Compute the actual in-control false-alarm probability of the conventional three-sigma range chart under the exact distribution of the relative range statistic \(W = R / \sigma\).
Value
A numeric vector with the same length and order as n.
Element \(i\) is
$$1 - \{F_W(D_2;n_i) - F_W(D_1;n_i)\},$$
where \(F_W\) is evaluated with stats::ptukey() using infinite
denominator degrees of freedom.
Details
The conventional limits are $$D_1\sigma = \max\{0, d_2(n) - 3d_3(n)\}\sigma$$ and $$D_2\sigma = \{d_2(n) + 3d_3(n)\}\sigma.$$ Because the relative range is not normally distributed for the small subgroup sizes commonly used in practice, the resulting false-alarm probability can be substantially larger than 0.0027.
Interpretation
The result is the probability that an in-control subgroup falls below the conventional lower limit or above the conventional upper limit. It is not the nominal value requested by an exact probability-limit design.
Errors
An error is raised when any subgroup size is smaller than 2. The underlying
d2() and d3() calculations also require finite admissible
subgroup sizes.
References
Barbosa, E. P., Gneri, M. A., and Meneguetti, A. (2013). Range control charts revisited: Simpler Tippett-like formulae, its practical implementation, and the study of false alarm. Communications in Statistics - Simulation and Computation, 42(2), 247–262. doi:10.1080/03610918.2011.639967 .
See also
cchart.R, d2, d3,
table.qtukey
Examples
alpha.risk(5)
#> [1] 0.004603048
alpha.risk(c(3, 5, 10, 15))
#> [1] 0.005842954 0.004603048 0.004367441 0.004493839