p.ewma.arl {spc}

Compute ARLs of binomial EWMA p control charts

Description

Computation of the (zero-state) Average Run Length (ARL) at given rate p.

Usage

p.ewma.arl(lambda, ucl, n, p, z0, sided="upper", lcl=NULL, d.res=1,
r.mode="ieee.round", i.mode="integer")

Arguments

lambda

smoothing parameter of the EWMA p control chart.

ucl

upper control limit of the EWMA p control chart.

n

subgroup size.

p

(failure/success) rate.

z0

so-called headstart (give fast initial response).

sided

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

lcl

lower control limit of the EWMA p control chart; needed for two-sided design.

d.res

resolution (see details).

r.mode

round mode – allowed modes are "gan.floor", "floor", "ceil", "ieee.round", "round", "mix".

i.mode

type of interval center – "integer" or "half" integer.

Details

The monitored data follow a binomial distribution with size n and failure/success probability p. The ARL values of the resulting EWMA control chart are determined by Markov chain approximation. Here, the original EWMA values are approximated by multiples of one over d.res. Different ways of rounding (see r.mode) to the next multiple are implemented. Besides Gan's paper nothing is published about the numerical subtleties.

Value

Return single value which resemble the ARL.

Author(s)

Sven Knoth

References

F. F. Gan (1990), Monitoring observations generated from a binomial distribution using modified exponentially weighted moving average control chart, J. Stat. Comput. Simulation 37, 45-60.

S. Knoth and S. Steinmetz (2013), EWMA p charts under sampling by variables, International Journal of Production Research 51, 3795-3807.

See Also

later.

Examples

## Gan (1990)

# Table 1

n <- 150
p0 <- .1
z0 <- n*p0

lambda <- c(1, .51, .165)
hu <- c(27, 22, 18)

p.value <- .1 + (0:20)/200

p.EWMA.arl <- Vectorize(p.ewma.arl, "p")

arl1.value <- round(p.EWMA.arl(lambda[1], hu[1], n, p.value, z0, r.mode="round"), digits=2)
arl2.value <- round(p.EWMA.arl(lambda[2], hu[2], n, p.value, z0, r.mode="round"), digits=2)
arl3.value <- round(p.EWMA.arl(lambda[3], hu[3], n, p.value, z0, r.mode="round"), digits=2)

arls <- matrix(c(arl1.value, arl2.value, arl3.value), ncol=length(lambda))
rownames(arls) <- p.value
colnames(arls) <- paste("lambda =", lambda)
arls

## Knoth/Steinmetz (2013)

n <- 5
p0 <- 0.02
z0 <- n*p0
lambda <- 0.3
ucl <- 0.649169922 ## in-control ARL 370.4 (determined with d.res = 2^14 = 16384)

res.list <- 2^(1:11)
arl.list <- NULL
for ( res in res.list ) {
  arl <- p.ewma.arl(lambda, ucl, n, p0, z0, d.res=res)
  arl.list <- c(arl.list, arl)
}
cbind(res.list, arl.list)