Page 194 - Special Topic Session (STS) - Volume 3

STS540 Zhi Song et al.
ℎ (0 ≤ ℎ ≤ 1) is the head-start parameter. When ℎ = 0, the EL-fir scheme is
the original EL scheme with fixed control limit.
2. EL scheme with FIR version of Rhoads et al. [6] [EL-fvacl] Rhoads et
al. [6] proposed an FIR feature for the EWMA in the manner suggested by
Lucas and Saccucci [1] but using the time-varying control limits. Following the
same idea, the FIR feature with the EL scheme based on the variance-adjusted
control limit, denoted as EL-fvacl, can be constructed in the following way:

4λ
C = inf { j ∈ N | Zj > 2+L √ (1 − (1 − λ) },
2
2−λ

4λ
2
with the starting value Z0h = 2+h×L √ (1 − (1 − λ) ) = 2+h×L
2−λ
4λ
√ (λ(2 − λ)) (cf. Knoth [3]).
2−λ

2.2 EC monitoring schemes
Mukherjee [7] proposed a single distribution-free EWMA scheme for
jointly monitoring the location and the scale parameters, which is based on
the Cucconi statistic and referred to as the EC procedure. This section provides
the structure of the EC monitoring scheme and further extends its design by
using FIR features. Before defining these structures, we first briefly review the
Cucconi statistic.
Define the following statistics:
N N
2
1, = ∑ kI 1, = ∑ I ,
k
k
k=1 k=1
where is an indicator variable, = 0 or 1 according as the kth order statistic

of the combined sample is an observation or a observation. is the

1,
WRS statistic for location, and similarly represents the sum of the squares
1,
of the ranks of the jth test sample in the combined sample. Further, the sum
of the squares of anti-ranks of the jth test sample in the combined sample, say
, is given by
2,
N
2
2
2, = ∑(N + 1 − k) = ( + 1) − 2( + 1) 1, + .
1,

k=1
Define the standardized statistics: = 1, − 1 , = 2, − 2 , and =

1 2
( , |), where ( , ) and , are the respective means and
1
2

1
2
standard deviations of 1, and , denotes the correlation coefficient
2,
between and . Detailed expressions for ( , ), , and are given in
2
1

1

2
the work of Chowdhury et al. [8] and hence are omitted here. Consequently,
the Cucconi statistic is defined by

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