xewma.arl.prerun: Compute ARLs of EWMA control charts in case of estimated...

In spc: Statistical Process Control -- Calculation of ARL and Other Control Chart Performance Measures

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 "one" and "two", respectively.
limits	distinguish between different control limits behavior.
q	change point position. For q=1 and μ=μ_0 and μ=μ_1, the usual zero-state ARLs for the in-control and out-of-control case, respectively, are calculated. For q>1 and μ!=0 conditional delays, that is, E_q(L-q+1|L≥ q), will be determined. Note that mu0=0 is implicitely fixed.
size	pre run sample size.
df	Degrees of freedom of the pre run variance estimator. Typically it is simply size - 1. If the pre run is collected in batches, then also other values are needed.
estimated	name the parameter to be estimated within the "mu", "sigma", "both".
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 L0 during applying the secant rule.
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)

spc documentation built on Oct. 24, 2022, 5:07 p.m.