exttol.int {tolerance} | R Documentation |
Provides 1-sided tolerance intervals for data distributed according to either a Weibull distribution or an extreme-value (also called Gumbel) distribution.
exttol.int(x, alpha = 0.05, P = 0.99, dist = c("Weibull", "Gumbel"), NR.delta = 1e-8)
x |
A vector of data which is distributed according to either a Weibull distribution or an extreme-value distribution. |
alpha |
The level chosen such that 1-alpha is the confidence level. |
P |
The proportion of the population to be covered by this tolerance interval. |
dist |
Select either dist = "Weibull" or dist = "Gumbel" if the data is distributed according
to the Weibull or extreme-value distribution, respectively. |
NR.delta |
The stopping criterion used for the Newton-Raphson algorithm when finding the maximum likelihood estimates of the Weibull or extreme-value distribution. |
Recall that the relationship between the Weibull distribution and the extreme-value distribution is that if the random variable X is distributed according to a Weibull distribution, then the random variable Y = ln(X) is distributed according to an extreme-value distribution.
If dist = "Weibull"
, then the natural logarithm of the data are taken so that a Newton-Raphson algorithm can
be employed to find the MLEs of the extreme-value distribution and then the data and MLEs are transformed back appropriately.
No transformation is performed if dist = "Gumbel"
. The Newton-Raphson algorithm is initialized by the method of moments
estimators for the parameters.
exttol.int
returns a data frame with items:
alpha |
The specified significance level. |
P |
The proportion of the population covered by this tolerance interval. |
shape.1 |
MLE for the shape parameter if dist = "Weibull" or for the location parameter if
dist = "Gumbel" . |
shape.2 |
MLE for the scale parameter if dist = "Weibull" or dist = "Gumbel" . |
1-sided.lower |
The 1-sided lower tolerance bound. |
1-sided.upper |
The 1-sided upper tolerance bound. |
Bain, L. J. and Engelhardt, M. (1981), Simple Approximate Distributional Results for Confidence and Tolerance Limits for the Weibull Distribution Based on Maximum Likelihood Estimators, Technometrics, 23, 15–20.
## 90%/90% 1-sided Weibull tolerance intervals for a sample ## of size 150. set.seed(100) x <- rweibull(150, 3, 75) out <- exttol.int(x = x, alpha = 0.15, P = 0.90, dist = "Weibull") out plottol(out, x, plot.type = "both", side = "lower", x.lab = "Weibull Data")