Performs a lifetime analysis based on the two-parameter Weibull distribution. The input for the lifetime analysis is one or more random samples with object service lives. The sample consists exclusively of the failed objects with corresponding failure times in the Y component of the data set. If you evaluate several samples, the object provides a list with one result per sample.
Parameters
The Number of right-censored data indicates how many objects had not yet failed by the end of the measurement (and are therefore not included in the sample). You can store this value as the NumberOfRightCensoredValues parameter directly in the sample data set. FlexPro then uses this in the lifetime analysis and ignores the value on the Options tab. You should especially consider this if you want to evaluate several samples. Only one number can be specified in the object, which is then used for all samples.
You can choose between the least squares and the maximum likelihood algorithms. For the maximum likelihood algorithm, you can apply a Correction factor that depends on the sample size. Background: One disadvantage of maximum likelihood parameter estimation is its bias when the sample size is small. To correct this disadvantage, appropriate correction factors (depending on the sample size) can be taken into account.
The maximum likelihood curve fitting of the two-parameter Weibull distribution is solved numerically by searching for zeros. The latter is implemented using fixed-point iteration. The Maximum number of iterations specifies the maximum number of iterations to be carried out. The fixed point iteration is otherwise terminated if the relative value change is smaller than the specified Tolerance.
Result
You can select the following results:
Result |
Description |
|---|---|
Weibull distribution |
The Weibull distribution estimates as a curve over time what proportion of objects will have failed at a point in time. |
Survival function |
The lifetime function estimates as a curve over time what proportion of objects will still be functional at a point in time. (Corresponds to 1 - Weibull distribution) |
Hazard function |
This is the quotient of the density function of the Weibull distribution and the lifetime function. |
Cumulative hazard function |
The integral of the failure rate. |
Empirical distribution |
The empirical distribution of the sample. |
Empirical survival function
|
Corresponds to 1 - empirical distribution. |
Statistics |
A list with the determined parameters of the Weibull distribution and further statistical parameters. |
You can output the frequencies of the distributions as relative frequencies normalized to one, as percentage frequencies or as absolute frequencies in relation to the size of the population, including censored values.
Here are the statistics in detail:
Result |
Description |
|---|---|
α |
He estimated parameter alpha of the Weibull distribution. |
β |
He estimated parameter beta of the Weibull distribution. |
T |
The characteristic lifetime of the distribution. In a Weibull distribution, the characteristic lifetime is the time at which 63.2 % of the population is likely to have failed. This parameter is also called the scale parameter. |
λ |
The parameter lambda of the Weibull distribution. This is the reciprocal of T. Lambda is also called the shape parameter. |
μ |
The expected value of the lifetime. |
σ² |
The variance of the lifetime. |
σ |
The standard deviation of the lifetime. |
If you select several results, the lifetime analysis provides a list.
Sampling
In order to numerically calculate the results of the Weibull distribution and survival function, the determined distribution function is sampled. Obtain from data set takes the X component of the distribution to be calculated from the specified data set. Calculate uses the parameters Start value, End value and Number of values to generate a data series with linearly increasing values for the X component of the distribution to be calculated. You can select Automatic for the End value. This is then derived from the determined parameter T as 5 * T.
Graphs (only in the Analysis Wizard)
The Survival plot shows the empirical survival function of the sample over the survival function determined by Weibull fit.
The Weibull plot shows the empirical distribution function of the sample over the Weibull distribution determined by Weibull fit. The Weibull distribution is shown here in linearized form. This is done using the transformation Ln(-Ln(1 - Y)), Ln(X).
Statistics Table (only in the Analysis Wizard)
The Optionla numeric results option on the third page of the Analysis Wizard creates a table with the statistical parameters specified above.
References
•Hartung, Joachim: Statistik (Statistics), 9th edition. Oldenbourg Verlag GmbH, Munich, 1993. ISBN 3-486-22055-1.
•Wilker, Holger: Weibull-Statistik in der Praxis (Weibull Statstics in Practice), 2nd edition. Books on Demand GmbH, Norderstedt, 2010. ISBN 3-8391-6241-5.
•T. Kernane and Z. A. Raizah: Fixed point iteration for estimating the parameters of extreme value distributions. In: Communications in Statistics - Simulation and Computation, Taylor and Francis, Vol. 38, No. 10, Pages 2161-2170. Prentice Hall, 2009.