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.
\(ARL(h^*) > \gamma \).
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, ...)
findH returns a vector and hValues returns a matrix with elements
in-control parameter
decision interval
reference value
ARL for a CUSUM with parameters k and h
corresponds to \(\Big| \frac{ARL(h) -\gamma}{\gamma} \Big|\)
desired in-control ARL \(\gamma\)
in-control parameter \(\theta_0\)
change to detect, see details
"poisson" or "binomial"
relative tolerance, i.e. the search for h* is
stopped if \(\Big| \frac{ARL(h^*) -\gamma}{\gamma} \Big| < \) rel.tol
the reference value k and the decision interval h
are rounded to digits decimal places
passed to findK
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
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 \(X \sim Bin(n, \pi) \)