xtewma.arl {spc}

Compute ARLs of EWMA control charts, t distributed data

Description

Computation of the (zero-state) Average Run Length (ARL) for different types of EWMA control charts monitoring the mean of t distributed data.

Usage

xtewma.arl(l,c,df,mu,zr=0,hs=0,sided="two",limits="fix",mode="tan",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.

df

degrees of freedom – parameter of the t distribution.

mu

true mean.

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.

mode

Controls the type of variables substitution that might improve the numerical performance. Currently, "identity", "sin", "sinh", and "tan" (default) are provided.

q

change point position. For q=1 and \mu=\mu_0 and \mu=\mu_1, the usual zero-state ARLs for the in-control and out-of-control case, respectively, are calculated. For q>1 and \mu!=0 conditional delays, that is, E_q(L-q+1|L\ge q), will be determined. 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

In case of the EWMA chart with fixed control limits, xtewma.arl determines the Average Run Length (ARL) by numerically solving the related ARL integral equation by means of the Nystroem method based on Gauss-Legendre quadrature. If limits is "vacl", then the method presented in Knoth (2003) is utilized. Other values (normal case) for limits are not yet supported.

Value

Except for the fixed limits EWMA charts it returns a single value which resembles the ARL. For fixed limits charts, it returns a vector of length q which resembles the ARL and the sequence of conditional expected delays for q=1 and q>1, respectively.

Author(s)

Sven Knoth

References

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

S. V. Crowder (1987), A simple method for studying run-length distributions of exponentially weighted moving average charts, Technometrics 29, 401-407.

J. M. Lucas and M. S. Saccucci (1990), Exponentially weighted moving average control schemes: Properties and enhancements, Technometrics 32, 1-12.

C. M. Borror, D. C. Montgomery, and G. C. Runger (1999), Robustness of the EWMA control chart to non-normality , Journal of Quality Technology 31, 309-316.

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.

See Also

xewma.arl for zero-state ARL computation of EWMA control charts in the normal case.

Examples

##  Borror/Montgomery/Runger (1999), Table 3
lambda <- 0.1
cE <- 2.703
df <- c(4, 6, 8, 10, 15, 20, 30, 40, 50)
L0 <- rep(NA, length(df))
for ( i in 1:length(df) ) {
  L0[i] <- round(xtewma.arl(lambda, cE*sqrt(df[i]/(df[i]-2)), df[i], 0), digits=0)
}
data.frame(df, L0)