Page 197 - Special Topic Session (STS) - Volume 3

STS540 Zhi Song et al.
m observations from a standard normal distribution for the reference sample
and observations from the same distribution for each test sample. When
several combinations of parameters yield the pre-specified CUFAP() value,
the CUTSP() criterion is used to select the optimal monitoring scheme
parameters. The combination (ℎ, ) that yields the maximum CUTSP() is
ℎ
selected to be the optimal combination. However, OOC run length distribution
depends on the underlying density and shift sizes. In other words, CUTSP()
values may vary with different densities and shift sizes. The exact value of
CUTSP() cannot be obtained since the underlying process distribution ()
is unknown in practice. Therefore, computation of CUTSP() requires special
attention and to this end we propose a Kernel density estimation (KDE)
approach. KDE is a nonparametric way to estimate the probability density
function (pdf) of a random variable, based on a pilot sample. We can further
draw the simulated reference sample of size m and the simulated test sample
of size from the fitted density and use the obtained to compute the
ℎ
estimates of CUFAP() and CUTSP(). Then the optimal estimate of ℎ, say ℎ,
̂
is selected for which the estimated CUTSP(W) is maximum. Inevitably, some
error may creep in during estimate of ℎ. Consequently, the estimated ℎ differs
̂
slightly from the theoretical ℎ, or called true ℎ. Note that the pure theoretical
ℎ is practically unobservable as the underlying process distribution () is
unknown. We consider this as a benchmark. The efficacy of the optimization
model depends on the closeness between the ℎ and ℎ. In other words, we
̂
expect the estimated optimal design scheme to be closer to the theoretical
optimal scheme.

4. Performance comparison
In the previous section, we frame the optimization model of nonparametric
FIR-based EWMA schemes to satisfy a desired value of CUFAP() and to
minimize CUTSP( ). We note that the exact underlying distribution is
unknown. Therefore, we propose a nonparametric approach, namely KDE for
estimating CUFAP and CUTSP. Clearly, the actual performance of the scheme
is affected by the accuracy of estimate of CUTSP. We here apply the
optimization model to the above FIR-based EL and EC schemes as described
in Section 2. In order to conduct a thorough investigation, our study includes
three typical distributions and considers thin-tailed, heavy-tailed, symmetric
and skewed distributions. Specifically, the distributions considered in the study
are: (a) the thin-tailed symmetric normal distribution abbreviated as (, ),
the IC sample is from (0,1), but the test samples are from a (, ); (b) the
heavy-tailed symmetric Cauchy distribution, denoted by Cauchy(, ). The IC
sample is taken from Cauchy(0,1), with the test samples coming from a

Cauchy (, ) distribution with pdf () = , ∈ (−∞, ∞); (c) the
2
2
( +(−) )
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