paretotol.int {tolerance}

Pareto (or Power Distribution) Tolerance Intervals

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 Y = ln(X) 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 Y = ln(1/W) is distributed according to a 2-parameter exponential distribution.

Value

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 \beta-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")