Optimal detection of homogeneous segment of observations in stochastic sequence

Optimal detecti on of inhomogen eous segmen t of observ ations in a sto c hastic sequen ce W o jciech Sarnowski a , ∗ Krzysztof S za jo wski a a Wr o c law University of T e chnolo gy, Institute of Mathematics and Computer Scienc e, Wyb rze ˙ ze W yspia´ nskie go 27, 50-370 Wr o c law, Poland Abstract W e register a random sequence c onstructed based on Mark ov pro cesses b y switching b et w een them. A t unobserv able random momen t a c h ange in distribution of observ ed sequence tak es place. Using probab ility maximizing approac h the optimal stopp in g rule for d etecting the disorder is iden tified. Some explicit solution for example is also obtained. The r esult is generalization of Bo jdec ki’s mo d el wh er e b efore and after the c hange indep end en t pro cesses are obs er ved. Keywords. Disorder pr oblem, sequen tial detection, optimal stopping, Marko v pro cess, c hange p oin t. 1 In tro duction This pap er d eals with a sp eci al p r oblem b elonging to the wide class of disorder problems. Supp ose that the pr o cess X = { X n , n ∈ N } , N = { 0 , 1 , 2 , . . . } , is obser ved sequen tially . It is obtained fr om Mark ov pro cesses by switc hing b et ween them at random moment θ in suc h a wa y that the pro c ess after θ starts from the state X θ − 1 . O u r ob jectiv e is to detect this moment based on observ ation of X . There ⋆ 27 Ma y 2018 ∗ Corresp ond ing author Email addr esses: Wojciech. Sarnowsk i@pwr.wroc.pl (W o jci ec h Sarn o ws k i), Krzyszto f.Szajow ski@pwr.wroc.pl (Krzysztof Sza jo wski). URLs: http ://www.i m.pwr.wr oc.pl/~sarnowski (W o jciec h Sarno wski), http://n eyman.im .pwr.wroc.pl/~szajow (Kr zysztof S za jo ws k i). 1 are s ome pap ers dev oted to th e d iscrete case of suc h disord er d etectio n whic h gen- eralize in v arious directions the basic problem s tated b y Shiryae v in [9] (see e.g. Bro dsky and Darkho vsky [5], Bo jdec ki [3], Bo jd ecki and Hosza [4], Y osh id a [15], Sza jo wski [11,12]). Suc h mo del of data app e ars in man y p ractical problems of th e qualit y con trol (see Bro dsky and Darkhovsky [5], Shewh art [8] and in the collection of the pap e rs [2]), traffic anomalies in n etw orks (in pap ers by Du b e and Mazumdar [6], T artak ovsky et al. [13]), epidemiology mo dels (see Baron [1]). T he aim is to recognize the moment of the c hange the p robabilistic c haracteristics of the phenomenon. T ypically , disorder problem is limited to the case of switching b et w een sequences of indep e ndent rand om v ariables (see Bo jd ec ki [3]). Some develo pments of basic mo del can b e f ound in [14] wher e the optimal d etectio n r ule of sw itc hing moment has b een obtained when the finite state-space Marko v chains is disord er ed . Mous- takides [7] f ormulates condition wh ic h helps to reduce p roblem of quick est detection for dep e ndent sequences b e fore and after the change to the case of ind ep endent p ro- cesses. Ou r result is generalizati on of results obtained b y Bo jdec ki in [3]. It admits Mark ovia n dep endence structure for sw itc hed sequen ces (w ith p ossibly u ncoun table state-space) . W e obtain an optimal r u le un der probab ility maximizing criterion. F ormulation of the p roblem can b e found in Section 2. T he main resu lt is presente d in Section 3. Section 4 pro vides example of application for considered mo del. In app end ix w e d eriv e us eful formulas for conditional p robabilities. 2 F orm ulation of the problem Let (Ω , F , P ) b e a pr obabilit y space w hic h sup p orts sequence of observ able r andom v ariables { X n } n ∈ N generating filtration F n = σ ( X 0 , X 1 , ..., X n ). The sequence tak es v alues in ( E , B ), where E is a sub s et of ℜ . Space (Ω , F , P ) su pp o rts also unobs erv- able (hence not measurable with resp ect to F n ) v ariable θ whic h has geometrical distribution: P ( θ = j ) = p j − 1 q , q = 1 − p ∈ (0 , 1) , j = 1 , 2 , ... (1) F or x ∈ E w e in tro d uce a lso t wo h omogeneous Marko v pro cesses ( X 0 n , G 0 n , P 0 x ), ( X 1 n , G 1 n , P 1 x ) (b oth indep e ndent on θ ), which are connected with { X n } and θ by the follo wing equation: X n = X 0 n · I { θ>n } + X 1 n · I { X 1 θ − 1 = X 0 θ − 1 ,θ ≤ n } . (2) 2 W e ha ve that: G i n = σ ( X i 0 , X i 1 , . . . , X i n ), i ∈ { 0 , 1 } , n ∈ { 0 , 1 , 2 , . . . } . On ( E , B ) for x ∈ E there are defined σ -additiv e measures µ ( . ) and µ i x ( i = 0 , 1) satisfying follo wing relatio ns: P i x ( { ω : X i 1 ∈ B } ) = P ( X i 1 ∈ B | X i 0 = x ) = Z B f i x ( y ) µ ( dy ) = Z B µ i x ( dy ) = µ i x ( B ) . for any B ∈ B . Let u s n o w define function S, G S ( x 0 ,n ) = n X i =1 p i − 1 q L n − i +1 ( x 0 ,n ) + p n L 0 ( x 0 ,n ) , (3) G ( x n − l − 1 ,n , α ) = αL l +1 ( x n − l − 1 ,n ) + (1 − α ) (4) × l X i =0 p l − i q L i +1 ( x n − l − 1 ,n ) + p l +1 L 0 ( x n − l − 1 ,n ) ! . where x 0 , x 1 , . . . , x n ∈ E n +1 , α ∈ [0 , 1] , 0 ≤ n − l − 1 < n . Here we u se the follo wing notation: x k ,n = ( x k , x k +1 , ..., x n − 1 , x n ) , k ≤ n, L m ( x k ,n ) = n − m Y r = k +1 f 0 x r − 1 ( x r ) n Y r = n − m +1 f 1 x r − 1 ( x r ) , A k ,n = × n i = k A i = A k × A k +1 × . . . × A n , A i ∈ B where th e con v en tion that Q j 2 i = j 1 x i = 1 for j 1 > j 2 holds. F u nction S ( x 0 ,n ) stands for j oin den s it y of v ector X 0 ,n . F or any D 0 ,n = { ω : X 0 ,n ∈ B 0 ,n , B i ∈ B } and an y x ∈ E we ha v e: P x ( D 0 ,n ) = P ( D 0 ,n | X 0 = x ) = Z B 0 ,n S ( x 0 ,n ) µ ( dx 0 ,n ) The m eanin g of function G ( x k ,n , α ) will b e clear in the sequel. Shortly sp eaking our mo del assu mes that pro cess { X n } is obtained by switc hing at random and unknown instant θ b et w een tw o Mark ov pro cesses { X 0 n } and { X 1 n } . Notice th at what w e assume here is that the first observ ation X θ after the c h ange dep end s on the previous sample X θ − 1 through the transition p d f f 1 X θ − 1 ( X θ ). During 3 on-line observ ation of { X n } w e aim in detection of sw itching time θ in optimal wa y , according to the maxim um pr ob ab ility criterium. F or an y fi xed d ∈ { 0 , 1 , 2 , ... } we lo ok f or the stopping time τ ∗ ∈ T su c h that P x ( | θ − τ ∗ | ≤ d ) = sup τ ∈ S X P x ( | θ − τ | ≤ d ) (5) where S X denotes the set of all stopp ing times with r esp ect to the filtration {F n } n ∈ N . Using p arameter d w e con trol the precision lev el of d etectio n. T he most rigorous case: d = 0 will b e s tu died in details. 3 Solution of the probblem Let u s d efine: Z n = P x ( | θ − n | ≤ d | F n ) , n = 0 , 1 , 2 , . . . , V n = ess sup { τ ∈ S X , τ ≥ n } P x ( | θ − n | ≤ d | F n ) , n = 0 , 1 , 2 , . . . τ 0 = inf { n : Z n = V n } (6) Notice that, if Z ∞ = 0, then Z τ = P x ( | θ − τ | ≤ d | F τ ) f or τ ∈ S X . Since F n ⊆ F τ (when n ≤ τ ) we ha v e V n = ess su p τ ≥ n P x ( | θ − τ | ≤ d | F n ) = ess su p τ ≥ n E x ( I {| θ − τ |≤ d } | F n ) = ess su p τ ≥ n E x (Z τ | F n ) The f ollo wing lemma ensur es existence of th e solution Lemma 1 The stopping time τ 0 define d by formula (6) is the solution of pr oblem (5). PR OO F. F rom the theorems presen ted in [3] it is enough to sho w that lim n →∞ Z n = 0. F or all natural num b ers n, k , wh ere n ≥ k w e hav e: Z n = E x ( I {| θ − n |≤ d } | F n ) ≤ E x (sup j ≥ k I {| θ − j |≤ d } | F n ) F r om Levy’s theorem lim sup n →∞ Z n ≤ E x (sup j ≥ k I {| θ − j |≤ d } | F ∞ ) where F ∞ = σ ( S ∞ n =1 F n ). It is true that: lim sup j ≥ k, k →∞ I {| θ − j |≤ d } = 0 a.s. and b y the domi- nated con verge nce theorem we get lim k →∞ E x (sup j ≥ k I {| θ − j |≤ d } | F ∞ ) = 0 a.s. what end s the pro of of the lemma. 4 Lemma 2 L et τ b e a stopping rule in the pr oblem (5). Then rule ˜ τ = m ax( τ , d + 1) is at le ast as go o d as τ . PR OO F. F or τ ≥ d + 1 r ules τ , ˜ τ are the same. Let us consider the case when τ < d + 1. W e h a ve ˜ τ = d and giv en th e fact that P x ( θ ≥ 1) = 1 we get: P x ( | θ − τ | ≤ d ) = P x ( τ − d ≤ θ ≤ τ + d ) = P x (1 ≤ θ ≤ τ + d ) ≤ P x (1 ≤ θ ≤ 2 d + 1) = P x ( ˜ τ − d ≤ θ ≤ ˜ τ + d ) = P x ( | θ − ˜ τ | ≤ d ) . In consequence w e can limit the class of p ossible stopping rules to S X d +1 i.e. stopping times equ al at least d + 1. F or f u rther considerations let us d efine p osterior p ro cess: Π 0 = 0 , Π n = P x ( θ ≤ n | F n ) , n = 1 , 2 , . . . whic h is d esigned for information ab out distr ibution of disord er instant θ . Next lemma tran s forms p a yoff fun ction to the m ore con v enien t form. Lemma 3 L et h ( x 1 ,d +2 , α ) = 1 − p d + q d +1 X m =1 L m ( x 1 ,d +2 ) p m L 0 ( x 1 ,d +2 ) ! (1 − α ) , (7) wher e x 1 , ..., x d +2 ∈ E , α ∈ (0 , 1) then P x ( | θ − n | ≤ d ) = E x  h ( X n − 1 − d,n , Π n )  PR OO F. W e rewrite initial criterion as the exp ectation P x ( | θ − n | ≤ d ) = E x [ P x ( | θ − n | ≤ d | F n )] = E x [ P x ( θ ≤ n + d | F n ) − P x ( θ ≤ n − d − 1 | F n )] Probabilities und er exp ect ation can b e transform ed to the con v enien t f orm using lemmata 9 and 6. Next, with the help of Lemma 10 (putting l = d ) we can express 5 P x ( θ ≤ n + d | F n ) in terms of Π n . Given this some straigh tforw ard calculations imply th at: P x ( | θ − n | ≤ d | F n ) = 1 − p d + q d +1 X m =1 L m ( X n − d − 1 ,n ) p m L 0 ( X n − d − 1 ,n ) ! (1 − Π n ) . Lemma 4 Pr o c ess { η n } n ≥ d +1 wher e η n = ( X n − d − 1 ,n , Π n ) forms a r andom M arkov function. PR OO F. Acc ording to Lemma 17 pp 102 -103 in [10] it is enough to sho w that η n +1 is a fu nction of p revious s tage η n and v ariable X n +1 and that conditional distribution of X n +1 giv en F n is a function of η n . F or x 1 , ..., x d +3 ∈ E , α ∈ (0 , 1) let us consid er a fun ction ϕ ( x 1 ,d +2 , α, x d +3 ) = x 2 ,d +3 , f 1 x d +2 ( x d +3 )( q + pα ) G ( x d +2 ,d +3 , α ) ! W e will show that η n +1 = ϕ ( η n , X n +1 ). Notice that b y Lemm a 10 ( l = 0) w e get Π n +1 = f 1 X n ( X n +1 )( q + p Π n ) G ( X n,n +1 , Π n ) . (8) Hence ϕ ( η n , X n +1 ) = ϕ ( X n − d − 1 ,n , Π n , X n +1 ) = X n − d,n , X n +1 , f 1 X n ( X n +1 )( q + p Π n ) G ( X n,n +1 , Π n ) ! =  X n − d,n +1 , Π n +1  = η n +1 . Define ˆ F n = σ ( θ, X 0 ,n ). T o see that conditional distrib ution of X n +1 giv en F n is a function of η n , for an y Borel fun ction u : E − → ℜ let us consider the conditional exp ectation of u ( X n +1 ) given F n : 6 E x ( u ( X n +1 ) | F n ) = E x ( u ( X n +1 )(1 − Π n +1 ) | F n ) + E x ( u ( X n +1 )Π n +1 | F n ) = E x  u ( X n +1 ) I { θ>n +1 } | F n  + E x  u ( X n +1 ) I { θ ≤ n +1 } | F n  = E x  E x ( u ( X n +1 ) I { θ>n +1 } | ˆ F n ) | F n  + E x  E x ( u ( X n +1 ) I { θ ≤ n +1 } | ˆ F n ) | F n  = E x  I { θ>n +1 } E x ( u ( X n +1 ) | ˆ F n ) | F n  + E x  I { θ ≤ n +1 } E x ( u ( X n +1 ) | ˆ F n ) | F n  = Z E u ( y ) f 0 X n ( y ) µ ( dy ) P x ( θ > n + 1 | F n ) + Z E u ( y ) f 1 X n ( y ) µ ( dy ) P x ( θ ≤ n + 1 | F n ) = Z u ( y )( p (1 − Π n ) f 0 X n ( y ) + ( q + p Π n ) f 1 X n ( y )) µ ( dy ) = Z u ( y ) G ( X n , y , Π n ) µ ( dy ) Here w e use Lemma A.1. Lemmata 3 and 4 are crucial for the solution of p osed problem (5). They sh o w that initial problem can b e reduced to the problem of stopping Mark o v r an d om fu nction η n = ( X n − d − 1 ,n , Π n ) w ith the pa yoff given by equ ation (7 ). In consequence w e can use to ols of optimal stopping theory for find in g stopping time τ ∗ suc h th at E x  h ( X τ ∗ − 1 − d,τ ∗ , Π τ ∗ )  = sup τ ∈ S X d +1 E x  h ( X τ − 1 − d,τ , Π τ )  (9) T o solv e r ed uced problem (9 ) for any Borel f u nction u : E d +2 × [0 , 1] − → ℜ let us define op erators: T u ( x 1 ,d +2 , α ) = E x  u ( X n − d,n +1 , Π n +1 ) | X n − 1 − d,n = x 1 ,d +2 , Π n = α  , Q u ( x 1 ,d +2 , α ) = max { u ( x 1 ,d +2 , α ) , T u ( x 1 ,d +2 , α ) } . Lemma 5 F or the p ayoff function h ( x 1 ,d +2 , α ) char acterize d by (7) and for se- quenc e { r k } ∞ k =0 : r 0 ( x 1 ,d +1 ) = p " 1 − p d + q d +1 X m =1 L m − 1 ( x 1 ,d +1 ) p m L 0 ( x 1 ,d +1 ) # , r k ( x 1 ,d +1 ) = p Z E f 0 x d +1 ( x d +2 ) max ( 1 − p d + q d +1 X m =1 L m ( x 1 ,d +2 ) p m L 0 ( x 1 ,d +2 ) ; r k − 1 ( x 2 ,d +2 ) ) µ ( dx d +2 ) . the fol lowing formulas hol d: Q k h 1 ( x 1 ,d +2 , α ) = (1 − α ) max ( 1 − p d + q d +1 X m =1 L m ( x 1 ,d +2 ) p m L 0 ( x 1 ,d +2 ) ; r k − 1 ( x 2 ,d +2 ) ) , k ≥ 1 , T Q k h 1 ( x 1 ,d +2 , α ) = (1 − α ) r k ( x 2 ,d +2 ) , k ≥ 0 . 7 PR OO F. By th e definition of op erator T and u sing Lemm a A.5 ( l = 0) giv en that ( X n − d − 1 ,n , Π n ) = ( x 1 ,d +2 , α ) we get T h ( x 1 ,d +2 , α ) = E x  h ( X n − d,n +1 , Π n +1 ) | X n − d − 1 ,n = x 1 ,d +2 , Π n = α  = E x "  1 − p d + q d +1 X m =1 L m ( X n − d,n +1 ) p m L 0 ( X n − d,n +1 )  (1 − Π n +1 ) | X n − d − 1 ,n = x 1 ,d +2 , Π n = α # = p (1 − α ) Z E 1 − p d + q d +1 X m =1 L m − 1 ( x 2 ,d +2 ) p m L 0 ( x 2 ,d +2 ) f 1 x d +2 ( x d +3 ) f 0 x d +2 ( x d +3 ) ! f 0 x d +2 ( x d +3 ) G ( x d +2 ,d +3 , α ) G ( x d +2 ,d +3 , α ) µ ( dx d +3 ) = p (1 − α ) " 1 − p d + q d +1 X m =1 Z E L m − 1 ( x 2 ,d +2 ) p m L 0 ( x 2 ,d +2 ) f 1 x d +2 ( x d +3 ) µ ( dx d +3 ) # = (1 − α ) p " 1 − p d + q d +1 X m =1 L m − 1 ( x 2 ,d +2 ) p m L 0 ( x 2 ,d +2 ) # = (1 − α ) r 0 ( x 2 ,d +2 ) . Directly fro m the definition o f Q r esults that Q h ( x 1 ,d +2 , α ) = max n h ( x 1 ,d +2 , α ); T h ( x 1 ,d +2 , α ) o = (1 − α ) max ( 1 − p d + q d +1 X m =1 L m ( x 1 ,d +2 ) p m L 0 ( x 1 ,d +2 ) ; r 0 ( x 2 ,d +2 ) ) . Supp ose now that Lemma 5 holds for TQ k − 1 h and Q k h for some k > 1. Then using similar transformat io n as in the case of k = 0 w e get T Q k h ( x 1 ,d +2 , α ) = E x h Q k h ( X n − d,n +1 , Π n +1 ) | X n − d − 1 ,n = x 1 ,d +2 , Π n = α i = Z E " max ( 1 − p d + q d +1 X m =1 L m ( x 2 ,d +3 ) p m L 0 ( x 2 ,d +3 ) ; r k − 1 ( x 3 ,d +3 ) ) (1 − α ) pf 0 x d +2 ( x d +3 ) # µ ( dx d +3 ) = (1 − α ) r k ( x 2 ,d +2 ) . Moreo ver Q k +1 h ( x 1 ,d +2 , α ) = max n h ( x 1 ,d +2 , α ); TQ k h ( x 1 ,d +2 , α ) o = (1 − α ) max ( 1 − p d + q d +1 X m =1 L m ( x 1 ,d +2 ) p m L 0 ( x 1 ,d +2 ) ; r k ( x 2 ,d +2 ) ) . 8 This completes the pro of. The f ollo wing theorem is the main result of the pap er. Theorem 1 (a) The solution of pr oblem (5) i s giv en by: τ ∗ = inf { n ≥ d + 1 : 1 − p d + q d +1 X m =1 L m ( X n − d − 1 ,n ) p m L 0 ( X n − d − 1 ,n ) ≥ r ∗ ( X n − d,n ) } (10) wher e r ∗ ( X n − d,n ) = lim k − →∞ r k ( X n − d,n ) (b) V alue of the pr oblem. Given X 0 = x maximal pr ob ability for (5) is e qual to P x ( | θ − τ ∗ | ≤ d ) = p d +1 Z E d +1 max ( 1 − p d + q d +1 X m =1 L m ( x, x 1 ,d +1 ) p m L 0 ( x, x 1 ,d +1 ) ; r ∗ ( x 1 ,d +1 ) ) × L 0 ( x, x 1 ,d +1 ) µ ( d ( x, x 1 ,d +1 )) . PR OO F. P art (a). According to L emma 2 w e lo ok for stoppin g time equ al at least d + 1. F rom optimal stopping th eory (c.f [10]) w e kno w that τ 0 defined by (6) can b e expressed as τ 0 = inf { n ≥ d + 1 : h ( X n − 1 − d,n , Π n ) ≥ Q ∗ h ( X n − 1 − d,n , Π n ) } where Q ∗ h ( X n − 1 − d,n , Π n ) = lim k − →∞ Q k h ( X n − 1 − d,n , Π n ). According to L emma 5: τ 0 = inf ( n ≥ d + 1 : 1 − p d + q d +1 X m =1 L m ( X n − d − 1 ,n ) p m L 0 ( X n − d − 1 ,n ) ≥ max { 1 − p d + q d +1 X m =1 L m ( X n − d − 1 ,n ) p m L 0 ( X n − d − 1 ,n ) ; r ∗ ( X n − d,n ) } ) = inf ( n ≥ d + 1 : 1 − p d + q d +1 X m =1 L m ( X n − d − 1 ,n ) p m L 0 ( X n − d − 1 ,n ) ≥ r ∗ ( X n − d,n ) ) = τ ∗ . P art (b ). Basing on kn o w n f acts from optimal stopping theory we can write: 9 P x ( | θ − τ ∗ | ≤ d ) = E x  h ⋆ 1 ( X 0 ,d +1 , Π d +1 )  = E x (1 − Π d +1 ) max ( 1 − p d + q d +1 X m =1 L m ( X 0 ,d +1 ) p m L 0 ( X 0 ,d +1 ) ; r ⋆ ( X 1 ,d +1 ) )! = E x E x ( I { θ>d +1 } | F d +1 ) max ( 1 − p d + q d +1 X m =1 L m ( X 0 ,d +1 ) p m L 0 ( X 0 ,d +1 ) ; r ⋆ ( X 1 ,d +1 ) )! = E x I { θ>d +1 } max ( 1 − p d + q d +1 X m =1 L m ( X 0 ,d +1 ) p m L 0 ( X 0 ,d +1 ) ; r ⋆ ( X 1 ,d +1 ) )! = P x ( θ > d + 1) Z E d +1 max ( 1 − p d + q d +1 X m =1 L m ( x, x 1 ,d +1 ) p m L 0 ( x, x 1 ,d +1 ) ; r ∗ ( x 1 ,d +1 ) ) × L 0 ( x, x 1 ,d +1 ) µ x d ( d ( x, x 1 ,d +1 )) What end s the pro of. 4 Example Let u s consider the case d = 0. Then, optimal ru le (10) reduces to simp ler form τ ∗ = inf { n ≥ 1 : f 1 X n − 1 ( X n ) pf 0 X n − 1 ( X n ) ≥ r ∗ ( X n ) } with r ∗ ( X n ) = p Z E f 0 X n ( u ) max { f 1 X n ( u ) pf 0 X n ( u ) , r ∗ ( u ) } dµ ( u ) Moreo ve r su pp ose that the state space E = { 0 , 1 } . Matrices of transition probabili- ties and conditional densities are as follo w  µ 0 i ( j )  i =0 , 1 j =0 , 1 =    0 . 1 0 . 9 0 . 8 0 . 2    ,  µ 1 i ( j )  i =0 , 1 j =0 , 1 =    0 . 7 0 . 3 0 . 4 0 . 6     f 0 i ( j )  i =0 , 1 j =0 , 1 =    1 1 1 1    ,  f 1 i ( j )  i =0 , 1 j =0 , 1 =    7 1 / 3 1 / 2 3    10 F or su c h mo d el we find threshold r ∗ ( i ), i = 0 , 1 solving the s y s tem of equations r ∗ ( i ) = X j =0 , 1 pf 0 i ( j ) max { f 1 i ( j ) pf 0 i ( j ) , r ∗ ( j ) } µ ( j ) ; i = 0 , 1 T reating r ∗ as a function of p arameter p we obtain: r ∗ p (0) = 1 [0 ,p 1 ] ( p ) + 7 + 9 p 10 1 ( p 1 ,p 2 ] ( p ) + 35 + 27 p 50 − 36 p 2 1 ( p 2 ,p 3 ] ( p ) + 35 − 7 p 50 − 10 p − 36 p 2 1 ( p 3 , 1] ( p ) r ∗ p (1) = 1 [0 ,p 2 ] ( p ) + 30 + 28 p 50 − 36 p 2 1 ( p 2 ,p 3 ] ( p ) + 14 p 25 − 50 − 18 p 2 1 ( p 3 , 1] ( p ) where: p 1 = 1 3 , p 2 = √ 229 − 7 18 , p 3 = √ 20625 − 15 136 . The most inte resting case tak es th e place when p > p 3 ≈ 0 , 946 b ecause then the av erage disorder time is not to o small. Obtained stopping rule τ ⋆ dep end s on observ ations collected at times τ ⋆ − 1 and τ ⋆ . Thus, to make optimal rule more clear we need to analyze all p ossible sequences of ( X τ ⋆ − 1 , X τ ⋆ ) i.e. { 0 , 0 } , { 0 , 1 } , { 1 , 0 } , { 1 , 1 } . Sequence { 0 , 0 } : In this case we stop if only 7 p ≥ 35 − 7 p 50 − 10 p − 36 p 2 . Solving the inequalit y for p , w e get that stopp ing time tak es the place for all p ∈ ( p 3 , 1). Sequence { 0 , 1 } : It reduces to inequalit y 1 3 p ≥ 14 p 25 − 50 p − 18 p 2 . T aking into accoun t that p ∈ ( p 3 , 1) a set of solutions is empt y . Sequence { 1 , 0 } : P air { 1 , 0 } implies the stopping time if 7 p ≥ 35 − 7 p 50 − 10 p − 36 p 2 . How ev er there is no solution for p ∈ ( p 3 , 1). Sequence { 1 , 1 } : This sequence rises the alarm if only 3 p ≥ 14 p 25 − 50 p − 18 p 2 . It turns out that the inequalit y is satisfied for an y p ∈ ( p 3 , 1). The analysis s ho w s that we obtain ve ry clear and simple optimal rule for case p > p 3 : stop at the first moment when tw o ”zeros” or tw o ”ones” o ccur in a ro w . A Lemmata Lemma 6 L et n > 0 , k ≥ 0 then: P x ( θ ≤ n + k | F n ) = 1 − p k (1 − Π n ) . (A.1) PR OO F. It is enough to sh ow that for D ∈ F n 11 Z D I { θ>n + k } d P x = Z D p k (1 − Π n ) d P x . Let u s d efine e F n = σ ( F n , I { θ>n } ). W e ha v e: Z D I { θ>n + k } d P x = Z D I { θ>n + k } I { θ>n } d P x = Z D ∩{ θ>n } I { θ>n + k } d P x = Z D ∩{ θ>n } E x ( I { θ>n + k } | e F n ) d P x = Z D ∩{ θ>n } E x ( I { θ>n + k } | θ > n ) d P x = Z D I { θ>n } p k d P x = Z D (1 − Π n ) p k d P x Lemma 7 F or n > 0 the fol lowing e qu ality holds: P x ( θ > n | F n ) = 1 − Π n = p n L 0 ( X 0 ,n ) S ( X 0 ,n ) . (A.2) PR OO F. Put D 0 ,n = { ω : X o,n ∈ A 0 ,n , A i ∈ B } . Then: P x ( D 0 ,n ) P x ( θ > n | D 0 ,n ) = Z D 0 ,n I { θ>n } d P x = Z D 0 ,n P x ( θ > n |F n ) d P x = Z A 0 ,n p n L 0 ( x 0 ,n ) S ( x 0 ,n ) S ( x 0 ,n ) µ ( dx 0 ,n ) = Z D 0 ,n p n L 0 ( X 0 ,n ) S ( X 0 ,n ) d P x Hence, by defi n ition of conditional exp ectation, we get the thesis. Lemma 8 F or x 0 ,l +1 ∈ E l +2 , α ∈ [0 , 1] and functions S, G g iven by e q uations (3) and (4) we have: S ( X 0 ,n ) = S ( X 0 ,n − l − 1 ) G ( X n − l − 1 ,n , Π n − l − 1 ) (A.3) PR OO F. By (A.2) w e h a ve 12 S ( X 0 ,n − l − 1 ) G ( X n − l − 1 ,n , Π n − l − 1 ) = S ( X 0 ,n − l − 1 )Π n − l − 1 L l +1 ( X n − l − 1 ,n ) + S ( X 0 ,n − l − 1 )(1 − Π n − l − 1 ) × l X k =0 p l − k q L k +1 ( X n − l − 1 ,n ) + p l +1 L 0 ( X n − l − 1 ,n ) ! ( A.2 ) = n − l − 1 X k =1 p k − 1 q L n − l − k ( X 0 ,n − l − 1 ) ! L l +1 ( X n − l − 1 ,n ) + p n − l − 1 L 0 ( X 0 ,n − l − 1 ) × l X k =0 p l − k q L k +1 ( X n − l − 1 ,n ) + p l +1 L 0 ( X n − l − 1 ,n ) ! = n − l − 1 X k =1 p k − 1 q L n − k +1 ( X 0 ,n ) + l X k =0 p n − k − 1 q L k +1 ( X 0 ,n ) + p n L 0 ( X 0 ,n ) = n − l − 1 X k =1 p k − 1 q L n − k +1 ( X 0 ,n ) + n X k = n − l p k − 1 q L n − k +1 ( X 0 ,n ) + p n L 0 ( X 0 ,n ) = n X k =1 p k − 1 q L n − k +1 ( X 0 ,n ) + p n L 0 ( X 0 ,n ) = S ( X 0 ,n ) . Lemma 9 F or n > l ≥ 0 the fol lo wing e quation i s satisfie d: P x ( θ ≤ n − l − 1 | F n ) = Π n − l − 1 L l +1 ( X n − l − 1 ,n ) G ( X n − l − 1 ,n , Π n − l − 1 ) . PR OO F. Let D 0 ,n = { ω : X o,n ∈ A 0 ,n , A i ∈ B } . Then P x ( D 0 ,n ) P x ( θ > n − l − 1 | D 0 ,n ) = Z D 0 ,n I { θ>n − l − 1 } d P x = Z D 0 ,n P x ( θ > n − 1 |F n ) d P x = Z A 0 ,n P n k = n − l P x ( θ = k ) L n − k +1 ( x 0 ,n ) + P x ( θ > n ) L 0 ( x 0 ,n ) S ( x 0 ,n ) S ( x 0 ,n ) µ ( dx 0 ,n ) = Z A 0 ,n p n − l − 1 L 0 ( x 0 ,n − l − 1 )  P l k =0 p l − k q L k +1 ( x n − l − 1 ,n ) + p l +1 L 0 ( x n − l − 1 ,n )  S ( x 0 ,n ) × S ( x 0 ,n ) µ ( dx 0 ,n ) = Z D 0 ,n p n − l − 1 L 0 ( x 0 ,n − l − 1 )  P l k =0 p l − k q L k +1 ( X n − l − 1 ,n ) + p l +1 L 0 ( X n − l − 1 ,n )  S ( X 0 ,n ) d P x ( A.3 ) = Z D 0 ,n p n − l − 1 L 0 ( x 0 ,n − l − 1 )  P l k =0 p l − k q L k +1 ( X n − l − 1 ,n ) + p l +1 L 0 ( X n − l − 1 ,n )  S ( X 0 ,n − l − 1 ) G ( X n − l − 1 ,n , Π n − l − 1 ) d P x ( A.2 ) = Z D 0 ,n (1 − Π n − l − 1 ) P l k =0 p l − k q L k +1 ( X n − l − 1 ,n ) + p l +1 L 0 ( X n − l − 1 ,n ) G ( X n − l − 1 ,n , Π n − l − 1 ) d P x 13 What imp lies th at: P x ( θ > n − l − 1 |F n ) (A.4) = (1 − Π n − l − 1 ) P l k =0 p l − k q L k +1 ( X n − l − 1 ,n ) + p l +1 L 0 ( X n − l − 1 ,n ) G ( X n − l − 1 ,n , Π n − l − 1 ) Simple tr an s formations of (A.4) lead to the thesis. Lemma 10 F or n > l ≥ 0 r e cursive e q uation holds: Π n = Π n − l − 1 L l +1 ( X n − l − 1 ,n ) + (1 − Π n − l − 1 ) q P d k =0 p l − k L k +1 ( X n − l − 1 ,n ) G ( X n − l − 1 ,n , Π n − l − 1 ) (A.5) PR OO F. Wit h the aid of (A.2) we get: 1 − Π n 1 − Π n − l − 1 = p n L 0 ( X 0 ,n ) S ( X 0 ,n ) S ( X 0 ,n − l − 1 ) p n − l − 1 L 0 ( X 0 ,n − l − 1 ) = p l +1 L 0 ( X n − l − 1 ,n ) G ( X n − l − 1 ,n , Π n − l − 1 ) Hence Π n = G ( X n − l − 1 ,n , Π n − l − 1 ) − p n − l − 1 L 0 ( X 0 ,n − l − 1 )(1 − Π n − l − 1 ) G ( X n − l − 1 ,n , Π n − l − 1 ) = Π n − l − 1 L l +1 ( X n − l − 1 ,n ) + (1 − Π n − l − 1 ) q P d k =0 p l − k L k +1 ( X n − l − 1 ,n ) G ( X n − l − 1 ,n , Π n − l − 1 ) . noinden t References [1] M. Baron, Early dete ction of epidemics as a se quential change-p oint pr oblem. , in L ongevity, aging and de gr adation mo dels in r eliability, public he alth, me dicine and biolo gy, LAD 2004. 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