np.order {tolerance}

Sample Size Determination for Tolerance Limits Based on Order Statistics

Description

For given values of m, alpha, and P, this function solves the necessary sample size such that the r-th (or (n-s+1)-th) order statistic is the [100(1-alpha)%, 100(P)%] lower (or upper) tolerance limit (see the Details section below for further explanation). This function can also report all combinations of order statistics for 2-sided intervals.

Usage

np.order(m, alpha = 0.05, P = 0.99, indices = FALSE)

Arguments

m

See the Details section below for how m is defined.

alpha

1 minus the confidence level attained when it is desired to cover a proportion P of the population with the order statistics.

P

The proportion of the population to be covered with confidence 1-alpha with the order statistics.

indices

An optional argument to report all combinations of order statistics indices for the upper and lower limits of the 2-sided intervals. Note that this can only be calculated when m>1.

Details

For the 1-sided tolerance limits, m=s+r such that the probability is at least 1-alpha that at least the proportion P of the population is below the (n-s+1)-th order statistic for the upper limit or above the r-th order statistic for the lower limit. This means for the 1-sided upper limit that r=1, while for the 1-sided lower limit it means that s=1. For the 2-sided tolerance intervals, m=s+r such that the probability is at least 1-alpha that at least the proportion P of the population is between the r-th and (n-s+1)-th order statistics. Thus, all combinations of r>0 and s>0 such that m=s+r are considered.

Value

If indices = FALSE, then a single number is returned for the necessary sample size such that the r-th (or (n-s+1)-th) order statistic is the [100(1-alpha)%, 100(P)%] lower (or upper) tolerance limit. If indices = TRUE, then a list is returned with a single number for the necessary sample size and a matrix with 2 columns where each row gives the pairs of indices for the order statistics for all permissible [100(1-alpha)%, 100(P)%] 2-sided tolerance intervals.

References

Hanson, D. L. and Owen, D. B. (1963), Distribution-Free Tolerance Limits Elimination of the Requirement That Cumulative Distribution Functions Be Continuous, Technometrics, 5, 518–522.

Scheffe, H. and Tukey, J. W. (1945), Non-Parametric Estimation I. Validation of Order Statistics, Annals of Mathematical Statistics, 16, 187–192.

See Also

nptol.int

Examples

 
## Only requesting the sample size.

np.order(m = 5, alpha = 0.05, P = 0.95)

## Requesting the order statistics indices as well.

np.order(m = 5, alpha = 0.05, P = 0.95, indices = TRUE)