# Calculation of Average Run Length for discrete CUSUM schemes

`arlCusum.Rd`

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 \(\frac{\code{h}}{2}\)