Computation of the critical values (similar to alarm limits) for different types of EWMA control charts (based on the log of the sample variance \(S^2\)) monitoring normal variance.
lns2ewma.crit(l,L0,df,sigma0=1,cl=NULL,cu=NULL,hs=NULL,sided="upper",mode="fixed",r=40)Returns the lower and upper control limit cl and cu.
smoothing parameter lambda of the EWMA control chart.
in-control ARL.
actual degrees of freedom, corresponds to subsample size (for known mean it is equal to the subsample size, for unknown mean it is equal to subsample size minus one.
in-control standard deviation.
deployed for sided="upper", that is, upper variance control chart with
lower reflecting barrier cl.
for two-sided (sided="two") and fixed upper control limit (mode="fixed"),
for all other cases cu is ignored.
so-called headstart (enables fast initial response) -- the default value (hs=NULL) corresponds to the in-control mean of ln \(S^2\).
distinguishes between one- and two-sided two-sided EWMA-\(S^2\) control charts by choosing
"upper" (upper chart with reflection at cl),
"lower" (lower chart with reflection at cu), and "two" (two-sided chart), respectively.
only deployed for sided="two" -- with "fixed" an upper control limit (see cu)
is set and only the lower is
calculated to obtain the in-control ARL L0, while with "unbiased" a certain unbiasedness of the ARL
function is guaranteed (here, both the
lower and the upper control limit are calculated). With "vanilla" limits symmetric around the in-control
mean of ln \(S^2\)
are determined, while for "eq.tails" the in-control ARL values of two single EWMA variance charts
(decompose the two-sided scheme into one lower and one upper scheme) are matched.
dimension of the resulting linear equation system: the larger the more accurate.
Sven Knoth
lns2ewma.crit determines the critical values (similar to alarm limits) for given in-control ARL L0
by applying secant rule and using lns2ewma.arl().
In case of sided="two" and mode="unbiased"
a two-dimensional secant rule is applied that also ensures that the
maximum of the ARL function for given standard deviation is attained
at sigma0. See Knoth (2010) and the related example.
C. A. Acosta-Mej\'ia and J. J. Pignatiello Jr. and B. V. Rao (1999), A comparison of control charting procedures for monitoring process dispersion, IIE Transactions 31, 569-579.
S. V. Crowder and M. D. Hamilton (1992), An EWMA for monitoring a process standard deviation, Journal of Quality Technology 24, 12-21.
S. Knoth (2005), Accurate ARL computation for EWMA-\(S^2\) control charts, Statistics and Computing 15, 341-352.
S. Knoth (2010), Control Charting Normal Variance -- Reflections, Curiosities, and Recommendations, in Frontiers in Statistical Quality Control 9, H.-J. Lenz and P.-T. Wilrich (Eds.), Physica Verlag, Heidelberg, Germany, 3-18.
lns2ewma.arl for calculation of ARL of EWMA ln \(S^2\) control charts.
## Knoth (2005)
## compare with 1.05521 mentioned on page 350 third line from below
L0 <- 200
lambda <- .05
df <- 4
limits <- lns2ewma.crit(lambda, L0, df, cl=0, hs=0)
limits["cu"]/sqrt( lambda/(2-lambda)*(2/df+2/df^2+4/3/df^3-16/15/df^5) )
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