \left(\frac{x-\mu }{\sigma }\right)\right)\right)\right] \]

See Lawless (2003) and Meeker and Escobar (1998) for a description of the generalized gamma and generalized log-gamma distributions.

When $\lambda =1$, the generalized log-gamma distribution reduces to the extreme value distribution with parameters $\mu $ and $\sigma $. In this case, the log lifetimes have the extreme value distribution, or, equivalently, the lifetimes have the Weibull distribution with parameters $\alpha =\exp (\mu )$ and $\beta =1/\sigma $. When $\lambda =0$, the generalized log-gamma reduces to the normal distribution with parameters $\mu $ and $\sigma $. In this case, the (unlogged) lifetimes have the lognormal distribution with parameters $\mu $ and $\sigma $. This chapter uses the notation $\mu $ for the location, $\sigma $ for the scale, and $\lambda $ for the shape parameters for the generalized log-gamma distribution.

ANALYZE Statement

You can use the ANALYZE statement to compute parameter estimates and other statistics for the distributions in Table 16.57. In addition, you can compute estimates for the binomial and Poisson distributions. The forms of these distributions are shown in Table 16.59.

Table 16.59: Binomial and Poisson Distributions

Distribution

Pr{Y=y}

Parameter

Parameter Name

Binomial

$\binom {n}{y}p^{y}(1-p)^{n-y}$

p

binomial probability

Poisson

$\frac{\mu ^{y}}{y!}\exp (-\mu )$

$\mu $

Poisson mean