K.factor {tolerance}

Estimating K-factors for Tolerance Intervals Based on Normality

Description

Estimates k-factors for tolerance intervals based on normality.

Usage

K.factor(n, f = NULL, alpha = 0.05, P = 0.99, side = 1, 
         method = c("HE", "HE2", "WBE", "ELL", "KM", "EXACT", 
         "OCT"), m = 50)

Arguments

n

The (effective) sample size.

f

The number of degrees of freedom associated with calculating the estimate of the population standard deviation. If NULL, then f is taken to be n-1.

alpha

The level chosen such that 1-alpha is the confidence level.

P

The proportion of the population to be covered by the 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".

Value

K.factor returns the k-factor for tolerance intervals based on normality with the arguments specified above.

Note

For larger sample sizes, there may be some accuracy issues with the 1-sided calculation since it depends on the noncentral t-distribution. The code is primarily intended to be used for moderate values of the noncentrality parameter. It will not be highly accurate, especially in the tails, for large values. See TDist for further details.

References

Ellison, B. E. (1964), On Two-Sided Tolerance Intervals for a Normal Distribution, Annals of Mathematical Statistics, 35, 762–772.

Howe, W. G. (1969), Two-Sided Tolerance Limits for Normal Populations - Some Improvements, Journal of the American Statistical Association, 64, 610–620.

Krishnamoorthy, K. and Mathew, T. (2009), Statistical Tolerance Regions: Theory, Applications, and Computation, Wiley.

Odeh, R. E. and Owen, D. B. (1980), Tables for Normal Tolerance Limits, Sampling Plans, and Screening, Marcel-Dekker.

Owen, D. B. (1964), Controls of Percentages in Both Tails of the Normal Distribution, Technometrics, 6, 377-387.

Wald, A. and Wolfowitz, J. (1946), Tolerance Limits for a Normal Distribution, Annals of the Mathematical Statistics, 17, 208–215.

Weissberg, A. and Beatty, G. (1969), Tables of Tolerance Limit Factors for Normal Distributions, Technometrics, 2, 483–500.

See Also

integrate, K.table, normtol.int, TDist

Examples

 
## Showing the k-factor under the Howe, Weissberg-Beatty, 
## and exact estimation methods.

K.factor(10, P = 0.95, side = 2, method = "HE")
K.factor(10, P = 0.95, side = 2, method = "WBE")
K.factor(10, P = 0.95, side = 2, method = "EXACT", m = 20)