Asymptotically best response-adaptive randomization
procedures
By FEIFANG HU
Department of Statistics, University of Virginia, Halsey Hall, Charlottesville, VA
22904-4135 U. S. A.
WILLIAM F. ROSENBERGER
Department of Mathematics and Statistics, University of Maryland, Baltimore
County, 1000 Hilltop Circle, Baltimore, MD 21250 U. S. A.
and
LI-XIN ZHANG
Department of Mathematics, Zhejiang University, Xixi Campus, Zhejiang, Hangzhou
310028 P. R. China
SUMMARY
We derive a lower bound on the asymptotic variance of the allocation proportions from responseadaptive randomization procedures when the allocation proportions are asymptotically normal. A
procedure that attains this lower bound is defined to be asymptotically best. We then compare the
asymptotic variances of five procedures to the lower bound. We find that a procedure by Zelen and a
procedure by Ivanova attain the lower bound and a procedure by Eisele and its extension to K > 2
treatments can attain the lower bound but are, in general, not asymptotically best. We discuss the
tradeoffs among the benefits of randomization, the benefits of attaining the lower bound, and the
benefits of targeting an optimal allocation. We conclude that none of these procedures possesses all
of these benefits.
Some key words:
Adaptive designs; Clinical trials; Doubly-adaptive biased coin design; Neyman
allocation; Rao-Cramér lower bound; Urn models.
i
1. Introduction
Response-adaptive randomization procedures for clinical trials incorporate sequentially accruing response data into future randomization probabilities (Rosenberger and
Lachin, 2002). Hu and Rosenberger (2003) have recently developed a template for a
careful evaluation of a response-adaptive randomization procedure for binary responses.
The template considers the competing goals of desiring high power for treatment comparisons while assigning fewer patients to the inferior treatment. It requires (1) a
target allocation, (2) a randomization procedure, and (3) the asymptotic distribution
of the allocation proportions to each treatment following that randomization procedure. In particular, if the observed allocation proportions are asymptotically normal,
the asymptotic power is an decreasing function of the asymptotic variance of the allocation proportions.
For requirement (2), Rosenberger and Lachin (2002) define three classes of responseadaptive randomization procedures: urn models, sequential estimation procedures, and
treatment effect mappings. There is, as yet, little information on the asymptotic distribution of allocation proportions from treatment effect mappings, and we will therefore
focus in this paper on specific randomization procedures from only the classes of urn
models and sequential estimation procedures. For K = 2 treatments, urn models
encompass Wei and Durham’s procedure (1978), often referred to as the “randomized
play-the-winner rule”, and Ivanova’s procedure (2003), which she refers to as the “dropthe-loser rule”. Sequential estimation procedures for two treatments can be described
in full generality for two treatments by Eisele’s procedure (1994), which he called the
“doubly-adaptive biased coin design”. Each of these procedures yields asymptotically
normal allocation proportions, and the asymptotic variance is known, thus satisfying
requirement (3) above. These procedures are generally analyzed according to a simple homogeneous parametric structure, with pA and pB being the success probabilities
for those patients assigned to treatment A and B, respectively. We can also define
qA = 1 − pA and qB = 1 − pB .
These procedures have analogs for K > 2 treatments, although the theoretical
1
developments and practical consequences are far more difficult to ascertain, and are
the subject of current research. For example, Eisele’s procedure was generalized by Hu
and Zhang (2003) to K > 2 treatments; the extension of Wei and Durham’s procedure
is the generalized Friedman’s urn model of Wei (1979); and the extension of Ivanova’s
procedure is the birth and death urn with immigration by Ivanova, et al. (2000).
For requirement (1), various target allocations ρ(pA , pB ) can be proposed, where ρ is
the proportion of patients assigned to treatment A in the limit. In all practical cases, ρ
will be a function of the unknown parameters, and hence the argument pA , pB . We refer
to the limiting allocation of urn models as “urn allocation”, which for both Wei and
Durham’s and Ivanova’s procedures we can write as ρ(pA , pB ) = qB /(qA + qB ). The urn
models have the inherent drawback that they can target only urn allocation, which is
not optimal in any formal sense (but allocates according to the relative risk of treatment
failure, which may be intuitively attractive). Various target allocations have been
proposed based on formal optimality criteria (Rosenberger, et al., 2001). If one wishes
to maximize the power of the usual test comparing two binomial probabilities, it is well√
√
√
known that Neyman allocation should be used, given by ρ = pA qA /( pA qA + pB qB ).
However, Neyman allocation may be inappropriate for ethical reasons because it assigns
more patients to the inferior treatment if pA + pB > 1. An alternative allocation that
minimizes the expected number of treatment failures for fixed power is ρ(pA , pB ) =
√
√
√
pA /( pA + pB ) (Rosenberger, et al., 2001), which we refer to as “RSIHR allocation”
as an acronym of the authors of the original paper. Other forms of optimal allocation
are given in that paper as well. An advantage of Eisele’s procedure over the urn
models is that we can target any desired allocation, including urn allocation, Neyman
allocation, and RSIHR allocation.
In this paper we derive a lower bound on the asymptotic variance of the allocation
proportions for general response-adaptive randomization procedures. The technique
used is an extension of the Rao-Cramér lower bound to dependent sequences of random
variables. We show that Ivanova’s procedure has minimum variance in the class of all
response-adaptive procedures targeting urn allocation. While Eisele’s procedure is not
2
minimum variance, we can tune certain parameters to obtain a deterministic procedure
that is minimum variance. We also discuss several tradeoffs that we encounter in
practice: the desire for high power, the desire for a fully randomized procedure, and
the desire to target any allocation of interest. In Section 2, we present the theorem
for the special case of binary response and K = 2 treatments, and then demonstrate
the results for the response-adaptive randomization procedures we have discussed so
far. In Section 3, we generalize the results to any response-adaptive randomization
procedure for general K treatments under mild regularity conditions. In Section 4, we
draw conclusions. Proofs are relegated to an Appendix.
2. Results for Two Treatments
In this section, we present the criterion for an asymptotically best response-adaptive
randomization procedure for K = 2 treatments. We first describe the response-adaptive
randomization procedures we will be examining and also give some important asymptotic properties. Let NA (n) be a random variable indicating the number of patients
assigned to treatment A in n patients (NB (n) = n = NA (n) is then determined).
Wei and Durham’s procedure. Wei and Durham’s (1978) famous randomized playthe-winner rule uses an urn model to allocate treatments. The urn starts with a fixed
number of type A balls and type B balls in an urn. To randomize a patient, a ball
is drawn, the corresponding treatment assigned, and the ball replaced. An additional
ball of the same type is added if the patient’s response is success and an additional ball
of the opposite type is added if the patient’s response is a failure. The urn can only
target one value, namely ρ(pA , pB ) = qB /(qA + qB ). Smythe and Rosenberger (1995)
showed that if pA + pB < 3/2,
n1/2 (NA (n)/n − qB /(qA + qB )) → N (0, v),
in law, where
v=
qA qB (5 − 2(qA + qB ))
.
(2(qA + qB ) − 1)(qA + qB )2
The limit is presumably non-normal for pA + pB > 3/2.
3
Ivanova’s procedure. Ivanova (2003) considered an urn containing balls of three
types, type A, type B, and type 0. A ball is drawn at random. If it is type A or type
B, the corresponding treatment is assigned and the patient’s response is observed. If it
is a success, the ball is replaced and the urn remains unchanged. If it is a failure, the ball
is not replaced. If a type 0 ball is drawn, no subject is treated, and the ball is returned
to the urn together with one ball of type A and one ball of type B. Ivanova (2003)
shows that the limiting allocation of the procedure is also ρ(pA , pB ) = qB /(qA + qB ),
and also that
n1/2 (NA (n)/n − qB /(qA + qB )) → N (0, v),
in law, where
v = qA qB (pA + pB )/(qA + qB )3 .
Eisele’s procedure. Eisele’s (1994) procedure is designed to target any desired allocation ρ(pA , pB ). He defines a function g(x, ρ) that represents the closeness of the current
allocation proportion to the target allocation, when replaced by the current values of
maximum likelihood estimators of pA and pB . At the jth allocation, the procedure
allocates patient j to treatment A with probability g(NA (j − 1)/(j − 1), ρ(p̂A , p̂B )). Hu
and Zhang (2003) define the following function g having nice interpretive properties.
For nonnegative integer γ,
ρ(ρ/x)γ
;
ρ(ρ/x)γ + (1 − ρ)((1 − ρ)/(1 − x))γ
g(0, ρ) = 1;
g(x, ρ) =
g(1, ρ) = 0.
Hu and Zhang also show that
n1/2 (NA (n)/n − ρ(pA , pB )) → N (0, v),
in law, where
v=
qA qB ((1 + 2γ)(pA + pB ) + 2)
.
(1 + 2γ)(qA + qB )3
Note that v is a decreasing function of γ.
4
(1)
The general theorem establishing a lower bound on the asymptotic variance of the
allocation proportions for K treatments and general parameter vector θ will be given
in Section 3 as Theorem 1 and proved in the Appendix. To illustrate the method, we
now state the theorem for binary responses and K = 2 treatments.
Let I(pA , pB , NA (n)) be the Fisher’s information, where the expectation is taken
conditional on NA (n), for estimating pA and pB . Suppose the following regularity
conditions hold:
1. pA , pB ∈ (0, 1).
2. NA (n)/n converges to ρ(pA , pB ) ∈ (0, 1) almost surely for the particular responseadaptive randomization procedure;
√
3. n(NA (n)/n − ρ(pA , pB )) converges in law to N (0, v) for the particular procedure.
Then a lower bound on the asymptotic variance of NA (n)/n is given by
µ
¶
µ
¶0
∂ρ(pA , pB ) ∂ρ(pA , pB )
∂ρ(pA , pB ) ∂ρ(pA , pB )
−1
I (pA , pB , nρ(pA , pB ))
.
∂pA
∂pB
∂pA
∂pB
(2)
We can compute


I −1 (pA , pB , nρ(pA , pB ) = 
pA qA
nρ(pA ,pB )
0
0
pB qB
n(1−ρ(pA ,pB ))
.
We refer to a response-adaptive allocation procedure that attains the lower bound as
asymptotically best for that particular target allocation ρ(pA , pB ). It is best in the sense
that, for a fixed allocation ρ(pA , pB ), it maximizes an asymptotic approximation to the
power of chi-square test for the difference of proportions
(p̂A − p̂B )2
pA qA /NA (n) + pB qB /NB (n)
as described in Hu and Rosenberger (2003), where p̂A and p̂B are the maximum likelihood estimators of pA and pB .
We now apply this result for the three allocation rules mentioned in Section 1: urn
allocation, Neyman allocation, and RSIHR allocation. For urn allocation, from (2),
the lower bound is given by
qA qB (pA + pB )
.
n(qA + qB )3
5
Since this is the asymptotic variance of Ivanova’s procedure, the procedure attains
the lower bound. Hence, Ivanova’s procedure is the asymptotically best procedure for
targeting urn allocation.
It is interesting to note that the Zelen’s (1969) deterministic procedure, termed
“play-the-winner rule”, which assigns the opposite treatment to the next patient if the
previous patient was a treatment failure, and the same treatment if the previous patient
was a treatment success, is also an asymptotically best procedure. However, unlike
Ivanova’s procedure, Zelen’s procedure is not randomized. That Zelen’s procedure
satisifies regularity condition 2 was known in the original paper. To our knowledge,
condition 3 along with v had not previously been known. This leads to the following
proposition:
Proposition 1. For Zelen’s (1969) procedure,
n1/2 (NA (n)/n − qB /(qA + qB )) → N (0, v),
in law, where
v = qA qB (pA + pB )/(qA + qB )3 .
For Eisele’s design with the g function given in (1), it can be seen that v ≥ qA qB (pA +
pB )/(qA + qB )3 , with equality holding when γ → ∞. Thus the lower bound can be
attained, but the result is a mostly deterministic procedure that assigns the next patient
to A with probability 1 if the current allocation is less than ρ(p̂A , p̂B ) and to B with
probability 1 if the current allocation is greater than ρ(p̂A , p̂B ). If the current allocation
is equal to ρ(p̂A , p̂B ), then allocation is according to a biased coin.
Wei and Durham’s procedure is not an asymptotically best procedure (except trivially when pA = pB ), as their asymptotic variance v is always larger than for Ivanova’s
procedure, a fact first pointed out by Ivanova (2003).
Complete randomization, which targets ρ(pA , pB ) = 1/2, satisfies regularity conditions 2 and 3 with v = 1/4, and this is not asymptotically optimal except in the trivial
case where pA = pB = 1/2. The asymptotic variance of Ivanova’s procedure is often
less than 1/4, except for very large values of pA and pB . So Ivanova’s procedure is fully
6
randomized, and assigns more patients to the better treatment (asymptotically) often
with smaller variability than complete randomization.
When the target allocation is Neyman allocation, the lower bound is computed
from (2) as
1
√
√
4n( pA qA + pB qB )3
µ
pB qB (qA − pA )2 pA qA (qB − pB )2
+
√
√
pA qA
pB qB
¶
.
Eisele’s procedure does not attain the lower bound (except as γ → ∞). When γ = 0, we
have a procedure explored by Page and Melfi (1998, 2000) targeting Neyman allocation,
and they discovered that this procedure was highly variable. The high variability can
reduce power substantially, which is undesirable when the target allocation is supposed
to maximize power. Essentially this procedure allocates to A with probability ρ(p̂A , p̂B )
at each stage.
Finally, when the target allocation is RSIHR allocation, the lower bound is computed from (2) as
1
√
√
4n( pA + pB )3
µ
pB qA pA qB
√ + √
pA
pB
¶
.
The use of Eisele’s procedure targeting RSIHR with γ = 2 was strongly recommended
in Hu and Rosenberger (2003), although we see that this is not a best asymptotic
procedure. We discuss this further in Section 4.
3. Main Results for K Treatments
We now establish a data structure which allows for simplification of the proof of the
main results for K treatments with general outcomes. The main results involve the
asymptotic properties of the maximum likelihood estimators from an exponential family
and then the Rao-Cramér lower bound based on the Fisher information matrix. For
the generalized Friedman’s urn model with K treatments, Rosenberger, Flournoy, and
Durham (1997) proved these results, but the conditions required are not satisfied for
many other response-adaptive randomization procedures.
Let Tij , i = 1, ..., n, j = 1, ..., K be the n treatment assignments, where Tij = 1 if
the ith patient is assigned to treatment j, 0 otherwise. Let N (n) = (N1 (n), ..., NK (n))
7
be the sample sizes on each treatment. Now define T 1 = (T11 , ..., T1K ), ..., T n =
(Tn1 , ..., TnK ). Note that each vector is a vector of K − 1 0s and one 1. For the
responses, let X 1 = (X11 , X12 , ..., X1K ), ..., X n = (Xn1 , Xn2 , ..., XnK ), where X i represents the sequence of responses that would be observed if each treatment were assigned
to the ith patient independently. However, only one element of X i will be observable.
We assume that X 1 , ..., X n are independent and identically distributed, with
X1j ∼ fj (·, θ j ), j = 1, ..., K,
where θ j ∈ Θj . We thus assume that X m is independent of X 1 , ..., X m−1 , T 1 , ..., T m ,
but that T m depends on X 1 , ..., X m−1 , T 1 , ..., T m−1 , m = 1, ..., n. The value of this
data structure for theoretical purposes is obscured by the complexity of the notation.
We therefore give a simple example to illustrate the notation before moving to our
main result.
Example. If K = 2 and we use Wei and Durham’s procedure, then f1 (·, θ 1 ) is
Bernoulli(pA ) and f2 (·, θ 2 ) is Bernoulli(pB ). Also T m = (1, 0) if treatment A was
assigned to the mth patient, and T m = (0, 1) if treatment B was assigned. Then
X m = (Xm1 , Xm2 ), where Xm1 ∼ f1 and Xm2 ∼ f2 . Clearly X m is independent
of X 1 , ..., X m−1 and also of T 1 , ..., T m , but we only observe the element of X m corresponding to the element of T m that is 1. However, T m depends on all previous
treatment assignments and responses according to the urn model.
We now state the regularity conditions for the main result:
1. The parameter space Θj is an open subset of Rd , d ≥ 1, for j = 1, ..., K.
2. The distributions f1 (·, θ 1 ), ..., fK (·, θ K ) follow an exponential family.
3. For target allocation ρ(θ) = (ρ1 (θ), ..., ρK (θ)) ∈ (0, 1)K ,
Nj (n)
→ ρj (θ)
n
almost surely for j = 1, ..., K.
4. For positive definite matrix V (θ),
¶
µ
√
N (n)
− ρ(θ) → N (0, V (θ))
n
n
8
in law.
We require the following lemma, which gives the asymptotic distribution of the
maximum likelihood estimator θ̂ of θ:
Lemma 1. Under regularity conditions 1 – 3, we have
√
n(θ̂ − θ) → N (0, I −1 (θ)),
where I(θ) = diag{ρ1 (θ)I 1 (θ 1 ), ..., ρK (θ)I K (θ K )} and I j (θ j ) is the Fisher’s information for a single observation on treatment j = 1, ..., K.
Remark. We can generalize to non-exponential families by imposing further regularity conditions, such as those given in Rosenberger, Flournoy, and Durham (1997).
However, they replace condition 3 essentially with a requirement that
E(Tij |Fi−1 ) → ρj (θ), j = 1, ..., K
almost surely, where Fn is the sigma algebra generated by the first n assignments and
responses. This condition is not satisifed for Ivanova’s procedure and Zelen’s procedure,
for example. However, our condition 3 is satisfied for all the procedures we examine.
Now we state the main result:
Theorem 1. Under regularity conditions 1 – 4, there exists a Θ0 ⊂ Θ = Θ1 ⊗· · ·⊗ΘK
with Lebesgue measure 0 such that for every θ ∈ Θ − Θ0 ,
∂ρ(θ) −1 ∂ρ(θ) 0
V (θ) ≥
I (θ)
.
∂θ
∂θ
We can now rigorously define an asymptotically best response-adaptive procedure as one
in which V (θ) attains the lower bound
B(θ) =
∂ρ(θ) −1 ∂ρ(θ) 0
I (θ)
∂θ
∂θ
for a particular target allocation ρ(θ).
We now show that an extension of Eisele’s procedure for K > 2 treatments (Hu
and Zhang, 2003) is not an asymptotically best procedure, in general, but can attain
the lower bound.
9
Hu and Zhang’s Procedure. The generalization of Eisele’s procedure to K > 2
treatments follows a similar allocation rule, where now g is a function mapping [0, 1]K ⊗
[0, 1]K to [0, 1]K . We allocate to treatment j with probability gj (N (n)/n, ρ(θ)), where
g(x, y) = (g1 (x, y), ..., gK (x, y)) and
gj (x, y) = PK
yj (yj /xj )γ ∧ c
j=1
{yj (yj /xj )γ ∧ c}
for some c > 1. The constant c is introduced for technical reasons, and if c is chosen
large it will have little influence on the allocation. Regularity conditions 1 – 4 hold
with
V (θ) =
1
2(1 + γ)
Σ1 (θ) +
B(θ),
1 + 2γ
1 + 2γ
where Σ1 (θ) = diag{ρ(θ)} − ρ(θ)0 ρ(θ) (see Hu and Zhang, 2003).
Note that, for any ρ(θ), the procedure is asymptotically best when γ → ∞, as in
Eisele’s procedure for K = 2 treatments. The form of the asymptotic variance is particularly interesting because it demonstrates the relationship between the asymptotic
variance and the lower bound for all choices of target allocations ρ(θ).
4. Conclusions
We have established a lower bound on the asymptotic variability of responseadaptive randomization procedures which provides a baseline for comparison of existing
procedures and further guidance in developing new procedures. As such, this paper
represents an evolving body of knowledge about these complex procedures; however, it
also presents numerous dilemmas regarding the tradeoffs among randomization, variability, and optimality. On the surface, Ivanova’s procedure would seem to give us
everything we want in a response-adaptive randomization procedure: it is fully randomized and it attains the lower bound. However, it can only target urn 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 pB and becomes
more deterministic for small values of pA and pB (see Hu and Rosenberger, 2003).
Eisele’s procedure and Hu and Zhang’s extension solve some of these deficiencies, in
10
that they can target any desired allocation, and can approach the lower bound for
large values of γ. However, the procedure becomes more deterministic as γ becomes
larger, and hence careful tuning of γ must be done in order to counter the tradeoff.
This leads to a challenge for researchers in the area: can a procedure be found that
preserves randomization, attains the lower bound, and targets any allocation? We also
note that Eisele’s and Hu and Zhang’s procedures are the only procedures mentioned
in this paper that can be used for more general (not multinomial) responses.
An issue not addressed in this paper, but an important one, is appropriate target
allocations for clinical trials of K > 2 treatments. This is not a straightforward extension of two treatments. Also the extension of Ivanova’s procedure to K > 2 treatments
will require a theoretical determination of regularity condition 4. We are currently
exploring these issues.
Acknowledgments
Professors Hu and Rosenberger were supported by a grant from the National Science
Foundation. Professor Zhang was supported by grants from the National Natural
Science Foundation of China and the National Natural Science Foundation of Zhejiang
Province. Professor Hu is also affiliated with the Department of Statistics and Applied
Probability, National University of Singapore. Professor Rosenberger is also affiliated
with the University of Maryland School of Medicine, Baltimore.
Appendix
Proof of Proposition 1. Zelen’s rule induces a Markov chain {Tn , n ≥ 1} with
states {0, 1} and transition matrix


P =
pA qA
qB pB
.
The stationary probabilities are π = (ρ, 1 − ρ), where ρ = qB /(qA + qB ). The n-step
transition matrix can then be computed as

(n)
P n = (pij ) = 
qB +qA (1−qA −qB )n
qA +qB
qB −qB (1−qA −qB )n
qA +qB
11
qA −qA (1−qA −qB )n
qA +qB
qA +qB (1−qA −qB )n
qA +qB

.
By the central limit theorem for Markov chains (cf. Theorem 17.0.1 of Meyn and
Tweedie, 1993), it follows that
√
µ
n
NA
−ρ
n
¶
n
1 X
L
=√
(Tm − Eπ (T1 )) → N (0, v),
n m=1
where
v = Eπ (T1 − Eπ (T1 ))2 + 2
∞
X
Eπ {(T1 − Eπ (T1 ))(Tm+1 − Eπ (T1 ))}.
m=1
We compute
Eπ (T1 − Eπ (T1 ))2 = π1 π2 =
qA qB
(qA + qB )2
and
Eπ {(T1 − Eπ (T1 ))(Tm+1 − Eπ (T1 ))} = Pπ (T1 = 1, Tm+1 = 1) − (Pπ (T1 = 1))2
(m)
= p11 Pπ (T1 = 1) − (Pπ (T1 = 1))2
qA qB (1 − qA − qB )m
=
.
(qA + qB )2
It follows that
qA qB
v=
(qA + qB )2
Ã
1+2
∞
X
!
(1 − qA − qB )m
=
m=1
qA qB (pA + pB )
.¤
(qA + qB )3
Proof of Lemma 1. Let
L(θ) =
n Y
K
Y
¡
K
¢Ti,j Y
fj (Xi,j , θ j )
=
Lj (θ j )
i=1 j=1
j=1
be the likelihood function based on the observed sample. It is sufficient to show that
1 ∂ log Lj (θ j )
+ op (n−1/2 )
Nj (n)
∂θ j
n
X
1
∂ log fj (Xi,j , θ j )
= −
+ op (n−1/2 ),
Ti,j
nρj (θ j ) i=1
∂θ j
bj − θ j )I j (θ j ) = −
(θ
j = 1, ..., K. Let
Yi,j = −Ti,j
∂ log fj (Xi,j , θ j )
∂θ j
12
(3)
and An = σ(X 1 , . . . , X n , T 1 , ..., T n+1 ). Then {Y i = (Yi,1 , . . . , Yi,K ), Ai ; i = 1, 2, . . .}
is a martingale sequence with
n
n
1X
1X
E (Y 0i Y i |Ai−1 ] =
diag (Ti,1 I 1 (θ 1 ), . . . , Ti,K I K (θ K ))
n i=1
n i=1
µ
¶
N1 (n)
NK (n)
p
= diag
I 1 (θ 1 ), . . . ,
I K (θ K ) → I(θ).
n
n
Conditions of the martingale central limit theorem (e.g., Billingsley, 1961) hold by
regularity conditions 2 and 3, and it follows that
n
1 X
L
√
Y i → N (0, I(θ)) .
n i=1
(4)
It remains to show (3). For fixed j = 1, 2, . . . , K, define τi (j) = min{k : Nj (k) =
i} = min{k > τi−1 (j) : Tk,j = 1}, where min{∅} = +∞. Let {ηi,j } be an independent
ei,j = Xτ (j),j I{τi (j) <
copy of {Xi,j }, which is also independent of {T i }. Define X
i
ei,j ; i = 1, 2, . . .} is a sequence of i.i.d.
∞} + ηi,j I{τi (j) = ∞}, i = 1, 2, . . .. Then {X
random variables, with the same distribution as X1,j . Let
ej (θ j ) = L
ej (θ j ; m) =
L
m
Y
ei,j , θ j ),
fj (X
i=1
em,j maximize L(θ
e j ). Then on the event {Nj (n) → ∞},
and let θ
bj = θ
eN (n),j .
ej (θ j ; Nj (n)) and θ
Lj (θ j ) = L
j
(5)
ei,j : i = 1, 2, . . . , n} are independent and identically distributed. Under
Notice that {X
regularity condition 2, we have
³
´
e
¡
¢
em,j − θ j I j (θ j ) = − 1 ∂ log Lj (θ j ; m) 1 + o(1) a.s.
θ
m
∂θ j
as m → ∞. By (5), on the event {Nj (n) → ∞},
³
´
¢
1 ∂ log Lj (θ j ) ¡
b
θ j − θ j I j (θ j ) = −
1 + o(1) a.s.
Nj (n)
∂θ j
By (4) and regularity condition 3, (3) follows. ¤
Proof of Theorem 1. Because N (n)/n is an asymptotically unbiased estimator
of ρ(θ) and satisfies regularity conditions 2 and 4, by Theorem 4.16 of Shao (1999, p.
249) the Theorem follows directly. ¤
13
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