spc (version 0.3)

xgrsr.ad: Compute steady-state ARLs of Shiryaev-Roberts schemes

Description

Computation of the steady-state Average Run Length (ARL) for Shiryaev-Roberts schemes monitoring normal mean.

Usage

xgrsr.ad(k, g, mu1, mu0 = 0, zr = 0, sided = "one", r = 30)

Arguments

k

reference value of the Shiryaev-Roberts scheme.

g

control limit (alarm threshold) of Shiryaev-Roberts scheme.

mu1

out-of-control mean.

mu0

in-control mean.

zr

reflection border to enable the numerical algorithms used here.

sided

distinguish between one- and two-sided schemes by choosing "one" and"two", respectively. Currently only one-sided schemes are implemented.

r

number of quadrature nodes, dimension of the resulting linear equation system is equal to r+1.

Value

• Returns a single value which resembles the steady-state ARL.

Details

xgrsr.ad determines the steady-state Average Run Length (ARL) by numerically solving the related ARL integral equation by means of the Nystroem method based on Gauss-Legendre quadrature.

References

S. Knoth (2006), The art of evaluating monitoring schemes -- how to measure the performance of control charts? S. Lenz, H. & Wilrich, P. (ed.), Frontiers in Statistical Quality Control 8, Physica Verlag, Heidelberg, Germany, 74-99.

G. Moustakides, A. Polunchenko, A. Tartakovsky (2009), Numerical comparison of CUSUM and Shiryaev-Roberts procedures for detectin changes in distributions, Communications in Statistics: Theory and Methods, to appear.

See Also

xewma.arl and xcusum-arl for zero-state ARL computation of EWMA and CUSUM control charts, respectively, and xgrsr.arl for the zero-state ARL.

Examples

## Small study to identify appropriate reflection border to mimic
## unreflected schemes
k <- .5
g <- log(390)
zrs <- -(0:10)
ZRxgrsr.ad <- Vectorize(xgrsr.ad, "zr")
ads <- ZRxgrsr.ad(k, g, 0, zr=zrs)
data.frame(zrs, ads)

## Table 2 from Knoth (2006)
## original values are
#  mu   arl
#  0    689
#  0.5  30
#  1    8.9
#  1.5  5.1
#  2    3.6
#  2.5  2.8
#  3    2.4
#
k <- .5
g <- log(390)
zr <- -5 # see first example
mus <- (0:6)/2
Mxgrsr.ad <- Vectorize(xgrsr.ad, "mu1")
ads <- round(Mxgrsr.ad(k, g, mus, zr=zr), digits=1)
data.frame(mus, ads)

## Table 4 from Moustakides et al. (2009)
## original values are
# gamma  A        STADD/steady-state ARL
# 50     28.02    4.37
# 100    56.04    5.46
# 500    280.19   8.33
# 1000   560.37   9.64
# 5000   2801.75  12.79
# 10000  5603.7   14.17
Gxgrsr.ad  <- Vectorize("xgrsr.ad", "g")
As <- c(28.02, 56.04, 280.19, 560.37, 2801.75, 5603.7)
gs <- log(As)
theta <- 1
zr <- -6
ads <- round(Gxgrsr.ad(theta/2, gs, theta, zr=zr, r=100), digits=2)
data.frame(As, ads)