tolerance (version 1.2.0)

bayesnormtol.int: Bayesian Normal Tolerance Intervals

Description

Provides 1-sided or 2-sided Bayesian tolerance intervals under the conjugate prior for data distributed according to a normal distribution.

Usage

bayesnormtol.int(x = NULL, norm.stats = list(x.bar = NA,  s = NA, n = NA), alpha = 0.05, P = 0.99,  side = 1, method = c("HE", "HE2", "WBE",  "ELL", "KM", "EXACT", "OCT"), m = 50, hyper.par = list(mu.0 = NULL,  sig2.0 = NULL, m.0 = NULL, n.0 = NULL))

Arguments

x

A vector of data which is distributed according to a normal distribution.

norm.stats

An optional list of statistics that can be provided in-lieu of the full dataset. If provided, the user must specify all three components: the sample mean (x.bar), the sample standard deviation (s), and the sample size (n).

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".

hyper.par

A list consisting of the hyperparameters for the conjugate prior: the hyperparameters for the mean (mu.0 and n.0) and the hyperparameters for the variance (sig2.0 and m.0).

Value

bayesnormtol.int returns a data frame with items:

alpha

The specified significance level.

P

The proportion of the population covered by this tolerance interval.

x.bar

The sample mean.

1-sided.lower

The 1-sided lower Bayesian tolerance bound. This is given only if side = 1.

1-sided.upper

The 1-sided upper Bayesian tolerance bound. This is given only if side = 1.

2-sided.lower

The 2-sided lower Bayesian tolerance bound. This is given only if side = 2.

2-sided.upper

The 2-sided upper Bayesian tolerance bound. This is given only if side = 2.

Details

Note that if one considers the non-informative prior distribution, then the Bayesian tolerance intervals are the same as the classical solution, which can be obtained by using normtol.int.

References

Aitchison, J. (1964), Bayesian Tolerance Regions, Journal of the Royal Statistical Society, Series B, 26, 161--175.

Guttman, I. (1970), Statistical Tolerance Regions: Classical and Bayesian, Charles Griffin and Company.

Young, D. S., Gordon, C. M., Zhu, S., and Olin, B. D. (2016), Sample Size Determination Strategies for Normal Tolerance Intervals Using Historical Data, Quality Engineering (to appear).

See Also

Normal, normtol.int, K.factor

Examples

## 95%/85% 1-sided Bayesian normal tolerance limits for 
## a sample of size 100. 

set.seed(100)
x <- rnorm(100)
out <- bayesnormtol.int(x = x, alpha = 0.05, P = 0.85, 
                   side = 1, method = "EXACT", 
                   hyper.par = list(mu.0 = 0, sig2.0 = 1, 
                   n.0 = 10, m.0 = 10))
out

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