Description Usage Arguments Details Value References See Also Examples

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

1 2 3 | ```
exttol.int(x, alpha = 0.05, P = 0.99, side = 1,
dist = c("Weibull", "Gumbel"), ext = c("min", "max"),
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 |

`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 |

`dist` |
Select either |

`ext` |
If |

`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 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.

`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 |

`shape.2` |
MLE for the scale parameter if |

`1-sided.lower` |
The 1-sided lower tolerance bound. This is given only if |

`1-sided.upper` |
The 1-sided upper tolerance bound. This is given only if |

`2-sided.lower` |
The 2-sided lower tolerance bound. This is given only if |

`2-sided.upper` |
The 2-sided upper tolerance bound. This is given only if |

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.

1 2 3 4 5 6 7 8 9 10 11 12 | ```
## 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")
``` |

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