xDshewhartrunsrules.arl: Compute ARLs of Shewhart control charts with and without runs rules under drift

Description

Computation of the zero-state Average Run Length (ARL) under drift for Shewhart control charts with and without runs rules monitoring normal mean.

Usage

xDshewhartrunsrules.arl(delta, c = 1, m = NULL, type = "12")
xDshewhartrunsrulesFixedm.arl(delta, c = 1, m = 100, type = "12")

Value

Returns a single value which resembles the ARL.

Arguments

delta: true drift parameter.
c: normalizing constant to ensure specific alarming behavior.
type: controls the type of Shewhart chart used, seed details section.
m: parameter of Gan's approach. If m=NULL, then m will increased until the resulting ARL does not change anymore.

Author

Sven Knoth

Details

Based on Gan (1991), the ARL is calculated for Shewhart control charts with and without runs rules under drift. The usual ARL function with mu=m*delta is determined and recursively via m-1, m-2, ... 1 (or 0) the drift ARL determined. xDshewhartrunsrulesFixedm.arl is the actual work horse, while xDshewhartrunsrules.arl provides a convenience wrapper. Note that Aerne et al. (1991) deployed a method that is quite similar to Gan's algorithm. For type see the help page of xshewhartrunsrules.arl.

References

F. F. Gan (1991), EWMA control chart under linear drift, J. Stat. Comput. Simulation 38, 181-200.

L. A. Aerne, C. W. Champ and S. E. Rigdon (1991), Evaluation of control charts under linear trend, Commun. Stat., Theory Methods 20, 3341-3349.

Examples

Run this code

## Aerne et al. (1991)
## Table I (continued)
## original numbers are
#     delta arl1of1 arl2of3 arl4of5  arl10
#  0.005623  136.67  120.90  105.34 107.08
#  0.007499  114.98  101.23   88.09  89.94
#  0.010000   96.03   84.22   73.31  75.23
#  0.013335   79.69   69.68   60.75  62.73
#  0.017783   65.75   57.38   50.18  52.18
#  0.023714   53.99   47.06   41.33  43.35
#  0.031623   44.15   38.47   33.99  36.00
#  0.042170   35.97   31.36   27.91  29.90
#  0.056234   29.21   25.51   22.91  24.86
#  0.074989   23.65   20.71   18.81  20.70
#  0.100000   19.11   16.79   15.45  17.29
#  0.133352   15.41   13.61   12.72  14.47
#  0.177828   12.41   11.03   10.50  12.14
#  0.237137    9.98    8.94    8.71  10.18
#  0.316228    8.02    7.25    7.26   8.45
#  0.421697    6.44    5.89    6.09   6.84
#  0.562341    5.17    4.80    5.15   5.48
#  0.749894    4.16    3.92    4.36   4.39
#  1.000000    3.35    3.22    3.63   3.52
c1of1 <- 3.069/3
c2of3 <- 2.1494/2
c4of5 <- 1.14
c10   <- 3.2425/3
DxDshewhartrunsrules.arl <- Vectorize(xDshewhartrunsrules.arl, "delta")
deltas <- 10^(-(18:0)/8)
arl1of1 <-
round(DxDshewhartrunsrules.arl(deltas, c=c1of1, type="1"), digits=2)
arl2of3 <-
round(DxDshewhartrunsrules.arl(deltas, c=c2of3, type="12"), digits=2)
arl4of5 <-
round(DxDshewhartrunsrules.arl(deltas, c=c4of5, type="13"), digits=2)
arl10 <- 
round(DxDshewhartrunsrules.arl(deltas, c=c10, type="SameSide10"), digits=2)
data.frame(delta=round(deltas, digits=6), arl1of1, arl2of3, arl4of5, arl10)

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