xgrsr.arl {spc}

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", q = 1, MPT = FALSE, 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 (enables fast initial response). If hs=NULL, then the classical headstart -Inf is used (corresponds to 0 for the non-log scheme).

sided

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

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.

MPT

switch between the old implementation (FALSE) and the new one (TRUE) that considers the complete likelihood ratio. MPT stands for the initials of G. Moustakides, A. Polunchenko and A. Tartakovsky.

r

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

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.

Value

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

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 detecting changes in distributions, Communications in Statistics: Theory and Methods 38, 3225-3239.

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)

XGRSR.arl  <- Vectorize("xgrsr.arl", "g")
zr <- -6

## Table 2 from Moustakides et al. (2009)
## original values are
# gamma   A     ARL/E_infty(L) SADD/E_1(L)
#   50   47.17      50.29        41.40
#  100   94.34     100.28        72.32
#  500  471.70     500.28       209.44
# 1000  943.41    1000.28       298.50
# 5000 4717.04    5000.24       557.87
#10000 9434.08   10000.17       684.17

theta <- .1
As2 <- c(47.17, 94.34, 471.7, 943.41, 4717.04, 9434.08)
gs2 <- log(As2)
arls0 <- round(XGRSR.arl(theta/2, gs2, 0, zr=-5, r=300, MPT=TRUE), digits=2)
arls1 <- round(XGRSR.arl(theta/2, gs2, theta, zr=-5, r=300, MPT=TRUE), digits=2)
data.frame(As2, arls0, arls1)

## Table 3 from Moustakides et al. (2009)
## original values are
# gamma   A     ARL/E_infty(L) SADD/E_1(L)
#   50   37.38      49.45        12.30
#  100   74.76      99.45        16.60
#  500  373.81     499.45        28.05
# 1000  747.62     999.45        33.33
# 5000 3738.08    4999.45        45.96
#10000 7476.15    9999.24        51.49

theta <- .5
As3 <- c(37.38, 74.76, 373.81, 747.62, 3738.08, 7476.15)
gs3 <- log(As3)
arls0 <- round(XGRSR.arl(theta/2, gs3, 0, zr=-5, r=70, MPT=TRUE), digits=2)
arls1 <- round(XGRSR.arl(theta/2, gs3, theta, zr=-5, r=70, MPT=TRUE), digits=2)
data.frame(As3, arls0, arls1)

## Table 4 from Moustakides et al. (2009)
## original values are
# gamma   A     ARL/E_infty(L) SADD/E_1(L)
#   50   28.02      49.78         4.98
#  100   56.04      99.79         6.22
#  500  280.19     499.79         9.30
# 1000  560.37     999.79        10.66
# 5000 2801.85    5000.93        13.86
#10000 5603.70    9999.87        15.24

theta <- 1
As4 <- c(28.02, 56.04, 280.19, 560.37, 2801.85, 5603.7)
gs4 <- log(As4)
arls0 <- round(XGRSR.arl(theta/2, gs4, 0, zr=-6, r=40, MPT=TRUE), digits=2)
arls1 <- round(XGRSR.arl(theta/2, gs4, theta, zr=-6, r=40, MPT=TRUE), digits=2)
data.frame(As4, arls0, arls1)