sewma.q.prerun {spc}

Compute RL quantiles of EWMA (variance charts) control charts under pre-run uncertainty

Description

Computation of quantiles of the Run Length (RL) for EWMA control charts monitoring normal variance.

Usage

sewma.q.prerun(l,cl,cu,sigma,df1,df2,alpha,hs=1,sided="upper",
r=40,qm=30,qm.sigma=30,truncate=1e-10)

sewma.q.crit.prerun(l,L0,alpha,df1,df2,sigma0=1,cl=NULL,cu=NULL,hs=1,
sided="upper",mode="fixed",r=40, qm=30,qm.sigma=30,truncate=1e-10,
tail_approx=TRUE,c.error=1e-10,a.error=1e-9)

Arguments

l

smoothing parameter lambda of the EWMA control chart.

cl

deployed for sided="Rupper", that is, upper variance control chart with lower reflecting barrier cl.

cu

for two-sided (sided="two") and fixed upper control limit (mode="fixed") a value larger than sigma0 has to been given, for all other cases cu is ignored.

sigma, sigma0

true and in-control standard deviation, respectively.

L0

in-control quantile value.

alpha

quantile level.

df1

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.

df2

degrees of freedom of the pre-run variance estimator.

hs

so-called headstart (enables fast initial response).

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 cl is not used), "Rupper" (upper chart with reflection at cl), "Rlower" (lower chart with reflection at cu), and "two" (two-sided chart), respectively.

mode

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).

r

dimension of the resulting linear equation system (highest order of the collocation polynomials).

qm

number of quadrature nodes for calculating the collocation definite integrals.

qm.sigma

number of quadrature nodes for convoluting the standard deviation uncertainty.

truncate

size of truncated tail.

tail_approx

controls whether the geometric tail approximation is used (is faster) or not.

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. Thereby the ideas presented in Knoth (2007) are used. sewma.q.crit.prerun determines the critical values (similar to alarm limits) for given in-control RL quantile L0 at level alpha by applying secant rule and using sewma.sf(). In case of sided="two" and mode="unbiased" a two-dimensional secant rule is applied that also ensures that the minimum of the cdf for given standard deviation is attained at sigma0.

Value

Returns a single value which resembles the RL quantile of order alpha and the lower and upper control limit cl and cu, 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.

K.-H. Waldmann (1986), Bounds for the distribution of the run length of geometric moving average charts, Appl. Statist. 35, 151-158.

See Also

sewma.q and sewma.q.crit for the version w/o pre-run uncertainty.

Examples

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