exttol.int {tolerance}

Weibull (or Extreme-Value) Tolerance Intervals

Description

Provides 1-sided or 2-sided tolerance intervals for data distributed according to either a Weibull distribution or extreme-value (also called Gumbel) distributions.

Usage

exttol.int(x, alpha = 0.05, P = 0.99, side = 1,
           dist = c("Weibull", "Gumbel"), ext = c("min", "max"), 
           NR.delta = 1e-8)

Arguments

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.

side

Whether a 1-sided or 2-sided tolerance interval is required (determined by side = 1 or side = 2, respectively).

dist

Select either dist = "Weibull" or dist = "Gumbel" if the data is distributed according to the Weibull or extreme-value distribution, respectively.

ext

If dist = "Gumbel", then select which extreme is to be modeled for the Gumbel distribution. The Gumbel distribution for the minimum (i.e., ext = "min") corresponds to a left-skewed distribution and the Gumbel distribution for the maximum (i.e., ext = "max") corresponds to a right-skewed distribution

NR.delta

The stopping criterion used for the Newton-Raphson algorithm when finding the maximum likelihood estimates of the Weibull or extreme-value distribution.

Details

Recall that the relationship between the Weibull distribution and the extreme-value distribution for the minimum 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 for the minimum.

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 for the minimum 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.

Value

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. This is given only if side = 1.

1-sided.upper

The 1-sided upper tolerance bound. This is given only if side = 1.

2-sided.lower

The 2-sided lower tolerance bound. This is given only if side = 2.

2-sided.upper

The 2-sided upper tolerance bound. This is given only if side = 2.

References

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.

Coles, S. (2001), An Introduction to Statistical Modeling of Extreme Values, Springer.

See Also

Weibull

Examples

 
## 85%/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, side = 1,
                  dist = "Weibull")
out

plottol(out, x, plot.type = "both", side = "lower", 
        x.lab = "Weibull Data")