Recall (1), (16) and (21). Comparing the P W rule, the RPW rule and the DABC design, one finds that, keeping the same limiting proportion
v = (92/(91 + Q2),QI/(QI + 92)) of assignments, the asymptotic variance in using PW rule and the drop-the-loss rule is the smallest, and the one in using RPW is the largest, i.e.,
apw < &DABC = Oa< &RPW, a > 1.
Hu and Rosenberger [16] points out that, if the observed allocation pro- portions are asymptotically normal, the asymptotic power is an decreasing function of the asymptotic variance of the allocation proportions.
Recently, Hu, Rosenberger and Zhang [17] obtains the lower bound of the asymptotic variance. Suppose the response sequence {£„} is an i.i.d.
sequence. And suppose the limiting proportion v = p ( 0 ) is a function of the parameter 0 = E£x. Assume that for each fc = 1,...,K, the response
£i,fc comes from a family of exponential distributions:
C(uk)exp{£lkUk}dn,uk G Uk C K and
6k = EÊi,fc = 0fc(ôk)>
and let Ik{uk) be the Fisher information of uk for this family. Then
^=Var(ei,fc) = ( ^ )2/ 4 K ) .
Hu, Rosenberger and Zhang [17] shows the following theorem.
Theorem 6.1. Suppose
n ^ N J n - v ) £ JV(0,V(u)).
Under some regularity conditions, there exists a Uo C U = U i ® • • • ® U # with Lebesgue measure 0 such that for every u G U — Uo,
0p(e(u))' ^P(e(u)) A
V(u) > gu I x(u,v) ^ u = B(u),
I(u, v) = diag{vih{u{),... ,vKIi{uK))-
Here, for two covariance matrices A and B , A > B means that A — B is non-negatively definite.
We can rigorously define an asymptotically best response-adaptive pro- cedure as one in which V(u) attains the lower bound B(u) for a particular target allocation p(6).
It is easily seen that
9P(0)'90(u)' , ae(u) dP(@y
B ( U ) = " 9 0 - ~ ^ r * ( U'V ) _^ 9 0 "
_ 9 p ( 0 ) ' / 1 . d6>i.2 1 td6K^dp{@) d@
A- ( 1 (aVi\Z x , a ^ , 2 \ o p ^ 5U1/1(u1)ldn1 j " ' " % / I M1 & K J / 90
= - ^ - ^ - ^ . . . — c r ^ - ^ - = £3( 0 ) , where E3 = E3( 0 ) is denned in (24).
In the case K = 2,
Further, if the responses are dichotomous (success and failure) and v = (92/(91 + 92),9i/(9i + 92)), then cr3 = apiy- It follows that the PW rule and the drop-the-loss rule are asymptotically best response-adaptive pro- cedure. However, the PW rule is too deterministic. On the surface, the drop-the-loss rule would seem to give us everything we want in a response- adaptive randomization procedure: it is fully randomized and its asymp- totic variance attains the lower bound. However, it can only target the PW rule allocation, which is not optimal in any formal sense, and previously reported simulations have shown that it can be slower to converge for large
values of PA and ps and becomes more deterministic for small values of PA and PB (see Hu and Rosenberger [16]). The DABC procedure solve some of these deficiencies, in that they can target any desired allocation, and can approach the lower bound for large values of a. However, the procedure becomes more deterministic as a becomes larger, and hence careful tuning of a must be done in order to counter the tradeoff.
When K > 3, no of the designs mentioned above is showed asymptot- ically best. However, in using the general DABC design, if the allocation rule is chosen as in (25), then the asymptotic variance of n1 , / 2(N„/n — v) satisfies
1 „ 2(1 + oA ^ ^
— — - S i + ) ;S3 \ S3 as a / oo.
1 + 2a 1 + 2a
If a is large enough, the asymptotic variance of n1 , / 2(N„/n—v) is very close to the lower bound S3. So, the DABC design is nearly asymptotically most powerful if a is chosen large.
Remark 6.1. Usually, many cases, in which the parameter 0 is not a mean of the response £n, can be transferred to the case we studied. In fact, if for each k, an estimate 9nk = 0nfc(£jfc : j = 1,2, • • • , n) of 0 can be written in the following form:
1 n
*nfc = - £ A&fc) + o(n-^2-6), for some S > 0, (27) then in the adaptive designs, we can define
#m-l,fc = 0JVm_ilfc,k(£jk : Xjk — l , j = 1 ) 2 , ••• ,J7l - 1)
and then
0m-i,k = j : - T E XM£jk)+°(N™-U5)-
JVm-l,fc + 1 ,
Many maximum likelihood estimators and moment estimators satisfy (27).
Appendix A. Strong approximations for martingale vectors The limiting theorems for martingales are main tools in studying the asymp- totic properties of adaptive designs. We study the asymptotic properties of adaptive designs by first approximating the models to certain multi- dimensional martingale vectors and then approximating the martingales
to Gaussian processes. In this appendix, we present some strong ap- proximations for martingale vectors. We assume that {Zn,.F„;n > 1} is a square-integrable sequence of Kd-valued martingale differences, denned on (fi,.F, P), IFO is the trivial <r-field. The probability space (£l,J-, P) is also assumed to be rich enough such that there is a uniformly dis- tributed random variable U, which is independent of { Zn; n > 1}. Denote Sn(m) = E r = m + i Zfc and S„ = Sn(0). Let
<rn = E[ZnZn\!Fn-i}, also let £ „ = £ £= 1 ak, S„(m) = YlT=m+i ak and
K = M£n) = E[||zfe||2|j-fc_1].
The first strong approximation theorem for martingales can be found in Strassen's fundamental paper [31], where the strong approximation is established for one-dimensional martingale via the Skorohod embedding theorem. For d > 2, the strong approximations are also studied by many authors, though it is impossible to embed a general Rd-valued martingale in an Revalued Gaussian process (c.f., Monrad and Philipp [25]). One can refer to Morrow and Philipp [27], Eberlein [9], Monrad and Philipp [26]
etc. However, in the approximations appeared in literatures, it is always assumed that the conditioned covariance matrix Sn is convergent in L\ with some convergence rates. For example, Eberlein [9] assumed the following condition: for some 6 > 0
||E[2n(m)|.Fm] - nTHi = Oin1'6), uniformly in m, and Monrad and Philipp [26] assumed that for some 0 < p < 1
Emax{||Sn - TVn\\/f(Vn)}p < oo.
n > l
For the martingales which can approximate our adaptive design models, it is usual hard to verify that Sn converges in L\ (or Lp) with the needed convergence rates. However, it is more easy to show that Sn converges almost surely with some convergence rates. So, we present the following approximation theorems for martingale vectors, whose proofs can be found in Zhang [35].
T h e o r e m A . l . Suppose that there exist constants 0 < 6, e < 1 and a covariance matrix T , measurable with respect to T^ for some k > 0, such that
E„ - n T = 0{nl~9) a.s. or ||S„ - n T ^ = 0{nl-e),
oo
£ E[||Zn||2/{||Zn||2 > n 1 " ^ ^ ] / ^ < oo a.s. (A.l)
n=l
Then there exists a sequence {Yn;n > 1} of i.i.d. Rd-valued standard normal random vectors, independent ofT, such that
Here K > 0 is a constant depending only on 6, e and d.
T h e o r e m A.2. Suppose that there exists constants 0 < e < 1 such that (A.l) is satisfied, and that T is a covariance matrix measurable with respect t° 3~k for some k > 0. Then for any 8 > 0, there exists an K > 0 and a sequence {Yn;n > 1} of i.i.d. M.d-valued standard normal random vectors, independent ofT, such that
S„ - £ YmT1/2 = 0(n1'2-*) + 0{alJ2+5) a.s.
Here an = m a x ^ , , | | Sm - mT||.
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A S Y M P T O T I C S A N D I N F E R E N C E S
S. N. C H I U
Department of Mathematics Hong Kong Baptist University
Kowloon Tong Hong Kong
E-mail: snchiu@hkbu.edu.hk
Seeds are randomly scattered in Rd according to a spatial-temporal point process.
Each seed has its own potential germination time. Each seed t h a t succeeds in germinating will be the centre of a growing spherical inhibited region t h a t prohibits germination of any seed with later potential germination time. The radius of an inhibited region at time t after t h e germination of t h e seed at its centre is vt. The set of locations first reached by the growth of the inhibited region originated from x is called the cell of x. T h e space will be partitioned into cells and this space-filling structure is called the Johnson-Mehl tessellation. We show that the time until a large cube is totally inhibited has an extreme value distribution. In particular, for d — 1, we obtain the exact distribution of this time by transforming the original process to a Markov process. Moreover, we prove a central limit theorem for the number of germinations within [0, L)d. Finally, the maximum likelihood estimation for v, a nonparametric estimation for the intensity measure and for its density, and the maximum likelihood estimation for the parameters of the intensity with known analytical form are proposed.
1. Introduction
Consider a set of n < oo distinct, isolated points {x,}, called seeds, in Rd. A seed at Xi will be stimulated by an internal or external stimulus after a time t{. A seed, once stimulated, immediately tries to germinate and at the same time to prohibit other seeds from germination by generating a spherical inhibited region the radius of which grows at a positive speed v.
A seed stimulated at time t* fails to germinate if and only if its location has been inhibited on or before U. Such a germination-growth process was motivated by applications in many diverse fields, such as crystal growth (Johnson and Mehl15, Kolmogorov16), DNA replication (Vanderbei and Shepp26, Cowan et al.10), synapses (Bennett and Robinson1, Quine and
136
Robinson23'24) and so on (see Okabe et al.22 for more details).
The set of locations
f llx — ii|I ||a; — Xj\\ ,,. , .1 ) x : J! lii + t < II 111 +1 Vj ^ i }
{ v v J first reached by the growth of the inhibited region originated from Xi is called the cell of xi. The space Rd will be partitioned into at most n cells, the interior of which are disjoint. Such a space-filling structure is known as the Johnson-Mehl tessellation. The most important special case of Johnson-Mehl tessellation is that all ij's are the same and so all seeds ger- minate simultaneously. As a result, all edges are segments of the bisectors between the seeds of two neighbouring cells and such a special Johnson- Mehl tessellation is the well-known Voronoi diagram (Okabe et al.22).
In this paper, seed locations and stimulation times are assumed to form a spatial-temporal Poisson process with intensity measure dxdA(t), where A is a non-decreasing function satisfying
A(t) = 0, for t < 0, A(t) < oo, for t < oo, A(oo) > 0.
That is to say, the number of seeds located in A C M.d and stimulated during [ti,^] follows a Poisson distribution with mean
/ didA(t).
The Lebesgue measure dx means that seed locations are spatially homoge- neous in Rd.
Characteristics of the typical cell of the Johnson-Mehl tessellation gen- erated by a spatial-temporal Poisson process have been studied in details by Evans11, Gilbert12, Meijering18 and M0ller20,21. In this paper limit theorems and statistics problems will be addressed.