# paretotol.int {tolerance}

### 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.

### See Also

`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)

## Comments

solutions are properly delivered so it's just worthy to have let the problem solver rest in the fauteuil relax