Immigrated urn models - asymptotic properties and applications

Submitted to the Annals of Statistics IMMIGRA TED URN MODELS – THEORETIC AL PR OPER TIES AND APPLICA TIONS By LI-XIN ZHANG ∗ , FEIF ANG HU † , SIU HUNG CHEUNG ‡ and W AI SUM CHAN Zhejiang University, University of Vir ginia and The Chinese U niversity of Hong Kong Urn mo dels hav e b een widely stu died and applied in b oth sci- entific and so cial science d isciplines. In clinical stu dies, the adoption of u rn mo dels in t reatmen t allocation sc hemes has b een pro ved to b e b eneficial to b oth researchers, by providing more efficient clini- cal trials, and patients, b y increasi ng the likelihood of receiving th e b etter treatment. In th is pap er, w e prop ose a new and general class of immigrated urn (IMU) models that incorp orates the immigration mec han ism into the urn pro cess. Theoretical prop erties are developed and the advan tages of th e IMU models are d iscussed. In general, the IMU mo dels hav e smaller v ariabilities than the classical urn mo dels, yielding more pow erful statistical inferences in applications. Illustra- tive ex amples are p resen ted to d emonstrate the wide app licabilit y of the I MU models. The p roposed IMU framework, includin g many p opular classical urn mo dels, not only offers a u nify p ersp ective for us to comprehend th e u rn p rocess, but also enables us to generate sever al nov el urn mo dels with d esirable properties. 1. In tro duction. 1.1. Urn Mo dels and their applic at ions. Urn mo dels ha ve long b een consid- ered p o w erfu l mathematical instruments in man y areas, in cluding the phys- ical sciences, biologica l sciences, so cial sciences, and engineering (Johnson and Kotz, 1977; Kotz and Balakrishnan, 1997). F or example, in medical sci- ence, Knoblauch, Neitz, and Neitz (2006) apply an urn mo del to stu d y cone ratios in human and macaque r etinas. In p opulation genetics, Hopp e (198 4) ∗ Researc h supp orted by grants from the National Natural Science F oundation of China (No. 11071214)), Natural Science F oundation of Zhejiang Province ( No. R6100119) and F undamental Research F unds for the Cen tral Universit y (No. 2010QNA3032). † Researc h supp orted by grants D MS -0349048 and DMS- 0907297 from the National Science F oundation (USA). ‡ Researc h sup p orted by a gran t from the R esearc h Grants Council of th e Hong Kong Sp ecial Administrative R egion (Pro ject no. CUHK400608). AMS 2000 subje ct classific ations: Primary 60F15, 62G10; secondary 60F05 , 60F10 Keywor ds and phr ases: adap t ive designs, asymptotic normalit y , clinical t rial, urn mod el, branching pro cess with immigratio n, birth and death urn, drop-the-loser ru le 1 imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: October 31, 2018 2 L. ZHANG ET AL. and Donnely and Kurtz (199 6) emplo y a P´ oly a-lik e urn m o del to s tu dy Ew en’s sampling distr ibution in n eu tral genetics m o dels. Bena ¨ ım, Sc hreib er, and T arr` es (2004) also mak e u s e of a class of generalized P´ oly a ur n mo dels to scrutinize evo lu tionary pro cesses. In economics, Beggs (2005) uses the mo dels to capture the m ec hanism of reinf orcemen t learning. In addition, n um erous examples of applications of u rn m o dels in the areas of physics, comm unication th eory , and computer s cience are p r o vided by Milenk o vic and Comp ton (2004). In statistics, an imp ortan t app lication of u rn mo dels is to r andomize treat- men ts to patien ts in a clinical trial (Hu and Rosen b erger, 2006). Consider an urn con taining balls of K t yp es, representing K treatment s. Patie nts nor- mally arriv e sequentiall y , and treatmen t assignment b ased on urn mo dels is usually an adaptiv e s c heme that dep ends on the urn comp osition and pre- vious treatmen t outcomes. Th e urn comp osition is also con tin uously r evised according to treatmen t outcomes. Early stud ies of urn mod els in statistic s include the generalized P´ oly a urn mo dels (GPU) of Athrey a and K arlin (1968 ), W ei and Durham (1978), and W ei (1979) . Another reno wn ed v ariatio n of th e P´ oly a ur n is the randomized P´ oly a urn (RPU) prop osed b y Du r ham, Flourn o y , and Li (1998). These clas- sic urn mo d els hav e a num b er of dra wb ac ks. (i) Th ey are u sually p rop osed for binary (multinomial) resp onses. (ii) The urn pro cess has a pr ed etermined limit of urn pr op ortions that do es n ot ha ve any connection with formal op- timal p rop erties (Hu and Rosenb erger, 2006). (iii) The ur n pro cess usually has higher v ariabilit y than other types of pro cedur es (Hu and Rosen b er ger, 2003) and is thus less p ow erful in statistical inferences. (iv) The f orm ulation of the asymptotic v ariabilit y is usu ally quite complex and it is intricate to deriv e a reasonable estimate. F or in stance, the asymptotic v ariabilities of the P´ oly a-urn-t yp e mo dels are relate d to the v ariance of a complicated Gaussian pro cess. In particular, for the multi-treat ment case, to derive the v ariabilit y requires extremely complicated calculations of matrices (c.f., Smythe (1996), Janson (2004 ), Bai and Hu (200 5), Zhang, Hu and Cheung (2006), Higueras et al. (2006)). (v) T he mo d els are designed mainly for the comparison of t w o tr eatments, so there is a shortage of metho d ology to handle cases with m ultiple treatmen ts. By em b edd ing the urn pro cess in a con tinuous-time birth and death pro cess (Iv anov a et al., 2000; Iv anov a and Flourn o y , 2001; Iv ano v a, 2006), Iv ano v a (2003) form ulates the drop-the-loser (DL) ru le for a clinical trial with t wo treatmen ts. T he DL rule utilizes th e idea of immigration and h as b een sho wn to yield a smaller v ariabilit y among v arious urn mo d els (Hu and Rosen b erger, 2003). Th e DL rule is generalized by Zhang et al. (2007 ) to imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 3 pro vide more flexible urn mo d els. Ho w eve r , these recen t prop osals are frag- men ted, offering only a partial solution to the aforementio ned drawbac ks of the classic u rn mo d els. T o supply a complete r esolution, w e seek to pro vid e a comprehensiv e paradigm through w hic h one w ill b e able to connect existing urn mo dels, dev elop useful theoretic results, and compare merits of differen t classes of urn mo dels. 1.2. Obj e ctive s and or ganization of the p ap er. In this pap er, w e prop ose the IMU f ramew ork that encompasses a wide sp ectrum of urn m o d els and incorp orates the immigration p ro cess, offering a greater fl exibilit y in the c hoice of appropr iate ur n mo dels in app lications. This framew ork includes man y urn mo d els in the literature and pro vides a basis for u s to deriv e sev eral new urn mo d els, toge ther w ith their desirable prop erties. These new urn mo dels are found to b e capable of solving the aforemen tioned problems of classic ur n mo dels. In the literature, the asym p totic prop erties of urn mo dels are u sually ob- tained by using Athrey a and Ney’s (1972) tec hn iqu e of em b edd ing the urn pro cess in a contin uous-time branching pro cess. Ho wev er, this tec h n ique re- lies on the assumption that th e transition of u rn composition is go verned b y the add ing r ules, which are iden tical and non -r andom (homogeneous). T his assumption is no longer v alid for the IMU m o dels in general due to the p os- sibilit y that the ur n comp osition ma y b e generated b y a n onhomogeneous immigration p ro cess. Hence, alternativ e mathematical appr oac hes hav e to b e utilized. Another ma jor theoretical intricacy regarding th e IMU pro cess is that it d ep ends on b oth the immigration rates and the adding r ules (refer to Section 2.1 for details). T o ov ercome these mathematical difficulties, w e put forw ard a f easible solution. First, the IMU pro cess is approxima ted b y us- ing m artingales, whic h can h andle b oth immigration r ates and add ing rules sim ultaneously; then, the IMU pro cess is appro ximated by the Wiener p ro- cess. Based on the Wiener pro cess, we will b e able to obtain the asymptotic prop erties of the IMU. T o summarize, th e ma jor con tributions of this p ap er are as follo ws. (a) It f orm ulates a general framewo r k of urn mo dels (IMU mo dels) th at not only encompasses most existing u rn mo d els for adaptive designs in the literature, but also enables us to deriv e n ew urn mo d els with d esir- able prop erties such as the freedom to design an ur n pro cess according to pr e-sp ecified optimalit y requ iremen ts. (b) Th e pap er d er ives asymptotic p rop erties of the IMU mod els, including strong consistency an d asymptotic n ormalit y of treatmen t allo cation prop ortions. Th ese asymptotic pr op erties co ve r many existing asymp- imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 4 L. ZHANG ET AL. totic pr op erties of u rn mo d els as sp ecial cases and form the basis for comparisons of different IMU mo dels. (c) Th e pap er p rop oses and d iscusses s everal n ew IMU mo dels that are useful in clinical trial app licatio ns . The general IMU mo d els and their asymptotic prop erties are pr o vided in Section 2. In addition, several p opular urn mo d els that are mem b er s of the IMU class are d iscussed. In Section 3, new IMU mod els are dev elop ed and their applications are giv en. Concluding r emarks are presented in Section 4. Finally , tec hnical pro ofs are pro vided in the app endix. 2. The immigrated urn mo del. 2.1. The b asic IMU fr amework. In a clinical trial, supp ose that su b jects arriv e sequentia lly to b e randomized to one the K av ailable treatmen ts, and resp onses are obtained immediately after treatmen t. An IMU mo d el is defined as follo ws. Consider an urn th at con tains balls of K + 1 t yp es. Balls of t yp es 1 , . . . , K r epresen t treatmen ts, and balls of typ e 0 are the immigration balls. T h e urn allo ws negativ e and fractional num b er of balls. Initially , there are Z 0 ,i ( ≥ 0) b alls of t yp e i , i = 0 , . . . , K . Let Z 0 = ( Z 0 , 0 , . . . , Z 0 ,K ) b e the initial ur n comp osition. Immediately b efore the m - th ( m > 0) sub ject arr iv es to b e randomized to a treatmen t, let the ur n comp osition b e Z m − 1 = ( Z m − 1 , 0 , . . . , Z m − 1 ,K ). T o av oid a negativ e lik eli- ho o d of selecting a treatmen t, w e adopt a sligh t adjustment to Z m − 1 and let Z + m − 1 ,i = m ax(0 , Z m − 1 ,i ), i = 1 , . . . , K , and Z + m − 1 = ( Z + m − 1 , 0 , . . . , Z + m − 1 ,K ). T o randomize the m -th su b ject, a ball is d ra wn at ran d om without re- placemen t. The p robabilit y of selecting a ball of t yp e i is Z + m − 1 ,i / | Z + m − 1 | , i = 0 , 1 , . . . , K . Here, | Z + m − 1 | = P K j = 0 Z + m − 1 ,j , and Z + m − 1 / | Z + m − 1 | is defined to b e (0 , 1 /K, . . . , 1 /K ) if | Z + m − 1 | = 0. Hence, the balls with negativ e v alues in Z m − 1 will h a v e no chance of b eing selected unless all Z + m − 1 ,k , k = 0 , . . . , K are zeros, and wh en | Z + m − 1 | = 0 (only for the particular case where the IMU mo del h as no imm igration ball), a tr eatment ball is dr a wn with an equal probabilit y of 1 /K . No w, consider the follo win g t w o p ossibilities. (a) If the selected ball is of t yp e 0 (i.e, an imm igration ball), n o treatment is assigned and th e ball is retur n ed to the u rn. A m − 1 = a m − 1 , 1 + . . . + a m − 1 ,K additional balls, a m − 1 ,k ( ≥ 0) of treatmen t t yp e k , k = 1 , . . . , K are add ed to the u rn. Then, a ball is drawn from this up dated ur n again unt il a treatmen t ball is dra wn . If the immigration ball is selected l times b efore a treatmen t ball is dra wn, the urn comp osition Z m − 1 is up dated to ( Z m − 1 , 0 , Z m − 1 , 1 + la m − 1 , 1 , . . . , Z m − 1 ,K + la m − 1 ,K ) and imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 5 the Z + m − 1 is u p dated to ( Z m − 1 , 0 , ( Z m − 1 , 1 + l a m − 1 , 1 ) + , . . . , ( Z m − 1 ,K + la m − 1 ,K ) + ) . (b) If a tr eatmen t ball is dr a wn (sa y , of t yp e k , k = 1 , . . . , K ), the m - th sub ject is giv en treatment k and the treatmen t outcome (resp onse) ξ m,k of th is su b ject on treatment k is observ ed. T he ball is not replaced. Instead, D m,k j = D k j ( ξ m,k ) balls of t yp e j are added to the ur n, j = 1 , . . . , K . D m,k j < 0 signifies the r emo v al of balls. With the IMU, the num b er of immigration balls r emains u nc hanged and a treatmen t ball is d ropp ed when it is dra wn. Th e n umb er of treatmen t balls that is added to the urn d ep ends on: (a) the v alue of a m,k when an immigration b all is dra wn from the urn; and (b) the v alue of D m,k j when a ball of treatment t yp e k is selected. Here, a m,k s represent the immigration r ates and D m,k j s represen t the adding rules. Both a m,k and D m,k j allo w fractional v alues, whic h enable us to define a design in a flexible manner for application. The IMU mo dels unify many existing urn mo dels in the lite ratur e. C lassic urn mo d els, mainly designed for b inary resp onses, are members of th e IMU family . Here we list a few p opu lar m o d els. (1) Th e randomized play- the-winn er (RPW) ru le (W ei and Durham, 1978). When K = 2, Z 0 , 0 = 0 or a m,k = 0 for all m and k . F urther, D m,k k = 2 if th e resp onse of th e m -th su b ject on tr eatmen t k is a success, and D m,k k = D m,k j = 1 ( j 6 = k ) otherwise. (2) Generalized P´ oly a urn m o d els (A threya and Karlin, 1968, also calle d the generalized F riedman’s urn). When a m,k = 0, we ob tain the GPU mo dels if one c ho oses the adding rule D m,k j as in Section 4.1 in Hu and Rosen b erger (2006) . If D m,k j is non-homogeneous, we ob tain the non-homogeneous GPU mod els d iscussed b y Bai and Hu (1999, 2005). (3) Th e birth and death u rn (BDU) (Iv ano v a et al., 2000). Su pp ose that a m,k ≡ 1, D m,k j = 0 for j 6 = k . I n addition, D m,k k = 2 if the resp onse of the m -th su b ject on treatmen t k is a success, and D m,k k = 0 otherwise. When K = 2, we obtain the birth and death urn (BDU) (Iv ano v a et al., 2000). When K > 2, we obtain generalized bir th and death u rn (BDU) for K treatmen ts. (4) Th e Drop-the-loser (DL) rule (Iv ano v a, 2003) . Supp ose that a m,k ≡ 1, D m,k j = 0 for j 6 = k . In ad d ition, D m,k k = 1 if the resp onse of the m -th s u b ject on treatmen t k is a success, and D m,k k = 0 otherwise. When K = 2, we obtain the DL rule (Iv ano v a, 2003). When K > 2, w e obtain DL rule for K treatmen ts. (5) Th e generalized drop-the-loser (GDL) rule (Zhang et al., 2007). S u p- imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 6 L. ZHANG ET AL. p ose that a m,k = a k (do es not dep end on m ) are constan ts and D m,k j = 0 for j 6 = k . When K = 2, we obtain th e GDL rule. When K > 2, w e obtain GDL rules for K treatmen ts. (6) Sequ en tial estimated urn (SEU) m o dels (Zhang, Hu and Cheun g, 2006 ). When Z 0 , 0 = 0 or a m,k = 0 for all m and k , and D m,k j dep ends on estimation, w e obtain the SEU mo dels prop osed by Zhang, Hu and Cheung (2006) and the urn mo dels in Bai, Hu and Shen (2002). In general, w e can select suitable a m,k and D m,k j to obtain the desirable I MU mo del for b oth b inary and con tin uous resp onses (see examples in S ection 3). In clinical trials, let N n,k b e the num b er of su b jects wh o hav e b een as- signed to treatment k , k = 1 , . . . , K . Denote N n = ( N n, 1 , . . . , N n,K ). In clinical studies, the prop ortions N n,k /n , k = 1 , . . . , K of patient s b eing as- signed to v arious treatmen ts are useful statistics. In fact, for urn mo del applications, there are sev eral imp ortant statistics, in cluding: (a) the urn pr op ortion Z n,k / P K k =1 Z n,k ; (b) the allocation prop ortion N n,k /n ; and (c) the estimation of the unknown parameters in the mo del. It is worth while noting that b oth a m,k and D m,k j dep end on m . This allo ws b oth the imm igration rates and the adding rules to b e expr essed as functions of all previous resp onses th us far in the clinical trial. Then , we are able to construct desirable IMU mo dels that can b e used to suit pre- sp ecified allocation prop ortion targets. T o reiterate, as b oth a m,k and D m,k j dep end on m , it is imp ossible to u se Athey a and Ney’s (197 2) tec hnique of em b edd ing the u rn pro cess in a con tinuous-time br anc hing p ro cess. It is also w orth noting that Hopp e’s urn (Hopp e, 1984) and its extensions (see for example Donnely and Kurtz, 1996) are not mem b ers of the IMU mo dels. F or Hopp e’s urn , the num b er of ball t yp es is increasing and random, but f or an IMU mo del the num b er of ball t yp es is fi xed ( K + 1). 2.2. Notation and assumptions. Before the d iscussion of ma jor asymptotic results regarding the IMU mo d els, we introd uce s ome basic n otatio n and the necessary assumptions. Supp ose that ξ m,k ( k = 1 , ..., K , m = 1 , 2 , 3 , ... ) is the r andom v ariable represen ting the resp onse of th e m -th sub j ect on treatmen t k . In practice, w e only observ e one ξ m,k for eac h m . Without loss of generalit y , w e assu m e that th e u nkno wn parameter θ k is the m ean of th e outcome ξ m,k and take the sample mean as its estimate. W rite ξ m = ( ξ m, 1 , . . . , ξ m,K ). F or the add ing rules, let D m = ( D m,k j ; k , j = 1 , . . . , K ), D ( k ) m = ( D m,k 1 , . . . , D m,k K ), k = 1 , . . . , K , and H m = ( h k j ( m )) = E D m . Let imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 7 b θ m − 1 ,k b e the sample mean of the resp onses b θ m − 1 ,k = c 1 + S m − 1 ,k c 2 + N m − 1 ,k , (2.1) where S m − 1 ,k is the s um of the resp onses on tr eatment k of all the previous m − 1 sub jects. Here, c 1 , c 2 > 0 are used to a voi d the nons en se case of 0 / 0. These t w o constants pla y a minor r ole, only in the earlier stages of the clinical trial when accumulate d obser v ations of the treatmen ts are still v ery sm all. In general, many estimators, suc h as the MLE, can b e w ritten in the form of ( 2.1 ) w ith S m − 1 ,k b eing r ep laced b y a sum of functions of th e resp onses plus a negligible remainder (see Hu an d Zhang (2004a) for detail discussion). As discu ssed in Section 2.1, the immigration rate a m,k pla ys an imp ortan t role in the IMU m o dels. Its significance w ill b e illustrated in the later part of this section wh en the theoretical p rop erties of the IMU mo dels are b eing review ed. In clinical trials, optimal allo cation prop ortions u s ually dep end on the u nkno wn parameters θ (See Rosen b erger, et al., 2001 and Tymofy ey ev, Rosen b erger and Hu, 2007). T o ac hieve these prop ortions, one can select the immigration rates a m,k as functions of θ . In practice, as θ is unkn own, one can use a m − 1 ,k = a k ( b θ m − 1 ) as the immigration rates. T h e guidelines for the selection of the fun ction a k will b e giv en in Section 3. In most app lica- tions, the add ing ru les D m = ( D m,k j ; k , j = 1 , . . . , K ) norm ally dep end on the resp ons e ξ m , similar to those in the GPU mo dels. Hence, we n eed the follo wing assumptions. Assumption 2.1 F unctions a k ( · ) > 0 ar e c ontinuous and twic e diffe r en- tiable at θ . Assumption 2.2 { ( ξ m,k , D m,k 1 , . . . , D m,k K ); m ≥ 1 } , k = 1 , . . . , K , ar e K se quenc es of i.i.d. r andom variables with sup m E | D m,k j | 2+ δ < ∞ , and sup m E | ξ m,k | 2+ δ < ∞ for some 0 < δ ≤ 2 , k = 1 , . . . , K . Henc e, let H m = H , which do es not dep e nd on m . F urther assume that D m,k k ≥ − C for some C , k = 1 , . . . , K , and also D m,k j ≥ 0 for k 6 = j . The con tinuit y of a k ( · ) in Assum ption 2.1 is n eeded to sh o w that 0 < min m,k a m,k ≤ max m,k a m,k < ∞ as giv en in Lemma A.5 . The differentia - bilit y of the function is required for the T a ylor expansion. T he moment condition in Assumption 2.2 is useful for applyin g the limit theorems and the appro ximation of r elated martingales. Finally , the lo wer b ound of D m,k j implies that w h en a ball is drawn, the maximum num b er of balls of that treatmen t t yp e wh ic h can b e r emo v ed is C + 1. Th is condition is used to imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 8 L. ZHANG ET AL. deriv e the lo w er b ound of Z n,k , as giv en in Lemma A.3 . 2.3. Main asympto tic r esults. W e no w discuss the asymp totic prop erties related to urn prop ortions and mod el p arameter estimato r s. Asymptotic results are classified into one of follo wing three p ossible cases, according to the exp ectatio n of the adding r ules. 1. H 1 ′ < 1 ′ where 1 = (1 , . . . , 1). Hence P K j = 1 h k j < 1 for all k = 1 , . . . , K . The u rn comp osition is m ainly up dated by the immigration balls b ecause, on a v erage, the n u m b er of added balls in eac h step according to the outcome of a treatmen t is less than th e num b er of dropp ed balls, whic h is 1. The d eriv ation of asymptotic results for this case is of the utmost imp ortance and pla ys a crucial role in this pap er. 2. H 1 ′ > 1 ′ . The total n u m b er of b alls in the ur n gradually increases to infinity . Hence, the pr obabilit y of drawing an immigration ball d rops to zero. F or this case, w e will pro v e that the IMU m o del is asymptotically equiv alent to the generalized P´ oly a urn m o del without immigration (refer to Theorem 2.1 ). 3. H 1 ′ = 1 ′ . This is the b ord er lin e case in whic h b oth the tr eatment balls and the immigration ball retain their roles in th e urn up d ating pro cess. These thr ee cases lead to very different asymptotic results. Let us fi rst consider the case of H 1 ′ > 1 ′ . T he follo wing theorem ensures that the IMU mo del b eha ve s asymptotically , the same as the generalized P´ oly a ur n mo del, when H 1 ′ > 1 ′ . The p ro of is giv en in th e app end ix. Based on this theorem, w e can obtain the asymptotic prop erties, in cluding the strong consistency , asymptotic n orm alit y and Gaussian app r o ximation, of the generalized P´ oly a urn mo del as discussed by J anson (2004), Bai and Hu (2005), Zh an g, Hu and Ch eung (2006) , Zh ang and Hu (2009 ), among others. Theorem 2.1 Supp ose that Assumption 2.2 is satisfie d, H 1 ′ = γ 1 ′ with γ > 1 , and 0 ≤ a m,k ≤ C m 1 / 2 − δ 0 for some δ 0 > 0 and al l m, k . L et v = ( v 1 , · · · , v K ) b e the left eige nv alue ve ctor of H that c orr e sp onds to the lar gest eigenvalue γ an d satisfies v 1 + · · · + v K = 1 , and denote f H = H − I γ − 1 − 1 ′ v . F urther, let λ 2 , · · · , λ K b e the other K − 1 e i genvalues of H and λ = max { Re ( λ 2 ) , · · · , Re ( λ K ) } . A ssume that λ − 1 < ( γ − 1) / 2 . Then, ther e exist two indep endent stand ar d K -dimensional W i ener pr o c esses B t 1 and B t 2 such that ( N n, 1 , · · · , N n,K ) − n v = G n 1 + 1 γ − 1 Z t 0 G x 2 x dx ( I − 1 ′ v ) + o ( n 1 / 2 − ǫ ) a.s., ( Z n, 1 , · · · , Z n,K ) − ( γ − 1) n v = ( γ − 1) G n 1 f H + G n 2 + o ( n 1 / 2 − ǫ ) a.s., imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 9 for some ǫ > 0 , wher e G ti is the solution of the e quation G ti = B ti Λ 1 / 2 i + Z t 0 G xi x dx f H , with Λ 1 = diag ( v ) − v ′ v and Λ 2 = P K k =1 v k V ar { D ( k ) 1 } . In p articular, Z n, 0 Z n, 0 + · · · + Z n,K → 0 a.s., Z n,k Z n, 0 + · · · + Z n,K → v k a.s., N n,k n → v k a.s., k = 1 , · · · , K , and n 1 / 2  Z n, 1 ( γ − 1) n − v 1 , · · · , Z n,K ( γ − 1) n − v K  D → N ( 0 , Γ 1 ) , n 1 / 2  N n, 1 n − v 1 , · · · , N n,K n − v K  D → N ( 0 , Γ 2 ) . Her e, the varianc e-c ovarianc e matric e s Γ 1 and Γ 2 c an b e sp e cifie d i n line with Bai and Hu (2005) and Zhang and Hu (2009) with D m − I γ − 1 and H − I γ − 1 r eplacing D m and H , r esp e ctively. F or details, one c an r efer to Pr op osition 3.4 of Zhang and Hu (2009). No w w e consider the case in which H 1 ′ < 1 ′ . Different from the case when H 1 ′ > 1 ′ in whic h the urn prop ortion and the sample allocation prop ortion hav e th e same limit, the urn prop ortion ma y not ha ve a limit in this case. F or the immigration rates, wr ite a k = a k ( θ ). Let a = ( a 1 , . . . , a K ), u = a ( I − H ) − 1 , s = a ( I − H ) − 1 1 ′ = P K k =1 u k and v = u /s . F urther, denote Σ k = V ar { D ( k ) 1 } , Σ 11 = P K k =1 v k Σ k , Σ 12 = ( Cov { D 1 ,k j , ξ k } ; j, k = 1 , . . . , K ), Σ 22 = diag ( Va r { ξ 1 , 1 } , . . . , Va r { ξ 1 ,K } ), and Λ =  Λ 11 Λ 12 Λ ′ 12 Λ 22  =  Σ 11 Σ 12 diag ( v ) diag ( v ) Σ ′ 12 Σ 22 diag ( v )  . (2.2) Theorem 2.2 Supp ose that Assumptions 2.1 - 2.2 ar e satisfie d, H 1 ′ < 1 ′ and Z 0 , 0 > 0 . Then Z n,k = o ( n 1 / 2 − ǫ ) a.s., k = 1 , . . . , K for some ǫ > 0 , and, one c an define a 2 K -dimensional Wiener pr o c esses ( W ( t ) , B ( t )) such that V ar { ( W ( t ) , B ( t )) } = t Λ (2.3) imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 10 L. ZHANG ET AL. and N n − n v = W ( n ) A + Z n 0 B ( x ) x dx diag  1 v  ∂ v ( θ ) ∂ θ + o ( n 1 / 2 − ǫ ) a.s. (2.4) for some ǫ > 0 , wher e A = ( I − H ) − 1 ( I − 1 ′ v ) , v = v ( θ ) = a ( θ )( I − H ) − 1 a ( θ )( I − H ) − 1 1 ′ and ∂ v ( θ ) ∂ θ =  ∂ v k ( θ ) ∂ θ j ; j, k = 1 , . . . , K  . Her e, 1 / v = (1 /v 1 , . . . , 1 /v K ) . Remark 2.1 Note that h ij ≥ 0 for i 6 = j . The existenc e of ( I − H ) − 1 is implie d by the assumption that H 1 ′ < 1 ′ . This assumption c an b e r eplac e d by a mor e gener al assumption in which ther e is a ve ctor e = ( e 1 , . . . , e K ) such that H e ′ < e ′ and e i > 0 , i = 1 , . . . , K . Based on Theorem 2.2 , we can see that the u rn comp osition ( √ n ) − 1 Z n,k con v erges to 0 almost surely . It is b ecause wh en H 1 ′ < 1 ′ , there will b e a net loss of balls fr om the u rn on a v erage if a tr eatmen t b all is dra wn . The pr o of of Theorem 2.2 is giv en in the app endix. Th e consistency and asymptotic normalit y of N n can b e deriv ed by u sing Equation ( 2.4 ) as follo ws. Corollary 2.1 U nder the assumptions in The or em 2.2 , N n − n v = O ( p n log log n ) a.s. and √ n  N n n − v  D → N ( 0 , Σ ) , (2.5) wher e Σ = Σ D + 2 Σ ξ + Σ D ξ + Σ ′ D ξ , and Σ D = A ′ Σ 11 A , Σ D ξ = A ′ Σ 12 ∂ v ( θ ) ∂ θ Σ ξ =  ∂ v ( θ ) ∂ θ  ′ diag  V ar { ξ 1 , 1 } v 1 , . . . , V ar { ξ 1 ,K } v K  ∂ v ( θ ) ∂ θ . In p articular, if D m ≡ cons t , then √ n  N n n − v  D → N ( 0 , 2 Σ ξ ); and if a m,k ≡ a k , k = 1 , . . . , K , do not dep end on the estimates, then √ n  N n n − v  D → N ( 0 , Σ D ) . imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 11 Pro of . Note that ( W ( n ) , R n 0 B ( x ) x dx ) is a cente r ed Gaussian vect or with W ( n ) = O ( p n log log n ) a.s ., Z n 0 B ( x ) x dx = O (1) + Z n e O ( √ x log lo g x ) x dx = O ( p n log log n ) a.s ., V ar { W ( n ) } = n Σ 11 , V ar  Z n 0 B ( x ) x dx  = Σ 22 diag ( v ) Z n 0 Z n 0 x ∧ y xy dxdy = 2 n Σ 22 diag ( v ) and Cov  W ( n ) , Z n 0 B ( x ) x dx  = Σ 12 diag ( v ) Z n 0 x ∧ n x dx = n Σ 12 diag ( v ) . ( 2.5 ) follo ws from ( 2.4 ) immediately .  Remark 2.2 In pr actic e, the r esp onses in clinic al trials ar e fr e quently not available imme diately b efor e the tr e atment al lo c ation of the next subje ct (de- laye d r esp onse). The p ar ameters c an b e estimate d and the urn c an b e up date d only by using al l available observe d r esp onses. In the delaye d r esp onse situ- ation, let µ k ( m, l ) b e the pr ob ability that the r esp onse of the m -th subje ct on tr e atment k o c cu rs after at le ast anoth er l subje cts arrive. If µ k ( m, l ) ≤ C l − γ for some γ ≥ 2 , then we c an show that the total sum of unobserve d outc omes up to the n - th assignment is with a high or der of √ n and thus the c onclusion in The or em 2.2 r emains true. It has b e en shown that the delay me chanism do es not effe ct the asymptotic pr op erties for many r esp onse-adaptive designs if the delay de c ays with a p ower r ate (c.f., Bai, Hu , and R osenb er ger, 2002; Hu and Zhang, 2004b; Zhang et al., 2007). In many IMU mo dels (suc h as, sp ecial cases (3), (4) and (5) in Section 2), th e additional rule, D m , is a diagonal matrix ( D m,k j = 0, j 6 = k ). F or this sp ecial case, we hav e the follo wing corollary that helps u s to obtain the asymptotical limits and co v ariance matrix of N n easily . Corollary 2.2 Supp ose that A ssu mptions 2.1 - 2.2 ar e satisfie d, D m,k j = 0 for j 6 = k , and h k = 1 − E D 1 ,k k > 0 . Write h = ( h 1 , . . . , h K ) , v k ( θ , h ) = a k ( θ ) /h k P K j = 1 a j ( θ ) /h j k = 1 , . . . , K , imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 12 L. ZHANG ET AL. v = v ( θ , h ) = ( v 1 ( θ , h ) , . . . , v K ( θ , h )) , and ∂ v ( θ , h ) ∂ θ =  ∂ v k ( θ , h ) ∂ θ j ; j, k = 1 , . . . , K  , ∂ v ( θ , h ) ∂ h =  ∂ v k ( θ , h ) ∂ h j ; j, k = 1 , . . . , K  . Then, N n n → v a.s. and √ n  N n n − v  D → N ( 0 , Σ ) , (2.6) wher e Σ = Σ D + 2 Σ ξ + Σ D ξ + Σ ′ D ξ , Σ D =  ∂ v ( θ , h ) ∂ h  ′ diag  σ 2 D 1 v 1 , . . . , σ 2 D K v K  ∂ v ( θ , h ) ∂ h , Σ ξ =  ∂ v ( θ , h ) ∂ θ  ′ diag  σ 2 ξ 1 v 1 , . . . , σ 2 ξ K v K  ∂ v ( θ , h ) ∂ θ , Σ D ξ = −  ∂ v ( θ , h ) ∂ h  ′ diag  σ D ξ 1 v 1 , . . . , σ D ξ K v K  ∂ v ( θ , h ) ∂ θ , and σ 2 D k = V ar { D 1 ,k k } , σ 2 ξ k = Va r { ξ 1 ,k } , σ ξ D k = Cov { D 1 ,k k , ξ k , 1 } , k = 1 , 2 , . . . , K . Pro of. It is easy to c h eck that Σ 11 = diag ( σ 2 D 1 v 1 , . . . , σ 2 D K v K ) , Σ 12 = diag ( σ ξ D 1 , . . . , σ ξ D K ) , Σ 22 = diag ( σ 2 ξ 1 , . . . , σ 2 ξ K ) , A = diag (1 / h )( I − 1 ′ v ) , and ∂ v ( θ , h ) /∂ h = − diag ( v ) A . Th en, the r esults follo w from Corollary 2.1 directly .  T o impro ve statistical efficiency , a suitable resp onse adaptiv e ran d om- ization pro cedur e should b e adopted b ecause of v ariabilit y (Hu and Rosen- b erger, 2003). Hu, Rosenberger and Zhang (2006) stud ied the v ariabilit y of a randomization p ro cedure that target s any giv en allocation prop ortion. They obtained a low er b oun d of the v ariabilit y . F or a large class of the IMU mo d- els in this pap er, the lo w er b oun d of the v ariabilit y is attained. When the v ariance of IMU mo d el attains the lo wer b ound, we can use the Cram ´ er-Rao form ula to compute the v ariance. In general, w e ha v e the follo wing theorem. Theorem 2.3 If e ach D m,k j is a line ar function of a r andom η m,k , j = 1 , . . . , K , wher e η m,k may b e a function of ξ m,k and for e ach k , η m,k , m = imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 13 1 , 2 , . . . , ar e i . i.d. r andom variables with finite varianc es, then we have Σ D =  ∂ v ∂ d  ′ diag  V ar { η 1 , 1 } v 1 , . . . , V ar { η 1 ,K } v K  ∂ v ∂ d (2.7) Σ D ξ =  ∂ v ∂ d  ′ diag  Cov { η 1 , 1 , ξ 1 , 1 } v 1 , . . . , Cov { η 1 ,K , ξ 1 ,K } v K  ∂ v ∂ θ , (2.8) wher e d = ( d 1 , . . . , d K ) = ( E η 1 , 1 , . . . , E η 1 ,K ) . F urther, if a ( · ) = cons t and V ar { η 1 ,k } is the inverse of the Fisher information of d k , then the asymptotic varianc e-c ovarianc e matrix of N n / √ n atta ins the fol lowing lower b ound,  ∂ v ∂ d  ′ diag  ( v 1 I 1 ) − 1 , . . . , ( v K I K ) − 1   ∂ v ∂ d  , (2.9) wher e I k is the Fisher information function of p ar ameter d k . Pro of. If we w r ite D ( k ) 1 = α k + β k η 1 ,k and K =   β 1 · · · β K   , th en Λ 11 = K X k =1 v k V ar { η 1 ,k } β ′ k β k =( diag ( v ) K ) ′ diag  V ar { η 1 , 1 } v 1 , . . . , V ar { η 1 ,K } v K  diag ( v ) K and Σ 12 = ( diag ( v ) K ) ′ diag  Cov { η 1 , 1 , ξ 1 , 1 } v 1 , . . . , Cov { η 1 ,K , ξ 1 ,K } v K  . Ho w eve r, ∂ H /∂ d k = diag ( 1 k ) K , w here 1 k has zero elements except th e k -th one wh ic h is 1. In addition, ∂ ( I − H ) − 1 ∂ d k = ( I − H ) − 1 ∂ H ∂ d k ( I − H ) − 1 = ( I − H ) − 1 diag ( 1 k ) K ( I − H ) − 1 . It f ollo ws that ∂ v ∂ d k = ∂ a ( I − H ) − 1 /∂ d k a ( I − H ) − 1 1 ′ − ∂ a ( I − H ) − 1 /∂ d k ( a ( I − H ) − 1 1 ′ ) 2 1 ′ a ( I − H ) − 1 = v diag ( 1 k ) K ( I − H ) − 1 ( I − 1 ′ v ) = v diag ( 1 k ) K A , i.e., ∂ v /∂ d = diag ( v ) K A . Hence, ( 2.7 ) and ( 2.8 ) are pro ve d by Corollary 2.1 .  Corollary 2.2 and Theorem 2.3 are useful for derivin g th e asymptotic v ari- ance. W e will illustrate this idea by in tro du cing several interesting examples in the next s ection. imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 14 L. ZHANG ET AL. Remark 2.3 In The or em 2.3 , for simplicity of notation we assume that the p ar ameter d k is a one- dimensional p ar ameter that c orr esp onds to tr e atment k . The the or em is stil l v alid if r eformulate d using a ve ctor p ar ameter d k , without extr a assumpt ions. Finally , w e consider the case wh en H 1 ′ = 1 ′ . Th e follo wing theorem, with pro of giv en in the app endix, can b e used to yield the consistency prop ert y of the allocation prop ortion. Ho w ever, it is still unkn own whether N n is asymptotically normal. Theorem 2.4 Supp ose that Assumptions 2.1 and 2.2 ar e satisfie d, and H 1 ′ = 1 ′ , Z 0 , 0 > 0 . Supp ose further that 1 is a single eigenvalue of H . Then N n − n v = O ( p n log lo g n ) a.s. and N n − n v = O P ( √ n ) , wher e v is the left eig e nvalue ve ctor of H that c orr esp onds to the eigenvalue 1 and satisfies v 1 + · · · + v K = 1 . These th eorems and corolla r ies are related to the sample allo cation pro- p ortion N n /n . Regarding the estimator ˆ θ n , we h a v e the follo wing theorem. Theorem 2.5 Supp ose that the assumptions in The or em 2.1 or 2.2 or 2.4 ar e satisfie d. We have √ n  ˆ θ n − θ  → N ( 0 , Σ θ ) , (2.10) wher e Σ θ = diag  V ar { ξ 1 , 1 } v 1 , . . . , V ar { ξ 1 ,K } v K  . Note that N n /n → v a.s. according to Theorem 2.1 or 2.2 or 2.4 , so the pro of of this T heorem is the same as that of Lemma 1 of Hu, Rosen b erger and Z h ang (2006) and is th us omitted h ere. 3. Examples a nd Applications. In th is section, we apply the gen- eral asymptotic results in Section 3 to selected IMU mo dels for illustrativ e purp oses. In Section 2.1, w e listed sev eral classic families of ur n mo dels as sp ecial cases of IMU mo d els. W e can apply d irectly the theoretical results in S ection 3 to these sp ecial cases and ob tain their asymp totic prop erties for b oth K = 2 (av ailable in the literature) and for the general v alue of K ≥ 3. In this section, w e fo cus on the generation of new families of urn mo dels from th e IMU framew ork and discuss their corresp on d ing p rop erties. Sev eral illustr ativ e examples are giv en. Firs t, we consider con tin uous -typ e imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 15 resp onses that are frequently en coun tered in clinical stu d ies, ev en though there h as b een a lac k of related studies in the literature. Example 1: Two tr e atments with c ontinuous r esp onses . S upp ose th at ξ m, 1 ( m = 1 , 2 , 3 , ... ) are i.i.d. rand om v ariables fr om N ( µ 1 , σ 2 1 ) and ξ m, 2 ( m = 1 , 2 , 3 , ... ) are i.i.d. r andom v ariables from N ( µ 2 , σ 2 2 ). Without the loss of generalit y , assume that the smaller the v alue of the resp onse, the b etter the treatmen t. W e n o w in tro du ce four IMU mo dels. (1.A) L et a m,k ≡ 1, D m,k j = 0 for j 6 = k . L et C b e a constan t suc h th at D m,k k = 1 if the resp onse of the m -th sub ject on treatment k , ξ m,k , is less than C , and D m,k k = 0 otherwise. (1.B) S upp ose th at there are tw o critical v alues C 1 < C 2 and if it is v ery desirable to hav e the v alue of the resp ons e fall b et w een C 1 and C 2 , then the follo wing IMU m o del is appr op r iate. T ak e a m,k ≡ 1, D m,k j = 0 for j 6 = k . F urther, let D m,k k = 1 if ξ m,k < C 1 , D m,k k = 0 if ξ m,k > C 2 and else D m,k k = 1 / 2. (1.C) If the p o w er of statistical inferences is an imp ortan t concern, the Ney- man allocation σ 1 / ( σ 1 + σ 2 ) can b e adopted to maximize the p ow er of testing. Then, consider the follo wing IMU mo d el. Let a m,k = b σ k , D m,k j = 0 for all j, k . Here, b σ 2 k is the curr en t sample v ariance of the resp onses on treatment k , k = 1 , 2, and can b e us ed as estimates in the Neyman allo catio n rule. (1.D) If the aim is to lo we r the pr op ortion of sub jects b eing assigned to the inferior treatment s for ethical reasons, the allo cation target √ µ 2 σ 1 / ( √ µ 2 σ 1 + √ µ 1 σ 2 ) where µ 1 , µ 2 > 0 (Zhang and Rosenb er ger (2006)) is an option. Let a m, 1 = p b µ 2 b σ 1 , a m, 2 = p b µ 1 b σ 2 , D m,k j = 0 for all j, k . Here, b µ k , b σ 2 k are the cur ren t samp le mean and s amp le v ariance of the resp onses on treatmen t k resp ectiv ely , k = 1 , 2. T o a vo id th e situation of b µ k ≤ 0, simply r eplace b µ 2 b y 1 /m when such an o ccasion arises. Designs (1.A) and (1.B) co ver a wid e sp ectrum of p oten tial app licatio n s . Note that Design (1.A) is equiv alen t to the DL r ule for bin ary r esp onse if the critical v alue C is used to classify resp onses into t wo categories. Designs (1.C) and (1.D) incorp orate p r e-sp ecified ob jectiv es of a clinical trial, d ep ending on w hether the ob jectiv e is to increase the testing p ow er (as in (1.C)), or reduce the num b er of patien ts b eing assigned to the in ferior tr eatments (as in (1.D)). F u rther, it w ould not difficult to generalize th ese four designs to studies with K > 2 treatmen ts. The asymptotic prop erties of the four designs can b e obtained u sing Th e- orem 2.2. F or illustrativ e pu rp oses, we discuss asymptotic norm alities for imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 16 L. ZHANG ET AL. Designs (1.C). It is easy to v erify that b σ 2 k =: b σ 2 m,k = 1 N m,k m X j = 1 X j,k ( ξ j,k − µ k ) 2 −  b µ k − µ k  2 = 1 N m,k m X j = 1 X j,k ( ξ j,k − µ k ) 2 + O  log log N m,k N m,k  a.s. By C orollary 2.1 , N n, 1 n → v 1 a.s. and n 1 / 2  N n, 1 n − v 1  D → N (0 , σ 2 ) , where v 1 = σ 1 / ( σ 1 + σ 2 ), and σ 2 equals to 2  ∂ v 1 ∂ ( σ 2 1 ) , ∂ v 1 ∂ ( σ 2 2 )  diag  V ar { ( ξ 1 , 1 − µ 1 ) 2 } v 1 , V ar { ( ξ 1 , 2 − µ 2 ) 2 } 1 − v 1   ∂ v 1 ∂ ( σ 2 1 ) , ∂ v 1 ∂ ( σ 2 2 )  ′ . After simp lification, we ha v e σ 2 = σ 1 σ 2 / ( σ 1 + σ 2 ) 2 . On e can also use Th e- orem 2.5 to der ive the asymp totic distribution of the estimators of the u n- kno wn parameters. F or example, in Design (1.C), √ n ( b σ 2 n,k − σ 2 ) D → N (0 , 2 σ 4 k /v k ). Example 2: Mo difie d DL (MDL) rule . W e prop ose th e MDL r ule, which is a mo dification of the DL rule. Th e pro cedure is similar to the DL rule in that w hen a treatmen t ball is d ra wn, this b all is replaced only wh en the resp onse is a success. Ho w ev er, when an immigration ball is dra wn, instead of adding an equal num b er of treatmen t b alls to the urn , we add C b p k ( C > 0) balls of t yp e k , k = 1 , . . . , K , where b p k is the cur ren t estimate of the successful p r obabilit y p k of tr eatment k , and C is a constan t. With this mo del, more balls are immigrated to treatmen ts with higher success rates, and subsequently , th e limit prop ortions will b e h igher for b etter treat ments. Regarding the asymp totic v ariance, it is straigh tforw ard to sh ow that a = ( p 1 C, . . . , p K C ) and H = diag ( p 1 , . . . , p K ). T he conditions in Corollary 2.2 are satisfied for all cases w ith 0 < p k < 1 and k = 1 , . . . , K . Hence, the limit prop ortions are v k = ( p k /q k ) / ( P K j = 1 p j /q j ) , k = 1 , . . . , K. The asymptotic v ariance-co v ariance can b e deriv ed by the formulae in Corollary 2.2 , in wh ic h θ = ( p 1 , . . . , p K ), h = ( q 1 , . . . , q K ), and σ 2 D k = σ 2 ξ k = σ D ξ k = p k q k , k = 1 , . . . , K . F or the tw o-treatmen t case, N n, 1 n → v 1 = p 1 /q 1 p 1 /q 1 + p 2 /q 2 a.s. and √ n ( N n, 1 /n − v 1 ) D → N (0 , σ 2 ) , imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 17 where σ 2 = q 1 q 2 [ p 2 1 (1 + q 2 2 ) + p 2 2 (1 + q 2 1 )] / ( p 2 q 1 + p 1 q 2 ) 3 . When the success probabilities p 1 and p 2 are b oth high, the v ariabilit y σ 2 is close to the lo wer b ound q 1 q 2 ( p 2 1 + p 2 2 ) / ( p 2 q 1 + p 1 q 2 ) 3 . Unlik e the generalized P´ oly a ur n mo d els without immigration in whic h the asymptotic normalit y holds only wh en a v ery strict condition on eigen v alues of a generating matrix is satisfied (c.f., Bai and Hu, 2005; Janson, 2004; Zhang, Hu, and Cheun g, 2006), the MDL rule allo ws asymptotic n ormalit y for all cases with 0 < p k < 1, k = 1 , . . . , K . In most IMU mo dels, th e adding rule D m is a diagonal matrix. Here w e giv e an example for the t wo -treatmen t case with dic hotomous resp onses in whic h the adding ru le D m is n ot a d iagonal matrix. Example 3: Two tr e atments with dichotom ous r esp onses . Consider the t w o- treatmen t case with dic hotomous resp onses, success or failure. Let p k b e the success pr ob ab ility of treatmen t k and q k = 1 − p k , k = 1 , 2. W e consider an immigrated u rn in w hic h a m, 1 = a m, 2 ≡ 1 and D m =  β ξ m, 1 α (1 − ξ m, 1 ) α (1 − ξ m, 2 ) β ξ m, 2  , where ξ m,k = 1 if the ou tcome of the m -th su b ject on treatmen t k is a success, and 0 otherw ise, k = 1 , 2, α ≥ 0. In this design, the draw of an immigration ball generates a b all of eac h treatmen t typ e; wh en a treatment t yp e b all is dropp ed, β balls of the same treatmen t t yp e are added if the outcome is a success and α balls of the alternate treatment typ e are add ed if the outcome is a failure. Hence, H =  β p 1 αq 1 αq 2 β p 2  . Based on Theorems 2.1-2 .5 of Section 2, we can der ive th e asymptotic prop- erties for the th r ee cases: (i) H 1 ′ > 1 ′ ; (ii) H 1 ′ < 1 ′ ; and (iii) H 1 ′ = 1 ′ . The tec hn ical details are omitted here. Nev ertheless, it is wo rth n oting that differen t c hoices of α and β generate v arious memb ers of th e IMU f amily . Remark 3.1 The GDL rule of Zhang et al. (2007) is a memb er of the IMU class with D m,k j = 0 , j 6 = k . In pr actic e, the values of θ k , k = 1 , ..., K ar e unknown and have to b e estimate d b y sample statistics. The derivation of the asymptot ic distributions of the tr e atment pr op ortions N n,k is usual ly difficult and is not include d by Zhang et al. (2007) if the estimates of θ k , k = 1 , ..., K ar e use d. However, by applying Cor ol lary 2.2 , one c an obtain the asymptotic pr op e rties of N n,k dir e ctly. imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 18 L. ZHANG ET AL. F or example, if the optimal pr op ortion v 1 = √ p 1 / ( √ p 1 + √ p 2 ) is use d for c omp aring two tr e atments, we c an sele ct an IMU mo del with D m ≡ 0 , a m,k = C p b p k , wher e b p k is the curr ent estimate of the suc c essful pr ob ability p k of tr e atment k , and C is a c onstant, k = 1 , 2 . By Cor ol lary 2.2 , we have √ n ( N n, 1 /n − v 1 ) D → N (0 , σ 2 ) , wher e σ 2 = 1 2( √ p 1 + √ p 2 ) 3  p 2 q 1 √ p 1 + p 1 q 2 √ p 2  . Zhang et al. (2006) pr op ose d the use of a GPU without immigr ation to tar- get this pr op ortion (c.f . , their Example 2). The c orr esp onding asymptotic varianc e is √ p 1 p 2 ( √ p 1 + √ p 2 ) 2 + 3 2( √ p 1 + √ p 2 ) 3  p 2 q 1 √ p 1 + p 1 q 2 √ p 2  , which is at le ast triple the varianc e of this IM U mo del. The IMU mo dels, suc h as th ose give n in the foregoing examples, can b e applied in clinical trials. W e discuss the app licatio n s in th r ee p ossible directions. (i) Th ere are n um erous applications of urn mo dels in clinical trials. One can app ly the prop osed IMU mo dels with multiple ob jectiv es, su c h as ethical concerns and design efficiency . F or instance, T amura et al. (1994 ) discussed the application of the RPW rule, a member of the IMU family , to study the treatmen t of out-patien ts s uffering from d e- pressiv e disord er . Later, in a sim u lation stu dy (using the same data), Bhattac hary a (2008) show ed that the DL ru le, another mem b er of the IMU family , has a smaller v ariabilit y and yields higher p o wer than the RPW rule. One can app ly the asymp totics of the IMU mo d el giv en in this pap er to compare v arious urn allo cation metho ds in stead of using only the sim u lation results giv en by Bhattac harya (2008). (ii) Urn mo dels are also frequently emplo yed in clinical studies to promote balance (see Matthews et al. (2010) and the references therein). In suc h circumstances, IMU mo dels s h ould b e considered as u seful candidates. The in tro du ction of the immigration urn will significan tly impro ve these allo cation sc hemes, mainly in r elation to the v ariabilit y of the u rn prop ortions. F urth ermore, asymp totic distributions of IMU mo dels can b e deriv ed, leading to a more compr ehensiv e und erstanding of these urn p r o cesses. (iii) F or comparin g K treatmen ts, Tymofy ey ev, Rosen b erger and Hu (2007), Zhu and Hu (2009 ) obtained optimal allo cation prop ortions for b oth binary and con tinuous resp ons es. Th e IMU mo d els are suitable c hoices imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 19 due to their lo w v ariabilit y and fl exibilit y in targeting these optimal allocation pr op ortions. 4. Conclusions. In this pap er, w e ha v e prop osed a general class of urn mo dels that incorp orates immigration. T he IMU framework unifi es many ex- isting classes of u rn mo dels and provides crucial link ages among these mo d els to enable us to hav e a more comprehensive un derstanding of different urn pro cesses and their imp ortan t p rop erties. F u rther, th is framew ork facilitat es the generation of new u rn mo d els with d esirable prop erties. Asymptotic prop erties of the IMU mo dels, with widely satisfied conditions, are giv en in Section 2. These imp ortan t results serve to connect existing asymptotic results ab out urn mo dels. More imp ortant ly , th e asymptotic n ormalit y for- m ula in this article can b e emplo ye d to ev aluate and compare differen t urn mo dels in terms of the distr ib utions of treatmen t allo cation prop ortions. Under very mild conditions, the suggested IMU mo dels alw a ys yield rela- tiv ely smaller asymp totic v ariances. In many cases, the asymptotic v ariance attains the lo w er b ound . Thus, the IMU mo d els ha ve s maller v ariabilities than the corresp ond ing generalized P´ oly a u rn mo dels. In clinical trials, resp onses ma y not b e av ailable immed iately after the patien ts ha ve b een treated. Ho wev er, there are n o logistical d ifficulties in incorp orating dela yed resp onses in to the IMU framewo rk . One can up date the ur n when resp ons es b ecome a v ailable. A m o d erate dela y in resp onse (see Hu and Zhang, 2004b) will n ot affect the asymp totic prop erties of the IMU. In fact, it is s traigh tforw ard to mo dify th e pro of in the app end ix to incorp orate d ela y ed resp onses. The discu s sion of clinical applications has b een the main fo cus of this article b ecause adaptive designs using u rn mo dels ha ve receiv ed m uch at- ten tion in statisti cs. Ho w ev er, it is necessary to emphasize that our results are ve ry general and should also pla y an im p ortan t role in other areas as w ell. F or example, in quantum mec hanics, Niven and Grendar (2009) use the P´ oly a ur n to understand the generalized probability d istr ibution for Maxw ell-Boltz man n , Bose-Einstein, and F ermi-Dirac statistics. With d iffer- en t colors in the urn, a b all is sampled, recorded and returned to the urn. Then, c balls of the same color are added to the urn. In their form ulation, the c hoices of c are c > 0, c = 0 and c < 0. As c < 0 implies a decrease of the n umb er of balls in the urn , it would b e in teresting to exp lore the p ossibilit y of using the IMU framew ork to av oid the distinction of balls of a p articular t yp e. App endix. Pro ofs. The outline of th e p ro ofs is as f ollo ws. First, w e pro ve Th eorem 2.2 , whic h is our main result, and then Theorem 2.4 . Finally imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 20 L. ZHANG ET AL. w e giv e a sk etc h of the pro of of Theorem 2.1 . Recall that Z m − 1 = ( Z m − 1 , 0 , Z m − 1 , 1 , . . . , Z m − 1 ,K ) represent the num- b ers of balls when the m -th sub j ect arriv es to b e rand omized, Z + m − 1 = ( Z + m − 1 , 0 , Z + m − 1 , 1 , . . . , Z + m − 1 ,K ) are the non-negativ e num b ers, and | Z + m − 1 | = Z + m − 1 , 0 + Z + m − 1 , 1 + . . . + Z + m − 1 ,K . W rite e Z m − 1 = ( Z m − 1 , 1 , . . . , Z m − 1 ,K ). Be- cause ev ery imm igration b all is replaced, Z + m − 1 , 0 = Z m − 1 , 0 = Z 0 , 0 for all m . Let X m b e th e result of the m -th assignment, where X m,k = 1 if the m -th su b ject is assigned to treatmen t k and 0 otherwise, k = 1 , . . . , K . Then, N n = ( N n, 1 , . . . , N n,K ) = P n m =1 X m . F urther, w e denote a m = ( a m, 1 , . . . , a m,K ), and ν m to b e the num b er of draws of t yp e 0 balls b et w een the ( m − 1)-th assignment and the m -th assignmen t. Note that b et we en the ( m − 1)-th assignmen t and the m -th assignmen t, w e ha ve d r a wn ν m balls of t yp e 0. Accordingly , w e ha ve add ed a m − 1 ,k ν m balls of typ e k to the ur n. Ho w ever, wh en a ball of type k is dr a wn, it is not replaced and another D m,k j balls of typ e j are added to the u rn. Hence, the c hange in the num b er of balls after the m -th assignment is e Z m − e Z m − 1 = a m − 1 ν m + X m ( D m − I ) . (A.1) It f ollo ws that e Z n − e Z 0 = n X m =1 a m − 1 ν m + n X m =1 X m ( D m − I ) = n X m =1 a m − 1 ν m − N n ( I − H ) + n X m =1 X m ( D m − E [ D m ]) = a N n, 0 + n X m =1 ( a m − 1 − a ) ν m − N n ( I − H ) + M n , (A.2) where N n, 0 = P n m =1 ν m is total n u m b er of draws of t yp e 0 b alls after the n -th assignment, and M n = P n m =1 X m ( D m − E [ D m ]) is a martin gale. T o pro ve T h eorem 2.2 we need t wo lemmas. Their pro ofs w ill b e giv en later. Lemma A.1 Supp ose that the assumptions in The or em 2.2 ar e satisfie d. Then, for 0 < δ 0 < 1 2 − 1 2+ δ , Z n,k = o ( n 1 / 2 − δ 0 ) a.s., k = 1 , . . . , K . (A.3) Lemma A.2 Supp ose that the assumptions in The or em 2.2 ar e satisfie d. Then, N n, 0 = n /s + O ( p n log log n ) a.s ., (A.4) imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 21 N n,k = n v k + O ( p n log log n ) a.s ., k = 1 , . . . , K, (A.5) wher e s = a ( I − H ) − 1 1 ′ . Also, for e ach k = 1 , . . . , K , b θ n,k → θ k a.s. (A.6) and b θ n,k − θ k = Q n,k nv k + o ( n − 1 / 2 − δ 0 ) a.s., (A.7) wher e Q n,k = P n m =1 X m,k ( ξ m,k − E ξ m,k ) is a martingale and Q n = ( Q n, 1 , . . . , Q n,K ) . No w w e b egin the pr o of of Theorem 2.2 . Consider th e 2 K -dimensional martingale { ( M n , Q n ) , A n ; n ≥ 1 } , wher e A n = σ ( X 1 , . . . , X n , ξ 1 , . . . , ξ n +1 ). According to ( A.5 ) w e ha ve n X i =1 E [(∆ M i ) ′ ∆ M i |A i − 1 ] = K X k =1 N n,k Σ k = n Σ 11 + O ( p n log log n ) a.s., (A.8) n X i =1 E [(∆ Q i ) ′ ∆ Q i |A i − 1 ] = Σ 22 diag ( N n ) = n Λ 22 + O ( p n log lo g n ) a.s., (A.9) n X i =1 E [(∆ M i ) ′ ∆ Q i |A i − 1 ] = Σ 12 diag ( N n ) = n Λ 12 + O ( p n log log n ) a.s. (A.10) By Corollary 1.1 of Zhang (200 4), we can define the 2 K -dimensional Wiener pro cesses ( W ( t ) , B ( t )) with v ariance-c ov ariance m atrix Λ suc h that for some ǫ > 0, M n = W ( n ) + o ( n 1 / 2 − ǫ ) a.s., Q n = B ( n ) + o ( n 1 / 2 − ǫ ) a.s . (A.11) Without loss of generalit y , w e assum e that ǫ ≤ δ 0 , where δ 0 is defined as it is in Lemm a A.1 . Next, w e need to sh o w that ( W ( t ) , B ( t )) s atisfies ( 2.4 ). Com binin g ( A.2 ) and ( A.3 ) yields N n ( I − H ) − a N n, 0 = M n + n X m =1 ( a m − 1 − a ) ν m + o ( n 1 / 2 − δ 0 ) a.s . (A.12) Recall that A = ( I − H ) − 1 ( I − 1 ′ v ), v = a ( I − H ) − 1 / ( a ( I − H ) − 1 1 ′ ) and n ote that N n 1 ′ = n , aA = s v ( I − 1 ′ v ) = 0 . According to ( A.12 ), N n − n v =  M n + n X m =1 ( a m − 1 − a ) ν m  A + o ( n 1 / 2 − δ 0 ) a.s. (A.13) imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 22 L. ZHANG ET AL. F or a m − a , due to ( A.7 ) and ( A.11 ), a m − a =( b θ m − θ ) ∂ a ( θ ) ∂ θ + O  k b θ m − θ k 2  = Q m m diag  1 v  ∂ a ( θ ) ∂ θ + o ( m − 1 / 2 − δ 0 ) (A.14) = B ( m ) m diag  1 v  ∂ a ( θ ) ∂ θ + o ( m − 1 / 2 − ǫ ) . Note that immigration o ccurs only when a t yp e 0 ball is dra wn. Let τ m b e the tota l n u m b er of draws when the m -th type 0 ball is drawn. A t that time, τ m − m sub j ects ha v e b een assigned and the ( τ m − m + 1)-th sub ject arriv es to b e rand omized. Hence, w e add a ( τ m − m +1) − 1 ,k balls of t yp e k to the ur n, k = 1 , . . . , K . It follo ws that n X j = 1 a j − 1 , k · ν j = N n, 0 X m =1 a τ m − m,k , i.e., n X m =1 ( a m − 1 − a ) ν m = N n, 0 X m =1 ( a τ m − m − a ) . It is easily seen that τ m = m in { n : N n, 0 ≥ m } + m . Due to ( A.4 ), τ m − m = min { n : N n, 0 ≥ m } = sm + O ( p m log lo g m ) a.s. It f ollo ws that a τ m − m − a = B ( τ m − m ) τ m − m diag  1 v  ∂ a ( θ ) ∂ θ + o ( m − 1 / 2 − ǫ ) = B ( sm ) sm diag  1 v  ∂ a ( θ ) ∂ θ + o ( m − 1 / 2 − ǫ ) a.s. Using ( A.4 ), we conclude that n X m =1 ( a m − 1 − a ) ν m = N n, 0 X m =1  B ( sm ) sm diag  1 v  ∂ a ( θ ) ∂ θ + o ( m − 1 / 2 − ǫ )  = Z n/s 0 B ( sx ) sx dx diag  1 v  ∂ a ( θ ) ∂ θ + o ( n 1 / 2 − ǫ ) = Z n 0 B ( x ) x dx diag  1 v  1 s ∂ a ( θ ) ∂ θ + o ( n 1 / 2 − ǫ ) a.s . (A.15) imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 23 Ho w eve r, it is easily c hec ked th at 1 s ∂ a ( θ ) ∂ θ A = ∂ v ( θ ) ∂ θ . (A.16) Com binin g ( A.11 )-( A.16 ) the p ro of of ( 2.4 ) is complete.  Three more lemmas are needed b efore we prov e Lemmas A.1 and A.2 . Lemma A.3 Under Assumption 2.2 and Z 0 , 0 > 0 , we have Z − n,k = O (1) a.s., k = 1 , . . . , K. Pro of . Note that | Z + m | ≥ Z 0 , 0 > 0 f or all m and so that th e balls with negativ e num b ers hav e no chance of b eing d ra wn. In add ition, at most C + 1 balls of eac h treatmen t type hav e the c hance of b eing r emo v ed only when a ball of the same t yp e is dra wn b ecause of the Assumption 2.2 . It follo ws that Z n,k ≥ − C − 1.  Lemma A.4 L et F n = σ ( X 1 , . . . , X n , Z 1 , . . . , Z n ) b e the history sigma field, and A m = P K k =1 a m,k . Supp ose that Assumption 2.2 is satisfie d. Then, A := min m A m > 0 implies E [ ν p n |F n − 1 ] ≤ c p   K X k =1 Z n − 1 ,k  − /A  p +1 Z 0 , 0 | Z + n − 1 | a.s., ∀ p ≥ 1 , (A.17) wher e c p > 0 is a r andom variable that is a function of Z 0 , 0 and min m A m . Particularly, min m A m > 0 implies E [ ν p n |F n − 1 ] = O (1) a.s. (A.18) Pro of . The ev en t { ν n = l } means that when the n -th sub ject is assigned, w e h a v e drawn l + 1 balls conti nuously in whic h the first l balls is of t yp e 0 and the last one is not. Hence, P ( ν n = 0 |F n − 1 ) = 1 − Z 0 , 0 / | Z + n − 1 | , and for l = 1 , 2 , . . . , P  ν n = l |F n − 1  = Z 0 , 0 | Z + n − 1 | l − 1 Y j = 1 Z 0 , 0 | ( Z n − 1 + j a n − 1 ) + | ·  1 − Z 0 , 0 | ( Z n − 1 + l a n − 1 ) + |  . (A.19) Ob viously , P  ν n = l |F n − 1  ≤ Z 0 , 0 / | Z + n − 1 | , l ≥ 1. Note that | ( Z n − 1 + j a n − 1 ) + | = Z 0 , 0 + K X k =1 ( Z n − 1 ,k + j a n − 1 ,k ) + ≥ Z 0 , 0 + K X k =1 Z n − 1 ,k + j A n − 1 . imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 24 L. ZHANG ET AL. It f ollo ws that A > 0 and P K k =1 Z n − 1 ,k ≥ − LA imply for l ≥ L , P  ν n = l |F n − 1  ≤ Z 0 , 0 | Z + n − 1 | l − 1 Y j = L Z 0 , 0 Z 0 , 0 + ( j − L ) A ≤ c 0 Z 0 , 0 | Z + n − 1 | e − 2( l − L ) , (A.20) where c 0 > 0 dep ends only on A and Z 0 , 0 . So E [ ν p n |F n − 1 ] ≤ L X l =1 l p Z 0 , 0 | Z + n − 1 | + ∞ X l = L +1 l p c 0 Z 0 , 0 | Z + n − 1 | e − 2( l − L ) ≤ c p L p +1 Z 0 , 0 | Z + n − 1 | . T aking L =  ( P K k =1 Z n − 1 ,k ) − /A  + 1 completes the pro of of ( A.17 ). ( A.18 ) follo ws from ( A.17 ) and Lemma A.3 .  Lemma A.5 Supp ose that Assumptions 2.1 - 2.2 ar e satisfie d. Then min m,k a m,k > 0 and max m,k a m,k < ∞ a.s . (A.21) Pro of . By Lemm a A.4 of Hu and Zh ang (2004a), we ha ve N n,k → ∞ imp lies b θ n,k → θ k a.s., k = 1 , . . . , K. (A.22) Then, a k ( y ) > 0 f or an y y on closure { b θ m ; m = 1 , 2 , . . . } = N K k =1 { θ k , b θ m,k ; m = 1 , 2 , . . . } . By the con tin u it y of a k ( · ), ( A.21 ) is satisfied.  Pro of of Le mma A.1 . By Lemm a A.5 , A =: min m A m > 0 and A =: max m A m < ∞ (A.23) Note that e Z n 1 ′ = P K k =1 Z n − 1 ,k . By ( A.17 ) and Lemma A.3 , E [ ν n |F n − 1 ] ≤ C 0 Z 0 , 0 / | Z + n − 1 | . S o, according to ( A.1 ) or ( A.2 ), we hav e e Z n 1 ′ = e Z n − 1 1 ′ + ν n A n − 1 − X n ( I − H ) 1 ′ + ∆ M n 1 ′ ≤ e Z n − 1 1 ′ + A n − 1 E [ ν n |F n − 1 ] − h + A n − 1 ( ν n − E [ ν n |F n − 1 ]) + ∆ M n 1 ′ ≤ e Z n − 1 1 ′ + C 0 A Z 0 , 0 | Z + n − 1 | − h + ∆ U n ≤ e Z n − 1 1 ′ + ∆ U n − h / 2 , if e Z n − 1 1 ′ ≥ 2 C 0 AZ 0 , 0 /h, (A.24) where h = min k (1 − P K j = 1 h k j ) > 0. Here, U n = P n m =1 A m − 1 ( ν m − E [ ν m |F n − 1 ])+ M n 1 ′ is a real martingale. Let S n = m ax { 1 ≤ j ≤ n : e Z j 1 ′ < 2 C 0 AZ 0 , 0 /h } , where max( ∅ ) = 0. Then, according to ( A.24 ), e Z n 1 ′ ≤ e Z n − 1 1 ′ + ∆ U n − h / 2 ≤ . . . ≤ e Z S n 1 ′ + ∆ U S n +1 + . . . + ∆ U n − ( n − S n ) h / 2 ≤| Z 0 | ∨  2 C 0 AZ 0 , 0 /h  + U n − U S n − ( n − S n ) h / 2 . (A.25) imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 25 F or the martingale { U n , F n ; n = 1 , 2 , . . . } , we ha ve E [ | ∆ U n | 2+ δ |F n − 1 ] ≤ C + C max j A 2+ δ j = O (1) due to Assump tion 2.2 and ( A.18 ). Accordingly , we can sho w that U n = O  p n log log n  a.s., (A.26) max m ≤ √ n log n | U n − [ √ n log n ]+ m − U n − [ √ n log n ] | = o ( n 1 2+ δ log n ) a.s. (A.27) If n − S n ≥ √ n log n , then f or n large enough U n − U S n − ( n − S n ) h / 2 ≤ O  p n log log n  − h p n log n/ 2 < 0 due to ( A.26 ). Note that n ≥ S n . If n − S n < √ n log n , then U n − U S n − ( n − S n ) h / 2 ≤ 2 max m ≤ √ n log n | U n − [ √ n log n ]+ m − U n − [ √ n log n ] | = o ( n 1 2+ δ log n ) a.s . b y ( A.27 ). I t follo ws that P K k =1 Z n,k ≤ o ( n 1 / 2 − δ 0 ) a.s. due to ( A.25 ). Ho w- ev er, Z − n,k = O (1) a.s. by Lemma A.3 . ( A.3 ) is pro ved.  Pro of of Lemma A.2 . Recall Q n,k = P n m =1 X m,k ( ξ m,k − θ k ) k = 1 , . . . , K , and b oth { M n,k , A n ; n ≥ 1 } and { Q n,k , A n ; n ≥ 1 } are martingales. Accord- ing to the la w of the iterated logarithm f or martingales, w e ha ve M n,k = O ( p n log lo g n ) and Q n,k = O ( p n log log n ) a.s. (A.28) Ho w eve r, for eac h k = 1 , . . . , K , b θ n,k − θ k = Q n,k + O (1) N n,k + c 2 a.s. (A.29) ( A.12 ) remains true by Lemmas A.1 . By ( A.12 ) and ( A.28 ) we h a v e N n ( I − H ) = n X m =1 a m − 1 ν m + o ( n ) a.s. (A.30) Note that all elemen ts of the vec tor P n m =1 a m − 1 ν m are b et ween a N n, 0 and aN n, 0 , where a = min m,k a m,k and a = max m,k a m,k . Hence, it is ob vious that lim inf n →∞ N n, 0 /n > 0 a.s., b ecause otherwise the limit of N n /n ma y b e 0 w hic h con tradicts to N n 1 ′ = n . On the other hand, the k -th elemen t of imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 26 L. ZHANG ET AL. N n ( I − H ) d o es n o exceed (1 − h k k ) N n,k . I t follo ws that lim in f n →∞ N n,k /n > 0 a.s. by ( A.30 ), whic h, together w ith ( A.29 ) and ( A.28 ), implies b θ n,k − θ k = O  Q m,k + O (1) n  = O  r log log n n  → 0 a.s. ( A.6 ) is p r o v ed and also a m,k − a k = a k ( b θ m ) − a k ( θ ) = O ( k b θ m − θ k ) = O  p (log log m ) /m  a.s. (A.31) Hence, b y Theorem 2.18 of Hall and Heyde (1980) it is easy to c hec k that P n m =1 ( a m − 1 ,k − a k )( ν m − E [ ν m |F m − 1 ]) = o ( √ n ) a.s. It follo ws that n X m =1 ( a m − 1 ,k − a k ) ν m = n X m =1 ( a m − 1 ,k − a k ) E [ ν m |F m − 1 ] + o ( √ n ) = n X m =1 O  r log log m m  O (1) + o ( √ n ) = O ( p n log log n ) a.s . (A.32) b y ( A.18 ) and ( A.31 ). Com bin in g ( A.12 ), ( A.28 ), an d ( A.32 ) yields N n − N n, 0 a ( I − H ) − 1 = O ( p n log log n ) a.s., whic h, together with N n 1 ′ = n , implies ( A.4 ) and ( A.5 ). Then , com bin ing ( A.5 ), ( A.28 ), and ( A.29 ) yields b θ n,k − θ k = Q n,k + O (1) nv k + O ( √ n log lo g n ) = Q n,k nv k + o ( n 1 / 2 − δ 0 ) a.s. ( A.7 ) is p r o v ed, and the pro of of Th eorem 2.2 is completed.  Pro of of Theorem 2.4 . Note that Ass umptions 2.1 and 2.2 are satisfied, and Z 0 , 0 > 0, H 1 ′ = 1 ′ . Similar to ( A.24 ), e Z n 1 ′ = e Z n − 1 1 ′ + ν n A n − 1 + ∆ M n 1 ′ ≤ e Z n − 1 1 ′ + C 0 A Z 0 , 0 | Z + n − 1 | + ∆ U n ≤ e Z n − 1 1 ′ + C 0 A/ √ n + ∆ U n , if e Z n − 1 1 ′ ≥ Z 0 , 0 √ n. It f ollo ws that e Z n 1 ′ ≤ e Z S n 1 ′ + ∆ U S n +1 + . . . + ∆ U n + C 0 A ( n − S n ) / √ n ≤ 2 C 0 A √ n + U n − U S n ≤ 2 C 0 A √ n + 2 max m ≤ n | U m | , imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 27 where S n = m ax { 1 ≤ j ≤ n : e Z j 1 ′ < Z 0 , 0 / √ n } and max( ∅ ) = 0. Hence, e Z n = O ( p n log log n ) a.s. and = O P ( √ n ) , b y the pr op erties of a martin gale and Lemma A.3 . So, b y ( A.2 ) and the la w of th e iterated logarithm of martingales, it f ollo ws that N n ( I − H ) = M n + n X m =1 a m − 1 ν m − e Z n + e Z 0 n X m =1 a m − 1 ν m + O ( p n log log n ) a.s . Multiplying b y 1 ′ yields P n m =1 ν m A m − 1 = O ( √ n log log n ) a.s. , and then N n, 0 = O ( √ n log lo g n ) a.s. and P n m =1 a m − 1 ν m = O ( √ n log log n ) a.s. b y ( A.21 ). So, ( N n − n v )( I − ( H − 1 ′ v )) = N n ( I − H ) = O ( p n log log n ) a.s . It follo ws that N n − n v = O ( √ n log log n ) a.s. b ecause ( I − ( H − 1 ′ v )) is in ve r tible. T he pro of of N n − n v = O P ( √ n ) is similar.  Pro of of T he orem 2.1 . Recall ( A.2 ); w e ha ve e Z n − e Z 0 = n X m =1 a m − 1 ν m + N n ( H − I ) + M n . (A.33) It follo ws th at | e Z n | = e Z n 1 ′ ≥ ( γ − 1) n + M n 1 ′ b y noticing H 1 ′ = γ 1 ′ . Hence, lim inf n →∞ | e Z + n | n ≥ lim in f n →∞ | e Z n | n ≥ γ − 1 > 0 a.s. Without loss of generalit y we can th us assum e that | e Z + n | ≥ cn > 0 for all n . Then , the conclusion of L emm a A.3 remains true. By Lemma A.3 , e Z m = e Z + m + O (1) a.s.. On the other hand, by ( A.19 ) w e ha ve P ( ν m = 1 |F m − 1 ) = Z 0 , 0 | Z + m − 1 |  1 − Z 0 , 0 | ( Z m − 1 + a m − 1 ) + |  ≤ c/m a.s., P ( ν m ≥ 2 |F m − 1 ) = Z 0 , 0 | Z + m − 1 | Z 0 , 0 | ( Z m − 1 + a m − 1 ) + | ≤  Z 0 , 0 | Z + m − 1 |  2 ≤ c/m 2 . It follo ws th at P ( ν m ≥ 2 i.o. ) = 0 and P n m =1 I { ν m = 1 } = O (log 2 n ) a.s. by Theorem 3.3.9 (ii) of Stout (197 4). So by the assump tion s tated in Theorem 2.1 th at 0 ≤ a m,k ≤ C m 1 / 2 − δ 0 , n X m =1 a m − 1 ν m = O (max m ≤ n A m − 1 )  n X m =1 I { ν m = 1 } + O (1)  = o ( n 1 / 2 − δ 0 / 2 ) a.s. imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 28 L. ZHANG ET AL. whic h means that the immigrated b alls can b e neglected. In addition, P ( X m,k = 1 |F m − 1 ) = Z + m − 1 ,k Z 0 , 0 + | e Z + m − 1 | 1 − Z + 0 , 0 Z 0 , 0 + | e Z + m − 1 | ! + P ( X m,k = 1 , ν m ≥ 1 |F m − 1 ) = Z + m − 1 ,k | e Z + m − 1 | + O  1 m  a.s. It f ollo ws that e Z + n = e Z n + O (1) = N n ( H − I ) + M n + o ( n 1 / 2 − δ 0 / 2 ) = n X m =1 ( X m − E [ X m |F m − 1 ])( H − I ) + M n + n X m =1 E [ X m |F m − 1 ]( H − I ) + o ( n 1 / 2 − δ 0 / 2 ) = n − 1 X m =0 ( X m − E [ X m |F m − 1 ])( H − I ) + M n + n X m =1 h e Z + m − 1 | e Z + m − 1 | + O  1 m i ( H − I ) + o ( n 1 / 2 − δ 0 / 2 ) =( γ − 1) n v + ( γ − 1) n X m =1 ( X m − E [ X m |F m − 1 ]) f H + M n + n − 1 X m =0 e Z + m | e Z + m | ( γ − 1) f H + o ( n 1 / 2 − δ 0 / 2 ) a.s. The expansion for e Z + n is similar to that for Y n in (6.2) of Zh ang and Hu (pp. 1421-1324 , 2009 ). Hence, the rest of the pro of is omitted. A CKNOWLEDG EMENTS Sp ecial thanks go to the anonymous r eferee and the asso ciate ed itor f or their constructiv e comments, whic h led to a muc h imp ro ve d ve rs ion of the p ap er. REFERENCES [1] A threy a, K. B. and Karlin, S. (1968 ). Embedding of urn sc hemes in to contin uous time branching p rocesses and related limit th eorems. Ann. Math. Statist. 39 1801– 1817. imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 29 [2] A threy a, K. B. and Ney, P. E. (1972). Br anching Pr o c esses . Spring-V erlag, Berlin. [3] Ba i , Z. D. and Hu, F. (1999). Asymp totic theorem for urn mo dels with nonhomo- geneous generating matrices. Sto chastic Pr o c ess Appl. 80 87–10 1. [4] Ba i , Z. D. and Hu, F. (2005). Asymptotics in randomized urn mo dels. Ann. Appl. Pr ob ab. 15 914–940 . [5] Ba i , Z. D., Hu, F. and Rosenber ger , W. F. (2002). Asymptotic prop erties of adaptive designs for clinical trials with delay ed resp onse. Ann. Statist. 30 122–139. [6] Ba i , Z. D., Hu, F. and She n, L. (2002). An Adaptive Design for Multi-Arm Clinical T rials. Journa l of Mul tivariate A nalysis, 81 , 1-18. [7] Bh a tt a char y a , R . (2008 ). Urn-b ased resp onse adaptive pro cedures and optimal- it y . Drug I nformation Journal 42 441–448. [8] Be ggs , A. W. (2005). O n the con vergence of reinforcemen t learning. Journal of Ec onomic The ory 122 1–36. [9] Be na ¨ ım M., Schre iber S. J. and T arr ` es , P . (2004). Generalized u rn mo dels of evol ut ionary pro cesses. A nn. Appl. Pr ob ab. 14 1455–1478. [10] Donnell y, P. and Kur tz, T. G. (1996). The asymptotic b ehavior of an u rn mod el arising in p opulation genetics. Sto chastic Pr o c ess Appl . 64 1–16. [11] Durham, S . D., Flourno y, N. and Li, W. ( 1998). Sequ entia l designs for max- imizing the probabilit y of a fav orable resp onse. Canadian Journal of Statistics 3 479–495 . [12] Hall, P. & Heyde , C. C. (1980). Martingale Limit The ory and its A ppl ic ations . Academic Press, London. [13] Higu e ras, I., Moler, J., Flo, F. and San Miguel, M. (2006), Central Limit theorems for generalized P´ oly a urn mo d els. J. Appl. Pr ob. 43 938–951. [14] Hoppe, F. M. (1984). P´ olya-lik e urns and the Ew ens’ sampling form ula. Journal of Mathematic al Biolo gy 20 91– 94. [15] Hu, F. and Ro senberger, W. F. (2003). O p timalit y , v ariabili ty , p ow er: eva luating response-ad aptive randomization pro cedures for t reatmen t comparisons. J. Amer. Statist. Asso c. 98 671–678. [16] Hu, F. and Rosenber ger, W . F. (2006). The The ory of R esp onse-A daptive R an- domization in Clini c al T rials . John Wiley and Sons. Wiley Series in Probabilit y and S tatistics. [17] Hu, F., Rosenber ger, W. F. and Zhang , L.-X. (2006). Asympt otically b est response-ad aptive rand omization proced ures. J. Statist. Plann. I nf. 136 1911– 1922. [18] Hu , F. and Zhang , L. X . (2004a). A symptotic prop erties of doub ly adaptive biased coin designs for multi-treatmen t clinical trials. Ann . Statist. 32 268–301. [19] Hu, F. and Zhang, L.-X. (2004b). Asympt otic normalit y of urn mod els for clinical trials with delay ed response. Bernoul li 10 447–463. [20] Iv anov a, A. (2003). A p la y-th e-winner typ e urn mo del with reduced v ariabilit y . Metrika 58 1–13. [21] Iv anov a, A. (2006). Urn d esigns with immigration: Useful connection with contin- uous time sto c hastic pro cesses. J. Statist. Plann. Inf. 136 1836-1844. imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 30 L. ZHANG ET AL. [22] Iv anov a, A. and Flournoy, N . (2001). A birth and death urn for ternary out- comes: stochastic pro cesses app lied to u rn mod els. In Pr ob ability and Statistic al Mo dels with Appli c ations (Charalam bides, Ch. A., Koutras, M. V. an d Balakrish- nan, N ., Eds.). Chapman & Hall/C RC, 583–6 00. [23] Iv anov a, A., Rosenber ger, W. F., Durham, S. D. and Flourno y, N. (2000). A birth and death urn for rand omized clinical trials: asymptotic metho ds. Sankhya B 62 104–118. [24] Janson, S. (2004). F unctional limit theorems for multit y p e branching pro cesses and generalized P´ olya urn s. Sto chastic Pr o c ess. Appl. 110 177 –245. [25] Johnson , N. L. and Kotz , S. (1977). Urn Mo dels and Their Applic ations . Wiley , New Y ork. [26] Kotz , S. and Balakrishnan , N. (1997). A d v ances in urn mo dels during the past tw o decades, I n A dvanc es in Combinatorial Metho ds and Applic ations to Pr ob ability and Statistics (Balakrishnan, N., Ed.) Birkh¨ auser, Boston. [27] Knoblauch, K. , Neitz, M. and Ne i tz, J. (2006). An urn mo del of the develop- ment of L/M cone ratios in human and macaque retinas. Visual Neur oscienc e 23 387–394 . [28] Ma tthews, E. E., Cook, P. F., Terada, M. and Aloia, M. S. ( 2010). Rand om- izing researc h participants: Promoting balance and concealmen t in small samples. R ese ar ch in Nursing and He alth 33 243– 253. [29] Milenko vic, O. and Compton, K. J. (2004). Probabili stic transforms for com bi- natorial urn mo dels. Combinatorics, Pr ob ability and Computing 13 645–675. [30] Nive n, R. K. and G rendar, M (2010 ). Generalized clas sical, quantum and in ter- mediate statistic s and the P´ olya urn model. Physics L etters A 373 621–62 6. [31] Rosenber ger, W. F., St allard, N ., Iv anov a, A., H arper, C . N. and Ricks, M. L. (2001). Opt imal adaptive designs for binary resp onse trials. Biometrics 57 909–913 . [32] Smythe , R.T. (1996). Central limit theorems for urn mo dels. Sto chastic Pr o c ess. Appl. 65 115–137. [33] Stout, W. F. (1974). A l m ost sur e c onver genc e . A cademic Press, New Y ork. [34] T amura , R . N., F a ries, D. E., A ndersen, J. S. and H eiligenstein, J. H. (1994). A case study of an adapt iv e clinical trial in th e treatment of out-patients with depressiv e disorder. J. Amer. Stat ist. Asso c. 89 768– 776. [35] Tymofyey ev, Y., Rosenberger, W.F. and Hu, F. (2007). Imp lemen ting optimal allocation in sequential binary resp onse exp eriments. J. Amer. Statist. Asso c. 102 224–234 . [36] Wei , L. J. (1979). The generalized P´ olya’s urn d esign for sequ entia l m ed ical trials. Ann . Statist. 7 291–296. [37] Wei , L. J. and Du rh am, S. D. (1978). The randomized play-the-winner rule in medical trials. J. Amer . Statist. Asso c. 73 , 840–843. [38] Zhang, L. J. an d Rosenber ger W. F. (2006). Resp onse-adaptive randomization for clinical trials with continuous outcomes. Biometrics 62 562C569. [39] Zhang, L.-X. (2004). Strong approximations of martingale vectors and their appli- cations in Mark ov-c hain adaptive designs. A cta Math. Appl. Sinic a, English Series 20 (2) 337 –352. imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018 IMMIGRA TED URN MODELS 31 [40] Zhang, L.-X. and Hu, F. (2009). The Gaussian approximation for m ulti-color generalized F riedmans urn mo d el. Scienc e in China Series A: Mathematics 52 (6) 1305C1 326. [41] Zhang, L.-X., Chan, W. S., Cheung, S. H. and Hu, F. (2007). A generalized urn model for clinical trials with d ela yed resp onses. Statistic a Sinic a 17 387–409. [42] Zhang, L.-X., H u , F. and C heung, S . H. (2006). A symptotic theorems of seq u en- tial estimation-adjusted u rn mo dels. A nn. Appl. Pr ob ab. 16 340–369. [43] Zhu, H. and Hu, F. (2009). I mplementi n g optimal allo cation in sequential contin- uous resp onse exp eriments. J. Statist. Pl ann. Inf. 139 2420–2430. L-X. ZHANG DEP AR TMENT OF MA THEMA TICS ZHEJIANG UNIVERSI TY HANGZHOU 310027 PEOPLE’S REPUBLIC OF CHINA E-mail: stazlx@zju.edu.cn F. HU DEP AR TMENT OF ST A TISTICS UNIVERSITY OF VIRGINA HALSEY HALL, CHARLOTTESVILLE VIRGINIA 22 9 04-4135, USA E-mail: fh6e@virginia.edu S. H. CHEUNG DEP AR TMENT OF ST A TISTICS THE CHINESE UNIVERSITY OF HONG K ONG SHA TIN, N.T., HONG KONG PEOPLE’S REPUBLIC OF CHINA E-mail: shc heung@sta.cuhk.edu.hk W. S . CHAN DEP AR TMENT OF FINANCE THE CHINESE UNIVERSITY OF HONG K ONG SHA TIN, N.T., HONG KONG PEOPLE’S REPUBLIC OF CHINA E-mail: c hanws@cuhk.edu.hk imsart-aos ver. 2009/08/13 file: annalsimu rev_sep_3.tex date: Oc tober 31, 2018