xewma.arl.prerun {spc} | R Documentation |
Compute ARLs of EWMA control charts in case of estimated parameters
Description
Computation of the (zero-state) Average Run Length (ARL) for different types of EWMA control charts monitoring normal mean if the in-control mean, standard deviation, or both are estimated by a pre run.
Usage
xewma.arl.prerun(l, c, mu, zr=0, hs=0, sided="two", limits="fix", q=1,
size=100, df=NULL, estimated="mu", qm.mu=30, qm.sigma=30, truncate=1e-10)
xewma.crit.prerun(l, L0, mu, zr=0, hs=0, sided="two", limits="fix", size=100,
df=NULL, estimated="mu", qm.mu=30, qm.sigma=30, truncate=1e-10,
c.error=1e-12, L.error=1e-9, OUTPUT=FALSE)
Arguments
l |
smoothing parameter lambda of the EWMA control chart. |
c |
critical value (similar to alarm limit) of the EWMA control chart. |
mu |
true mean shift. |
zr |
reflection border for the one-sided chart. |
hs |
so-called headstart (give fast initial response). |
sided |
distinguish between one- and two-sided EWMA control chart
by choosing |
limits |
distinguish between different control limits behavior. |
q |
change point position. For |
size |
pre run sample size. |
df |
Degrees of freedom of the pre run variance estimator. Typically it is simply |
estimated |
name the parameter to be estimated within
the |
qm.mu |
number of quadrature nodes for convoluting the mean uncertainty. |
qm.sigma |
number of quadrature nodes for convoluting the standard deviation uncertainty. |
truncate |
size of truncated tail. |
L0 |
in-control ARL. |
c.error |
error bound for two succeeding values of the critical value during applying the secant rule. |
L.error |
error bound for the ARL level |
OUTPUT |
activate or deactivate additional output. |
Details
Essentially, the ARL function xewma.arl
is convoluted with the
distribution of the sample mean, standard deviation or both.
For details see Jones/Champ/Rigdon (2001) and Knoth (2014?).
Value
Returns a single value which resembles the ARL.
Author(s)
Sven Knoth
References
L. A. Jones, C. W. Champ, S. E. Rigdon (2001), The performance of exponentially weighted moving average charts with estimated parameters, Technometrics 43, 156-167.
S. Knoth (2003), EWMA schemes with non-homogeneous transition kernels, Sequential Analysis 22, 241-255.
S. Knoth (2004), Fast initial response features for EWMA Control Charts, Statistical Papers 46, 47-64.
S. Knoth (2014?), tbd, tbd, tbd-tbd.
See Also
xewma.arl
for the usual zero-state ARL computation.
Examples
## Jones/Champ/Rigdon (2001)
c4m <- function(m, n) sqrt(2)*gamma( (m*(n-1)+1)/2 )/sqrt( m*(n-1) )/gamma( m*(n-1)/2 )
n <- 5 # sample size
m <- 20 # pre run with 20 samples of size n = 5
C4m <- c4m(m, n) # needed for bias correction
# Table 1, 3rd column
lambda <- 0.2
L <- 2.636
xewma.ARL <- Vectorize("xewma.arl", "mu")
xewma.ARL.prerun <- Vectorize("xewma.arl.prerun", "mu")
mu <- c(0, .25, .5, 1, 1.5, 2)
ARL <- round(xewma.ARL(lambda, L, mu, sided="two"), digits=2)
p.ARL <- round(xewma.ARL.prerun(lambda, L/C4m, mu, sided="two",
size=m, df=m*(n-1), estimated="both", qm.mu=70), digits=2)
# Monte-Carlo with 10^8 repetitions: 200.325 (0.020) and 144.458 (0.022)
cbind(mu, ARL, p.ARL)
## Not run:
# Figure 5, subfigure r = 0.2
mu_ <- (0:85)/40
ARL_ <- round(xewma.ARL(lambda, L, mu_, sided="two"), digits=2)
p.ARL_ <- round(xewma.ARL.prerun(lambda, L/C4m, mu_, sided="two",
size=m, df=m*(n-1), estimated="both"), digits=2)
plot(mu_, ARL_, type="l", xlab=expression(delta), ylab="ARL", xlim=c(0,2))
abline(v=0, h=0, col="grey", lwd=.7)
points(mu, ARL, pch=5)
lines(mu_, p.ARL_, col="blue")
points(mu, p.ARL, pch=18, col ="blue")
legend("topright", c("Known", "Estimated"), col=c("black", "blue"),
lty=1, pch=c(5, 18))
## End(Not run)