spc (version 0.3)

xgrsr.arl: Compute (zero-state) ARLs of Shiryaev-Roberts schemes

Description

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

Usage

xgrsr.arl(k, g, mu, zr = 0, hs=NULL, sided = "one", r = 30)

Arguments

k

reference value of the Shiryaev-Roberts scheme.

g

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

mu

true mean.

zr

reflection border to enable the numerical algorithms used here.

hs

so-called headstart (give fast initial response). If hs=NULL, then the classical headstart -Inf is used (corresponds to 0 for the non-log scheme).

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 ARL.

Details

xgrsr.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.

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.ad for the steady-state ARL.

Examples

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

## Table 2 from Knoth (2006)
## original values are
#  mu   arl
#  0    697
#  0.5  33
#  1    10.4
#  1.5  6.2
#  2    4.4
#  2.5  3.5
#  3    2.9
#
k <- .5
g <- log(390)
zr <- -5 # see first example
mus <- (0:6)/2
Mxgrsr.arl <- Vectorize(xgrsr.arl, "mu")
arls <- round(Mxgrsr.arl(k, g, mus, zr=zr), digits=1)
data.frame(mus, arls)

## Table 4 from Moustakides et al. (2009)
## original values are
# gamma  A        ARL/E_infty(L) SADD/E_1(L)
# 50     28.02    50.79          5.46
# 100    56.04    100.79         6.71
# 500    280.19   500.8          9.78
# 1000   560.37   1000.79        11.14
# 5000   2801.75  5001.75        14.34
# 10000  5603.7   10000.78       15.73
Gxgrsr.arl  <- Vectorize("xgrsr.arl", "g")
As <- c(28.02, 56.04, 280.19, 560.37, 2801.75, 5603.7)
gs <- log(As)
theta <- 1
zr <- -6
arls0 <- round(Gxgrsr.arl(theta/2, gs, 0, zr=zr, r=100), digits=2)
arls1 <- round(Gxgrsr.arl(theta/2, gs, theta, zr=zr, r=100), digits=2)
data.frame(As, arls0, arls1)