Exact Group Sequential Methods for Estimating a Binomial Proportion

�e �rst review existing sequential methods for estimating a binomial proportion. A�erward, we propose a new family of group sequential sampling schemes for estimating a binomial proportion with prescribed margin of error and con�dence level. In particular, we establish the uniform controllability of coverage probability and the asymptotic optimality for such a family of sampling schemes. Our theoretical results establish the possibility that the parameters of this family of sampling schemes can be determined so that the prescribed level of con�dence is guaranteed with little waste of samples. Analytic bounds for the cumulative distribution functions and expectations of sample numbers are derived. Moreover, we discuss the inherent connection of various sampling schemes. Numerical issues are addressed for improving the accuracy and efficiency of computation. Computational experiments are conducted for comparing sampling schemes. Illustrative examples are given for applications in clinical trials.


Introduction
Estimating a binomial proportion is a problem of ubiquitous signi�cance in many areas of engineering and sciences.�or economical reasons and other concerns, it is important to use as fewer as possible samples to guarantee the required reliability of estimation.To achieve this goal, sequential sampling schemes can be very useful.In a sequential sampling scheme, the total number of observations is not �xed in advance.e sampling process is continued stage by stage until a prespeci�ed stopping rule is satis�ed.e stopping rule is evaluated with accumulated observations.In many applications, for administrative feasibility, the sampling experiment is performed in a group fashion.Similar to group sequential tests [1,Section 8], [2], an estimation method based on taking samples by groups and evaluating them sequentially is referred to as a group sequential estimation method.It should be noted that group sequential estimation methods are general enough to include �xed-sample-size and fully sequential procedures as special cases.Particularly, a �xedsample-size method can be viewed as a group sequential procedure of only one stage.If the increment between the sample sizes of consecutive stages is equal to 1, then the group sequential method is actually a fully sequential method.
It is a common contention that statistical inference, as a unique science to quantify the uncertainties of inferential statements, should avoid errors in the quanti�cation of uncertainties, while minimizing the sampling cost.at is, a statistical inferential method is expected to be exact and efficient.e conventional notion of exactness is that no approximation is involved, except the round-off error due to �nite word length of computers.Existing sequential methods for estimating a binomial proportion are dominantly of asymptotic nature (see, e.g., [3][4][5][6][7] and the references therein).Undoubtedly, asymptotic techniques provide approximate solutions and important insights for the relevant problems.However, any asymptotic method inevitably introduces unknown error in the resultant approximate solution due to the necessary use of a �nite number of samples.In the direction of nonasymptotic sequential estimation, the primary goal is to ensure that the true coverage probability is above the prespeci�ed con�dence level for any value of the associated parameter, while the required sample size is as low as possible.In this direction, Mendo and Hernando [8] developed an inverse binomial sampling scheme for estimating a binomial proportion with relative precision.Tanaka [9] developed a rigorous method for constructing �xed-width sequential con�dence intervals for a binomial proportion.Although no approximation is involved, Tanaka's method is very conservative due to the bounding techniques employed in the derivation of sequential con�dence intervals.Franzén [10] studied the construction of �xed-width sequential con�dence intervals for a binomial proportion.However, no e�ective method for de�ning stopping rules is proposed in [10].In his later paper [11], Franzén proposed to construct �xed-width con�dence intervals based on sequential probability ratio tests (SPRTs) invented by Wald [12].His method can generate �xed-sample-size con�dence intervals based on SPRTs.Unfortunately, he made a fundamental �aw by mistaking that if the width of the �xed-samplesize con�dence interval decreases to be smaller than the prespeci�ed length as the number of samples is increasing, then the �xed-sample-size con�dence interval at the termination of sampling process is the desired �xed-width sequential con�dence interval guaranteeing the prescribed con�dence level.More recently, Frey published a paper [13] in e American Statistician (TAS) on the classical problem of sequentially estimating a binomial proportion with prescribed margin of error and con�dence level.Before Frey submitted his original manuscript to TAS in July 2009, a general framework of multistage parameter estimation had been established by Chen [14][15][16][17][18], which provides exact methods for estimating parameters of common distributions with various error criterion.is framework is also proposed in [19].e approach of Frey [13] is similar to that of Chen [14][15][16][17][18] for the speci�c problem of estimating a binomial proportion with prescribed margin of error and con�dence level.
In this paper, our primary interests are in the exact sequential methods for the estimation of a binomial proportion with prescribed margin of error and con�dence level.We �rst introduce the exact approach established in [14][15][16][17][18].In particular, we introduce the inclusion principle proposed in [18] and its applications to the construction of concrete stopping rules.We investigate the connection among various stopping rules.Aerward, we propose a new family of stopping rules which are extremely simple and accommodate some existing stopping rules as special cases.We provide rigorous �usti�cation for the feasibility and asymptotic optimality of such stopping rules.We prove that the prescribed con�dence level can be guaranteed uniformly for all values of a binomial proportion by choosing appropriate parametric values for the stopping rule.We show that as the margin of error tends to be zero, the sample size tends to the attainable minimum as if the binomial proportion were exactly known.We derive analytic bounds for distributions and expectations of sample numbers.In addition, we address some critical computational issues and propose methods to improve the accuracy and efficiency of numerical calculation.We conduct extensive numerical experiment to study the performance of various stopping rules.We determine parametric values for the proposed stopping rules to achieve unprecedentedly efficiency while guaranteeing prescribed con�dence levels.We attempt to make our proposed method as user-friendly as possible so that it can be immediately applicable even for layer persons.
e remainder of the paper is organized as follows.In Section 2, we introduce the exact approach proposed in [14][15][16][17][18].In Section 3, we discuss the general principle of constructing stopping rules.In Section 4, we propose a new family of sampling schemes and investigate their feasibility, optimality, and analytic bounds of the distribution and expectation of sample numbers.In Section 5, we compare various computational methods.In particular, we illustrate why the natural method of evaluating coverage probability based on gridding parameter space is neither rigorous nor efficient.In Section 6, we present numerical results for various sampling schemes.In Section 7, we illustrate the applications of our group sequential method in clinical trials.Section 8 is the conclusion.e proofs of theorems are given in appendices.roughout this paper, we shall use the following notations.e empty set is denoted by ∅. e set of positive integers is denoted by ℕ. e ceiling function is denoted by ⌈⋅⌉.e notation Pr{   denotes the probability of the event  associated with parameter .e expectation of a random variable is denoted by ⋅.e standard normal distribution is denoted by Φ(⋅).For   ( ), the notation   denotes the critical value such that Φ(  ) =  − .For   ℕ, in the case that    …    are i.i.d.samples of , we denote the sample mean (∑  =   )/ by   , which is also called the relative frequency when  is a Bernoulli random variable.e other notations will be made clear as we proceed.

How Can It Be Exact?
In many areas of scienti�c investigation, the outcome of an experiment is of dichotomy nature and can be modeled as a Bernoulli random variable , de�ned in probability space (Ω Pr ℱ), such that Pr{ =  =  − Pr{ =  =   ( )  (1) where  is referred to as a binomial proportion.In general, there is no analytic method for evaluating the binomial proportion .A frequently used approach is to estimate  based on i.i.d.samples     2  … of .To reduce the sampling cost, it is appropriate to estimate  by a multistage sampling procedure.More formally, let   ( ) and  − , with   ( ), be the prespeci�ed margin of error and con�dence level, respectively.e ob�ective is to construct a sequential estimator   for  based on a multistage sampling scheme such that Pr  −  <    ≥  −  (2) for any    .roughout this paper, the probability Pr{|   |     is referred to as the coverage probability.Accordingly, the probability Pr{|   |     is referred to as the complementary coverage probability.
Clearly, a complete construction of a multistage estimation scheme needs to determine the number of stages, the sample sizes for all stages, the stopping rule, and the estimator for .roughout this paper, we let  denote the number of stages and let  ℓ denote the number of samples at the ℓth stages.at is, the sampling process consists of  stages with sample sizes     2  ⋯    .For ℓ =  2 …  , de�ne  ℓ = ∑  ℓ =   and   ℓ =  ℓ / ℓ .e stopping rule is to be de�ned in terms of   ℓ  ℓ =  …  .Of course, the index of stage at the termination of the sampling process, denoted by , is a random number.Accordingly, the number of samples at the termination of the experiment, denoted by , is a random number which equals   .Since for each ℓ,   ℓ is a maximum-likelihood and minimum-variance unbiased estimator of , the sequential estimator for  is taken as In the above discussion, we have outlined the general characteristics of a multistage sampling scheme for estimating a binomial proportion.It remains to determine the number of stages, the sample sizes for all stages, and the stopping rule so that the resultant estimator   satis�es (2) for any    .
Actually, the problem of sequential estimation of a binomial proportion has been treated by Chen [14][15][16][17][18] in a general framework of multistage parameter estimation.e techniques of [14][15][16][17][18] are sufficient to offer exact solutions for a wide range of sequential estimation problems, including the estimation of a binomial proportion as a special case.e central idea of the approach in [14][15][16][17][18] is the control of coverage probability by a single parameter , referred to as the coverage tuning parameter, and the adaptive rigorous checking of coverage guarantee by virtue of bounds of coverage probabilities.It is recognized in [14][15][16][17][18] that, due to the discontinuity of the coverage probability on parameter space, the conventional method of evaluating the coverage probability for a �nite number of parameter values is neither rigorous not computationally efficient for checking the coverage probability guarantee.
As mentioned in the introduction, Frey published an article [13] in TAS on the sequential estimation of a binomial proportion with prescribed margin of error and con�dence level.For clarity of presentation, the comparison of the works of Chen and Frey is given in Section 5.4.In the remainder of this section, we shall only introduce the idea and techniques of [14][15][16][17][18], which had been precedentially developed by Chen before Frey submitted his original manuscript to TAS in July 2009.We will introduce the approach of [14][15][16][17][18] with a focus on the special problem of estimating a binomial proportion with prescribed margin of error and con�dence level.
(i) Stopping rules parameterized by the coverage tuning parameter    such that the associated coverage probabilities can be made arbitrarily close to  by choosing    to be a sufficiently small number.
(ii) Recursively computable lower and upper bounds for the complementary coverage probability for a given  and an interval of parameter values.
(iii) Adapted branch and bound algorithm.
Without looking at the technical details, one can see that these four components are sufficient for constructing a sequential estimator so that the prescribed con�dence level is guaranteed.e reason is as follows.As lower and upper bounds for the complementary coverage probability are available, the global optimization technique, branch and bound (B&B) algorithm [20], can be used to compute exactly the maximum of complementary coverage probability on the whole parameter space.us, it is possible to check rigorously whether the coverage probability associated with a given  is no less than the prespeci�ed con�dence level.Since the coverage probability can be controlled by , it is possible to determine  as large as possible to guarantee the desired con�dence level by a bisection search.is process is referred to as bisection coverage tuning in [14][15][16][17][18]. Since a critical subroutine needed for bisection coverage tuning is to check whether the coverage probability is no less than the prespeci�ed con�dence level, it is not necessary to compute exactly the maximum of the complementary coverage probability.erefore, Chen revised the standard B&B algorithm to reduce the computational complexity and called the improved algorithm as the adapted B&B Algorithm.e idea is to adaptively partition the parameter space as many subintervals.If for all subintervals, the upper bounds of the complementary coverage probability are no greater than , then declare that the coverage probability is guaranteed.If there exists a subinterval for which the lower bound of the complementary coverage probability is greater than , then declare that the coverage probability is not guaranteed.Continue partitioning the parameter space if no decision can be made.e four components are illustrated in the sequel under the headings of stopping rules, interval bounding, adapted branch and bound, and bisection coverage tuning.

Stopping
where  and  are integers such that 0 ≤  ≤  ≤ .Assume that 0 <  < 1.For the purpose of controlling the coverage probability Pr{|  − p| <    by the coverage tuning parameter, Chen has proposed four stopping rules as follows.
To avoid unnecessary checking of the stopping criterion and thus reduce administrative cost, there should be a possibility that the sampling process is terminated at the �rst stage.Hence, the minimum sample size  1 should be chosen to ensure that { =  1  ≠ ∅. is implies that the sample size  1 for the �rst stage can be taken as  mi .On the other hand, since the sampling process must be terminated at or before the th stage, the maximum sample size   should be chosen to guarantee that {     = ∅.is implies that the sample size   for the last stage can be taken as  max .If the number of stages  is given, then the sample sizes for stages in between 1 and  can be chosen as  − 2 integers between  mi and  max .Particularly, if the group sizes are expected to be approximately equal, then the sample sizes can be taken as Since the stopping rule is associated with the coverage tuning parameter , it follows that the number of stages  and the sample sizes  1   2  …    can be expressed as functions of .In this sense, it can be said that the stopping rule is parameterized by the coverage tuning parameter .e above method of parameterizing stopping rules has been used in [14][15][16][17] (9).It should be pointed out that such lower and upper bounds of Pr{|   |    } can also be computed by the recursive path-counting method of Franzén [10, page 49].
2.4.Adapted Branch and Bound.e third component for the exact sequential estimation of a binomial proportion is the adapted B&B algorithm, which was proposed in [15, Section 2.8], for quick determination of whether the coverage probability is no less than    for any value of the associated parameter.Such a task of checking the coverage probability is also referred to as checking the coverage probability guarantee.Given that lower and upper bounds of the complementary coverage probability on an interval of parameter values can be obtained by the interval bounding techniques, this task can be accomplished by applying the B&B algorithm [20] to compute exactly the maximum of the complementary coverage probability on the parameter space.However, in our applications, it suffices to determine whether the maximum of the complementary coverage probability Pr{|   |    } with respect to  ∈ ( ) is greater than the con�dence parameter .For fast checking whether the maximal complementary coverage probability exceeds , Chen proposed to reduce the computational complexity by revising the standard B&B algorithm as the Adapted B&B Algorithm in [15,Section 2.8].To describe this algorithm, let ℐ init denote the parameter space ( ).For an interval ℐ  ℐ init , let max Ψ(ℐ) denote the maximum of the complementary coverage probability Pr{|   |    } with respect to  ∈ ℐ.Let Ψ lb (ℐ) and Ψ ub (ℐ) be, respectively, the lower and upper bounds of Ψ(ℐ), which can be obtained by the interval bounding techniques introduced in Section 2.3.Let  >  be a prespeci�ed tolerance, which is much smaller than .e adapted B&B algorithm of [15] is represented with a slight modi�cation as in Algorithm 1.
It should be noted that for a sampling scheme of symmetrical stopping boundary, the initial interval ℐ init may be taken as ( /2) for the sake of efficiency.In Section 5.1, we will illustrate why the adapted B&B algorithm is superior than the direct evaluation based on gridding parameter space.As will be seen in Section 5.2, the objective of the adapted B&B algorithm can also be accomplished by the Adaptive Maximum Checking Algorithm due to Chen [21, Section 3.3] and rediscovered by Frey [13,Appendix].An explanation is given in Section 5.3 for the advantage of working with the complementary coverage probability.

Bisection Coverage
Tuning.e fourth component for the exact sequential estimation of a binomial proportion is Bisection Coverage Tuning.Based on the adaptive rigorous checking of coverage probability, Chen proposed in [14, Section 2.7] and [15, Section 2.6] to apply a bisection search method to determine maximal  such that the coverage probability is no less than    for any value of the associated parameter.Moreover, Chen has developed asymptotic results in [15, page 21, eorem 8] for determining the initial interval of  needed for the bisection search.Speci�cally, if the complementary coverage probability Pr{|   |    } associated with     tends to  as   , then the initial interval of  can be taken as   2     2 + , where  is the largest integer such that the complementary coverage probability associated with     2  is no greater than  for all  ∈ ( ).By virtue of a bisection search, it is possible to obtain  * ∈   2     2 +  such that the complementary coverage probability associated with    * is guaranteed to be no greater than  for all  ∈ ( ).

Principle of Constructing Stopping Rules
In this section, we shall illustrate the inherent connection between various stopping rules.It will be demonstrated that a lot of stopping rules can be derived by virtue of the inclusion principle proposed by Chen [18, Section 3].

Inclusion
Principle.e problem of estimating a binomial proportion can be considered as a special case of parameter estimation for a random variable  parameterized by   , where the objective is to construct a sequential estimator   for  such that Pr{|    |         for any   .Assume that the sampling process consists of  stages with sample sizes     2  ⋯    .For ℓ = , … , , de�ne an estimator   ℓ for  in terms of samples   , … ,   ℓ of .Let [ ℓ ,  ℓ ], ℓ = , 2, … ,  be a sequence of con�dence intervals such that for any ℓ, [ ℓ ,  ℓ ] is de�ned in terms of   , … ,   ℓ and that the coverage probability Pr{ ℓ ≤  ≤  ℓ   can be made arbitrarily close to  by choosing    to be a sufficiently small number.In eorem 2 of [18], Chen proposed the following general stopping rule: At the termination of the sampling process, a sequential estimator for  is taken as   =    , where  is the index of stage at the termination of sampling process.
Clearly, the general stopping rule (10) provided that Pr{ ℓ     ℓ       for ℓ = , … ,  and   .is demonstrates that if the number of stages  is bounded respective to , then the coverage probability Pr{|   |     associated with the stopping rule derived from the inclusion principle can be controlled by .Actually, before explicitly proposing the inclusion principle in [18], Chen had extensively applied the inclusion principle in [14][15][16][17] to construct stopping rules for estimating parameters of various distributions such as binomial, Poisson, geometric, hypergeometric, and normal distributions.A more general version of the inclusion principle is proposed in [19,Section 2.4].For simplicity of the stopping rule, Chen had made effort to eliminate the computation of con�dence limits.
In the context of estimating a binomial proportion , the inclusion principle immediately leads to the following general stopping rule: Consequently, the sequential estimator for  is taken as   according to (3).It should be pointed out that the stopping rule (12) had been rediscovered by Frey in Section 2, the 1st paragraph of [13].e four stopping rules considered in his paper follow immediately from applying various con�dence intervals to the general stopping rule (12).
In the sequel, we will illustrate how to apply (12) to the derivation of Stopping Rules A, B, C, and D introduced in Section 2.2 and other speci�c stopping rules.

�.�. �to���n� �u�e from ���son�s �on�den�e �nter�a�s.
Making use of the interval estimation method of Wilson [25], one can obtain a sequence of con�dence intervals [ ℓ ,  ℓ ], ℓ = 1, … ,  for  such that and that Pr{ ℓ ≤  ≤  ℓ |   1−2 for ℓ = 1, … ,  and  ∈ (0, 1).It should be pointed out that the sequence of Wilson's con�dence intervals has been applied by Frey [13, Section 2, page 243] to the general stopping rule (12) for estimating a binomial proportion.Since a stopping rule directly involves the sequence of Wilson's con�dence intervals is cumbersome, it is desirable to eliminate the computation of Wilson's con�dence intervals in the stopping rule.For this purpose, we need to use the following result.eorem 1. Assume that 0 <  < 1 and 0 <  < 1/2.en, ���son�s �on�den�e �nter�a�s sat�sf� See Appendix A for a proof.As a consequence of eorem 1 and the fact that for any  ∈ (0, 1/), there exists a unique number  ′ ∈ (0, 1/) such that   =  2 ln(1/ ′ ), applying the sequence of Wilson's con�dence intervals to (12) leads to the following stopping rule.
Actually, the con�dence intervals of Chen et al. [29] are derived from Massart's inequality [31] on the tailed probabilities of the sample mean of Bernoulli random variable.For this reason, Stopping Rule B is also referred to as the stopping rule from Massart's inequality in [21, Section 4..].

Double-Parabolic Sequential Estimation
From Sections 2.2, 3.2, and 3.7, it can be seen that, by introducing a new parameter   [0, ] and letting  take values 3 and 0, respectively, Stopping Rules B and D can be accommodated as special cases of the following general stopping rule.Continue the sampling process until for some ℓ  {, , … , , where   0, .Moreover, as can be seen from ( 16), the stopping rule derived from applying �ilson's con�dence intervals to (12) can also be viewed as a special case of such general stopping rule with   .
From the stopping condition (21), it can be seen that the stopping boundary is associated with the double-parabolic function      [4 −  −  −   ] such that  and  correspond to the sample mean and sample size, respectively.For   0.,   0.0, and   , stopping boundaries with various  are shown by Figure 1.
For �xed  and , the parameters  and  affect the shape of the stoping boundary in a way as follows.As  increases, the span of stopping boundary is increasing in the axis of sample mean.By decreasing , the stopping boundary can be dragged toward the direction of increasing sample size.Hence, the parameter  is referred to as the dilation coefficient.e parameter  is referred to as the coverage tuning parameter.Since the stopping boundary consists of two parabolas, this approach of estimating a binomial proportion is referred to as the double-parabolic sequential estimation method.

Parametrization of the Sampling Scheme.
In this section, we shall parameterize the double-parabolic sequential sampling scheme by the method described in Section 2.2.From the stopping condition (21), the stopping rule can be restated as follows.Continue sampling until   ℓ ,  ℓ    for some ℓ ∈ {1, … , , where the function ,  is de�ned by Clearly, the function ,  associated with the doubleparabolic sequential sampling scheme depends on the design parameters , ,  and .Applying the function ,  de�ned by (22) to (6) yields for some nonnegative integer  not exceeding  .

(23)
Since  is usually small in practical applications, we restrict  to satisfy 0 <   14.As a consequence of 0    14 and the fact that |  12|  12 for any  ∈ 0, 1, it must be true that |  12|   2  12   2 for any  ∈ 0, 1.It follows from ( 23) that 12   2 ≥ 14 +  2  min 2 ln, which implies that the minimum sample size can be taken as On the other hand, applying the function ,  de�ned by (22) to (7) gives for all nonnegative integer  not exceeding  .(25) Since |  12|   2 ≥ 0 for any  ∈ 0, 1, it follows from (25) that 14 +  2  max 2 ln  0, which implies that maximum sample size can be taken as erefore, the sample sizes  1 , … ,   can be chosen as functions of , , , and  which satisfy the following constraint: In particular, if the number of stages  is given and the group sizes are expected to be approximately equal, then the sample sizes,  1 , … ,   , for all stages can be obtained by substituting  min de�ned by (24) and  max de�ned by ( 26) into (8).For example, if the values of design parameters are  = 0.0,  = 0.0,  = 4,  = 2. and  = , then the sample sizes of this sampling scheme are calculated as e stopping rule is completely determined by substituting the values of design parameters into (21).

Uniform
See Appendix B for a proof.For eorem 2 to be valid, the choice of sample sizes is very �exible.Particularly, the sample sizes can be arithmetic or geometric progressions or any others, as long as the constraint ( 27) is satis�ed.It can be seen that for the coverage probability to be uniformly controllable, the dilation coefficient  must be greater than 0. eorem 2 asserts that there exists   0 such that the coverage probability is no less than 1  , regardless of the associated binomial proportion .For the purpose of reducing sampling cost, we want to have a value of  as large as possible such that the prespeci�ed con�dence level is guaranteed for any  ∈ 0, 1.is can be accomplished by the technical components introduced in Sections 2.1, 2.3, 2.4, and 2.5.Clearly, for every value of , we can obtain a corresponding value of  (as large as possible) to ensure the desired con�dence level.�owever, the performance of resultant stopping rules are different.erefore, we can try a number of values of  and pick the best resultant stopping rule for practical use.

Asymptotic Optimality of Sampling
Schemes.Now we shall provide an important reason why we propose the sampling scheme of that structure by showing its asymptotic optimality.Since the performance of a group sampling scheme will be close to its fully sequential counterpart, we investigate the optimality of the fully sequential sampling scheme.In this scenario, the sample sizes  1 ,  2 , … ,   are consecutive integers such that e fully sequential sampling scheme can be viewed as a special case of a group sampling scheme of  =   − for any   , 1.
See Appendix C for a proof.From (32), it can be seen that lim    Pr{|  − | <    = 1 −  for any   , 1 if  = 1 −12 2 2 .Such value can be taken as an initial value for the coverage tuning parameter .In addition to providing guidance on the coverage tuning techniques, eorem 3 also establishes the optimality of the sampling scheme.To see this, let , ,  denote the minimum sample size  required for a �xed-sample-size procedure to guarantee that Pr{|  − | <     1 −  for any   , 1, where   = ∑  =1   .It is well known that from the central limit theorem, Applying (33), (34), and letting  = 1 −12 2 2 , we have lim    , , , , ,  = 1 for   , 1 and   , 1, which implies the asymptotic optimality of the double-parabolic sampling scheme.By virtue of (33), an approximate formula for computing the average sample number is given as follows: for   , 1 and   , 1.From (34), one obtains , ,  ≈ 1 −  2  2 , which is a well-known result in statistics.In situations that no information of  is available, one usually uses as the sample size for estimating the binomial proportion  with prescribed margin of error  and con�dence level 1 − .Since the sample size formula (36) can lead to undercoverage, researchers in many areas are willing to use a more conservative but rigorous sample size formula which is derived from the Chernoff-Hoeffding bound [32,33].Comparing ( 35) and (37), one can see that under the premise of guaranteeing the prescribed con�dence level 1 − , the double-parabolic sampling scheme can lead to a substantial reduction of sample number when the unknown binomial proportion  is close to  or 1.

Bounds on Distribution and Expectation of Sample
Number.We shall derive analytic bounds for the cumulative distribution function and expectation of the sample number  associated with the double-parabolic sampling scheme.In this direction, we have obtained the following results.See Appendix D for a proof.By the symmetry of the double-parabolic sampling scheme, similar analytic bounds for the distribution and expectation of the sample number can be derived for the case that   12, 1.

Comparison of Computational Methods
In this section, we shall compare various computational methods.First, we will illustrate why a frequently used method of evaluating the coverage probability based on gridding the parameter space is not rigorous and is less efficient as compared to the adapted B&B algorithm.Second, we will introduce the Adaptive Maximum Checking Algorithm of [21] which has better computational efficiency as compared to the adapted B&B algorithm.ird, we will explain that it is more advantageous in terms of numerical accuracy to work with the complementary coverage probability as compared to direct evaluation of the coverage probability.Finally, we will compare the computational methods of Chen [14][15][16][17][18] and Frey [13] for the design of sequential procedures for estimating a binomial proportion.

Verifying Coverage Guarantee without Gridding Parameter Space.
For purpose of constructing a sampling scheme so that the prescribed con�dence level 1 −  is guaranteed, an essential task is to determine whether the coverage probability Pr{|  − |     associated with a given stopping rule is no less than 1 − .In other words, it is necessary to compare the in�mum of coverage probability with 1 − .To accomplish such a task of checking coverage guarantee, a natural method is to evaluate the in�mum of coverage probability as follows: (i) choose  grid points  1  …    from parameter space  1; (ii) compute    Pr{|  − |       for   1 …  ; (iii) Take min{ 1  …     as inf 1 Pr{|  − |    .
is method can be easily mistaken as an exact approach and has been frequently used for evaluating coverage probabilities in many problem areas.
It is not hard to show that if the sample size  of a sequential procedure has a support , then the coverage probability Pr{|  − |     is discontinuous at      1, where   {     is a nonnegative integer no greater than   .e set  typically has a large number of parameter values.Due to the discontinuity of the coverage probability as a function of , the coverage probabilities can di�er signi�cantly for two parameter values which are extremely close.is implies that an intolerable error can be introduced by taking the minimum of coverage probabilities of a �nite number of parameter values as the in�mum of coverage probability on the whole parameter space.So, if one simply uses the minimum of the coverage probabilities of a �nite number of parameter values as the in�mum of coverage probability to check the coverage guarantee, the sequential estimator   of the resultant stopping rule will fail to guarantee the prescribed con�dence level.
In addition to the lack of rigorousness, another drawback of checking coverage guarantee based on the method of gridding parameter space is its low efficiency.A critical issue is on the choice of the number, , of grid points.If the number  is too small, the induced error can be substantial.On the other hand, choosing a large number for  results in high computational complexity.
In contrast to the method based on gridding parameter space, the adapted B&B algorithm is a rigorous approach for checking coverage guarantee as a consequence of the mechanism for comparing the bounds of coverage probability with the prescribed con�dence level.e algorithm is also efficient due to the mechanism of pruning branches.

Adaptive Maximum Checking Algorithm.
As illustrated in Section 2, the techniques developed in [14][15][16][17][18] are sufficient to provide exact solutions for a wide range of sequential estimation problems.However, one of the four components, the adapted B&B algorithm, requires computing both the lower and upper bounds of the complementary coverage probability.To further reduce the computational complexity, it is desirable to have a checking algorithm which needs only one of the lower and upper bounds.For this purpose, Chen had developed the Adaptive Maximum Checking Algorithm (AMCA) in [21, Section 3.3] and [19, Section 2.7].In the following introduction of the AMCA, we shall follow the description of [21].e AMCA can be applied to a wide class of computational problems dependent on the following critical subroutine.
Determine whether a function  is smaller than a prescribed number  for every value of  contained in interval [ .
Particularly, for checking the coverage guarantee in the context of estimating a binomial proportion, the parameter  is the binomial proportion  and the function  is actually the complementary coverage probability.In many situations, it is impossible or very difficult to evaluate  for every value of  in interval [ , since the interval may contain in�nitely many or an extremely large number of values.Similar to the adapted B&B algorithm, the purpose of AMCA is to reduce the computational complexity associated with the problem of determining whether the maximum of  over [  is less than .e only assumption required for AMCA is that, for any interval [   [ , it is possible to compute an upper bound   such that     for any   [  and that the upper bound converges to  as the interval width  −  tends to . e backward AMCA proceeds as in Algorithm 2.
e output of the backward AMCA is a binary variable  such that "  " means "  " and "  " means "  ." An intermediate variable  is introduced in the description of AMCA such that "  " means that the le endpoint of the interval is reached.e backward AMCA starts from the right endpoint of the interval (i.e.,   ) and attempts to �nd an interval [  such that    .If such an interval is available, then, attempt to go backward to �nd the next consecutive interval with twice width.If doubling the interval width fails to guarantee    , then try to repeatedly cut the interval width in half to ensure that    .If the interval width becomes smaller than a prescribed tolerance , then AMCA declares that "  ." For our relevant statistical problems, if    for some   [ , it is sure that "  " will be declared.On the other hand, it is possible that "  " is declared even though    for any   [ .However, such situation can be made extremely rare and immaterial if we choose  to be a very small number.Moreover, this will only introduce negligible conservativeness in the evaluation of  if  is chosen to be sufficiently small (e.g.,    −5 ).Clearly, the backward AMCA can be easily modi�ed as forward AMCA.Moreover, the AMCA can also be easily modi�ed as Adaptive Minimum Checking Algorithm (forward and backward).For checking the maximum of complementary coverage probability Pr{|  − |    , one can use the AMCA with   Pr{|  − |     over interval [ /2.We would like to point out that, in contrast to the adapted B&B algorithm, it seems difficult to generalize the AMCA to problems involving multidimensional parameter spaces.

Working with Complementary Coverage Probability.
We would like to point out that, instead of evaluating the coverage probability as in [13], it is better to evaluate the complementary coverage probability for purpose of reducing numerical error.e advantage of working on the complementary coverage probability can be explained as follows.Note that, in many cases, the coverage probability is very close to  and the complementary coverage probability is very close to .Since the absolute precision for computing a number close to  is much lower than the absolute precision for computing a number close to , the method of directly evaluating the coverage probability will lead to intolerable numerical error for problems involving small .As an example, consider a situation that the complementary coverage probability is in the order of  −5 .Direct computation of the coverage probability can easily lead to an absolute error of the order of  −5 .However, the absolute error of computing the complementary coverage probability can be readily controlled at the order of  −9 .

Comparison of Approaches of Chen and Frey.
As mentioned in the introduction, Frey published a paper [13] in e American Statistician (TAS) on the sequential estimation of a binomial proportion with prescribed margin of error and con�dence level.e approaches of Chen and Frey are based on the same strategy as follows.First, construct a family of stopping rules parameterized by  (and possibly other design parameters) so that the associated coverage probability Pr{|  − |     can be controlled by parameter  in the sense that the coverage probability can be made arbitrarily close to  by increasing .Second, apply a bisection search method to determine the parameter  so that the coverage probability is no less than the prescribed con�dence level  −  for any    .
For the purpose of controlling the coverage probability, Frey [13] applied the inclusion principle previously proposed in [18, Section 3] and used in [14][15][16][17].As illustrated in Section 3, the central idea of inclusion principle is to use a sequence of con�dence intervals to construct stopping rules so that the sampling process is continued until a con�dence interval is included by an interval de�ned in terms of the estimator and margin of error.Due to the inclusion relationship, the associated coverage probability can be controlled by the con�dence coefficients of the sequence of con�dence intervals.e critical value  used by Frey plays the same role for controlling coverage probabilities as that of the coverage tuning parameter  used by Chen.Frey [13] stated stopping rules in terms of con�dence limits.is way of expressing stopping rules is straightforward and insightful, since one can readily see the principle behind the construction.For convenience of practical use, Chen proposed to eliminate the necessity of computing con�dence limits.
Similar to the AMCA proposed in [21, Section 3.3], the algorithm of Frey [13,Appendix] for checking coverage guarantee adaptively scans the parameter space based on interval bounding.e adaptive method used by Frey for updating step size is essentially the same as that of the AMCA.Ignoring the number . in Frey's expression "   min{.2 − −  −2 , " which has very little impact on the computational efficiency, Frey's step size   can be identi�ed as the adaptive step size  in the AMCA.e operation associated with "   min{.2 − −  i−2 " has a similar function as that of the command "Let st ←  and ℓ ← 2" in the outer loop of the AMCA.e operation associated with Frey's expression " − +   /2  ,   " is equivalent to that of the command "Let ℓ ← ℓ −  and  ← 2 ℓ " in the inner loop of the AMCA.Frey proposed to declare a failure of coverage guarantee if "the distance from  − to the candidate value for   falls below  −4 ." e number " −4 " actually plays the same role as "" in the AMCA, where "   −5 " is recommended by [21].

Numerical Results
In this section, we shall illustrate the proposed doubleparabolic sampling scheme through examples.As demonstrated in Sections 2.2 and 4, the double-parabolic sampling scheme can be parameterized by the dilation coefficient  and the coverage tuning parameter .Hence, the performance of the resultant stopping rule can be optimized with respect to     and  by choosing various values of  from interval   and determining the corresponding values of   by the computational techniques introduced in Section 2 to guarantee the desired con�dence interval.

Asymptotic Analysis May Be Inadequate.
For fully sequential cases, we have evaluated the double-parabolic sampling scheme with   ,   ,   , and       ≈ .e stopping boundary is displayed in the le side of Figure 2. e function of coverage probability with respect to the binomial proportion is shown in the right side of Figure 2, which indicates that the coverage probabilities are generally substantially lower than the prescribed con�dence level     .By considering    as a small number and applying the asymptotic theory, the coverage probability associated with the sampling scheme is expected to be close to .is numerical example demonstrates that although the asymptotic method is insightful and involves virtually no computation, it may not be adequate.
In general, the main drawback of an asymptotic method is that there is no guarantee of coverage probability.Although an asymptotical method asserts that if the margin of error  tends to , the coverage probability will tend to the prespeci�ed con�dence level   , it is difficult to determine how small the margin of error  is sufficient for the asymptotic method to be applicable.Note that    implies the average sample size tends to ∞.However, in reality, the sample sizes must be �nite.Consequently, an asymptotic method inevitably introduces unknown statistical error.Since an asymptotic method does not necessarily guarantee the prescribed con�dence level, it is not fair to compare its associated sample size with that of an exact method, which guarantees the prespeci�ed con�dence level.
is example also indicates that, due to the discrete nature of the problem, the coverage probability is a discontinuous and erratic function of , which implies that Monte Carlo simulation is not suitable for evaluating the coverage performance.

Parametric Values of Fully Sequential Schemes.
For fully sequential cases, to allow direct application of our doubleparabolic sequential method, we have obtained values of coverage tuning parameter , which guarantee the prescribed con�dence levels, for double-parabolic sampling schemes with   4 and various combinations of   as shown in Table 1.We used the computational techniques introduced in Section 2 to obtain this table.
To illustrate the use of Table 1, suppose that one wants a fully sequential sampling procedure to ensure that Pr{| |   |    for any    .is means that one can choose   ,    and the range of sample size is given by (30)   ,   , and    into its de�nition.e stopping boundary of this sampling scheme is displayed in the le side of Figure 3. e function of coverage probability with respect to the binomial proportion is shown in the right side of Figure 3.

Parametric Values of Group Sequential
Schemes.In many situations, especially in clinical trials, it is desirable to use group sequential sampling schemes.In Tables 2 and 3, assuming that sample sizes satisfy (8) for the purpose of having approximately equal group sizes, we have obtained parameters for concrete schemes by the computational techniques introduced in Section 2.
For dilation coefficient    and con�dence parameter   , we have obtained values of coverage tuning parameter , which guarantee the prescribed con�dence level , for double-parabolic sampling schemes, with the number of stages  ranging from  to , as shown in Table 2.
For dilation coefficient    and con�dence parameter   , we have obtained values of coverage tuning parameter , which guarantee the prescribed con�dence level , for double-parabolic sampling schemes, with the number of stages  ranging from  to , as shown in Table 3.
To illustrate the use of these tables, suppose that one wants a ten-stage sampling procedure of approximately equal group sizes to ensure that Pr{|   |       for any    .is means that one can choose     ,    and sample sizes satisfying (8).To obtain appropriate parameter values for the sampling procedure, one can look at Table 3 to �nd the coverage tuning parameter  corresponding to    and   .From Table 3, it can be seen that  can be taken as .Consequently, the stopping rule is completely determined by substituting the values of design parameters   ,   ,   ,   , and    into its de�nition and (8).e stopping boundary of this sampling scheme and the function of coverage probability with respect to the binomial proportion are displayed, respectively, in the le and right sides of Figure 4.

Comparison of Sampling Schemes.
We have conducted numerical experiments to investigate the impact of dilation coefficient  on the performance of our double-parabolic sampling schemes.Our computational experiences indicate that the dilation coefficient    is frequently a good choice in terms of average sample number and coverage probability.For example, consider the case that the margin of error is given as    and the prescribed con�dence level is    with   .For the double-parabolic sampling scheme with the dilation coefficient  chosen as  , and , we have determined that, to ensure the prescribed con�dence level     , it suffices to set the coverage tuning parameter  as   and , respectively.e average sample numbers of these sampling schemes and the coverage probabilities as functions of the binomial proportion are shown, respectively, in the le and right sides  of Figure 5. From Figure 5, it can be seen that a doubleparabolic sampling scheme with dilation coefficient    has better performance in terms of average sample number and coverage probability as compared to that of the doubleparabolic sampling scheme with smaller or larger values of dilation coefficient.
We have investigated the impact of con�dence intervals on the performance of fully sequential sampling schemes constructed from the inclusion principle.We have observed that the stopping rule derived from Clopper-Pearson intervals generally outperforms the stopping rules derived from other types of con�dence intervals.�owever, via appropriate choice of the dilation coefficient, the double-parabolic sampling scheme can perform uniformly better than the stopping rule derived from Clopper-Pearson intervals.To illustrate, consider the case that    and   .For stopping rules derived from Clopper-Pearson intervals, Fishman's intervals, Wilson's intervals, and revised Wald intervals with   , we have determined that to guarantee the prescribed con�dence level  −   , it suffices to set the coverage tuning parameter  as , , 2, and , respectively.For the stopping rule derived from Wald intervals, we have determined    to ensure the con�dence level, under the condition that the minimum sample size is taken as ⌈( (.Recall that for the double-parabolic sampling scheme with   , we have obtained   2 for purpose of guaranteeing the con�dence level.e average sample numbers of these sampling schemes are shown in Figure 6.From these plots, it can be seen that as compared to the stopping rule derived from Clopper-Pearson intervals, the stopping rule derived from the revised Wald intervals performs better in the region of  close to  or , but performs worse in the region of  in the middle of (, .e performance of stopping rules from Fishman's intervals (i.e., from Chernoff bound) and Wald intervals are obviously inferior as compared to that of the stopping rule derived from Clopper-Pearson intervals.It can be observed that the double-parabolic sampling scheme uniformly outperforms the stopping rule derived from Clopper-Pearson intervals.
In some situations, we need to estimate a binomial proportion with a high con�dence level.For e�ample, one might want to construct a sampling scheme such that, for    and    − , the resultant sequential estimator   satis�es Pr{|  − |       −  for any   (, .By working with the complementary coverage probability, we determined that it suffices to let the dilation coefficient    and the coverage tuning parameter   .e stopping boundary and the function of coverage probability with respect to the binomial proportion are displayed, respectively, in the le and right sides of Figure 7.As addressed in Section 5.3, it should be noted that it is impossible to obtain such a sampling scheme without working with the complementary coverage probability.

Illustrative Examples for Clinical Trials
In this section, we shall illustrate the applications of our double-parabolic group sequential estimation method in clinical trials.
An example of our double-parabolic sampling scheme can be illustrated as follows.Assume that      is given and that the sampling procedure is expected to have 7 stages with sample sizes satisfying (8).Choosing   , we have determined that it suffices to take   7 to guarantee that the coverage probability is no less than 1 −    for all    1.Accordingly, the sample sizes of this sampling scheme are calculated as  11 17 1 88 , and .is sampling scheme, with a sample path, is shown in the le side of Figure 8.In this case, the stopping rule can be equivalently described by virtue of Figure 8 as the following: continue sampling until   ℓ   ℓ  hit a green line at some stage.e coverage probability is shown in the right side of Figure 8.
To apply this estimation method in a clinical trial for estimating the proportion  of a binomial response with margin of error  and con�dence level %, we can have seven groups of patients with group sizes  7 7 8 7 7, and 8.In the �rst stage, we conduct experiment with       99.Since the stopping rule is not satis�ed with the values of (   ,   )  (99, ), we need to conduct the �h stage of experiment with the  patients of the �h group.Suppose we observe that  patients among this group have positive responses.en, we add  with , the number of positive responses before the �h stage, to get  positive responses among    9 +  +  + 8 +   88 patients.So, at the �h stage, we get the relative frequency     88  8.It can be seen that the stopping rule is satis�ed with the values of (   ,   )  (8, 88).erefore, we can terminate the sampling experiment and take    88  8 as an estimate of the proportion of the whole population having positive responses.With a 9% con�dence level, one can believe that the difference between the true value of  and its estimate    8 is less than .
In this experiment, we only use 88 samples to obtain the estimate for .Except the round-off error, there is no other source of error for reporting statistical accuracy, since no asymptotic approximation is involved.As compared to �xed-sample-size procedure, we achieved a substantial save of samples.To see this, one can check that using the rigorous formula (37) gives a sample size 8, which is overly conservative.From the classical approximate formula (35), the sample size is determined as 8, which has been known to be insufficient to guarantee the prescribed con�dence level 9%.e exact method of [34] shows that at least 9 samples are needed.As compared to the best-�xed-sample size obtained by the method of [34], the reduction of sample sizes resulted from our double-parabolic sampling scheme is 9 − 88  .It can be seen that the �xed-sample-size procedure wastes 88  % samples as compared to our group sequential method, which is also an exact method.is percentage may not be serious if it were a save of a number of simulation runs.However, as the number count is for patients, the reduction of samples is important for ethical and economical reasons.Using our group sequential method, the worst-case sample size is equal to , which is only  more than the minimum sample size of �xed-sample procedure.However, a lot of samples can be saved in the average case.
As  or  become smaller, the reduction of samples is more signi�cant.For example, let    and   , we have a double-parabolic sample scheme with  stages.e sampling scheme, with a sample path, is shown in the le side of Figure 9. e coverage probability is shown in the right side of Figure 9.

Conclusion
In this paper, we have reviewed recent development of group sequential estimation methods for a binomial proportion.We have illustrated the inclusion principle and its applications to various stopping rules.We have introduced computational techniques in the literature, which suffice for determining parameters of stopping rules to guarantee desired con�dence levels.Moreover, we have proposed a new family of sampling schemes with stopping boundary of double-parabolic shape, which are parameterized by the coverage tuning parameter and the dilation coefficient.ese parameters can be determined by the exact computational techniques to reduce the sampling cost, while ensuring prescribed con�dence levels.e new family of sampling schemes are extremely simple in structure and asymptotically optimal as the margin of error tends to .We have established analytic bounds for the distribution and expectation of the sample number at the termination of the sampling process.We have obtained parameter values via the exact computational techniques for the proposed sampling schemes such that the con�dence levels are guaranteed and that the sampling schemes are generally more efficient as compared to existing ones.

F 5 :F 6 :
Double-parabolic sampling with various dilation coefficients.Comparison of average sample numbers.
eorem 2. Let ,  ∈ 0, 1 and  ∈ 0, 1 be ��ed.Assume that the number of stages  and the sample sizes  1 , … ,   are functions of  ∈ 0, 1 such that the constraint (27) is satis�ed.�en� Pr{|   | <    is no less than 1   for any  ∈ 0, 1 provided that 1 +1 stages and group size 1.Clearly, if ,  and  are �xed, the sampling scheme is dependent only on .Hence, for any   , 1, if we allow  to vary in , 1, then the coverage probability Pr{|  − | <    and the average sample number  are functions of .We are interested in knowing the asymptotic behavior of these functions as   , since  is usually small in practical situations.e following theorem provides us the desired insights.
. From Table 1, it can be seen that the value of  corresponding to    and    is 474.Consequently, the stopping rule is completely determined by substituting the values of design parameters   , Double-parabolic sampling with   ,   ,   , and   .
Consider function       2 /   for     and    .It can be checked that  /            2, which shows that for any �xed    ,   is a unimodal function of    , with a maximum attained at   .B such a property of   and the de�nition of �ilson�s con�dence intervals, we have  ℓ     ℓ    <  ℓ     ℓ  ∪    ℓ     <  ℓ     ℓ    ℓ   ℓ   ℓ    2  , where we have used the fact that {  ℓ >   { ℓ >  {  ℓ <     { ℓ <  and    ℓ    ℓ   ℓ  .Recall that  <  < /2.It follows that    ℓ     ℓ   ℓ    ℓ   By the assumption that   ≥ /2 2  ln/, we have /4   2   /2 ln   and, consequently, Pr{|    /2|   2 ≥ /4   2   /2 ln  .It follows from the de�nition of the sampling scheme that the sampling process must stop at or before the th stage.In other words, Pr{    .is allows one to write exp(4(   (    exp(2 2 ].is inequality can be written as 4(   (  (2    exp(2 2 ] or, equivalently,   ( exp(((2    exp(2 2 ]4(  .e proof of the theorem is thus completed.First, we need to show that Pr{im    ((          for any   ( .Clearly, the sample number  is a random number dependent on .Note that for any   , the sequences { ( ( ( and { ( ( ( are subsets of {  ( ∞  .By the strong law of large numbers, for almost every   , the sequence {  ( ∞  converges to .Since every subsequence of a convergent sequence must converge, it follows that the sequences { ( ( ( and { ( ( ( converge to  as    provided that (  ∞ as   .Since it is certain that   2((   (  ∞ as   , we have that {im    ((     is a sure event.It follows that   {im        im        im    ((     is an almost sure event.By the de�nition of the sampling scheme, we have    is an almost sure event. ��ne   {im    ((     .We need to show that  is an almost sure event.For this purpose, we let      and expect to show that   .As a consequence of     , Next, we need to show that im    Pr{|   | <     2Φ(2 (   for any   ( .For simplicity of notations, let   (   and   2 (.Note that Pr{| | <     Pr{|  | <     Pr{√|  | < √.Clearly, for any   ( , Recall that we have established that , , ,    almost surely as   .is implies that √   and , , ,    in probability as  tends to zero.It follows from Anscombe's random central limit theorem [35] that as  tends to zero, √  −  converges in distribution to a Gaussian random variable with zero mean and unit variance.Hence, from (C.6), for any    .erefore, to guarantee that Pr{|   | <    ≥    for any    , it is sufficient to choose  such that 2 Combining (C.3) and (C.5) yields im    (((      and thus   .is implies that  is an almost sure event and thus Pr{im    ((          for   ( . C.16) Since lim sup    ⌈  , , ,  =   , for the purpose of establishing lim sup    , , ,  ≤   , it remains to show that Consider functions           and      for    .Note that                       ≤  +   (C.18) for all    .For    , there exists a positive number       such that        for any       + , since  is a continuous function of .From now on, let    be sufficiently small such that  +     .en,   .Since +  +   l ≤    l for all  ≥  + , it follows that  ≥  + .erefore, we have shown that if  is sufficiently small, then there exists a number    such that Using this inclusion relationship and the Chernoff-Hoeffding bound [32, 33], we have Pr   ≤ Pr  ∉     +  ≤  exp    (C.24) for all  ≥  +  provided that    is sufficiently small.Letting    +  and using (C.24), we have    and    as   .So, we have established (C.11).Since the argument holds for arbitrarily small   , it must be true that l          for any    .is completes the proof of the theorem.Recall that  denotes the index of stage at the termination of the sampling process.Observing that      Pr      Pr ≤     Pr ≤ ⊆   ∉     +   (C.23) for all  ≥  + .D. Proof of Theorem 4 ℓ  ℓ+   ℓ  Pr ≤ ℓ  (D.1)