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

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. $\Big| \frac{ARL(h^*) -\gamma}{\gamma} \Big| < \epsilon$, or larger, i.e. $ARL(h^*) > \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, ...)

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 $\Big| \frac{ARL(h^*) -\gamma}{\gamma} \Big| < $ 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 $\frac{\code{h}}{2}$ is returned
...: further arguments for the distribution function, i.e. number of trials n for binomial cdf

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

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) $