Description Usage Arguments Details Value Author(s) References See Also Examples

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

1 2 3 |

`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 |

`H` |
A given decision interval, which is given by |

`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. |

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`

.

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. |

Douglas M. Hawkins, David H. Olwell, and Boxiang Wang

Maintainer: Boxiang Wang boxiang-wang@uiowa.edu

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

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)
``` |

```
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
```

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