tolerance (version 0.4.0)

paretotol.int: 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 met

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.

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.

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.

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")