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Early Detection Statistics With Online Multivariate Data
Keh-Shin Lii1; Ashis SenGupta2
Department of Statistics, University of Californai, Riverside, CA 92521.
Applied Statistics Unit, Indian Statistical Institute, Klkata, India.
E-mail: email@example.com; firstname.lastname@example.org
Suppose that observations on an underlying process are being recorded temporally. It is suspected that at
some unknown point of time t0 , a shift in the process mean has taken place. This shift could be of a slowly
emerging or an abrupt jump type. The change (type) continues till there is the onset of another change (type).
We consider the problem of detecting the occurrence of the change as soon as possible after its onset, subject
to a pre-designated level on the rate of false alarm. Here we discuss only the jump or step change, i.e.
processes or systems whose mean subject to “random step changes". The detection statistic sought for should
be optimal, in some rich family of distributions, in the sense that it maximizes the chance of early detection
for a prefixed false alarm rate. A solution to the above problem for the univariate uniparameter framework in
the exponential family was derived by Shiryayev (1963) and also independently by Roberts (1966), yielding
a detection statistic, which we will refer to as the Shiryayev-Roberts (SR) statistic. Since then there has been
a flurry of papers on this topic, but almost all in the univariate uniparameter framework. While the need
obviously exists, there is a paucity of techniques for multi-characteristic variables or for several variables
T being observed simultaneously. This situation is somewhat comparable with that for quality control
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techniques, where also the need has been long felt, see e.g. Jackson (1959), Ghare and Torgerson (1968),
Mohebbi and Hayre (1989), etc. Some approaches have been suggested for control charts, e.g., Woodall and
Ncube (1985) suggest simultaneously operating separate control charts for each variable being monitored.
For multivariate procedures see e.g., Yeh et al (2003) who suggest a single control chart based on an one-
dimensional summary measure | S | , the generalized variance; Healy (1985) for multivariate CUSUM
procedures and Lowry and Montgomery (1995) for a review.
We consider here the case of multivariate observations and demonstrate its remarkable advantages compared
to separate univariate analyses for the early detection of change-point.
Univariate Uniparameter Mean Case:
If the average detection delay is h( k, Δ ) , then h(k , Δ ) = h(4l k , Δ 2− l ) where Δ > 0 is the shift in the mean
to be detected, k is the number of independent predictors (variables) and l is any arbitrary positive integer.
Multivariate Multiparameter Mean Case:
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The SR statistic for testing for a change-point retains its optimality property for this multiparameter problem
when the underlying population is a member of the multiparameter exponential family in the same sense that
it does for the one-parameter problem in the one-parameter exponential family.
Fisher Information and ADD:
We now establish a crucial relationship between the FI and the ADD for two possibly different dimensional
normal distributions. By the uniparameterization theorem 2 above, it suffices for our setup to confine to
X N k (m1, Σ), where m is the (mean) parameter of interest and Σ is known. We then have
The ADD for a change in the mean m is the same as long as the FI for two normal distributions, even with
possibly different dimensions, are the same.
 Bhattacharya, P.K. and Zhou, H. (1994). A rank-CUSUM procedure for detecting small changes in a
symmetric distribution. In Change-point Problems, IMS Lecture Notes , 57-65.
 Ghare, P.M. and Torgerson, P.E. (1968). The Multicharacteristic control charts. J. Industr. Engr., , 269-
T  Healy, D.J. (1987). Multivariate CUSUM procedures. Technometrics, , 409-412.
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 Holmes, D.S. and Mergen, A.E. (1993). Improving the performance of the T 2 control chart. Quality
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mixed models and unbalanced one-way random models. Technometrics, , 323-335.
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