CUSUM AND EWMA MULTI-CHARTS FOR DETECTING A RANGE OF MEAN SHIFTS

Transcription

1 tatitia inia 17(007), UUM AND EWMA MULTI-HART FR DETETING A RANGE F MEAN HIFT Dong Han 1, Fugee Tung, Xijian Hu 3 and Kaibo Wang 1 hanghai Jiao Tong Univerity Hong Kong Univerity of iene and Tehnology and 3 Xinjiang Univerity Abtrat: The multi-hart onit of everal UUM or EWMA hart with different referene value that are ued imultaneouly to detet antiipated proe hange. We not only prove that the hart an quikly ahieve the aymptoti optimal bound, but alo give an integral equation to determine the referene value to arrive at optimality. imulation reult are ued to verify the theoretial optimal propertie and to how that the UUM multi-hart i uperior on the whole to ingle UUM, ingle EWMA, and EWMA multi-hart in term of run length and robutne, and an ompete with GLR ontrol hart in deteting a range of variou mean hift. We invetigate the deign of both UUM and EWMA multi-hart. ome pratial guideline are provided for determining multi-hart parameter, uh a the number of ontituent hart and the alloation of their referene value. Key word and phrae: average run length, hange point detetion, tatitial proe ontrol. 1. Introdution tatitial proe ontrol (P) tehnique are widely ued in monitoring and ontrolling both manufaturing and ervie proee. Variou P heme have been extenively tudied in the literature, among then the umulative um (UUM) and exponential weighted moving average (EWMA) heme (ee Montgomery (1996), Lai (1995), and referene therein). The performane of thee heme, however, motly depend on the pre-peified ize of the hift in the variable that one wihe to detet. For example, it ha been hown by Moutakide (1986) and Ritov (1990) that the performane in deteting the mean hift of the one-ided UUM ontrol hart with the referene value δ i optimal in term of average run length (ARL) if the atual mean hift i δ. rivatava and Wu (1993, 1997) and Wu (1994) provided a deign of the optimal EWMA by hooing an optimal weighting parameter in an EWMA ontrol hart uh that it an minimize the out of ontrol ARL for a given referene value, and

2 1140 DNG HAN, FUGEE TUNG, XIJIAN HU AND KAIB WANG they illutrated that the optimal EWMA perform almot a well a the UUM hart in term of ARL. Lua and aui (1990) alo provided the optimal deign parameter of an EWMA hart that depend on a pre-peified ize of mean hift a well a a given in-ontrol ARL. ome heme do not depend on a peifi hift ize δ, for example the GLR hart of iegmund and Venkatraman (1995). Their imulation reult how that the GLR hart i better than the UUM ontrol hart in deteting a mean hift that i larger or maller than δ, and i only lightly inferior in deteting a mean hift of ize δ. Alo, by taking the maximum weighting parameter in the EWMA ontrol hart, Han and Tung (004) propoed a generalized EWMA (GEWMA) ontrol hart that doe not depend on the referene value, and proved that the GEWMA ontrol hart i better than the optimal EWMA in deteting a mean hift of any ize when the in-ontrol ARL i large. However, thee method uually require omplex omputing and have not been regularly applied to real on-line problem. Beaue we rarely know the exat hift value of a proe before it i deteted, it may be more important to look at a range of known or unknown mean hift. Example inlude a emiondutor wafer manufaturing proe that need to monitor and detet a range of antiipated hange in the poition and ize of erial marking, and a wire manufaturing proe that require ontinuou diameter monitoring uing laer mirometer for a wide range of unknown hift. To handle thi problem, an alternative approah i to onider a multiple model or a mixture of everal ontrol hart. In fat, Lorden (1971) ha already onidered and tudied uh a model. ine then, Lorden and Eienberger (1973), Lua (198), Rowland et al. (198), Dragalin (1993, 1997) and park (000) have further invetigated and tudied a ombination of everal UUM hart and a ombined hewhart-uum to detet mean hift in a range. They have hown the effiieny of the ombined UUM and hewhart-uum hart, and provided variou deign for thee proedure, baed on numerial imulation. The ombination of everal UUM hart mentioned above an be alled UUM multi-hart, to onit of multiple UUM hart with different referene value that are ued imultaneouly to detet the mean hift. For example, let the antiipated interval of the mean hift, µ, be [a,b]. Then, we an reate a UUM multi-hart with a number of UUM hart, T (δ 1 )),...,T (δ m ) (ee (.1) in etion for the definition of T (δ 1 )), by hooing the parameter value, δ 1,...,δ m, in the interval. If one of the UUM hart, T (δ k ), trigger a ignal of having a mean hift, the multi-hart would end an out-of-ontrol warning. The UUM multi-hart ha it root in onventional ontrol hart and ha muh redued omputational omplexity ompared with GLR and GEWMA. Although the referene δ an be defined in a more general ene (e.g., a dynami,

3 UUM AND EWMA MULTI-HART FR DETETING A RANGE F MEAN HIFT 1141 non-ontant mean hange) in uing the multi-hart, we fou on the ae with ontant mean hift at firt. Alo, the ontituent hart of the multi-hart have great flexibility in taking variou form of hart, but in the paper we mainly invetigate the UUM and EWMA multi-hart. Although the general theoretial reult regarding aymptoti optimality have been given by Lorden (1971), it i not lear whether the UUM multihart ha ome peial aymptoti optimal propertie. Although Rowland et al. (198) and park (000) have hown that two or three UUM were uffiient to almot ahieve the optimal envelope, it i not lear whether there exit an optimal deign of the UUM multi-hart that an be arried out by theoretial alulation and Monte arlo imulation. The primary goal of thi paper i to deal with thee two problem. It will be hown that the UUM multi-hart annot only quikly ahieve the aymptotially optimal bound but alo ha better performane (quiker and more robut) than that of a ingle UUM and EWMA ontrol hart in deteting a range of variou mean hift. An optimal deign of the UUM multi-hart i provided for determining the multi-hart parameter, uh a the number of ontituent hart and the alloation of their referene value. The remainder of the paper i organized a follow. In the next etion, we diu ome propertie related to the UUM multi-hart, EWMA multihart and GLR hart. A novel harting performane index i propoed in etion 3 for the ituation with a range of known or unknown hift. Baed on that, the performane of the UUM multi-hart and the EWMA multi-hart are ompared with their ontituent hart and the GLR hart in etion 4. Alo in that etion, the fat aymptoti optimality of the UUM multi-hart and the integral equation to determine the optimal hoie of the referene value are preented. etion 5 provide an optimal deign of the UUM multi-hart and ome pratial guideline for both UUM multi-hart and EWMA multi-hart to determine the number of ontituent hart and alloation of their referene value. onluion and problem for further tudy are diued in etion 5, with the proof of three theorem given in the Appendix.. The UUM and EWMA Multi-hart Let X i, i = 1,..., be N(µ 0,σ). uppoe that at ome time period τ, uually alled a hange point, the probability ditribution of X i hange from N(µ 0,σ) to N(µ,σ). In other word, from time period τ onward, X i ha the ommon ditribution N(µ,σ). Thu, the mean of X i undergoe a peritent hift of ize µ µ 0, where µ 0 and σ are known and, without lo of generality, aumed to be µ 0 = 0 and σ = 1.

4 114 DNG HAN, FUGEE TUNG, XIJIAN HU AND KAIB WANG Let m = {δ k : 1 k m} and R m = {r k : 1 k m} be two et of number, where δ k > 0 and 0 < r k 1 are known referene value. Let k > 0 and d k > 0 be two number that uually depend on δ k and r k, repetively. Then the one-ided UUM and EWMA multi-hart, T ( m, m ) and T E (R m,d m ), are T = min δ i m {T (δ i, i )} and T E (R m,d m ) = min ri R m {T E (r i,d i )} where T (δ i, i ) = min{n : max 1 k n δ i[x n + + X n k+1 δ i k/] > i }, (.1) n 1 T E (r i,d i ) = min{n : r i (1 r i ) k X n k > d i }. (.) k=0 Here, T (δ i, i ) and T E (r i,d i ) are, repetively the one-ided UUM and EWMA hart. A an be een, for the obervation X 1,...,X n, one require mn alulation for the UUM multi-hart to detet a mean hift. The GLR and GEWMA hart (ee iegmund and Venkatraman (1995) and Han and Tung (004)) are T GL () = min{n : max 1 k n [X n + + X n k+1 ]/k 1/ > } and T GE () = inf{n 1 : max 1 k n W n ( 1 k ) }, where ( 1 k ) W n ( 1 k ) = 1 k [1 (1 1 k )n ] n 1 i=0 1 k (1 1 k )i X n i, and require n(n+1)/ alulation. In partiular, when n i large, e.g., 1,000, the omputational burden for the GLR hart i very heavy. Thu, the multi-hart ha an advantage in reduing omputational omplexity ompared with the GLR and GEWMA hart. In addition to it omputational advantage, we will demontrate it performane in deteting a wide range of antiipated hange, and it flexibility in deign for variou ituation. 3. harting Performane Index for a Range of Mean hift The average run length (ARL) ha been extenively ued in evaluating different harting method. For omparion, the in-ontrol ARL (ARL 0 ) of all andidate hart are fored to be equal, and to orrepond to the ame level of type I error. The hart that ha the lowet out-of-ontrol ARL at the deired mean hift ize preent the highet power to detet the pre-peified hift. Although the ARL i a popular riterion, it ha a defiieny in evaluating a harting performane for a range of antiipated mean hift. For example, Figure 1 how the ARL urve of two UUM hart, one deigned for deteting a mean hift ize of 0.1, the other of.0. The ARL urve interet at a mean hift of about Thu, the hart deigned for 0.1 outperform the hart for.0 in

5 UUM AND EWMA MULTI-HART FR DETETING A RANGE F MEAN HIFT 1143 the range of (0,0.77), while the hart for.0 outperform the hart for 0.1 in the range of (0.77,4.0]. It would be diffiult to evaluate their performane if the whole range of mean hift i of interet. Figure 1. ARL urve of two UUM hart deigned for mean hift of 0.1 and.0. To handle uh a ituation, we propoe an verall harting Performane Index (PI) a follow. ( b PI u f a w(µ) ARL(µ) ARL ) r(µ) dµ, (3.1) ARL r (µ) where µ [a,b] i a hift ize in the antiipated range within whih the performane i evaluated, ARL(µ) i the ARL of the hart to be evaluated, and ARL r (µ) i a referene or baeline with the lowet ARL value at the hift ize µ. It i known that the UUM hart with the parameter δ ahieve the lowet ARL at the hift µ = δ among all UUM heme, o that the ARL value at eah hift ize µ within the range [a,b] will be ued a a lower bound ARL r in our later tudy. The referene urve that i a ompoite of a olletion of the lowet poible ARL at eah hift ize i denoted a an ptimal ARL urve (A). Note that w(µ) i a weighting funtion to emphaize variou mean hift within a range baed on prior knowledge and experiene with the proe, given preferential onideration to ertain mean hift. For example, if the large mean hift (e.g., µ ) i onidered to be more important than the mall one (e.g., µ < ), thi an be aknowledged. Thu, we an ompare the performane of hart by the PI to know whih i better in deteting large mean hift. If no

6 1144 DNG HAN, FUGEE TUNG, XIJIAN HU AND KAIB WANG prior information or preferene i provided, we ue w(µ) = (b a) 1 throughout the range. A an be een, the range of PI value i from 0 to +. If we hooe f(x) = e x, the range of the PI i (0,1]. In addition, the PI with f(x) = e x i more ommuniable and omparable by denoting 0 a the wort detetion performane and 1 a the bet performane for a hart. Moreover, the omparion reult do not hange a long a the eleted funtion are all tritly monotoni dereaing (or inreaing). Here we take f(x) = e x. If the peifi ize of the antiipated mean hift within a range are known, we modify the PI in (4.1) to the following form: ( n ARL i ARL ) ri PI k = f w i, (3.) ARL ri i=1 where i = 1 to n repreent the n ize of antiipated mean hift, and w i = 1/n. 4. Aymptoti Analyi of the UUM Multi-hart The UUM hart i popular and ha attrative theoretial propertie in that it i the optimal tet for a known mean hift, but it doe le well for a range of hift away from it deigned hift (ee Hawkin and lwell (1998)). The GLR hart, on the other hand, i good for unknown mean hift. However, it i le popular due to it exeive omputational effort. In thi etion, we tart with the invetigation of the UUM multi-hart by proving it aymptoti optimality in deteting a range of known and unknown mean hift, and then ompare the detetion performane of the UUM multi-hart and the EWMA multi-hart with their ontituent UUM hart, EWMA hart, and GLR hart The antiipated mean hift, µ k, are known Here, we uppoe antiipated mean hift ize, µ k (1 k n), in a range are known from prior knowledge and experiene. For a topping time, T, a the alarm time in a deteting proedure, we have the in-ontrol ARL 0 (T) = E 0 (T), and the out-of-ontrol ARL µ (T) = E µ (T). Let T(µ k, k ) and T(µ k, k ) be two one-ided UUM hart with different ontrol limit k and k. Denote T(µ k, k ) and T(µ k, k ), repetively, a T k and T k. onider the UUM multi-hart T = min 1 k n{(t k )}. Take the ontrol limit, 1,..., n, uh that k > k, 1 k n, and E 0 (T 1) = = E 0 (T n) = L > E 0 (T 1 ) = = E 0 (T n ) = L = E 0 (T ). (4.1)

7 UUM AND EWMA MULTI-HART FR DETETING A RANGE F MEAN HIFT 1145 Thi mean that the UUM hart, T 1,...,T n and the UUM multi-hart, T, have a ommon ARL 0, i.e., E 0 (.) = L. It ha been hown by Lorden (1971), Moutakide (1986), rivatava and Wu (1997) and Wu (1994), that the UUM hart, T k, i optimal and the optimal lower boundary i log L/µ + ontant. That i, for an arbitrary ontrol hart T ubjet to the ontraint E 0 (T) L, E µk (T) E µk (T k ) hold for 1 k n and E µk (T k ) = log L µ k + M(µ k ) + o(1) for 1 k n a L, where M(µ k ) = /µ k + µ k ln(µ k /) and o(1) = ( L 1 ln(µ L/) ). Thu, the expetation of the optimal lower boundarie an be written a B(L) = n k=1 ( log L ) P(Z = µ k ) µ + M(µ k ), k where P(Z = µ k ) denote the probability that the mean hift ize Z i µ k and n k=1 P(Z = µ k) = 1. Aymptoti optimality of the UUM multi-hart reue from the following theorem. Theorem 1. Let P(Z = µ k ) = π k for 1 k n. A L, or min 1 i n { i }, we have log L log L 0 and n π k E µk (T) (ln L)3 B(L) ( L ) = (3 1e 1 ) 0. (4.) k=1 Remark 1. Let L = nl, that i, eah UUM hart T k (1 k n) ha the ame in-ontrol ARL 0 = nl for ome large L. Note that T T k for 1 k n. It follow from Theorem 1 of Lorden (1971), in onjuntion with Lorden remark following hi Theorem, that the UUM multi-hart ha in-ontrol ARL 0 L and log m + log L µ k for 1 k n a L. That i, + M(µ k ) + o(1) = E µk (T k ) E µ k (T ) E µk (T k ) = log L µ k + M(µ k ) + o(1) n log m π k µ k=1 k n + o(1) π k E µk (T) B(L) o(1) k=1

8 1146 DNG HAN, FUGEE TUNG, XIJIAN HU AND KAIB WANG a L, o (4.) an not be dedued diretly from Theorem 1 of Lorden (1971). The proof of Theorem 1 i given in the appendix. orollary 1. If a ontrol hart, T, i ubjet to the ontraint E 0 (T) L and E µk (T) E µk (T),1 k n, then n π k E µk (T) k=1 n π k E µk (T) (4.3) k=1 a L, where E µ (T) = up τ 1 e upe µ [(T τ + 1) + X 1,...,X τ 1 ], τ i hange time (ee Lorden (1971)). peifially, for eah UUM hart T j atifying (4.1) (1 j n) and n > 1, we have, a L, n π k E µk (T j ) k=1 n k=1 π k E µk (T ). (4.4) It follow from (4.4) that the UUM multi-hart perform better than any ingle UUM hart in deteting more than one antiipated mean hift when L = ARR 0. Thi property will be een later in Monte arlo imulation. 4.. The antiipated mean hift, µ, i unknown Now we invetigate the ituation where we know the antiipated range to monitor but the peifi ize of an antiipated mean hift i unknown. Let a > 0. Here we hooe the referene value δ k in [a,b] uh that a δ k < δ k+1 b for 0 k m, where δ 0 = 0 and δ m+1 = b. Let I k = {µ : (δ k 1 +δ k )/ < µ (δ k + δ k+1 )/} for 1 k m. Denote by PI u (δ 1,...,δ m ) the PI of the UUM multi-hart, T = min 1 k m{(t k )}, where T k = T(δ k, k ) i the UUM hart with the referene value δ k ( 1 k m) atifying (4.1). Here we take w(µ) = (b a) 1 and f(x) = e x in PI u. Theorem. Let µ I k, 1 k m. A L, E µ (T ) E µ(t k ) k δ k (µ δ k ). (4.5) Furthermore, a L, there exit the number, δ 1 < δ < < δ m, uh that PI u (δ 1,...,δ m ) = max {PI u(δ 1,...,δ m )}, (4.6) {δ k,1 k m}

9 UUM AND EWMA MULTI-HART FR DETETING A RANGE F MEAN HIFT 1147 and δ 1 and δ k are, repetively, the unique olution to I 1 (x) = I k (x) = x+δ a x+δ k+1 δ k 1 +x (µ x)µ x (µ x )dµ = 0, (4.7) (µ x)µ x (µ x )dµ = 0 for k m, where δ 0 = 0, a < δ 1 < a and δ m < δ m+1 = b. Remark. It follow from Theorem 1 and etion 3 of Lorden (1971) that for k 1, and PI u (δ 1,...,δ k ) < PI u (δ 1,...,δ k+1 ) (4.8) lim L lim PI u(δ 1,...,δ m ) = 1 (4.9) m for δ k = a + k(b a)/(m + 1), 1 k m. The proof of Theorem i in the appendix. By uing (4.6) and (4.7) we an get an optimal deign of the UUM multi-hart. The inequality (4.8) mean that the PI u will inreae if one more referene value that i greater than the exitent referene value i added to T. From (4.9) it follow that the ARL of the UUM multi-hart, ARL µ (T ), an approximate the optimal ARL urve, ARL r (µ), if there are many referene value evenly ditributed in the range [a,b]. Let T k and T E denote, repetively, a one-ided UUM hart with the referene value δ k (1 k m) and a one-ided EWMA hart with the referene value r (0 < r 1). Theorem 3. Let the number p k atify p k > 0 and m k=1 p k = 1. If the UUM multi-hart, UUM and EWMA have a ommon ARL 0 = L, a L, and for µ > δ 1 /. m k=1 p k E µ (T k ) > E µ (T ), (4.10) E µ (T E ) > E µ (T ), (4.11) Remark 3. Let T GL be the one-ided GLR hart with ARL 0 = L. It follow from etion 3 of Lorden (1971), or Theorem 6 of Han and Tung (004), that E µ (T GL ) E µ (T ). a L. The proof of Theorem 3 i in the appendix. From Theorem 3 we find that the UUM multi-hart ha better performane than any ingle ontituent

10 1148 DNG HAN, FUGEE TUNG, XIJIAN HU AND KAIB WANG UUM hart in deteting an unknown mean hift. The UUM multi-hart i alo better than the EWMA exept in deteting the mean hift of a ize le than δ 1 /. Although the GLR i better than the UUM multi-hart, by Remark 3, when the ARL 0 goe to infinity, the imulation reult given in Table 1 how that the UUM multi-hart atually outperform GLR in deteting mall mean hift when the ARL 0 i not large. A imulation reult how, the good property of the multi-hart alo hold when the ARL 0 i et at ome typial value (e.g., 500) that i not large. Table 1. ARL and their tandard error (in parenthee) of the UUM hart with ARL 0 = 500. HIFT δ 1 = 0.1 δ = 0.5 δ 3 = 1 δ 4 = 1.5 δ 5 = (µ) 1 = = = = = (414) 500(491) 500(50) 500(490) 500(498) (169) 301(84) 369(366) 417(416) 439(433) (4.7) 94.(77.) 144(135) 0(198) 5(50) (14.3) 31.0(17.7) 38.9(31.8) 58.1(53.7) 81.9(78.8) (7.46) 17.5(7.55) 17.(11.1).0(17.9) 30.7(7.6) 1 1.5(4.74) 1.(4.45) 10.5(5.56) 11.6(7.9) 14.6(11.9) (3.43) 9.34(.96) 7.5(3.36) 7.53(4.37) 8.56(6.04) (.59) 7.59(.16) 5.83(.9) 5.50(.7) 5.80(3.59) 10.8(1.69) 5.55(1.3) 4.07(1.30) 3.56(1.41) 3.43(1.63) 3 7.7(0.93) 3.68(0.7).60(0.66).19(0.64) 1.95(0.71) (0.64).84(0.51).03(0.38) 1.64(0.51) 1.39(0.51) PI k PI u imulation reult imulation reult were baed on a 10,000-repetition experiment. The ommon ARL 0 wa hoen to be 500. We ompare the imulation reult for the ten mean hift (µ 1 = 0.1, µ = 0.5,..., µ 10 = 4) lited in the firt olumn of the table with hange point τ = 1. The following table illutrate the numerial reult of ARL of the two-ided UUM, EWMA, UUM multi-hart, EWMA multi-hart, GLR and the optimal UUM multi-hart. In order to ompare the average of ARL of the UUM and EWMA hart with thoe of the UUM and EWMA multi-hart, we at firt lit the imulation reult of the UUM hart with the parameter, {δ 1 = 0.1, δ = 0.5, δ 3 = 1, δ 4 = 1.5, δ 5 = } and EWMA hart with {r 1 = 0.1, r = 0.3, r 3 = 0.5, r 4 = 0.7, r 5 = 0.9} in Table 1 and. In the firt two row, denote variou value of the width of the ontrol limit, and δ i the parameter of the UUM hart. The ize of the mean hift (µ) are lited in the firt olumn of the table. The value in parenthee are the tandard deviation of the ARL.

11 UUM AND EWMA MULTI-HART FR DETETING A RANGE F MEAN HIFT 1149 Table. ARL and their tandard error (in parenthee) of the EWMA ontrol hart with ARL 0 = 500. HIFT r 1 =0.1 r =0.3 r 3 =0.5 r 4 =0.7 r 5 =0.9 (µ) 1 =.818 = = = = (497) 500(495) 500(49) 500(504) 500(50) (316) 403(398) 438(431) 455(458) 470(473) (95.8) 187(181) 56(55) 308(311) 354(355) (.) 55.4(51.6) 88.7(87.4) 18(18) 176(178) (8.85).5(18.9) 36.0(33.9) 55.5(54.6) 84.6(85.3) (4.78) 11.9(8.61) 17.4(15.3) 6.9(5.5) 4.7(4.1) (3.05) 7.65(4.73) 10.0(8.01) 14.7(13.4) 3.5(.9) (.15) 5.55(3.00) 6.53(4.64) 8.90(7.7) 13.7(13.1) 4.36(1.5) 3.55(1.48) 3.64(.04) 4.30(3.1) 5.80(5.01) 3.87(0.67).16(0.66) 1.9(0.78) 1.86(0.95) 1.98(1.8) 4.19(0.4) 1.61(0.5) 1.33(0.49) 1.3(0.45) 1.1(0.48) PI k PI u Table 3. omparion of the average of ARL of the UUM and EWMA hart with the ARL of the multi-hart and GLR ontrol hart with ARL 0 = 500 (with their tandard error hown in parenthee). HIFT Ave.UUM UUM pt.uum Ave.EWMA EWMA GLR(T G ) (µ) Multi-hart Multi-hart Multi-hart = (479) 500(460) 500(477) 500(498) 500(499) 500(49) (334) 6(01) 7(9) 417(415) 381(374) 34(88) (141) 97.0(60.5) 96.3(60.1) 4(40) 146(135) 114(83.1) (39.3) 35.(0.9) 35.8(0.4) 95.8(93.5) 40.1(31.0) 37.4(3.8) (14.3) 18.(9.73) 18.8(10.0) 4.9(40.3) 18.(11.3) 18.6(10.8) (6.9) 11.6(5.98) 11.86(6.16) 1.8(19.) 11.(6.08) 11.4(6.4) (4.03) 8.08(3.98) 8.(4.11) 1.7(10.4) 7.81(3.95) 7.83(4.11) (.67) 6.03(.8) 6.11(.95) 8.15(6.13) 5.85(.91) 5.77(.9) 5.48(1.47) 3.83(1.61) 3.80(1.75) 4.33(.58) 3.68(1.77) 3.58(1.66) (0.73).0(0.73).01(0.84).16(0.87) 1.9(0.89) 1.94(0.81) 4.69(0.51) 1.58(0.53) 1.34(0.51) 1.5(0.47) 1.8(0.49) 1.31(0.49) PI u PI k In Table 3, we ompare the imulation reult of the ARL µ for the GLR, UUM and EWMA multi-hart, and the average of the ARL µ for five ontituent UUM hart orreponding to the ae {δ 1 = 0.1, δ = 0.5, δ 3 = 1, δ 4 = 1.5, δ 5 = }. The Ave. UUM in the eond olumn how the average of ARL for the ontituent UUM hart from Table 1. In the third olumn,

12 1150 DNG HAN, FUGEE TUNG, XIJIAN HU AND KAIB WANG to obtain the ARL 0 (T ) = 500 for the UUM multi-hart T, we take the ontrol limit 1 =.71, = 5., 3 = 6.09, 4 = 6.8 and 5 = uh that ARL 0 (T(δ 1, 1 )) = 1,97.4, ARL 0(T(δ, )) = 1,98.5, ARL 0(T(δ 3, 3 )) = 1,98.6, ARL 0 (T(δ 4, 4 )) = 1,97. and ARL 0(T(δ 5, 5 )) = 1,98.1.The imulation reult for the optimal UUM multi-hart are lited in the fourth olumn with the referene value δk hoen aording to (4.6) and (4.7), that i, δ1 = 0.166, 1 = 3.64; δ = 0.458, = 5.4; δ 3 = 0.997, 3 = 6.177; δ 4 = 1.86, 4 = 6.458; δ 5 = 3.16, 5 = 6.0, where the ontrol limit k are taken for ARL 0 (T ) = 500. More diuion on the optimal UUM multi-hart i in etion 5. Alo, in the fifth and the ixth olumn, we have Ave. EWMA, whih give the average of ARL for the ontituent EWMA hart from Table, and the EWMA multi-hart TE. Moreover, we lit the imulation reult of the GLR (T GL ) in the lat olumn with the ontrol limit = 3.494, whih lead to the ame ARL 0 value. The bottom two row of eah table lit the PI k and PI u value for different hart, where we take f(x) = e x in the PI. Thee repreent the PI value under known and unknown hift, repetively. PI k i alulated baed on the five antiipated hift ize of 0.1, 0.5, 1, 1.5 and, auming that the atual mean hift are onitent with the antiipated mean hift. Here the referene optimal ARL urve, ARL r (µ), i taken a the ARL of the UUM hart, that i, ARL r (0) = 500,ARL r (0.1) = 39,ARL r (0.5) = 8.95,ARL r (0.5) = 31.0, ARL r (0.75) = 16.54,ARL r (1) = 10.53,ARL r (1.5) = 7.386,ARL r (1.5) = 5.496, ARL r () = 3.43, ARL r (3) = 1.793, ARL r (4) = PI u hould be alulated baed on all poible mean hift in a range. Here, the implified alulation i baed on all the mean hift lited in the firt olumn to repreent the performane for a range of unknown mean hift. ur finding baed on the omparion of the numerial reult are ummarized a follow. The reult in Table 1 how that eah of the five ontituent UUM hart i good for it deigned optimal hift a expeted, while the PI k and PI u value that repreent the overall performane over a range vary aording to the deigned parameter. In both the ituation with known and unknown mean hift in an antiipated range, the optimal UUM multihart i onitently better than any of the ingle ontituent UUM hart in term of the PI k and PI u value, a hown in Table 3. Table 3 alo indiate that the performane of UUM multi-hart i onitently better than the average performane of the ontituent UUM hart in the ene that the average ARL of the ontituent hart i alway larger than the ARL of the

13 UUM AND EWMA MULTI-HART FR DETETING A RANGE F MEAN HIFT 1151 multi-hart. ompared with GLR hart, the UUM multi-hart ha a higher PI value when the mean hift are known, and i only lightly better when the hift are unknown. Moreover, Table and 3 how that the EWMA multi-hart i not a good a the UUM multi-hart and the GLR hart, and one EWMA hart with r=0.1 eem to perform partiularly well for a range of known hift. However, the performane of the EWMA multi-hart i onitently better than the average performane of the ontituent EWMA hart a the average ARL of the EWMA hart are alway larger than the ARL of the EWMA multi-hart. Finally, an intereting reult in Table 3 i that the tandard deviation of the ARL for the UUM multi-hart are the mallet among the ix hart, exept when deteting the mean hift µ = Deign of a Multi-hart Although we have proved the optimal property of the UUM multi-hart, the uperior performane of the multi-hart till require an effetive deign. For the ituation with known antiipated mean hift, we an deign the multi-hart by ombining thoe ontituent hart peifially deigned for eah antiipated mean hift ize. With an unknown mean hift in a range, we an alo determine how many ontituent hart to ombine and where to loate them aording to a deirable PI. Thi etion will examine thi problem via theoretial alulation and Monte arlo imulation, and provide a general guideline for the deign of a UUM multi-hart and an EWMA multi-hart Deign of a UUM Multi-hart We firt fou on the deign of the UUM multi-hart. Here, four deign heme are propoed. Denote the antiipated range a [a,b], and uppoe n ontituent UUM hart are to be ued. Let p i be the proportion of the poition of eah ontituent hart within the range and δ i be the plaement loation. The deign heme are deribed a the following. 1. An optimal plaement heme. Let a = 0.1,b = 4. By uing (4.7) we an obtain the theoretial optimal referene value δk for n =,3,4 and 5, uh that PI u attain it maximum value, PIu. Thu for n=, δ 1 =0.1948, δ =1.607, PI u =0.715; n=3, δ 1 =0.184, δ =0.85, δ 3 =.474, PI u =0.897; n=4, δ 1 =0.17, δ =0.585, δ 3 =1.433, δ 4 =.886, PI u =0.9438; n=5, δ 1 =0.166, δ =0.458, δ 3 =0.997, δ 4 =1.86, δ 5 =3.16, PI u =

14 115 DNG HAN, FUGEE TUNG, XIJIAN HU AND KAIB WANG The theoretial value PI u = of the optimal UUM multi-hart for n = 5 i high. The imulation reult in Table 3 onfirm that the optimal UUM multi-hart with the five UUM hart ha the bet performane among the ix ontrol hart in term of PI k, even though the ARL 0 i not large.. Even plaement heme. Take p i = i n + 1, δ i = a + (b a) p i. (5.1) A hown in Figure (a), the ontituent hart are evenly ditributed in the antiipated range. 3. ide-onentrated plaement heme. Take p i = τi 1 τ (n+1) 1, δ i = a + (b a) p i. (5.) A hown in Figure (b) and (), the emphai of the multi-hart i on the extreme. If τ > 1, the hart onentrate on the lower end, while if τ < 1, the hart will onentrate on the higher end. 4. enter-onentrated plaement heme. Take τ i 1 if n i even and i n p i = [τ n 1]+τ n (τ 1) τ n i+1 1 [τ n 1]+τ n (τ 1) τ i 1 [τ n+1 1] τ n i+1 1 [τ n+1 1] δ i = a + (b a) p i. if n i even and i > n if n i odd and i n if n i odd and i > n (5.3) Figure (d) and (e) how a heme that emphaize the enter or both end. If τ > 1, the hart will onentrate on the end; if τ < 1, the hart onentrate on the enter of the antiipated range. () () () () Figure. Plaement of four UUM hart. () Even plaement. () ide-onentrated, τ = 0.5. () ide-onentrated, τ =.0. () enteronentrated, τ = 0.5. () enter-onentrated, τ =.0. ()

15 UUM AND EWMA MULTI-HART FR DETETING A RANGE F MEAN HIFT 1153 Here the imulation i onduted baed on the three different deign heme. The antiipated hift range i eleted a (0,3), and the PI i ued a a riterion for performane evaluation. The reult are hown in Figure 3. PI Num of hart Figure 3. PI urve of multi-uum. () ide-onentrated, τ = 0.5. () ide-onentrated, τ =.0. () enter-onentrated, τ = 0.5. () enteronentrated, τ =.0. () Even plaement. Note: An unfinihed urve exit beaue it plaement i too loe to zero. ne an ee that when the number of ontituent UUM hart i no more than three, the enter-onentrated heme with τ =.0 give the highet PI value. When the number of hart i larger than three, the even plaement heme how the bet performane. Another notable phenomenon i that the PI doe not alway inreae with the number of hart. For.0 and.0 plaement heme, the PI atually dereae when uing more than four hart. For the even plaement heme, the PI value i tabilized with minor flutuation when more than five hart are ued. Alo, the ARL urve of 0.5 i alway far below the other, whih turn out to be the wort heme. Thu, for the ituation with an antiipated hift range (0, 3), we onlude that plaing the ontituent hart evenly along the antiipated range hould give a reaonably good reult with more than three hart. If no more than three ontituent hart are ued, putting ome on the lower ide and ome on the higher ide hould generate good performane. Alo, the PI urve provide an indiation for the hart number determination. In thi ae, we ugget no more than five ontituent hart, a the PI urve beome flat after that. With different antiipated mean hift range, we ondut more extenive imulation by hooing everal uual antiipated mean hift range for the deign,

16 1154 DNG HAN, FUGEE TUNG, XIJIAN HU AND KAIB WANG with the atual mean hift either within or outide the deign range, a in Figure 4. For example, if the deign range i [0., 1] and expeted hift fall into the ame range, the even plaement heme with three UUM hart hould be ued ine it indiate the highet PI value and the urve goe flat or downward after that. (a) (b) PI PI Num of hart Num of hart () (d) PI PI Num of hart Num of hart (e) (f) PI PI Num of hart Num of hart

17 UUM AND EWMA MULTI-HART FR DETETING A RANGE F MEAN HIFT 1155 (g) (h) PI PI Num of hart (i) Num of hart (j) PI PI Num of hart Num of hart Figure 4. imulation reult of multi-uum. (a) Deign: 0.-1, hift: (b) Deign: 0.-1, hift: 0-3. () Deign: 0.-, hift: 0.-. (d) Deign: 0.-, hift: 0-3. (e) Deign: 0.-3, hift: (f) Deign: 0.-3, hift: 0-3. (g) Deign: 1-, hift: 1- (h) Deign: 1-, hift: 1-3. (i) Deign: 1-3, hift: 1-3. (j) Deign: 1-3, hift: 1-3. Legend: () ide-onentrated, τ = 0.5; () ide-onentrated, τ =.0; () enter-onentrated, τ = 0.5; () enter-onentrated, τ =.0; () even plaement. 5.. Deign of an EWMA Multi-hart Here, we invetigate the deign of EWMA multi-hart. To deign an EWMA multi-hart that ombine everal ontituent EWMA hart, we need to look into the moothing oeffiient of EWMA, r. We all the moothing oeffiient the loation of an EWMA hart. The ame plaement heme an then be applied a in the UUM multi-hart, exept that now the range of plaement i (0,1) ine r annot exeed 1. Figure 5 how the PI of different deign heme for EWMA multihart. ne an ee that the.0 heme how the bet ARL performane when the number of hart i le than even, the.0 heme athe up when

18 1156 DNG HAN, FUGEE TUNG, XIJIAN HU AND KAIB WANG the number of hart i more than even, while the 0.5 heme give the wort performane. PI Num of hart Figure 5. PI urve of multi-ewma. Legend: () ide-onentrated, τ = 0.5; () ide-onentrated, τ =.0; () enter-onentrated, τ = 0.5; () enter-onentrated, τ =.0; () even plaement. Note: An unfinihed urve exit beaue it plaement i too loe to zero. We onlude that in thi ae we hould hooe the moothing parameter r loe to 0 if fewer than even hart are to be ued. If more than even hart are ued, ome r hould be loe to 0, and ome loe to 1. The imulation reult alo how that the PI urve flatten and goe downward after three hart, whih indiate that no more than three hart i ueful Diuion on Deign Guideline From the reult of Monte arlo experiment for the UUM multi-hart and the EWMA multi-hart, we an ee that the performane doe not alway inreae by adding more hart if the referene value of the added UUM hart are le than the exitent referene value. We may reommend the number of ontituent hart by finding the initial flat or downward point on the PI urve. The alloation of the ontituent hart then follow the orreponding plaement heme that generate the bet PI value. The ARL alulation of a multi-hart an alo be done by numerial method. The ingle UUM ARL numerial method by Brook and Evan (197) and the ingle EWMA ARL numerial method by Lua and aui (1990) an be eaily extended to a multi-hart enario. For eah ingle hart, we diretize the range between ontrol limit. A Markov hain an be formed by taking the

19 UUM AND EWMA MULTI-HART FR DETETING A RANGE F MEAN HIFT 1157 tate a a multidimenional vetor (E 1,...,E n ), where E i i the tate of the ith hart, n i the total number of hart in ue. If eah hart i diretized into t interval, the tranition matrix of the Markov hain i a (t+1) n (t+1) n matrix. Brook and Evan (197) reommened t = 5 for a reaonably good reult. For a multi-hart, the dimenion of thi matrix grow exponentially. If n = 4 hart with t = 5, the dimenion of the tranition matrix will be 1, 96 1, 96, whih will be very diffiult to manipulate. Thu, in thi paper, all reult are obtained by Monte arlo experiment only. 6. onluion We have mainly diued the UUM and EWMA multi-hart heme to handle the ituation with an antiipated range of known or unknown proe hange by ombining the trength of multiple hart. We how that the multihart ha the merit of quik detetion of a range of mean hift, eay and flexible deign for variou ituation, and great redution in omputational omplexity. In partiular, we have proved the aymptoti optimality of the UUM multihart in deteting more than one poible mean hift in a range. Alo, the numerial imulation reult how that the UUM multi-hart i more effiient and robut on the whole than the UUM, EWMA and EWMA multi-hart in term of PI, and an perform a well a the GLR hart in deteting variou mean hift when the in-ontrol ARL i not large. The harting performane of a multi-hart depend on the deign of the multi-hart parameter inluding the number of ontituent hart and the alloation of their referene value. We have provided an optimal deign of the UUM multi-hart and ome pratial guideline for both UUM and EWMA multi-hart baed on the PI urve with different plaement heme. Note that the multi-hart ha great flexibility in taking variou form of it ontituent hart to further improve it performane. The deign and analyi of the multihart with mixed form of hart warrant future reearh. A an be een that the reult onidered in the paper are from the initial tate, µ 0 = 0. It would be intereting to invetigate whether the reult are imilar if the hift are generated from a teady tate, e.g., hift are generated after the UUM i allowed to run through everal in-ontrol value. The intuitive idea i that the reult hould be imilar if all the mean hift µ and the referene value δ k are greater than µ 0 = max{everal in ontrol value} when the ARL 0 i large. However, it eem diffiult to prove the intuitive idea ine it i not eay to hooe a proper ARL 0 for the everal in-ontrol value. imilarly, it hould be anew onidered whether the optimality propertie of Moutakeide (1986) in Lorden ene (1971) till hold when the UUM hart i allowed to run through everal in-ontrol value.

20 1158 DNG HAN, FUGEE TUNG, XIJIAN HU AND KAIB WANG Moreover, in reent year, adaptive UUM (park (000)) and adaptive EWMA (apizzi and Maarotto (003)) have been propoed in the literature to ahieve the ame aim a in the paper. It i worthwhile to ompare thee hart on whih i more effiient in deteting a group of unknown mean hift. Aknowledgement We thank the Editor and two referee for their valuable omment and uggetion that have improved thi work. Thi work wa upported by RG ompetitive Earmarked Reearh Grant HKUT63/04E and HKUT604/05E. Appendix Proof of Theorem 1. ine T T i for all 1 i n, it follow that log L +M(µ i )+o(1) = E µi (T i) E µi (T) E µi (T i )= log L +M(µ i )+o(1) µ i for all 1 i n a L, where o(1) = ( L 1 ln(µ L/) ). Thu, Theorem 1 i true if log L log L ((ln L) 3 /L) = ( 3 1 e 1 ) a L, or a min 1 i n { i } +. It i known that (ee rivatava and Wu (1997)) L = E 0 (T i ) = e( i +δ i ρ) 1 ( i + δ iρ) δ i + (δ i ) (A.1) µ i for large i, 1 i n, where ρ Denote by ϕ and Φ the tandard normal denity and ditribution funtion, repetively. Let U m (k) = [X m + + X m k+1 ]/k 1/,1 k m. Then the topping time T i { [ T i = min n : max U n (k) > 1 k n From (A.1) and (A.), it follow that i + δ ] } i k. δ i k an be written (A.) 1 δ 1 > δ > > m δ m, (A.3) [ i j = (1 + o(1))( i j ) = (1 + o(1)) ln( δ ] i ) + (δ i δ j )ρ, (A.4) δ j for large min 1 k n { k }. We firt how that 0 E 0 (T 1) E 0 (T ) A( 1) 3 + B (A.5)

21 UUM AND EWMA MULTI-HART FR DETETING A RANGE F MEAN HIFT 1159 for large 1, where A and B are two ontant not depending on i,1 i n. The left inequality of (A.5) i obviou ine T T 1. Note that + E 0 (T 1 ) = P 0 (T 1 n) = E 0 (T ) = = n=1 + n=1 + n=1 + n=1 P 0 (U n (k) < 1 /(δ 1 k) + (δ1 /) ) k,1 k l,1 l n, P 0 (T n) P 0 (U n (k) < min 1 i m { i/(δ i k) + (δi /) ) k},1 k l,1 l n. ine i /(δ i k) + (δi /) k attain it minimum value, i, at k = i /(δ i), it follow from (A.3) that P 0 (T 1 n) P 0(T n) n [1 Φ( 1 )] for 1 n 1 /(δ 1), and P 0 (T 1 n) P 0(T n) ( 1 ) /(δ 1 ) 4 [1 Φ( 1 )] for 1 /(δ 1) < n n, where n = 1 exp{ 1 }. Note that 1 Φ( 1 ) = (( 1 exp{ 1 }) 1 ) for large 1. Thu, n P 0 (T 1 n) P 0 (T n) A( 1) 3 n=1 (A.6) for large 1. n the other hand (ee iegmund (1985, p.5)), the topping time T 1 = N 1+ +N K = K(( K i=1 N i)/k), where {N i } i dependent and identially ditributed with mean E(N 1 ) = b ( 1 ) and K i geometrially ditributed with mean E(K) = (b 1 exp{ 1 }) for large 1. Hene, we have + n=n +1 ( + P 0 (T 1 n) P 0 (K n ) ) n=n +1 1 ( = exp{ 1}(1 b 1 n ) n ) b B (A.7) for large 1. By uing (A.6) and (A.7), we ee that (A.5) hold. ine E 0(T 1 ) = L = E 0 (T ), it follow from (A.5) that E 0(T 1 )) E 0(T ) = e ( 1 1) 1 E 0 (T 1 ) (δ 1 ) [A( 1) 3 + B]e ( 1+δ 1 ρ) 0 a L, otherwie we have a ontradittion. Thi mean that a L +. Note that e ( 1 1) 1 = o( 1 1) and log L log L =

22 1160 DNG HAN, FUGEE TUNG, XIJIAN HU AND KAIB WANG ( 1 e 1 ) + (1 e 1 ). Thu log L log L (δ 1 ) [A( ) 3 + B]e ( 1+δ 1 ρ) ( = ( 3 1 e 1 (ln L) 3) ) =, L and thi omplete the proof. Proof of Theorem. Let j k and µ I k. Note that µ > δ k / ine δ k 1 + δ k )/ < µ (δ k + δ k+1 )/ and δ k 1 0. Thu, the number µ mut atifie one of the following: (i) δ k / < µ δ j /; (ii) δ k / < δ j / < µ; (iii) µ > δ k / > δ j /. It follow from (4.1) and (A.1) that j = k + jk + o(1) for large L, where jk = log[δj /δ k ]. By the trong Law of Large Number we have 1 max 1 i n n n l=n i+1 for 1 k m a n. Note that δ k [X l (ω) δ k ] max{0,δ k(µ δ k )}, a.. P µ (A.8) T(δ l ) = T(δ l,ω) a.. P µ (A.9) for all 1 k m a L. Thu, without lo of generality, we aume that (A.8) and (A.9) hold for all ω Ω, where P µ (Ω) = 1. Aume that there i a ω Ω uh that T k = T(δ k,ω) T j = T(δ j,ω). Thi mean that 1 max 1 i T j T j T j l=t j i+1 jk + o(1) T j + T k 1 T j δ j [X l δ j ] > j T j = k + jk + o(1) T j max 1 i T k 1 1 T k 1 T k 1 l=t k 1 i+1 δ k [X l δ k ], (A.10) ine max Tk 1 1 i Tk 1 l=t k 1 i+1 δ Tj k[x l δ k /] k and max 1 i Tj l=t j i+1 δ j [X l δ j /] > j Thu, it follow from (A.8), (A.9) and (A.10) that max{0,δ j (µ δ j )} max{0,δ k(µ δ k )} (A.11) a L. Thi ontradit, δ j / µ > δ k /, ae (i). Thi mean that the aumption T k = T(δ k,ω) T j = T(δ j,ω) i not true. imilarly, for the ae (ii) and (iii), it follow from (A.11) that δ j (µ δ j /) δ k (µ δ k /), that i, µ (δ j + δ k )/, ae (ii), and µ (δ j + δ k )/, ae (iii). Note that µ I k, It follow that µ > δ k + δ k 1 = δ k + δ j + δ k 1 δ j δ k + δ j

23 UUM AND EWMA MULTI-HART FR DETETING A RANGE F MEAN HIFT 1161 for ae (iii), ine δ k 1 δ j. Thi ontradit µ (δ j + δ k )/. imilarly, by µ I k we have µ (δ k + δ j )/ + (δ k+1 δ j )/ (δ k + δ j )/ for ae (ii) ine δ k+1 δ j. But µ (δ k + δ j )/ for ae (ii), o δ k+1 δ j and µ = (δ k + δ k+1 )/. In thi ae, δ k+1 (µ δ k+1 /) = δ k (µ δ k /). Thu, we have T k+1 k+1 /[δ k+1 (µ δ k+1 /)] and T k k /[δ k (µ δ k /)] a L. In fat, max 1 l Tk (T k ) 1 T k i=t k l+1 δ k[x i δ k /] > k /T k and max 1 l Tk 1(T k 1) 1 Tk 1 i=t k l δ k[x i δ k /] k /T k. Thi implie that T k k /[δ k (µ δ k /)] for all ω Ω a L. imilar reult an be obtained for T k+1. o T j = T k+1 > T k a L ine k+1 > k. Thi ontradit the aumption T j = T k+1 T k a L. Thu we have T(δ j ) > T(δ k ), a.. P µ, for j k and µ I k a L. imilarly, T (δ j ) > T (δ k ), a.. P µ, for j k and µ I k a L. Hene T = T k, a.. P µ, for µ I k a L. ine the family {T k / k, k > 0} i uniformly integrable with repet to P µ, o i {T / k, k > 0}. Hene, a L, E µ (T ) E µ(t k ) k /[δ k(µ δ k /)] for µ I k. Thu, (4.5) of Theorem i etablihed. ine k k 0, APL r (µ) k /µ and APL µ (T ) = E µ(t ) E µ(t k ) k /[δ k(µ δ k /)] a L for µ I k,1 k m, { 1 PI u (δ 1,...,δ k ) = exp b a a L, where F(δ 1,...,δ k ) = 1 It follow that and b a = (1 + o(1))exp [ δ 1 +δ a + b PI u (δ 1,...,δ k,...,δ m ) δ k F δ 1 δ 1 +δ a (µ δ 1 )µ APL µ (T ) ARL } r(µ) dµ ARL r (µ) { 1 } b a F(δ 1,...,δ k ) + 1 µ δ 1 (µ δ 1 ) dµ + δ k 1 +δ k δ 1 (µ δ 1 ) dµ, k 1 i= µ δ k (µ δ k ) dµ ]. δ i +δ i+1 δ i 1 +δ i µ δ i (µ δ i ) dµ { 1 } = exp b a F + 1 F(δ1,...,δ k,...,δ m ) δ k F δ k δ k +δ k+1 δ k 1 +δ k (µ δ k )µ δ k (µ δ k ) dµ. for k m. It an be heked that F/ δ k < 0 a δ k approximate δ k 1 and F/ δ k > 0 a δ k approximate δ k+1 ; imilarly, it i true for F/ δ 1, and F (δ k+1 δ k 1 )[δk 3 + δ k 1δ k+1 (δ k 1 + δ k + δ k+1 )] δ k 8δ k 1 δ k+1 δk 3 > 0

24 116 DNG HAN, FUGEE TUNG, XIJIAN HU AND KAIB WANG for k m. Hene, there exit a unique erie of number, δk,1 k m, uh that a < δ1 < a, δ k < δ k+1 < b for 1 k m 1, and F(δ 1,...,δ k,...,δ m ) attain it minimum value at δ1,...,δ k,...,δ m, that i, PI u attain it maximum value at δ1,...,δ k,...,δ m. Thi omplete the proof of Theorem. Proof of Theorem 3. Let T(δ,) denote the UUM hart with referene value δ and ontrol limit. It i known that (ee rivatava and Wu (1997)), a L, E µ (T(δ,)) = (1 + o(1)) e(δ µ)(+δρ)δ 1 1 (δ µ)( + δρ)δ 1 ) (µ δ ) for δ > µ, E µ (T(δ,)) = (1 + o(1))( /δ ) a δ µ, and E µ (T(δ,)) = (1 + o(1))[/δ(µ δ)] for δ < µ. Hene, by log L log L 0, or 0 a L, we have E µ (T i)) = (1 + o(1))e µ (T i )) (A.1) for 1 i m. (i). If δ i > µ,1 i m, then (δ i µ)/δ i > (δ j µ)/δ j for i > j, and therefore, by (A.1), E µ (T i )) = (1 + o(1))[ e(δ i µ)( i +δ i ρ)/δ i 1 (δ i µ)( i + δ i ρ)/δ) (µ δ ] i ) ( > E µ (T 1)) p1 ) + o E µ (T 1 )) 1 p 1 a L for i. Hene, m k=1 p ke µ (T k )) > E µ (T 1 )) E µ(t ) a L ine (1 + o(1))e µ (T 1 )) = E µ (T 1 ). (ii). If δ m > µ,δ 1 µ or δ m = µ,δ 1 < µ, then E µ (T m ))/E µ (T 1 )) + a L +. (iii). If δ m < µ, then k E µ (T k )) = (1 + o(1)) δ k (µ δ k ) i > (1 + o(1)) δ i (µ δ i ) = E µ(t i )) for k i, where the parameter δ i atifie δ i (µ δ i ) = max 1 k m δ k (µ δ k ). Thu, m k=1 p ke µ (T k )) > E µ (T i )) E µ(t ) a L. By (i), (ii) and (iii), (4.10) of Theorem 3 follow. Let T E denote the optimal EWMA hart with the referene value r = a δ1 /b (0 < r 1), where a , b > 0 i the ontrol limit uh that E 0 (T E ) = L. It ha been hown by Wu (1994) and rivatava and Wu (1997) that E µ (T E ) E µ (T E ) a L, and E 0 (T E ) e0.834δ 1 e b / 0.408δ1 b, (A.13)

25 UUM AND EWMA MULTI-HART FR DETETING A RANGE F MEAN HIFT 1163 E µ (T E ) = 1 δ 1 δ 1 [ ln(1 a δ 1 µ ) a (b ε(b)) a δ 1 (1 (1 µ ) ) 1 4µ + o( )] (A.14) (1 a δ 1 µ ) (b ε(b)) for µ > a δ 1, where 0 < ε(b) < D/b and D i a ontant. We an further how that E µ (T E ) πb(1 µ a δ 1 )exp{ b (1 µ a δ 1 ) } for µ < a δ 1 and E µ (T E ) πb ln b for µ = a δ 1 a L. Thu, to prove (4.11) of Theorem 3, we need only prove E µ (T E ) > E µ (T ). ine E 0 (T E ) = L = E 0 (T k ), 1 k m, it follow from (4.5) and (4.13) that b = 1 + o(1). By µ > δ 1 /, we may aume µ I k, where k 1. We have proved in the proof of Theorem that E µ (T ) E µ (T k ) k δ k (µ δ k ) (A.15) a L. Note that δ k (µ δ k /) δ j (µ δ j /) for j k, and i / j 1 a L for all 1 i,j m. Thu we have E µ (T j ) E µ (T k ) for j k and E µ (T k ) E µ (T k ). n the other hand, we have E µ(t E ) > E µ (T 1 ) for µ > a δ 1 ine b = 1 + o(1) and (δ1 ) 1 ln(1 a δ 1 /µ)/a > (δ 1 (µ δ 1 )) 1 a L. Thu, it follow from (A.14) and (A.15) that E µ (T E ) > E µ (T ) a L, proving (4.11) of Theorem 3. Referene Brook, D. and Evan, D. A. (197). An approah to the probability ditribution of uum run length. Biometrika 59, apizzi, G. and Maarotto, G. (003). An adaptive exponentially weighted moving average ontrol hart. Tehnometri 45, Dragalin, V. (1993). The optimality of generalized UUM proedure in quiket detetion problem. Proeeding of teklov Math. Int.: tatiti. and ontrol of tohati Proee 0, Dragalin, V. (1997). The deign and analyi of -UUM proedure. omm. tatit. imulation omput. 6, Han, D. and Tung, F. G. (004). A generalized EWMA ontrol hart and it omparion with the optimal EWMA, UUM and GLR heme. Ann. tatit. 3, Hawkin, D. M. and lwell, D. H. (1998). umulative um hart and harting for Quality Improvement. pringer-verlag, New York.

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