xewma.sf.prerun {spc}

Compute the survival function of EWMA run length in case of estimated parameters

Description

Computation of the survival function of the Run Length (RL) for EWMA control charts monitoring normal mean if the in-control mean, standard deviation, or both are estimated by a pre run.

Usage

xewma.sf.prerun(l, c, mu, n, zr=0, hs=0, sided="one", limits="fix", q=1,
size=100, df=NULL, estimated="mu", qm.mu=30, qm.sigma=30,
truncate=1e-10, tail_approx=TRUE, bound=1e-10)

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.

n

calculate sf up to value n.

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 \mu=\mu_0 and \mu=\mu_1, the usual zero-state situation for the in-control and out-of-control case, respectively, are calculated. 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.

tail_approx

Controls whether the geometric tail approximation is used (is faster) or not.

bound

control when the geometric tail kicks in; the larger the quicker and less accurate; bound should be larger than 0 and less than 0.001.

Details

The survival function P(L>n) and derived from it also the cdf P(L<=n) and the pmf P(L=n) illustrate the distribution of the EWMA run length...

Value

Returns a vector which resembles the survival function up to a certain point.

Author(s)

Sven Knoth

References

F. F. Gan (1993), An optimal design of EWMA control charts based on the median run length, J. Stat. Comput. Simulation 45, 169-184.

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.

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.

K.-H. Waldmann (1986), Bounds for the distribution of the run length of geometric moving average charts, Appl. Statist. 35, 151-158.

See Also

xewma.sf for the RL survival function of EWMA control charts w/o pre run uncertainty.

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

# Figure 6, subfigure r=0.1
lambda <- 0.1
L <- 2.454

CDF0 <- 1 - xewma.sf(lambda, L, 0, 600, sided="two")
m <- 10 # pre run size
CDF1 <- 1 - xewma.sf.prerun(lambda, L/c4m(m,n), 0, 600, sided="two",
size=m, df=m*(n-1), estimated="both")
m <- 20
CDF2 <- 1 - xewma.sf.prerun(lambda, L/c4m(m,n), 0, 600, sided="two",
size=m, df=m*(n-1), estimated="both")
m <- 50
CDF3 <- 1 - xewma.sf.prerun(lambda, L/c4m(m,n), 0, 600, sided="two",
size=m, df=m*(n-1), estimated="both")

plot(CDF0, type="l", xlab="t", ylab=expression(P(T<=t)), xlim=c(0,500), ylim=c(0,1))
abline(v=0, h=c(0,1), col="grey", lwd=.7)
points((1:5)*100, CDF0[(1:5)*100], pch=18)
lines(CDF1, col="blue")
points((1:5)*100, CDF1[(1:5)*100], pch=2, col="blue")
lines(CDF2, col="red")
points((1:5)*100, CDF2[(1:5)*100], pch=16, col="red")
lines(CDF3, col="green")
points((1:5)*100, CDF3[(1:5)*100], pch=5, col="green")

legend("bottomright", c("Known", "m=10, n=5", "m=20, n=5", "m=50, n=5"),
       col=c("black", "blue", "red", "green"), pch=c(18, 2, 16, 5), lty=1)