sewma.sf.prerun {spc}

Compute the survival function of EWMA run length

Description

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

Usage

sewma.sf.prerun(n, l, cl, cu, sigma, df1, df2, hs=1, sided="upper",
qm=30, qm.sigma=30, truncate=1e-10, tail_approx=TRUE)

Arguments

n

calculate sf up to value n.

l

smoothing parameter lambda of the EWMA control chart.

cl

lower control limit of the EWMA control chart.

cu

upper control limit of the EWMA control chart.

sigma

true standard deviation.

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.

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.

Details

The survival function P(L>n) and derived from it also the cdf P(L<=n) and the pmf P(L=n) illustrate the distribution of the EWMA run length. For large n the geometric tail could be exploited. That is, with reasonable large n the complete distribution is characterized. The algorithm is based on Waldmann's survival function iteration procedure and on results in Knoth (2007)...

Value

Returns a vector which resembles the survival function up to a certain point.

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.sf for the RL survival function of EWMA control charts w/o pre-run uncertainty.

Examples

## will follow