Page 193 - Special Topic Session (STS) - Volume 3

STS540 Zhi Song et al.
IC if during the course of Phase-II monitoring we observe F = G, that is when
θ = 0 and δ = 1. When the process is OOC, either θ ≠ 0 or δ ≠ 1 or both.
2.1 EL monitoring shemes
This section provides the structure of traditional EL monitoring sheme and
further extends its design by using FIR features. Mukherjee [4] proposed some
single distribution-free EWMA schemes for monitoring the location and the
scale parameters of an unknown but continuous univariate process at Phase-
II. These schemes are based on the well-known Lepage statistic and referred
to as the EL procedures. The Lepage statistic is the sum of squares of the
standardized Wilcoxon rank-sum (WRS) statistic for location and the
standardized Ansari-Bradley (AB) statistic for scale. The WRS statistic, denoted
as TW, j, is interpreted as the sum of ranks of the jth test sample in the combined
sample of size N(=m+n). Further, the AB statistic, denoted as TAB, j, is defined
by the sum of the absolute deviation of ranks of the jth test sample in the
combined sample from the average rank, that is, (N +1)/2. The standardized
WRS and AB statistics for the jth inspection state are = , − and =

, respectively, where (µ , σ ) and (µAB,σAB) are the means and
, −

standard deviations of TW, j and TAB, j , under the IC case: θ = 0 and δ = 1.
Detailed expressions for (µW ,σW) and (µAB,σAB) are given in the work of
Mukherjee and Chakraborti [5] and hence are omitted here. Thereafter, the
2
2
2
Lepage statistic is defined as = + .

The plotting statistic of the EL procedure is given by = max{2,λ +
2

2
(1−λ) −1 }, j = 1,2,... and with the starting value Z0 = 2, as E( |IC) = 2. Here, 0

< λ ≤ 1 is the smoothing parameter. Next, we look on some FIR-based EL
schemes, which allow to improve the detection performance at early time
points.
1. EL scheme with FIR version of Lucas and Saccucci [1] [EL-fir] Lucas
and Saccucci [1] used an FIR feature for the EWMA to improve its performance
at start-up. We propose using the FIR feature with the fixed-width control limit
EL scheme (denoted as EL-fir) in the line of Lucas and Saccucci [1], which is
formulated as follows:

4λ
= { ∈ ℕ| > 2 + √ }

2 − λ

4λ
The EL-fir scheme triggers a signal whenever exceeds the 2 + √ 2−λ .

The stopping variable C is the number of samples until the scheme first
4λ
generates a signal. We set the starting value ℎ = 2 + ℎ × √ 2−λ , where

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