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 Functions

## CUSUMARL Function

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

### Syntax

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. is the shift to be detected, expressed as a multiple of the process standard deviation . 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 , 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 ,where n is the subgroup sample size). headstart is the headstart value (optional) expressed in standard units (a multiple of ,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 , 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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