spc (version 0.3)

xDgrsr.arl: Compute ARLs of Shiryaev-Roberts schemes under drift

Description

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

Usage

xDgrsr.arl(k, g, delta, zr = 0, hs = NULL, sided = "one", m = NULL,
mode = "Gan", q = 1, r = 30, with0 = FALSE)

Arguments

k

reference value of the Shiryaev-Roberts scheme.

g

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

delta

true drift parameter.

zr

reflection border for the one-sided chart.

hs

so-called headstart (give fast initial response).

sided

distinguish between one- and two-sided Shiryaev-Roberts schemes by choosing "one" and "two", respectively. Currentlly, the two-sided scheme is not implemented.

m

parameter used if mode="Gan". m is design parameter of Gan's approach. If m=NULL, then m will increased until the resulting ARL does not change anymore.

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, $E_q(L-q+1|L\geq)$, will be determine

mode

decide whether Gan's or Knoth's approach is used. Use "Gan" and "Knoth", respectively. "Knoth" is not implemented yet.

r

number of quadrature nodes, dimension of the resulting linear equation system is equal to r+1 (one-sided) or r (two-sided).

with0

define whether the first observation used for the ARL calculatio follows already 1*delta or 0*delta. With q additional flexibility is given.

Value

• Returns a single value which resembles the ARL.

Details

Based on Gan (1991) or Knoth (2003), the ARL is calculated for Shiryaev-Roberts schemes under drift. In case of Gan's framework, the usual ARL function with mu=m*delta is determined and recursively via m-1, m-2, ... 1 (or 0) the drift ARL determined. The framework of Knoth allows to calculate ARLs for varying parameters, such as control limits and distributional parameters. For details see the cited papers.

References

F. F. Gan (1991), EWMA control chart under linear drift, J. Stat. Comput. Simulation 38, 181-200.

S. Knoth (2003), EWMA schemes with non-homogeneous transition kernels, Sequential Analysis 22, 241-255.

See Also

xewma.arl and xewma.ad for zero-state and steady-state ARL computation of EWMA control charts for the classical step change model.

Examples

## Monte Carlo example with 10^8 replicates
#   delta      arl    s.e.
#   0.0001 381.8240   0.0304
#   0.0005 238.4630   0.0148
#   0.001  177.4061   0.0097
#   0.002  125.9055   0.0061
#   0.005   75.7574   0.0031
#   0.01    50.2203   0.0018
#   0.02    32.9458   0.0011
#   0.05    18.9213   0.0005
#   0.1     12.6054   0.0003
#   0.5      5.2157   0.0001
#   1        3.6537   0.0001
#   3        2.0289   0.0000
k <- .5
L0 <- 500
zr <- -7
r <- 50
g <- xgrsr.crit(k, L0, zr=zr, r=r)
DxDgrsr.arl <- Vectorize(xDgrsr.arl, "delta")
deltas <- c(0.0001, 0.0005, 0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1, 0.5, 1, 3)
arls <- round(DxDgrsr.arl(k, g, deltas, zr=zr, r=r), digits=4)
data.frame(deltas, arls)