Probability Limits for u Charts

Compute normal or Cornish-Fisher probability limits for a u chart without constructing a plot. The model is \(X \sim Poisson(lambda n)\) and the monitored rate is \(U = X/n\).

Usage

uchart_limits(
  lambda,
  n,
  alpha = ALPHA,
  type = c("normal", "cf1", "cf2"),
  sides = c("two.sided", "upper"),
  truncate = TRUE
)

Arguments

lambda

In-control defect rate per inspection unit. A positive scalar.

n

Positive inspection size or vector of inspection sizes.

alpha

Nominal false-alarm probability. Defaults to 0.0027.

type

Limit method: "normal", "cf1", or "cf2".

sides

Either "two.sided" or "upper".

truncate

Logical. If TRUE, negative lower limits are set to 0.

Value

A list containing center, lcl, ucl, n, alpha, type, sides, and mean_count.

Details

For \(U\), the standardized third and fourth cumulants are \(gamma_1 = 1/sqrt(lambda n)\) and \(gamma_2 = 1/(lambda n)\). Consequently, the first Cornish-Fisher correction is \((z^2 - 1)/(6n)\) and the second-order correction reduces to \(z(1-z^2)/(72 n sqrt(lambda n))\). Each tail is evaluated at its signed normal quantile: \(z_{\mathrm{upper}} > 0\) for the UCL and \(z_{\mathrm{lower}} < 0\) for the LCL.

With \(z=3\), the CF2 limits recover $$UCL = lambda + 3 sqrt(lambda/n) + 4/(3n) - 1/(3 n sqrt(lambda n))$$ and $$LCL = lambda - 3 sqrt(lambda/n) + 4/(3n) + 1/(3 n sqrt(lambda n)),$$ before optional truncation at zero.

Examples

uchart_limits(1.4, 10, type = "normal")
#> $center
#> [1] 1.4
#> 
#> $lcl
#> [1] 0.2775114
#> 
#> $ucl
#> [1] 2.522489
#> 
#> $n
#> [1] 10
#> 
#> $alpha
#> [1] 0.0027
#> 
#> $type
#> [1] "normal"
#> 
#> $sides
#> [1] "two.sided"
#> 
#> $mean_count
#> [1] 14
#> 
uchart_limits(1.4, 10, type = "cf1")
#> $center
#> [1] 1.4
#> 
#> $lcl
#> [1] 0.4108424
#> 
#> $ucl
#> [1] 2.65582
#> 
#> $n
#> [1] 10
#> 
#> $alpha
#> [1] 0.0027
#> 
#> $type
#> [1] "cf1"
#> 
#> $sides
#> [1] "two.sided"
#> 
#> $mean_count
#> [1] 14
#> 
uchart_limits(1.4, 10, type = "cf2")
#> $center
#> [1] 1.4
#> 
#> $lcl
#> [1] 0.4197509
#> 
#> $ucl
#> [1] 2.646911
#> 
#> $n
#> [1] 10
#> 
#> $alpha
#> [1] 0.0027
#> 
#> $type
#> [1] "cf2"
#> 
#> $sides
#> [1] "two.sided"
#> 
#> $mean_count
#> [1] 14
#>