getARL {CUSUMdesign}

compute average run length (ARL) for CUSUM charts

Description

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

Usage

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 [e-mail]

References

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

See Also

getH

Examples

# 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)