tolerance (version 1.2.0)

gamtol.int: Gamma (or Log-Gamma) Tolerance Intervals

Description

Provides 1-sided or 2-sided tolerance intervals for data distributed according to either a gamma distribution or log-gamma distribution.

Usage

gamtol.int(x, alpha = 0.05, P = 0.99, side = 1,  method = c("HE", "HE2", "WBE", "ELL", "KM", "EXACT",  "OCT"), m = 50, log.gamma = FALSE)

Arguments

x

A vector of data which is distributed according to either a gamma distribution or a log-gamma 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 calculating the k-factors. The k-factor for the 1-sided tolerance intervals is performed exactly and thus is the same for the chosen method. "HE" is the Howe method and is often viewed as being extremely accurate, even for small sample sizes. "HE2" is a second method due to Howe, which performs similarly to the Weissberg-Beatty method, but is computationally simpler. "WBE" is the Weissberg-Beatty method (also called the Wald-Wolfowitz method), which performs similarly to the first Howe method for larger sample sizes. "ELL" is the Ellison correction to the Weissberg-Beatty method when f is appreciably larger than n^2. A warning message is displayed if f is not larger than n^2. "KM" is the Krishnamoorthy-Mathew approximation to the exact solution, which works well for larger sample sizes. "EXACT" computes the k-factor exactly by finding the integral solution to the problem via the integrate function. Note the computation time of this method is largely determined by m. "OCT" is the Owen approach to compute the k-factor when controlling the tails so that there is not more than (1-P)/2 of the data in each tail of the distribution.

m

The maximum number of subintervals to be used in the integrate function. This is necessary only for method = "EXACT" and method = "OCT". The larger the number, the more accurate the solution. Too low of a value can result in an error. A large value can also cause the function to be slow for method = "EXACT".

log.gamma

If TRUE, then the data is considered to be from a log-gamma distribution, in which case the output gives tolerance intervals for the log-gamma distribution. The default is FALSE.

Value

gamtol.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 log-gamma distribution, then the random variable $Y = ln(X)$ is distributed according to a gamma distribution.

References

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

GammaDist, K.factor

Examples

## 99%/99% 1-sided gamma tolerance intervals for a sample
## of size 50. 

set.seed(100)
x <- rgamma(50, 0.30, scale = 2)
out <- gamtol.int(x = x, alpha = 0.01, P = 0.99, side = 1,
                  method = "HE")
out

plottol(out, x, plot.type = "both", side = "upper", 
        x.lab = "Gamma Data")