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

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.