Trace-Statistic Control Limits

Compute one-sided upper control limits for the auxiliary trace statistic \(T = (n - 1)\operatorname{tr}(\Sigma_0^{-1}S)\).

Usage

trv_limits(
  n,
  p,
  alpha = 0.0027,
  type = c("chisq", "simulation"),
  nsim = 1e+05,
  seed = NULL
)

Arguments

n

Integer subgroup sample size, at least 2.

p

Integer process dimension, at least 2.

alpha

Nominal upper-tail false-alarm probability, strictly between 0 and 1.

type

Character string selecting the calculation:

"chisq"

Exact chi-square quantile with \(p(n - 1)\) degrees of freedom.

"simulation"

Monte Carlo quantile from the same chi-square distribution. This is mainly a diagnostic and teaching option.

nsim

Integer number of Monte Carlo draws, at least 1000. Used only when type = "simulation".

seed

NULL or a finite numeric scalar used as the Monte Carlo seed. A supplied seed makes simulated limits reproducible and restores the caller's existing random-number state on exit.

Value

A list with components lcl, ucl, center, type, alpha, n, p, df, nsim, and seed. The lower control limit is zero because the chart is one-sided upper.

Details

Table 3 of Barbosa, Gneri, and Meneguetti uses \(N\) as the row label, although its numerical values correspond to subgroup size \(n\) and \(p(n - 1)\) degrees of freedom. Its printed probability headings also appear reversed: the column headed 0.9980 matches the 0.9973 quantile and vice versa. This function follows the mathematical definition and evaluates qchisq(1 - alpha, df = p * (n - 1)); it does not interchange tail probabilities to reproduce the apparent table-label error.

Examples

trv_limits(n = 8, p = 2)
#> $lcl
#> [1] 0
#> 
#> $ucl
#> [1] 33.19518
#> 
#> $center
#> [1] 14
#> 
#> $type
#> [1] "chisq"
#> 
#> $alpha
#> [1] 0.0027
#> 
#> $n
#> [1] 8
#> 
#> $p
#> [1] 2
#> 
#> $df
#> [1] 14
#> 
#> $nsim
#> [1] 1e+05
#> 
#> $seed
#> NULL
#> 
trv_limits(n = 8, p = 2, type = "simulation", nsim = 5000, seed = 2026)
#> $lcl
#> [1] 0
#> 
#> $ucl
#> [1] 33.97193
#> 
#> $center
#> [1] 14
#> 
#> $type
#> [1] "simulation"
#> 
#> $alpha
#> [1] 0.0027
#> 
#> $n
#> [1] 8
#> 
#> $p
#> [1] 2
#> 
#> $df
#> [1] 14
#> 
#> $nsim
#> [1] 5000
#> 
#> $seed
#> [1] 2026
#>