# paretotol.int {tolerance}

Pareto (or Power Distribution) Tolerance Intervals
Package:
tolerance
Version:
1.2.0

### Description

Provides 1-sided or 2-sided tolerance intervals for data distributed according to either a Pareto distribution or a power distribution (i.e., the inverse Pareto distribution).

### Usage

```paretotol.int(x, alpha = 0.05, P = 0.99, side = 1,
method = c("GPU", "DUN"), power.dist = FALSE)
```

### Arguments

x
A vector of data which is distributed according to either a Pareto distribution or a power 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).
method
The method for how the upper tolerance bound is approximated when transforming to utilize the relationship with the 2-parameter exponential distribution. `"GPU"` is the Guenther-Patil-Upppuluri method. `"DUN"` is the Dunsmore method, which was empirically shown to be an improvement for samples greater than or equal to 8. More information on these methods can be found in the "References".
power.dist
If `TRUE`, then the data is considered to be from a power distribution, in which case the output gives tolerance intervals for the power distribution. The default is `FALSE`.

### Details

Recall that if the random variable X is distributed according to a Pareto distribution, then the random variable is distributed according to a 2-parameter exponential distribution. Moreover, if the random variable W is distributed according to a power distribution, then the random variable X = 1/W is distributed according to a Pareto distribution, which in turn means that the random variable is distributed according to a 2-parameter exponential distribution.

### Values

`paretotol.int` returns a data frame with items:

alpha
The specified significance level.
P
The proportion of the population covered by this tolerance interval.
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

Dunsmore, I. R. (1978), Some Approximations for Tolerance Factors for the Two Parameter Exponential Distribution, Technometrics, 20, 317--318.

Engelhardt, M. and Bain, L. J. (1978), Tolerance Limits and Confidence Limits on Reliability for the Two-Parameter Exponential Distribution, Technometrics, 20, 37--39. Guenther, W. C., Patil, S. A., and Uppuluri, V. R. R. (1976), One-Sided β-Content Tolerance Factors for the Two Parameter Exponential Distribution, Technometrics, 18, 333--340.

Krishnamoorthy, K., Mathew, T., and Mukherjee, S. (2008), Normal-Based Methods for a Gamma Distribution: Prediction and Tolerance Intervals and Stress-Strength Reliability, Technometrics, 50, 69--78.

`TwoParExponential`, `exp2tol.int`

### Examples

```## 95%/99% 2-sided Pareto tolerance intervals
## for a sample of size 500.

set.seed(100)
x <- exp(r2exp(500, rate = 0.15, shift = 2))
out <- paretotol.int(x = x, alpha = 0.05, P = 0.99, side = 2,
method = "DUN", power.dist = FALSE)
out

plottol(out, x, plot.type = "both", side = "two",
x.lab = "Pareto Data")```

Documentation reproduced from package tolerance, version 1.2.0. License: GPL (>= 2)