neghypertol.int {tolerance}

Negative Hypergeometric Tolerance Intervals

Description

Provides 1-sided or 2-sided tolerance intervals for negative hypergeometric random variables. When sampling without replacement, these limits are on the total number of expected draws in a future sample in order to achieve a certain number from group A (e.g., "black balls" in an urn).

Usage

neghypertol.int(x, n, N, m = NULL, alpha = 0.05, P = 0.99,
                side = 1, method = c("EX", "LS", "CC"))

Arguments

x

The number of units drawn in order to achieve n successes. Can be a vector, in which case the sum of x is used.

n

The target number of successes in the sample drawn (e.g., the number of "black balls" you are to draw in the sample).

N

The population size (e.g., the total number of balls in the urn).

m

The target number of successes to be sampled from the universe for a future study. If m = NULL, then the tolerance limits will be constructed assuming n for this quantity.

alpha

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

P

The proportion of units from group A in future samples of size m 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 lower and upper confidence bounds, which are used in the calculation of the tolerance bounds. The default method is "EX", which is an exact-based method. "LS" is the large-sample method. "CC" gives a continuity-corrected version of the large-sample method.

Value

neghypertol.int returns a data frame with items:

alpha

The specified significance level.

P

The proportion of units from group A in future samples of size m.

rate

The sampling rate determined by x/N.

p.hat

The proportion of units in the sample from group A, calculated by n/x.

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.

Note

As this methodology is built using large-sample theory, if the sampling rate is less than 0.05, then a warning is generated stating that the results are not reliable.

References

Khan, R. A. (1994), A Note on the Generating Function of a Negative Hypergeometric Distribution, Sankhya: The Indian Journal of Statistics, Series B, 56, 309–313.

Young, D. S. (2014), Tolerance Intervals for Hypergeometric and Negative Hypergeometric Variables, Sankhya: The Indian Journal of Statistics, Series B, 77(1), 114–140.

See Also

acc.samp, NegHypergeometric

Examples

## 90%/95% 2-sided negative hypergeometric tolerance
## intervals for a future number of 20 successes when
## the universe is of size 100.  The estimates are 
## based on having drawn 50 in another sample to achieve 
## 20 successes.

neghypertol.int(50, 20, 100, m = 20, alpha = 0.05, 
                P = 0.95, side = 2, method = "LS")