xsewma.q {spc}

Compute critical values of simultaneous EWMA control charts (mean and variance charts) for given RL quantile

Description

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.

Usage

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)

Arguments

lx

smoothing parameter lambda of the two-sided mean EWMA chart.

cx

control limit of the two-sided mean EWMA control chart.

ls

smoothing parameter lambda of the variance EWMA chart.

csu

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.

L0

in-control RL quantile at level alpha.

df

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.

alpha

quantile level.

mu

true mean.

sigma

true standard deviation.

mu0

in-control mean.

sigma0

in-control standard deviation.

hsx

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.

Nx

dimension of the approximating matrix of the mean chart.

csl

lower control limit of the variance EWMA control chart; default value is 0; not considered if sided is "upper".

hss

headstart (enables fast initial response) of the variance chart.

Ns

dimension of the approximating matrix of the variance chart.

sided

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).

mode

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).

qm

number of quadrature nodes used for the collocation integrals.

c.error

error bound for two succeeding values of the critical value during applying the secant rule.

a.error

error bound for the quantile level alpha during applying the secant rule.

Details

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.

Value

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.

Author(s)

Sven Knoth

References

S. Knoth (2007), Accurate ARL calculation for EWMA control charts monitoring simultaneously normal mean and variance, Sequential Analysis 26, 251-264.

See Also

xsewma.arl for calculation of ARL of simultaneous EWMA charts and xsewma.sf for the RL survival function.

Examples

## will follow