surveillance (version 1.21.1)

findH: Find decision interval for given in-control ARL and reference value

Description

Function to find a decision interval h* for given reference value k and desired ARL $\gamma$ so that the average run length for a Poisson or Binomial CUSUM with in-control parameter ${\theta }_0$, reference value k and is approximately $\gamma$, i.e. $|\frac{ARL(h^{\ast })-\gamma }{\gamma }|<ϵ$, or larger, i.e. $ARL(h^{\ast })>\gamma$.

Usage

findH(ARL0, theta0, s = 1, rel.tol = 0.03, roundK = TRUE,
      distr = c("poisson", "binomial"), digits = 1, FIR = FALSE, ...)
hValues(theta0, ARL0, rel.tol=0.02, s = 1, roundK = TRUE, digits = 1,
        distr = c("poisson", "binomial"), FIR = FALSE, ...)

Value

findH returns a vector and hValues returns a matrix with elements

theta0

in-control parameter

h

decision interval

k

reference value

ARL

ARL for a CUSUM with parameters k and h

rel.tol

corresponds to $|\frac{ARL(h)-\gamma }{\gamma }|$

Arguments

ARL0

desired in-control ARL $\gamma$

theta0

in-control parameter ${\theta }_0$

s

change to detect, see details

distr

"poisson" or "binomial"

rel.tol

relative tolerance, i.e. the search for h* is stopped if $|\frac{ARL(h^{\ast })-\gamma }{\gamma }|<$ rel.tol

digits

the reference value k and the decision interval h are rounded to digits decimal places

roundK

passed to findK

FIR

if TRUE, the decision interval that leads to the desired ARL for a FIR CUSUM with head start $\frac{\text{code}h}{2}$ is returned

...

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

Details

The out-of-control parameter used to determine the reference value k is specified as: $${\theta }_1={\lambda }_0+s\sqrt{{\lambda }_0}$$ for a Poisson variate $X\sim Po(\lambda )$

$${\theta }_1=\frac{s{\pi }_0}{1+(s-1){\pi }_0}$$ for a Binomial variate $X\sim Bin(n,\pi )$