There are numerous papers on dynamic control problems, ail of which deal in one way or another with the problem of shift at unknown time points. In parti-cular we mention hère the papers of Girshick and Rubin [2], Bather [7,8]
Lorden [43], Yadin and Zacks [68], Shiryaev [60,61], and Zacks and Barzily [69].
We présent first the Bayesian theory, followed by discussed of the CUSUM procé-dure. Again, we consider a séquence of independent random variables
X.,X.,...,X t,X ,... Let T be the point of shift, T = 0,1,... If T < 1 , ail i 2. m— 1 m
the observations are from F (x). If T = t (t = 2,3,...) then the first t-1 observations are from F (x) and X , X .,... are from F.(x). Let f (x) and O t T+1 1 O f.(x) be the p.d.f. corresponding to F (x) and F (x), respectively.
The random variables X. ,X.,... are observed sequentially and we wish to apply a stopping rule which will stop soon after the shift occurs, without too many
Mfalse alarms". The following objectives are considered in the sélection of a stopping variable N :
1°) If ÏÏ(T) dénotes the prior distribution of T, then the prior risk (5.1) R(ïï,N) = Pn(N < T ) + c Pn(N > T ) En(N-T|N > T }
is minimized, with respect to ail stopping rules.
2°) To minimize E {N-T|N > T } subject to the constraint PÏÏ(N < T) < a, 0 < a < 1 .
5.1. - The Bayesian Procédures
The shift index, T, is considered a random variable, having a prior p.d.f.
îl(t), concentrated on the non-negative integers. Shiryaev [60] postulated the following prioir distribution
(5.2) n(t)
H , if t = 0 (î-ïDpa-p)*"1 , if t = 1,2,,
for 0 < H < 1, 0 < p < 1. (TÎ+(l-ÎT)p) is the prior probability that the shift has occurred before the first observation, and p is the prior probability of a shift occurring between any two observations.
After observing X] ,. .. ,X , the prior p.d.f. îî(t) is converted to a posterior probability function on fn,n+l,...}, namely,
(5.3) nn(t)
nn ; t - n
d - n ^ p d - p )6"1, t = n+I,,
where Iïn is the posterior probability that the shift took place before the n-th observation. This posterior probability is given by II - 1-q , where
and n 1 = l
n
Dn = (n+(i-n)P) n f^x,.) + i=l
(5.5) (i-n)p z (i-p)J n f (x.) n f.(x.) + J=l 1=1 l'=J+l
n
(i-n)(i-P)n n fo(x.) . i=l
.72.
Let R(X.) = f.(X.)/f (X.), i = 1,2,..., then i I i o i
(5.6)
V l= a-oo-p)*"
n + 1 R(X)[D-(l-n)(I-p)n]+B ..
n+1 n n+i where
B
n + 1= R(x
n+1)(,-ii)(,-p)
nP+ (,-n)(,-
P)
n+1But (,-n)(,-p) - q D . Hence,
(5.7)
qn(1-p)
or (5.8)
n+1 " R(Xn+])(,-qn(l-p))+qn(,-p)
(n
n+(i-it
n)p-R(x
n+1)
[
n+i " (n +(,-n )p)R(x_
I)+(i-n„)(i-p) '
n n n+i nn = 0,1,... with II = IT and q = 1-ÎI. Accordingly, the séquence of posterior probabilities (îî ; n > 0} is Markovian, i.e., the conditional distribution of
n
II . dépends on the first n observations X.,...,X , only through II . This can n+i I n n lead immédiatly to the construction of recursive détermination of the distri-bution of any stopping variable depending only on II (see Zacks [71]). Shiryaev
[60] has shown that when F and F. are known, the optimal stopping variable, vith respect to the above objectives, is to stop at the smallest n for which îï £ A , for some 0 < A < I.
Bather [7] has shown that for the constraint of bounding the expected number of false alarms by n, A = (n+1) is the optimal stopping boundary.
When the distributions F and F are not completely specified, the above problem of finding optimal stopping variables becomes much more complicated. Zacks and Barzily [69] studied Bayes procédures for detecting shifts in the probability of success, 0, of Bernouilli trials, when the values 6 , before the shift, and the value 9, after it, are unknown. The Bayesian model assumed that 0 and 8.
1 J o 1
hâve a uniform prior distribution over the simplex { ( 8 , 0f) ; O < 0 < 6. < 1}
o 1 o 1 and the point of shift, T , has the prior distribution (5.2). In this case, the posterior probability ÏÏ dépends on the whole vector of observations X ,...,X , and not only on ÏÏ _. and X . It is shown that this posterior probability is a function of X = (X.,...,X ) ' given by
-n i n
(5.9) H (X ) = I-(l-ïï)(l-p)n"1B(T +1, n-T +2)/D (X ) n _n n n n -n where B(p,q) is the beta-function ;
J
T. = S X. and
J i=l L
(5.10)
D (X ) = ÏÏ B(T +2, n-T +1) + n „n n n
(l-ïï)p Z (l-p)j"! B(T(n^+! , n-j-T(n^+l).
T(n)
n-j /n-j-1 \
.£ ( J B(T +1, n-T +2)
i=o V i/
n+ ( l - m a - p )1 1'1 B(T +1, n-T +2).
n n
Hère, T ". = T -T. (j = 0,...,n). The séquence {ïï (X ) ; n > 1 } is not Markovian,
11 j n j n ^n
but is submartingale. Zacks and Barzily considered the problem of determining the optimal stopping rule under the following cost conditions :
After each observation we bave the option to stop observations and déclare that a shift has occurred. The process is then inspected. If the shift has not yet
occurred a penalty of 1 unit is imposed.lf, on the other hand, the shift has already occurred, a penalty of C units per delayed observation (or time unit) is imposed.
It is shown then that the optimal stopping variable is (5. M ) N° = least n > 1, such that n (X ) > b (X ) ,
n „n n _n
where the stopping boundary b (X ) is given implicitly, as the limit for j •+ », of
(5.,2) b<J>(x
n)-,i,
(n*-
n c,
p-
n, , ) ,
with IT = p/(C+p) are the functions M J'(X ) can be determined recursively, n _n
according to the formula
(5.13) M<j)(X ) - E{min (0,C ÏÏ . (X ,X .)}
n _n n+l _n n+1
It is very difficult, if not impossible, to détermine thèse functions explicitly, for large values of j. The authors therefore considered a suboptimal procédure bases on b (X ) only. Numerical simulations illustrate the performance of the (2)
n ,n r
suboptimal procédure.
.74.
5.2 - Asymptotically Minimax Rules and the CUSUM Control
Lorden [42,43] considered the sequential détection procédure from a non-Bayesian point of view and proved that the well known CUSUM procédures of Page [47,48,49]
are asymptotically minimax.
Let X.,X_,... be a séquence of independent random variables. The distributions of X.,...,X . is F (x) and that of X ,X _ ,.. . is F.(x). The point of shift m
i m—i o m m+1 i
is unknown , F (x) and F.(x) are known. The family of probability measures is {P ; m = 1,2,...}, where P (X ) is the joint p.d.f. of X = (X. ,...,X ) , in m m .n ,n i n which X is the first random variable with a c.d.f. F.(x).
m l It is desired to devise a sequential procédure with a (possibly) extended
stopping variable, N, (i.e., lim P [N>n] > d > 0, m = 0,1 , ...) which minimizes n-*» m
the largest possible expectation of delayed action, and does not lead to too
many false alarms. More precisely, if P (X) dénotes the c.d.f. under the assumption that ail observations hâve F (X) as a c.d.f. ; and if E {.} dénotes expectation under P (.), the objectife is to minimize
m _ _ ^ _ ( 5 . 1 4 ) E.{N} = sup e s s sup E {(N-m-l) + | F . }
l ^. m m— I m>l
subject to the constraint
( 5 . 1 5 ) E {N} > y * , I < Y* < » •
E {.IF ,} dénotes the conditional expectation given the a-field generated by m m—i
(X ,...,X ) . It is proven by Lorden [43] that an asymptotically minimax procé-dure, as y -*• œ» is provided by PageTs procédure, which is described below.
Let RfX^ = f (X.)/f (X.), i = 1,2,... where f.(x) is the p.d.f. corresponding to F.(X), i = 0,1. Let
k
S. = Z log R(X.), k=l,2,... and T = S - min St . Then for Y = log Y
k i-l x n n k<n k
(5.16) N* = least n > 1 such that T > Y n is Page1s (extended) stopping variable.
The statistic T can be computed recursively by the formula (5.17) T = (T + log R(X . ) ) \ n = 0,1,...
T = 0.
o
iûe above détection p^oceaure eau be consiaerea as a séquence of one-aidea Wald's SPRT with boundaries ( 0 , Y ) . Whenever the T statistic hits the lower boun-dary, 0, the SPRT is recycled, and ail the previous observations can be discarded.
On the other hand, for the first time T > Y t h e sampling process is stopped.
The repeated cycles are independent and identically distributed. Thus, WaldTs theory of SPRT can be used to obtain the main results of the présent theory.
Let a and 6 be the error probabilities in each such independent cycle of Wald's SPRT ; i.e., a = P [T > Y ] and 0 - P.[T =0]. Let N. be the length of a
o n i n i cycle. Accordingly,
(5.18) E (N*} = - E {N.}
o a o I and
M
N*> =R? V V
-Set Y s* 1 — » then the constraint (5.15) is satisfied, since E {N } > 1.
Moreover, Lorden proved that E.{N } = Ê,{N }. Finally, applying well known results on the expected sample size in Wald's SPRT, we obtain
(5.19) E.{N*} = ^ | - 2 . , as a ^ 0,
il f.(X)
where I. = E. {log J V } is the Kull back-Leibler information for discrimi-na ting between F and F,.
o i
The right hand size of (5.19) vas shown to be the asymptotically minimum expected sample size. Thus, Page's procédure is asymptotically minimax.
in [42] Lorden and Eisenberg applied the theory presented hère to solve a problem of life testing for a reliability System. It is assumed that the life length of the System is distributed exponentially, with intensity (failure-rate) A. At an unknown time point, 0, the failure rate shifts from X to A(l+n), 0 < n <n^n2 < œ -Approximations to the formulae of E {N } and E {N } are given, assuming that X
o n
is known. By proper transformations of the statistics the détection procédure can be applied also to cases of unknown A. It is interesting to présent some of the numerical results of this studv. For the case of X =1 and a =1/Y . The expected number of observations required is
n Y E {N} Ê {N}
o n .4 20 422 48 .6 50 676 36 .9 40 342 20
Page's CUSUM procédure is thus very conservative, relative to the Bayes procédures, which detect the shifts fast, but bave also small EQ {N}.
.76.
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