CUSUMARL Function

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Functions

CUSUMARL Function

computes the average run length for a cumulative sum control chart scheme.

Syntax

CUSUMARL(type, $\delta,h,k\lt,$ headstart>)

where

type indicates a one-sided or two-sided scheme. Valid values are `ONESIDED' or `O' for a one-sided scheme, and `TWOSIDED' or `T' for a two-sided scheme.

$\delta$ is the shift to be detected, expressed as a multiple of the process standard deviation $(\sigma)$ .

h is the decision interval (one-sided scheme) or the vertical distance between the origin and the upper arm of the V-mask (two-sided scheme), each time expressed as a positive value in standard units (a multiple of $\sigma/\sqrt{n}$ , where n is the subgroup sample size).

k is the reference value (one-sided scheme) or the slope of the lower arm of the V-mask (two-sided scheme), each time expressed as a positive value in standard units (a multiple of $\sigma/\sqrt{n}$ ,where n is the subgroup sample size).

headstart is the headstart value (optional) expressed in standard units (a multiple of $\sigma/\sqrt{n}$ ,where n is the subgroup sample size). The default headstart is zero. For details, refer to Lucas and Crosier (1982).

Description

The CUSUMARL function returns the average run length of one-sided and two-sided cumulative sum schemes with parameters as described above. The notation is consistent with that used in the CUSUM procedure.

For a one-sided scheme, the average run length is calculated using the integral equation method (with 24 Gaussian points) described by Goel and Wu (1971) and Lucas and Crosier (1982).

For a two-sided scheme with no headstart, the average run length (ARL) is calculated using the fact that

( ARL)^-1 = ( ARL₊)^-1 + ( ARL_-)^-1

where ARL₊ and ARL_- denote the average run lengths of the equivalent one-sided schemes for detecting a shift of the same magnitude in the positive direction and in the negative direction, respectively.

For a two-sided scheme with a nonzero headstart, the ARL is calculated by combining average run lengths for one-sided schemes as described in Appendix A.1 of Lucas and Crosier (1982, 204).

For a specified shift $\delta$ , you can use the CUSUMARL function to design a cusum scheme by first calculating average run lengths for a range of values of h and k and then choosing the combination of h and k that yields a desired average run length.

You can also use the CUSUMARL function to interpolate published tables of average run lengths.

Examples

The following three sets of statements result in the values 4.1500826715, 4.1500836225, and 4.1061588131, respectively.

   data;
      arl=cusumarl('twosided',2.5,8,0.25);
      put arl;
   run;


   data;
      arl=cusumarl('onesided',2.5,8,0.25);
      put arl;
   run;


   data;
      arl=cusumarl('o',2.5,8,0.25,0.1);
      put arl;
   run;

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type	indicates a one-sided or two-sided scheme. Valid values are `ONESIDED' or `O' for a one-sided scheme, and `TWOSIDED' or `T' for a two-sided scheme.
$\delta$	is the shift to be detected, expressed as a multiple of the process standard deviation $(\sigma)$ .
h	is the decision interval (one-sided scheme) or the vertical distance between the origin and the upper arm of the V-mask (two-sided scheme), each time expressed as a positive value in standard units (a multiple of $\sigma/\sqrt{n}$ , where n is the subgroup sample size).
k	is the reference value (one-sided scheme) or the slope of the lower arm of the V-mask (two-sided scheme), each time expressed as a positive value in standard units (a multiple of $\sigma/\sqrt{n}$ ,where n is the subgroup sample size).
headstart	is the headstart value (optional) expressed in standard units (a multiple of $\sigma/\sqrt{n}$ ,where n is the subgroup sample size). The default headstart is zero. For details, refer to Lucas and Crosier (1982).