Calculates the average run length (ARL) for an upward CUSUM scheme for discrete distributions (i.e. Poisson and binomial) using the Markov chain approach.

## Usage

arlCusum(h=10, k=3, theta=2.4, distr=c("poisson","binomial"),
W=NULL, digits=1, ...)

## Arguments

h

decision interval

k

reference value

theta

distribution parameter for the cumulative distribution function (cdf) $$F$$, i.e. rate $$\lambda$$ for Poisson variates or probability $$p$$ for binomial variates

distr

"poisson" or "binomial"

W

Winsorizing value W for a robust CUSUM, to get a nonrobust CUSUM set W > k+h. If NULL, a nonrobust CUSUM is used.

digits

k and h are rounded to digits decimal places

...

further arguments for the distribution function, i.e. number of trials n for binomial cdf

## Value

Returns a list with the ARL of the regular (zero-start) and the fast initial response (FIR) CUSUM scheme with reference value k, decision interval h for $$X \sim F(\theta)$$, where F is the Poisson or binomial CDF.

ARL

one-sided ARL of the regular (zero-start) CUSUM scheme

FIR.ARL

one-sided ARL of the FIR CUSUM scheme with head start $$\frac{\code{h}}{2}$$

## Source

Based on the FORTRAN code of

Hawkins, D. M. (1992). Evaluation of Average Run Lengths of Cumulative Sum Charts for an Arbitrary Data Distribution. Communications in Statistics - Simulation and Computation, 21(4), p. 1001-1020.