xewma.sf {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 mean.

Usage

xewma.sf(l, c, mu, n, zr=0, hs=0, sided="one", limits="fix", q=1, r=40)

Arguments

l

smoothing parameter lambda of the EWMA control chart.

c

critical value (similar to alarm limit) of the EWMA control chart.

mu

true mean.

n

calculate sf up to value n.

zr

reflection border for the one-sided chart.

hs

so-called headstart (enables fast initial response).

sided

distinguishes between one- and two-sided EWMA control chart by choosing "one" and "two", respectively.

limits

distinguishes between different control limits behavior.

q

change point position. For q=1 and \mu=\mu_0 and \mu=\mu_1, the usual zero-state situation for the in-control and out-of-control case, respectively, are calculated. Note that mu0=0 is implicitely fixed.

r

number of quadrature nodes, dimension of the resulting linear equation system is equal to r+1 (one-sided) or r (two-sided).

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. For varying limits and for change points after 1 the algorithm from Knoth (2004) is applied. Note that for one-sided EWMA charts (sided="one"), only "vacl" and "stat" are deployed, while for two-sided ones (sided="two") also "fir", "both" (combination of "fir" and "vacl"), and "Steiner" are implemented. For details see Knoth (2004).

Value

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

Author(s)

Sven Knoth

References

F. F. Gan (1993), An optimal design of EWMA control charts based on the median run length, J. Stat. Comput. Simulation 45, 169-184.

S. Knoth (2003), EWMA schemes with non-homogeneous transition kernels, Sequential Analysis 22, 241-255.

S. Knoth (2004), Fast initial response features for EWMA Control Charts, Statistical Papers 46, 47-64.

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

See Also

xewma.arl for zero-state ARL computation of EWMA control charts.

Examples

## Gan (1993), two-sided EWMA with fixed control limits
## some values of his Table 1 -- any median RL should be 500

G.lambda <- c(.05, .1, .15, .2, .25)
G.h <- c(.441, .675, .863, 1.027, 1.177)/sqrt(G.lambda/(2-G.lambda))

for ( i in 1:length(G.lambda) ) {
  SF <- xewma.sf(G.lambda[i], G.h[i], 0, 1000)
  if (i==1) plot(1:length(SF), SF, type="l", xlab="n", ylab="P(L>n)")
  else lines(1:length(SF), SF, col=i)
}