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\name{xDshewhartrunsrules.arl}
\alias{xDshewhartrunsrules.arl}
\alias{xDshewhartrunsrulesFixedm.arl}
\title{Compute ARLs of Shewhart control charts with and without runs rules
under drift}
\description{Computation of the zero-state Average Run Length (ARL)
under drift for Shewhart control charts with and without runs rules
monitoring normal mean.}
\usage{xDshewhartrunsrules.arl(delta, c = 1, m = NULL, type = "12")
xDshewhartrunsrulesFixedm.arl(delta, c = 1, m = 100, type = "12")
}
\arguments{
\item{delta}{true drift parameter.}
\item{c}{normalizing constant to ensure specific alarming behavior.}
\item{type}{controls the type of Shewhart chart used, seed details section.}
\item{m}{parameter of Gan's approach. If \code{m=NULL}, then \code{m} will increased until
the resulting ARL does not change anymore.}
}
\details{
Based on Gan (1991), the ARL is calculated for
Shewhart control charts with and without runs rules
under drift. The usual ARL function with mu=m*delta is determined and recursively via
m-1, m-2, ... 1 (or 0) the drift ARL determined.
\code{xDshewhartrunsrulesFixedm.arl} is the actual work horse, while
\code{xDshewhartrunsrules.arl} provides a convenience wrapper.
Note that Aerne et al. (1991) deployed a method that is
quite similar to Gan's algorithm. For \code{type} see
the help page of \code{xshewhartrunsrules.arl}.
}
\value{Returns a single value which resembles the ARL.}
\references{
F. F. Gan (1991),
EWMA control chart under linear drift,
\emph{J. Stat. Comput. Simulation 38}, 181-200.
L. A. Aerne, C. W. Champ and S. E. Rigdon (1991),
Evaluation of control charts under linear trend,
\emph{Commun. Stat., Theory Methods 20}, 3341-3349.
}
\author{Sven Knoth}
\seealso{
\code{xshewhartrunsrules.arl} for zero-state ARL computation of
Shewhart control charts with and without runs rules
for the classical step change model.
}
\examples{
## Aerne et al. (1991)
## Table I (continued)
## original numbers are
# delta arl1of1 arl2of3 arl4of5 arl10
# 0.005623 136.67 120.90 105.34 107.08
# 0.007499 114.98 101.23 88.09 89.94
# 0.010000 96.03 84.22 73.31 75.23
# 0.013335 79.69 69.68 60.75 62.73
# 0.017783 65.75 57.38 50.18 52.18
# 0.023714 53.99 47.06 41.33 43.35
# 0.031623 44.15 38.47 33.99 36.00
# 0.042170 35.97 31.36 27.91 29.90
# 0.056234 29.21 25.51 22.91 24.86
# 0.074989 23.65 20.71 18.81 20.70
# 0.100000 19.11 16.79 15.45 17.29
# 0.133352 15.41 13.61 12.72 14.47
# 0.177828 12.41 11.03 10.50 12.14
# 0.237137 9.98 8.94 8.71 10.18
# 0.316228 8.02 7.25 7.26 8.45
# 0.421697 6.44 5.89 6.09 6.84
# 0.562341 5.17 4.80 5.15 5.48
# 0.749894 4.16 3.92 4.36 4.39
# 1.000000 3.35 3.22 3.63 3.52
c1of1 <- 3.069/3
c2of3 <- 2.1494/2
c4of5 <- 1.14
c10 <- 3.2425/3
DxDshewhartrunsrules.arl <- Vectorize(xDshewhartrunsrules.arl, "delta")
deltas <- 10^(-(18:0)/8)
arl1of1 <-
round(DxDshewhartrunsrules.arl(deltas, c=c1of1, type="1"), digits=2)
arl2of3 <-
round(DxDshewhartrunsrules.arl(deltas, c=c2of3, type="12"), digits=2)
arl4of5 <-
round(DxDshewhartrunsrules.arl(deltas, c=c4of5, type="13"), digits=2)
arl10 <-
round(DxDshewhartrunsrules.arl(deltas, c=c10, type="SameSide10"), digits=2)
data.frame(delta=round(deltas, digits=6), arl1of1, arl2of3, arl4of5, arl10)
}
\keyword{ts}
|
