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Gregory Gurevich and Albert Vexler 1 2
The Department of Industrial Engineering and Management, SCE- Shamoon College of Engineering, Beer-Sheva 84100, Israel
Department of Biostatistics, The New York State University at Buffalo, Buffalo, NY 14214, USA
1
2
כנס האיכות הרביעי בגליל "איכות – הלכה ומעשה"2011 במאי, 12
On Optimality of the Shiryaev-Roberts Procedure for
Retrospective and Sequential Change Point Detections
This paper demonstrates that criteria for which a given test is optimal can be declared by
the structure of this test, and hence almost any reasonable test is optimal. In order to establish
this conclusion, the principle idea of the fundamental lemma of Neyman and Pearson is
applied to interpret the goodness of tests, as well as retrospective and sequential change point
detections are considered in the context of the proposed technique. We demonstrate that the
Shiryaev-Roberts approach applied to the retrospective change point detection yields the
averagely most powerful procedure. We also prove non-asymptotic optimality properties for
the Shiryayev-Roberts sequential procedure.
1 .INTRODUCTION
Without loss of generality, we can say that the principle idea of the proof of the fundamental
lemma of Neyman and Pearson is based on the trivial inequality
0 BAIBA (1.1 )
for all A , B and ]1,0[ ( I is the indicator function(. Thus, for example ,
if we would like to classify i.i.d. observations nXXX ,...,, 21 regarding the hypotheses:
10 X :H is from a density function 0f
versus
11 X :H is from a density function 1f ,
then the likelihood ratio test )i.e., we should reject 0H if CXfXfn
i ii 1 01 /
where C is a threshold( is uniformly most powerful. This classical proposition directly follows
from the expected value under 0H of )1.1( with n
i ii XfXfA1 01 / ,
CB and that is considered as any decision rule based on the observed sample .
CXf
XfEC
Xf
XfIC
Xf
XfE
n
i i
iH
n
i i
in
i i
iH
1 0
1
1 0
1
1 0
100 ,
n
n
ii
n
i i
in
i i
i dxdxxfCxf
xfI
xf
xf...... 1
10
1 0
1
1 0
1
n
n
ii
n
i i
i dxdxxfCxf
xfIC ...... 1
10
1 0
1
n
n
ii
n
i i
i dxdxxfxf
xf...... 1
10
1 0
1
n
n
ii dxdxxfC ...... 1
10 .
Therefore
11
01011 0
1
1 0
1
HH
n
i i
iH
n
i i
iH CPPC
Xf
XfCPC
Xf
XfP .
However, in the case when we have a given test-statistic nS instead of the likelihood ratio
n
i ii XfXf1 01 / , the inequality )1.1( yields
CSCSICS nnn ,
for any decision rule ]1,0[ . Therefore, the test-rule CSn has also an optimal
property following the application of )1.1(. The problem is to obtain a reasonable interpretation
of the optimality.
The present paper proposes that the structure of any given retrospective or sequential test
can substantiate type-)1.1( inequalities that provide optimal properties of that test. Although
this approach can be applied easily to demonstrate an optimality of tests, presentation of that
optimality in the classical operating characteristics )e.g., the power/significance level of tests(
is the complicated issue.
In order to represent the proposed methodology, we consider different kinds of hypothesis
testing. Firstly, we consider the retrospective change point problem. We demonstrate that the
Shiryaev-Roberts approach applied to the change point detection yields the average most
powerful procedure. Secondly, we consider the sequential testing. Here the focus on the
inequality )1.1( leads us to a non-asymptotic optimality of a well known sequential
procedure, for which only an asymptotic result of optimality has been proved in the literature .
2 .RETROSPECTIVE CHANGE POINT DETECTION
In this section we will focus on a sequence of previously obtained independent observations
in an attempt to detect whether they all share the same distribution. Note that, commonly, the
change point problem is evaluated under assumption that if a change in the distribution did
occur, then it is unique and the observations after the change all have the same distribution ,
which differs from the distribution of the observations before the change.
Let nXX ,...,1 be independent observations with density functions ngg ,...,1 ,
respectively. We want to test the null hypothesis
0i0 g : fH for all ni ,...,1
versus the alternative
10111 ......g : fggfgH n , is unknown.
The maximum likelihood estimation of the change point parameter applied to the
likelihood ratio
n
i ii XfXf 01 / leads us to the well accepted in the change point
literature CUSUM test: we should reject 0H if
CXf
Xfn
ki i
i
nk
0
1
1max ,
where 0C is a threshold .
Alternatively, following Gurevich and Vexler )2010(, the test based on the Shiryaev-Roberts
approach has the following form: we reject 0H if
CXf
Xf
nR
n
n
k
n
ki i
in
1 0
111 . (2.1 )
Consider an optimal meaning of the test )2.1(. Let kP and kE ( nk ,...,0 ) respectively
denote probability and expectation conditional on k (the case 0k corresponds to
0H .)By virtue of )1.1( with nRA n / and CB ,it follows that
CR
nCR
nICR
n nnn111
. (2.2 )
Without loss of generality, we assume that 1,0 is any decision rule based on the observed
sample nXX ,...,1 such that if 1 , then we reject 0H .Since
n
k
n
ki i
in Xf
XfE
nRE
n 1 0
100
11
n
k
n
ki
n
i
n
iii
i
i dxxfxf
xf
n 1 1 10
0
1...1
n
k
k
i
n
ii
n
kiii dxxfxfI
n 1
1
1 1101...
1
n
kkP
n 1
11
derivation of 0H -expectation of the left and right side of )2.2( directly provides the following
proposition.
Proposition 2.1 The test )2.1( is the averagely most powerful test, i.e.
n
kk
n
knnk CPP
nCR
nCPCR
nP
n 10
10 11
1111 ,
for any decision rule ]1,0[ based on the observations nXXX ,...,, 21 .
3 .SEQUENTIAL SHIRYAEV-ROBERTS CHANGE POINT DETECTION
In this section, we suppose that independent observations ,..., 21 XX are surveyed
sequentially. Let 121 ,...,, XXX be each distributed according to a density function 0f ,
whereas ,..., 1 XX have a density function 1f , where 1 is unknown .
The case corresponds to the situation when all observations are distributed according
to 0f .The formulation of sequential change point detection conforms to raising an alarm as
soon as possible after the change and to avoid false alarms. Efficient detection methods for this
stated problem are based on CUSUM and Shiryaev-Roberts stopping rules. The CUSUM
policy is: we stop sampling of X -s and report that a change in distribution of X has been
detected at the first time 1n that
CXfXfn
ki iink
01
1/max ,
for a given threshold C . The Shiryaev-Roberts procedure is defined by the stopping time
CnNC nR : 1inf ,(3.1 )
where the Shiryayev-Roberts test-statistic nR is
n
k
n
ki i
in Xf
XfR
1 0
1 (3.2) .
The sequential CUSUM detection procedure has a non-asymptotic optimal property
Moustakides, 1986(. At that, for the Shiryaev-Roberts procedure an asymptotic )as C )
optimality has been shown )Pollak, 1985(. )Although Yakir )1997( attempted to prove a non -
asymptotic optimality of the Shiryaev-Roberts procedure, Mei )2006( has pointed out the
inaccuracy of the Yakir's proofs.( In order to demonstrate optimality of the Shiryaev-Roberts
detection scheme, Pollak )1985( has proved an asymptotic closeness of the expected loss
using a Bayes rule for the considered change problem with a known prior distribution of
to that using the rule CN .However, in the context of simple application of the proposed
methodology, the procedure )3.1( itself declares loss functions for which that detection policy
is optimal. That is, setting
nNCRA ,min and CB in )1.1( leads to
0,min,min CRICR nNnN CC , (3.3)
for all ]1,0[ .Since nNCR CnNC,min ,
CRICR nNnN CC ,min,min(3.4)
011
nNICRkNICR Cn
n
kCk .
It is clear that )3.4( can be utilized to present an optimal property of the detection rule CN .
However, here, for simplicity, noting that every summand in the left side of the inequality
(3.4 )is non-negative, we focus only on
0 nNICR Cn .
Thus, if is a stopping time )assuming that for all k ,the event k is measurable
in the -algebra generated by kXX ,...,1
)and is defined by nI
then
0N , C nnIRCE n . (3.5)
By virtue of the definition )3.2(, we obtain from )3.5( that
nNPnNPC CC ,min
0,min1
n
kCkCk nNPnNP .
(3.6)
Therefore,
1
,minn
CC nNPnNPC
0,min1 1
n
n
kCkCk nNPnNP ,
where
(3.7)
1 1
,minn
n
kCkCk nNPnNP
1
,mink kn
CkCk nNPnNP
1
1,min1k
CkCk kNEkNE ,
(3.8)
0 aaIa .Let us summarize )3.7( with )3.8( in the next proposition.
Proposition 3.1 The Shiryaev-Roberts stopping time )3.1( satisfies
Cn
Cn NCEnNE
1
1
Cn
Cn NCEnNE ,min1,minmin1
.
Note that , E corresponds to the average run length to false alarm of a stopping rule
,and hence small values of E are privileged, whereas small values of
1nEn are also preferable )because 1nEn relates to fallibility
of the sequential detection in the case n .)Obviously, if then
CN,min detects that faster than the stopping time CN .
4 .REMARK AND CONCLUSION
In the context of statistical testing, when distribution functions of observations are known up
to parameters, one of the approaches dealing with estimation of unknown parameters is the
mixture technique. To be specific, consider the change point problem introduced in Section 2 .
Assume that the density function ;11 ufuf ,where is the unknown
parameter. In this case, we can transform the test )2.1( in accordance with the mixture
methodology )e.g. Krieger et al., 2003(. That is, we have to choose a prior and pretend that
~ .Thus, the mixture Shiryaev-Roberts statistic has form
n
k
n
ki i
in ud
uXf
uXf
nR
1 0
1(1) ();
;1
and hence the consequential transformation of the test )2.1( has the following property
CRn
CPuduXXCRn
Pn n
n
kikjjnk
(1)0
11
(1) 1();f from are
11
();f from are H ofrejection declares 1
110 uduXXP
n
n
kikjjk
00 H ofrejection declares CP ,
for any decision rule ]1,0[ based on the observations nXX ,...,1 .In this case the
optimality formulated in Proposition 2.1 is integrated over values of the unknown parameter .
Finally, note that for the case where one wishes to investigate the optimality of a given test
in an unfixed context, inequalities similar to )1.1( can be obtained by focusing on the
structure of a test. These inequalities report an optimal property of the test. However ,
translation of that property in terms of the quality of tests is the issue.
QuestionsQuestions??
Thank youThank you!!