Probability Distributions

The RELIABILITY Procedure

Probability Distributions

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 30.37. 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 those distributions for which a threshold parameter is indicated in Table 30.37. If you do not specify a threshold parameter, the threshold is set to 0.

Note that you should not interpret the parameters $\mu$ and $\sigma$ as representing the means and standard deviations for all of the distributions in Table 30.37. The normal is the only distribution in Table 30.37 for which this is the case.

Table 30.37: Distributions and Parameters for PROBPLOT and RELATIONPLOT Statements

		Parameters
Distribution	Density Function	Location	Scale	Shape	Threshold

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

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

Lognormal	$\frac{\log(10)}{\sqrt{2\pi}\sigma(x-\theta)}\exp(-\frac{(\log_{10}(x-\theta)-\mu)^2}{2\sigma^2})$	$\mu$	$\sigma$		$\theta$
(base 10)

Extreme Value	$\frac{1}{\sigma}\exp(\frac{x-\mu}{\sigma}) \exp(-\exp(\frac{x-\mu}{\sigma}))$	$\mu$	$\sigma$

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

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

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

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

The exponential distribution shown in Table 30.37 is a special case of the Weibull distribution with $\beta=1$ .The remaining distributions in Table 30.37 are related to one another as shown in Table 30.38. The threshold parameter, $\theta$ , is assumed to be 0 in Table 30.38.

Table 30.38: Relationship among Life Distributions

Distribution of T	Parameters		Distribution of Y=logT	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 of the distributions in Table 30.37 are available for regression model estimation using the MODEL statement. In addition, the generalized gamma distribution with the following probability density function f(t) is available for regression model estimation in the MODEL statement.

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

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.

$g(x) = \frac{|\lambda|}{\sigma\Gamma(\lambda^{-2})}(\lambda^{-2})^{(\lambda^{-2... ...lambda^{-2}(\lambda(\frac{x-\mu}{\sigma})- \exp(\lambda(\frac{x-\mu}{\sigma})))]$

Refer to Lawless (1982, p. 240) 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 30.37. In addition, you can compute estimates for the binomial and Poisson distributions. The forms of these distributions are shown in Table 30.39.

Table 30.39: Binomial and Poisson Distributions

Distribution	Pr{Y=y}	Parameter	Parameter Name

Binomial	$(n \ y )p^y(1-p)^{n-y}$	p	binomial probability

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

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