# getARL: compute average run length (ARL) for CUSUM charts In CUSUMdesign: Compute Decision Interval and Average Run Length for CUSUM Charts

## Description

Compute average run lengths for CUSUM charts based on the Markov chain algorithm.

## Usage

 ```1 2 3``` ```getARL(distr=NULL, K=NULL, H=NULL, Mean=NULL, std=NULL, prob=NULL, Var=NULL, mu=NULL, lambda=NULL, samp.size=NULL, is.upward=NULL, winsrl=NULL, winsru=NULL) ```

## Arguments

 `distr` Integer valued from 1 to 6: 1 refers to “normal mean", 2 refers to “normal variance", 3 refers to “Poisson", 4 refers to “binomial", 5 refers to “negative binomial", and 6 refers to “inverse Gaussian mean". `K` A reference value, which is given by `getH`. `H` A given decision interval, which is given by `getH`. `Mean` Mean value, which has to be provided when distr = 1 (normal mean), 3 (Poisson), and 5 (negative binomial). The value must be positive when distr = 3 or distr = 5. `std` Standard deviation, which has to be provided when distr = 1 (normal mean) and 2 (normal variance). The value must be positive. `prob` Success probability, which has to be provided when distr = 4 (binomial); 0 < prob <= 1. `Var` Variance, which has to be provided when distr = 5 (negative binomial). The value has to be larger than Mean when distr = 5. `mu` A positive value representing the mean of inverse Gaussian distribution. The argument 'mu' has to be provided when distr = 6 (inverse Gaussian mean). `lambda` A positive value representing the shape parameter for inverse Gaussian distribution. The argument 'lambda' has to be provided when distr = 6 (inverse Gaussian mean). `samp.size` Sample size, an integer which has to be provided when distr = 2 (normal variance) or distr = 4 (binomial). `is.upward` Logical value, whether to depict a upward or downward CUSUM. `winsrl` Lower Winsorizing constant. Use NULL or -999 if Winsorization is not needed. `winsru` Upper Winsorizing constant. Use NULL or 999 if Winsorization is not needed.

## Details

Computes ARL when the reference value and decision interval are given. For each case, the necessary parameters are listed as follows.

Normal mean (distr = 1): `Mean`, `std`, `K`, `H`.
Normal variance (distr = 2): `samp.size`, `std`, `K`, `H`.
Poisson (distr = 3): `Mean`, `K`, `H`.
Binomial (dist = 4): `samp.size`, `prob`, `K`, `H`.
Negative binomial (distr = 5): `Mean`, `Var`, `K`, `H`.
Inverse Gaussian mean (distr = 6): `mu`, `lambda`, `K`, `H`.

## Value

A list including three variables:

 `ARL_Z` The computed zero-start average run length for CUSUM. `ARL_F` The computed fast-initial-response (FIR) average run length for CUSUM. `ARL_S` The computed steady-state average run length for CUSUM.

## Author(s)

Douglas M. Hawkins, David H. Olwell, and Boxiang Wang
Maintainer: Boxiang Wang boxiang-wang@uiowa.edu

## References

Hawkins, D. M. and Olwell, D. H. (1998) “Cumulative Sum Charts and Charting for Quality Improvement (Information Science and Statistics)", Springer, New York.

`getH`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```# normal mean getARL(distr=1, K=11, H=5, Mean=10, std=2) # normal variance getARL(distr=2, K=3, H=1, std=2, samp.size=5, is.upward=TRUE) # Poission getARL(distr=3, K=3, H=1, std=2, Mean=5, is.upward=TRUE) # Binomial getARL(distr=4, K=0.8, H=1, prob=0.2, samp.size=100, is.upward=TRUE) # Negative binomial getARL(distr=5, K=3, H=6, Mean=2, Var=5, is.upward=TRUE) # Inverse Gaussian mean getARL(distr=6, K=2, H=4, mu=3, lambda=0.5, is.upward=TRUE) ```

### Example output

```The cusum is assumed upward.
( k =    0.5000,  h =    2.5000).
zero start, FIR, steady state ARLs
68.19      61.68      64.93
\$ARL_Z
[1] 68.18596

\$ARL_F
[1] 61.67604

\$ARL_S
[1] 64.93084

( k =    0.7500,  h =    0.2500).
zero start, FIR, steady state ARLs
2.40       2.22       1.37
\$ARL_Z
[1] 2.398765

\$ARL_F
[1] 2.223546

\$ARL_S
[1] 1.372227

( k =    3.0000,  h =    1.0000).
zero start, FIR, steady state ARLs
1.36       1.36       0.36
\$ARL_Z
[1] 1.360592

\$ARL_F
[1] 1.360592

\$ARL_S
[1] 0.3605922

( k =    0.8000,  h =    1.0000).
zero start, FIR, steady state ARLs
1.00       1.00       0.00
\$ARL_Z
[1] 1

\$ARL_F
[1] 1

\$ARL_S
[1] 5.296294e-09

( k =    3.0000,  h =    6.0000).
zero start, FIR, steady state ARLs
32.82      29.80      31.23
\$ARL_Z
[1] 32.8202

\$ARL_F
[1] 29.80318

\$ARL_S
[1] 31.22837

( k =    2.0000,  h =    4.0000).
zero start, FIR, steady state ARLs
7.93       7.39       6.82
\$ARL_Z
[1] 7.926828

\$ARL_F
[1] 7.385704

\$ARL_S
[1] 6.819859
```

CUSUMdesign documentation built on Feb. 25, 2020, 1:06 a.m.