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\name{xewma.q.prerun}
\alias{xewma.q.prerun}
\alias{xewma.q.crit.prerun}
\title{Compute RL quantiles of EWMA control charts in case of estimated parameters}
\description{Computation of quantiles 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.q.prerun(l, c, mu, p, 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, bound=1e-10)
xewma.q.crit.prerun(l, L0, mu, p, zr=0, hs=0, sided="two", limits="fix", size=100,
df=NULL, estimated="mu", qm.mu=30, qm.sigma=30, truncate=1e-10, bound=1e-10,
c.error=1e-10, p.error=1e-9, OUTPUT=FALSE)}
\arguments{
\item{l}{smoothing parameter lambda of the EWMA control chart.}
\item{c}{critical value (similar to alarm limit) of the EWMA control chart.}
\item{mu}{true mean shift.}
\item{p}{quantile level.}
\item{zr}{reflection border for the one-sided chart.}
\item{hs}{so-called headstart (give fast initial response).}
\item{sided}{distinguish between one- and two-sided EWMA control chart
by choosing \code{"one"} and \code{"two"}, respectively.}
\item{limits}{distinguish between different control limits behavior.}
\item{q}{change point position. For \eqn{q=1} and
\eqn{\mu=\mu_0} and \eqn{\mu=\mu_1}, the usual
zero-state ARLs for the in-control and out-of-control case, respectively,
are calculated. For \eqn{q>1} and \eqn{\mu!=0} conditional delays, that is,
\eqn{E_q(L-q+1|L\geq)}, will be determined.
Note that mu0=0 is implicitely fixed.}
\item{size}{pre run sample size.}
\item{df}{Degrees of freedom of the pre run variance estimator. Typically it is simply \code{size} - 1.
If the pre run is collected in batches, then also other values are needed.}
\item{estimated}{name the parameter to be estimated within the \code{"mu"}, \code{"sigma"},
\code{"both"}.}
\item{qm.mu}{number of quadrature nodes for convoluting the mean uncertainty.}
\item{qm.sigma}{number of quadrature nodes for convoluting the standard deviation uncertainty.}
\item{truncate}{size of truncated tail.}
\item{bound}{control when the geometric tail kicks in; the larger the quicker and less accurate; \code{bound} should be larger than 0 and less than 0.001.}
\item{L0}{in-control quantile value.}
\item{c.error}{error bound for two succeeding values of the critical value during
applying the secant rule.}
\item{p.error}{error bound for the quantile level \code{p} during applying the secant rule.}
\item{OUTPUT}{activate or deactivate additional output.}
}
\details{
Essentially, the ARL function \code{xewma.q} 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 RL quantile of order \code{q}.}
\references{
L. A. Jones, C. W. Champ, S. E. Rigdon (2001),
The performance of exponentially weighted moving average charts
with estimated parameters,
\emph{Technometrics 43}, 156-167.
S. Knoth (2003),
EWMA schemes with non-homogeneous transition kernels,
\emph{Sequential Analysis 22}, 241-255.
S. Knoth (2004),
Fast initial response features for EWMA Control Charts,
\emph{Statistical Papers 46}, 47-64.
S. Knoth (2014?),
tbd,
\emph{tbd}, tbd-tbd.
K.-H. Waldmann (1986),
Bounds for the distribution of the run length of geometric moving
average charts, \emph{Appl. Statist. 35}, 151-158.
}
\author{Sven Knoth}
\seealso{
\code{xewma.q} for the usual RL quantiles computation of EWMA control charts.
}
\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.Q <- Vectorize("xewma.q", "mu")
xewma.Q.prerun <- Vectorize("xewma.q.prerun", "mu")
mu <- c(0, .25, .5, 1, 1.5, 2)
Q1 <- ceiling(xewma.Q(lambda, L, mu, 0.1, sided="two"))
Q2 <- ceiling(xewma.Q(lambda, L, mu, 0.5, sided="two"))
Q3 <- ceiling(xewma.Q(lambda, L, mu, 0.9, sided="two"))
cbind(mu, Q1, Q2, Q3)
\dontrun{
p.Q1 <- xewma.Q.prerun(lambda, L/C4m, mu, 0.1, sided="two",
size=m, df=m*(n-1), estimated="both")
p.Q2 <- xewma.Q.prerun(lambda, L/C4m, mu, 0.5, sided="two",
size=m, df=m*(n-1), estimated="both")
p.Q3 <- xewma.Q.prerun(lambda, L/C4m, mu, 0.9, sided="two",
size=m, df=m*(n-1), estimated="both")
cbind(mu, p.Q1, p.Q2, p.Q3)
}
## original values are
# mu Q1 Q2 Q3 p.Q1 p.Q2 p.Q3
# 0.00 25 140 456 13 73 345
# 0.25 12 56 174 9 46 253
# 0.50 7 20 56 6 20 101
# 1.00 4 7 15 3 7 18
# 1.50 3 4 7 2 4 8
# 2.00 2 3 5 2 3 5
}
\keyword{ts}
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