Calculates values related to Extended Hanson--Koopmans tolerance bounds as described by Vangel (1994).

hk_ext_z(n, i, j, p, conf)

hk_ext_z_j_opt(n, p, conf)

## Arguments

n

the sample size

i

the first order statistic (1 <= i < j)

j

the second order statistic (i < j <= n)

p

the content of the tolerance bound (normally 0.90 or 0.99)

conf

the confidence level (normally 0.95)

## Value

For hk_ext_z, the return value is a numeric value representing the parameter z (denoted as k in CMH-17-1G).

For hk_ext_z_j_opt, the return value is named list containing z and k. The former is the value of z, as defined by Vangel (1994), and the latter is the corresponding order statistic.

## Details

Hanson (1964) presents a nonparametric method for determining tolerance bounds based on consecutive order statistics. Vangel (1994) extends this method using non-consecutive order statistics.

The extended Hanson--Koopmans method calculates a tolerance bound (basis value) based on two order statistics and a weighting value z. The value of z is based on the sample size, which order statistics are selected, the desired content of the tolerance bond and the desired confidence level.

The function hk_ext_z calculates the weighting variable z based on selected order statistics i and j. Based on this value z, the tolerance bound can be calculated as:

$$S = z X_{(i)} + (1 - z) X_{(j)}$$

Where $$X_{(i)}$$ and $$X_{(j)}$$ are the i-th and j-th ordered observation.

The function hk_ext_z_j_opt determines the value of j and the corresponding value of z, assuming i=1. The value of j is selected such that the computed tolerance limit is nearest to the desired population quantile for a standard normal distribution when the order statistics are equal to the expected value of the order statistics for the standard normal distribution.

basis_hk_ext()

## Examples

# The factors from Table 1 of Vangel (1994) can be recreated
# using the hk_ext_z function. For the sample size n=21,
# the median is the 11th ordered observation. The factor
# required for calculating the tolerance bound with a content
# of 0.9 and a confidence level of 0.95 based on the median
# and first ordered observation can be calculated as follows.
hk_ext_z(n = 21, i = 1, j = 11, p = 0.9, conf = 0.95)
#>  1.204806

##  1.204806

# The hk_ext_z_j_opt function can be used to refine this value
# of z by finding an optimum value of j, rather than simply
# using the median. Here, we find that the optimal observation
# to use is the 10th, not the 11th (which is the median).
hk_ext_z_j_opt(n = 21, p = 0.9, conf = 0.95)
#> $z #>  1.217717 #> #>$j
#>  10
#>

## $z ##  1.217717 ## ##$j
##  10