Non-Normal Distributions - Weibull Distribution
The Weibull distribution has the probability density function:
f(x) = c/b*[(x-θ)/b]c-1 * e^{-[(x-θ)/b]c}
θ < x, b > 0, c > 0
where
| b | is the scale parameter of the distribution |
| c | is the shape parameter of the distribution |
| θ | is the location parameter of the distribution |
| e | is the base of the natural logarithm, sometimes called Euler's e (2.71...) |
Threshold (location) parameter
The valid range for the Weibull distribution is from 0 to infinity. Instead of 0 (zero), Statistica allows you to enter a different value for the lowest threshold (location) parameter; that value will be subtracted from the data values before the
Weibull distribution is fitted. Thus, the threshold value must be less than the smallest observed value.
Applications
As described earlier, the exponential distribution is often used as a model of time-to-failure measurements, when the failure (hazard) rate is constant over time. When the failure probability varies over time, then the
Weibull distribution is appropriate. Thus, the
Weibull distribution is often used in
reliability testing (e.g., of electronic relays, ball bearings, etc.; see Hahn and Shapiro, 1967).
Estimation
Statistica will compute
maximum likelihood estimates for the scale and shape parameters of the
Weibull distribution (see Evans, Hastings, and Peacock, 1993).
Weibull distribution in Survival Analysis
Note that in Survival Analysis, instead of the scale parameter
b, the inverse 1/b = Lambda is often estimated. Also, if you use the life table analysis facilities to estimate the parameters of the Weibull distribution (using weighted least squares methods), the program will estimate and report the parameter
L' = Lc (Lambda to the power of
c). Therefore, when comparing the results computed by the Survival Analysis module with those computed by the
Process Analysis module, the estimates for the scale parameter will not be directly compatible.
Weibull and reliability/failure time analysis
Maximum likelihood estimates for the two-and three-parameter
Weibull distribution can also be computed via the Weibull & Reliability/Failure Time Analysis options. Those options will also handle data sets with censored observations and grouped (life table) data.