Computation of the critical values (similar to alarm limits) for different types of simultaneous EWMA control charts (based on the sample mean and the sample variance \(S^2\)) monitoring normal mean and variance.
xsewma.q(lx, cx, ls, csu, df, alpha, mu, sigma, hsx=0,
Nx=40, csl=0, hss=1, Ns=40, sided="upper", qm=30)xsewma.q.crit(lx, ls, L0, alpha, df, mu0=0, sigma0=1, csu=NULL,
hsx=0, hss=1, sided="upper", mode="fixed", Nx=20, Ns=40, qm=30,
c.error=1e-12, a.error=1e-9)
Returns a single value which resembles the RL quantile of order alpha and
the critical value of the two-sided mean EWMA chart and
the lower and upper controls limit csl and csu of the
variance EWMA chart, respectively.
smoothing parameter lambda of the two-sided mean EWMA chart.
control limit of the two-sided mean EWMA control chart.
smoothing parameter lambda of the variance EWMA chart.
for two-sided (sided="two") and fixed upper
control limit (mode="fixed", only for xsewma.q.crit)
a value larger than sigma0
has to been given, for all other cases cu is ignored.
It is the upper control limit of the variance EWMA control chart.
in-control RL quantile at level alpha.
actual degrees of freedom, corresponds to subgroup size (for known mean it is equal to the subgroup size, for unknown mean it is equal to subgroup size minus one.
quantile level.
true mean.
true standard deviation.
in-control mean.
in-control standard deviation.
so-called headstart (enables fast initial response) of the mean chart -- do not confuse with the true FIR feature considered in xewma.arl; will be updated.
dimension of the approximating matrix of the mean chart.
lower control limit of the variance EWMA control chart; default value is 0;
not considered if sided is "upper".
headstart (enables fast initial response) of the variance chart.
dimension of the approximating matrix of the variance chart.
distinguishes between one- and two-sided two-sided
EWMA-\(S^2\) control charts by choosing "upper" (upper chart without
reflection at cl -- the actual value of of cl is not used).
only deployed for sided="two" -- with "fixed"
an upper control limit (see cu) is set and only the lower is
determined 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).
number of quadrature nodes used for the collocation integrals.
error bound for two succeeding values of the critical value during applying the secant rule.
error bound for the quantile level alpha during
applying the secant rule.
Sven Knoth
Instead of the popular ARL (Average Run Length) quantiles of the EWMA
stopping time (Run Length) are determined. The algorithm is based on
Waldmann's survival function iteration procedure and on Knoth (2007).
xsewma.q.crit determines the critical values (similar to alarm limits)
for given in-control RL quantile L0 at level alpha by applying secant
rule and using xsewma.sf().
In case of sided="two" and mode="unbiased"
a two-dimensional secant rule is applied that also ensures that the
maximum of the RL cdf for given standard deviation is attained at sigma0.
S. Knoth (2007), Accurate ARL calculation for EWMA control charts monitoring simultaneously normal mean and variance, Sequential Analysis 26, 251-264.
xsewma.arl for calculation of ARL of simultaneous EWMA charts and
xsewma.sf for the RL survival function.