On-Line Selection of c-Alternating Subsequences from a Random Sample

Abstract

A sequence 𝑋₁,𝑋₂,…,𝑋_𝑛 is a 𝑐-alternating sequence if any odd term is less than or equal to the next even term −𝑐 and the any even term is greater than or equal to the next odd term +𝑐, where 𝑐 is a nonnegative constant. In this paper, we present an optimal on-line procedure to select a 𝑐-alternating subsequence from a symmetric distributed random sample. We also give the optimal selection rate when the sample size goes to infinity.

1. Introduction

Given a finite (or infinite) sequence 𝑥={𝑥₁,𝑥₂,…,𝑥_𝑛,…} of real numbers, we say that a subsequence 𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑘,… with 1≤𝑖₁<𝑖₂<⋯<𝑖_𝑘<⋯ is 𝑐-alternating if we have 𝑥_𝑖1+𝑐<𝑥_𝑖2>𝑥_𝑖3+𝑐<𝑥_𝑖4⋯, where 𝑐 is a nonnegative real number. When 𝑐=0, 𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑘,… is alternating. We are mainly concerned with the length 𝛼(𝑥) of the longest 𝑐-alternating subsequence of 𝑥. Here, we study the problem of making on-line selection of a 𝑐-alternating subsequence. That is now we regard the sequence 𝑥₁,𝑥₂,… as being available to us sequentially, and, at time 𝑖 when 𝑥_𝑖 is available, we must choose to include 𝑥_𝑖 as a term of our subsequence or reject 𝑥_𝑖 as a member of our subsequence.

We will consider the sequence to be given by independent, identically distributed, symmetric random variables over the interval [0,1]. In [1], Arlotto et al. studied the case that the sequence to be given by independent, identically distributed, uniform random variables over the interval [0,1] and 𝑐=0. So this paper can be considered as an extension of their paper.

Now we need to be more explicit about the set 𝐼𝐼 of feasible strategies for on-line selection. At time 𝑖, when 𝑋_𝑖 is presented to us, we must decide to select 𝑋_𝑖 based on its value, the value of earlier members of the sequence, and the actions we have taken in the past. All of this information can be captured by saying that 𝑖_𝑘, the index of the 𝑘th selection, must be a stopping time with respect to the increasing sequence of 𝜎-fields, 𝐹_𝑖=𝜎{𝑋₁,𝑋₂,…,𝑋_𝑖}, 𝑖=1,2,…. Given any feasible policy 𝜋 in 𝐼𝐼 the random variable of most interest here is 𝐴^𝑜_𝑛(𝜋), the number of selections made by the policy 𝜋 up to and including time 𝑛. In other words, 𝐴^𝑜_𝑛(𝜋) is equal to the largest 𝑘 for which there are stopping times 1≤𝑖₁<𝑖₂<⋯<𝑖_𝑘≤𝑛 such that {𝑋_𝑖1,𝑋_𝑖2,…,𝑋_𝑖𝑘} is a 𝑐-alternating subsequence of the sequence {𝑋₁,𝑋₂,…,𝑋_𝑛}. In this paper, we are interested in the optimal selection and the asymptotic rate of the optimal selection. That is, we have a selection policy 𝜋^∗_𝑛 in 𝐼𝐼 such that lim_𝑛→∞𝐸𝐴^𝑜_𝑛𝜋^∗_𝑛=lim_𝑛→∞sup_{𝜋∈𝐼𝐼}𝐸𝐴^𝑜_𝑛(𝜋). (1)

2. Main Results

For each 0≤𝑐<1 and each 0≤𝜆≤(1−𝑐)/2, we define a threshold function 𝑓^∗ as follows: 𝑓^∗(𝑦)=max{𝑐+𝜆,𝑐+𝑦} for all 0≤𝑦≤1−𝑐. We now recursively define random variables {𝑌_𝑖∶𝑖=1,2,…} by setting 𝑌₀=𝑦 and taking 𝑌_𝑖=𝑌_𝑖−1 if 𝑋_𝑖<𝑓^∗(𝑌_𝑖−1), 𝑌_𝑖=1−𝑋_𝑖 if 𝑋_𝑖≥𝑓^∗(𝑌_𝑖−1). Introduce a value function 𝑉(𝜆,𝑦,𝜌)=𝐸{∑^∞_𝑖=1𝜌^𝑖−1𝐼[𝑋_𝑖≥𝑓^∗(𝑌_𝑖−1)]∣𝑌₀=𝑦}, where 0<𝜌<1 is a constant and 𝐼[𝑋_𝑖≥𝑓^∗(𝑌_𝑖−1)] is the indicator function of the event [𝑋_𝑖≥𝑓^∗(𝑌_𝑖−1)].

Let 𝐺 be the distribution function and 𝑔 the probability density function of 𝑋_𝑖. If 𝑋_𝑖 is not a uniform random variable over the interval [0,1], then we will assume that 𝑔^′ exists and is nonzero. Since 𝑋_𝑖 is symmetric over the interval [0,1], 𝐺(1−𝑥)=1−𝐺(𝑥) and 𝑔(𝑥)=𝑔(1−𝑥) for all 0≤𝑥≤1.

It is easy to see that 𝑉𝑉+𝜆,𝑦,𝜌=𝜌𝐺max𝜆,𝑦+𝑐𝜆,𝑦,𝜌_{1max𝜆,𝑦+𝑐}𝑔(𝑥)𝑉+1+𝜌𝑉𝜆,1−𝑥,𝜌𝑑𝑥=𝜌𝐺max𝜆,𝑦+𝑐𝜆,𝑦,𝜌_{01−max(𝜆,𝑦)−𝑐}1+𝜌𝑉𝜆,𝑥,𝜌𝑔(𝑥)𝑑𝑥 (2) since 𝑔(𝑥)=𝑔(1−𝑥) for all 0≤𝑥≤1. For simplicity, we will let 𝑉(𝑦) denote 𝑉(𝜆,𝑦,𝜌) for fixed 𝜆 and 𝜌. It is easy to check that for all 0≤𝑦≤𝜆, 𝑉(𝑦)=𝜌𝐺(𝜆+𝑐)𝑉(𝑦)+∫₀^{1−𝜆−𝑐}[1+𝜌𝑉(𝑥)]𝑔(𝑥)𝑑𝑥, for all 𝜆<𝑦≤1−𝑐, 𝑉(𝑦)=𝜌𝐺(𝑦+𝑐)𝑉(𝑦)+∫₀^{1−𝑦−𝑐}[1+𝜌𝑉(𝑥)]𝑔(𝑥)𝑑𝑥, and for all 1−𝑐<𝑦≤1, 𝑉(𝑦)=0. Since for all 0≤𝑦≤𝜆, 𝑉(𝑦)=𝑉(𝜆), 𝑉(𝑦)=𝜌𝐺(𝑦+𝑐)𝑉(𝑦)+[1+𝜌𝑉(𝜆)]𝐺(1−𝑦−𝑐) for all 1−𝜆−𝑐≤𝑦≤1−𝑐. Since 𝐺(1−𝑦−𝑐)=1−𝐺(𝑦+𝑐), 𝑉(𝑦)=𝜌𝐺(𝑦+𝑐)𝑉(𝑦)+[1+𝜌𝑉(𝜆)][1−𝐺(𝑦+𝑐)] for all 1−𝜆−𝑐≤𝑦≤1−𝑐. From now on, we will let 𝑉^′(𝜆) denote the right derivative of the function 𝑉 at 𝜆 and 𝑉^′(1−𝑐) denote the left derivative of the function 𝑉 at 1−𝑐.

For 𝜆<𝑦<1−𝑐, when we differentiate 𝑉(𝑦) we have the following differential equation: 𝑉^′𝑦𝑉𝑦−𝑔.1−𝜌𝐺𝑦+𝑐=𝜌𝑔𝑦+𝑐1+𝜌𝑉1−𝑦−𝑐1−𝑦−𝑐 (3) Since 𝑔(1−𝑦−𝑐)=𝑔(𝑦+𝑐) and 𝑔(1−𝑦)=𝑔(𝑦). Now we have the following differential equations: 𝑉^′𝑦𝑉𝑦−𝑔𝜌𝑉𝑦,𝑉1−𝜌𝐺𝑦+𝑐=𝜌𝑔𝑦+𝑐1+𝜌𝑉1−𝑦−𝑐𝑦+𝑐=𝑔𝑦+𝑐−𝑉1−𝑦−𝑐−1_′𝑦𝑉−𝑦𝑔𝑦𝑦𝜌𝑉𝑦.1−𝑦−𝑐1−𝜌𝐺1−𝑦=𝜌𝑔1−𝑦−𝑐1+𝜌𝑉=𝑔1−𝑦−𝑐−𝑉−1 (4)(5) Add (3) and (4) together, we have 𝑉^′1−𝑦−𝑐1−𝜌𝐺1−𝑦𝑔𝑦=−𝑉^′𝑦1−𝜌𝐺𝑦+𝑐𝑔𝑦+𝑐−2. (6) In summary, we have the following equations: 𝑉((𝜆=(𝜆𝐺(𝜆),𝑉1−𝜆−𝑐)1−𝜌+𝜌𝐺)1+𝜌𝑉)_′(𝜆((𝜌[𝑉(𝜆)(,𝑉)1−𝜌𝐺𝜆+𝑐)=𝑔𝜆+𝑐)−𝑉1−𝜆−𝑐)]−1_′((𝜆(𝜆𝜌[𝑉()(𝜆.1−𝜆−𝑐)1−𝜌+𝜌𝐺)=𝑔)1−𝜆−𝑐−𝑉)]−1 (7) For all 𝜆≤𝑦≤1−𝑐, let us define ℎ𝑦=1−𝜌𝐺𝑦+𝑐²1−𝜌+𝜌𝐺𝑦𝑔.𝑦+𝑐 (8) Then we have the following equation: 𝑉^′(1−𝜆−𝑐)=𝑉^′(𝜆)ℎ(𝜆)ℎ()−1−𝜆−𝑐2𝜌ℎ(1−𝜆−𝑐)_𝜆^{1−𝜆−𝑐}1−𝜌+𝜌𝐺(𝑥−𝑐)𝑔(𝑥)𝑑𝑥. (9)

Proof of (9). Differentiate (4). Again, we have 𝑉^′′𝑦1−𝜌𝐺𝑦+𝑐=𝜌𝑔𝑦+𝑐2𝑉^′𝑦+𝑉^′1−𝑦−𝑐+𝑔^′𝑦+𝑐𝜌𝑉𝑦−𝑉1−𝑦−𝑐−1. (10) Replace 𝜌[𝑉(𝑦)−𝑉(1−𝑦−𝑐)]−1 by 𝑉^′(𝑦)((1−𝜌𝐺(𝑦+𝑐))/𝑔(𝑦+𝑐)) and replace 𝑉^′(1−𝑦−𝑐) by {−2−𝑉^′(𝑦)((1−𝜌𝐺(𝑦+𝑐))/𝑔(𝑦+𝑐))}{𝑔(𝑦)/(1−𝜌+𝜌𝐺(𝑦))}. And after the simplification, we have the following equation: 𝑉^′′𝑦𝑦1−𝜌𝐺𝑦+𝑐=𝜌𝑔𝑦+𝑐−2𝑔1−𝜌+𝜌𝐺𝑦+𝑉^′𝑦×𝑔^′𝑦+𝑐1−𝜌𝐺𝑦+𝑐𝑔𝑔𝑦𝑦+𝑐+𝜌𝑔𝑦+𝑐2−1−𝜌𝐺𝑦+𝑐𝑔𝑦.𝑦+𝑐1−𝜌+𝜌𝐺+𝜌𝑔𝑦+𝑐 (11) Multiplying both sides of (11) by [1−𝜌+𝜌𝐺(𝑦)][1−𝜌𝐺(𝑦+𝑐)]/𝑔(𝑦+𝑐), we obtain the following equation: 𝑉^′′𝑦1−𝜌𝐺𝑦+𝑐²1−𝜌+𝜌𝐺𝑦𝑔𝑦𝑦+𝑐=−2𝜌𝑔1−𝜌𝐺𝑦+𝑐+𝑉^′𝑦𝑔^′𝑦+𝑐1−𝜌𝐺𝑦+𝑐𝑔𝑦+𝑐²1−𝜌+𝜌𝐺𝑦+𝑉^′𝑦2𝑦𝜌𝑔𝑦+𝑐1−𝜌+𝜌𝐺1−𝜌𝐺𝑦+𝑐𝑔^′𝑦𝑦+𝑐−𝑔1−𝜌𝐺𝑦+𝑐𝑔𝑦+𝑐². (12) Notice that −ℎ^′𝑦=𝑔^′𝑦+𝑐1−𝜌𝐺𝑦+𝑐𝑔𝑦+𝑐²𝑦2𝑦1−𝜌+𝜌𝐺+𝜌𝑔𝑦+𝑐1−𝜌+𝜌𝐺1−𝜌𝐺𝑦+𝑐𝑔^′𝑦𝑦+𝑐−𝑔1−𝜌𝐺𝑦+𝑐𝑔𝑦+𝑐². (13) Equation (12) can be rewritten as 𝑉^′′𝑦ℎ𝑦+𝑉^′𝑦ℎ^′𝑦=−2𝜌1−𝜌𝐺𝑦+𝑐𝑔𝑦. (14) By integrating both sides of (14), we have 𝑉^′𝑦=𝑉^′(𝜆)ℎ(𝜆)ℎ𝑦−2𝜌ℎ𝑦^𝑦_𝜆1−𝜌𝐺(𝑧+𝑐)𝑔(𝑧)𝑑𝑧. (15)

Therefore, we have the following theorem.

Theorem 1. (i)𝑉((𝜆=(𝜆𝐺(𝜆),()𝑉1−𝜆−𝑐)1−𝜌+𝜌𝐺)1+𝜌𝑉)ii^′(𝜆((𝜌[𝑉(𝜆)(,()𝑉)1−𝜌𝐺𝜆+𝑐)=𝑔𝜆+𝑐)−𝑉1−𝜆−𝑐)]−1iii^′((𝜆(𝜆𝜌[𝑉()(𝜆,()𝑉1−𝜆−𝑐)1−𝜌+𝜌𝐺)=𝑔)1−𝜆−𝑐−𝑉)]−1iv^′(1−𝜆−𝑐)=𝑉^′(𝜆)ℎ(𝜆)ℎ()−1−𝜆−𝑐2𝜌ℎ()×1−𝜆−𝑐_{1−𝜆𝜆+𝑐}1−𝜌+𝜌𝐺(𝑥−𝑐)𝑔(𝑥)𝑑𝑥. (16)

Now we have four unknown variables 𝑉(𝜆), 𝑉(1−𝜆−𝑐), and 𝑉^′(𝜆), 𝑉^′(1−𝜆−𝑐) and also have four linear equations involving these four unknown variables. We solve these four linear equations and obtain the following solutions.

Theorem 2. (i)𝑉(𝜆)=𝜌𝐺(𝜆)1−𝜌𝐺(𝜆+𝑐)+𝜌^1−𝜆_𝜆+𝑐(𝑔(𝑥)×𝜌(1−𝜌+𝜌𝐺𝑥−𝑐)𝑑𝑥1−𝜌1−𝜌𝐺𝜆+𝑐)⁻¹,(ii)𝑉^′(𝜆)=𝑔()×𝜌𝜆+𝑐_{1−𝜆𝜆+𝑐}(𝑔(𝑥)−(×(𝜆×(1−𝜌+𝜌𝐺𝑥−𝑐)𝑑𝑥1−𝜌𝐺𝜆+𝑐)1−𝜌+𝜌𝐺)1−𝜌𝐺𝜆+𝑐)²1−𝜌+𝜌𝐺(𝜆)⁻¹,()𝑉()(𝜆(𝜆(iii1−𝜆−𝑐=𝐺)1−𝜌+𝜌𝐺)1−𝜌𝐺𝜆+𝑐)−𝜌^1−𝜆_𝜆+𝑐(𝑔(𝑥)×(×(𝜆1−𝜌+𝜌𝐺𝑥−𝑐)𝑑𝑥1−𝜌1−𝜌𝐺𝜆+𝑐)1−𝜌+𝜌𝐺)⁻¹. (17)

By Theorem 2, 𝑉^′(𝜆)=0 if and only if 𝜌^1−𝜆_𝜆+𝑐(𝑔(𝑥)=((𝜆1−𝜌+𝜌𝐺𝑥−𝑐)𝑑𝑥1−𝜌𝐺𝜆+𝑐)1−𝜌+𝜌𝐺) (18)since 𝑔(𝜆+𝑐) and [1−𝜌𝐺(𝜆+𝑐)]²[1−𝜌+𝜌𝐺(𝜆)] are positive. For each 0<𝜌<1, let 𝜆(𝜌) denote a solution of 𝑉′(𝜆)=0. The next theorem indicates that when 𝜌<1 but close enough to 1, 𝜆(𝜌) is unique and 0≤𝜆(𝜌)<(1−𝑐)/2.

Theorem 3. When 𝜌<1 but close enough to 1, 𝜆(𝜌) is unique and 0≤𝜆(𝜌)<(1−𝑐)/2.

Proof. For all 0≤𝜆<(1−𝑐)/2, let 𝐾(𝜆)=𝜌^1−𝜆_𝜆+𝑐(𝑔(𝑥)−((𝜆.1−𝜌+𝜌𝐺𝑥−𝑐)𝑑𝑥1−𝜌𝐺𝜆+𝑐)1−𝜌+𝜌𝐺) (19) Then 𝐾^′(𝜆)𝑔(((𝜆(𝐾=−𝜌1−𝜆)1−𝜌+𝜌𝐺1−𝜆−𝑐)+𝑔)1−𝜌𝐺𝜆+𝑐)<0,1−𝑐2=−1−𝜌+𝜌𝐺1−𝑐21−𝜌𝐺1+𝑐2𝐾(0)<0,=𝜌²¹_𝑐𝐺(𝑥−𝑐)𝑔(𝑥)𝑑𝑥−1−𝜌²>0 (20) if 𝜌 is close enough to 1. Therefore, 0≤𝜆(𝜌)<(1−𝑐)/2 and 𝜆(𝜌) is unique if 𝜌 is close enough to 1. This completes the proof of Theorem 3.

A routine calculation is as follows: 𝑉^′′𝜆𝜌=−2𝜌𝑔𝜆𝜌𝑔𝜆𝜌+𝑐1−𝜌+𝜌𝐺𝜆𝜌1−𝜌𝐺𝜆𝜌+𝑐<0. (21) So we have found the maximum of the function 𝑉. After the simplification, 𝑉𝜆𝜌𝜌=𝜆𝜌,𝜆,𝜌1−𝜌+2𝜌𝐺𝜌1−𝜌 (22) if 𝜌 is close enough to 1. Therefore, (1−𝜌)𝑉(𝜆(𝜌),𝜆(𝜌),𝜌)→2𝐺(𝜆(1)) as 𝜌↑1, where _𝜆(1)^1−𝜆(1)_+𝑐𝐺(𝑥−𝑐)𝑔(𝑥)𝑑𝑥=𝐺(𝜆(1))[1−𝐺(𝜆(1)+𝑐)]. (23)

Example 4. When 𝑐=0, then 2𝐺(𝜆(1))=2−√2.

Example 5. When 𝐺(𝑥)=𝑥 for all 0≤𝑥≤1, then 2𝐺(𝜆(1))=(1−𝑐)(2−√2).

In [1], the following two strategies are mentioned:(I) the maximally timid strategy which can be described as follows: at the start, accept the first observation which is less than 1/2, then accept the next one which is greater than 1/2, then accept the next one which is less than 1/2. Continue this way until we observe 𝑛 observations, then we stop;(II) the purely greedy strategy which can be described as follows: at the start, accept the first observation, then accept the next one which is greater than the first one, accept the next one which is less than the second selected one, then accept the next one which is greater than the third selected one. Continue this way until we observe 𝑛 observations, then we stop.

Now we define these two strategies for the 𝑐-alternating subsequence as follows:(I^′) the maximally 𝑐-timid strategy which can be described as follows: at the start, accept the first observation which is less than (1−𝑐)/2, accept the next one which is greater than (1+𝑐)/2, accept the next one which is less than (1−𝑐)/2, then accept the next one which is greater than (1+𝑐)/2. Continue this way until we observe 𝑛 observations, then we stop;(II^′) the purely 𝑐-greedy strategy which can be described as follows: at the start, accept the first observation which is less than 1−𝑐, accept the next one which is greater than the first selected one +𝑐, accept the next one which is less than the second selected one −𝑐, then accept the next one which is greater than the third selected one +𝑐. Continue this way until we observe 𝑛 observations, then we stop.

When 𝑐=0, the maximally 𝑐-timid strategy is the maximally timid strategy and the purely 𝑐-greedy strategy is the purely greedy strategy. In fact, the maximally 𝑐-timid strategy is the case when 𝜆=(1−𝑐)/2 and the purely 𝑐-greedy strategy is the case when 𝜆=0.

The asymptotic selection rate for the maximally 𝑐-timid strategy is 𝐺((1−𝑐)/2) and the asymptotic selection rate for the purely 𝑐-greedy strategy is ∫₀^1−𝑐𝐺(𝑥)𝑔(𝑥+𝑐)𝑑𝑥/(1−𝐺(𝑐)).

Example 6. When 𝑐=0, then the asymptotic selection rate for both the maximally timid strategy and the purely greedy strategy is 1/2. These results are the same as those in [1].

Example 7. When 𝐺(𝑥)=𝑥 for all 0≤𝑥≤1, then the asymptotic selection rate for both the maximally 𝑐-timid strategy and the purely 𝑐-greedy strategy is (1/2)(1−𝑐). It is easy to see these results are consistent with the result of Example 5.

If the random variables 𝑋₁,𝑋₂,…,𝑋_𝑛 are independent, identically distributed symmetric random variables over the interval [𝑎,𝑏], where 𝑎<𝑏 and 𝑎,𝑏 are finite, then we can change 𝑋_𝑖 into 𝑍_𝑖 by 𝑍_𝑖=(𝑋_𝑖−𝑎)/(𝑏−𝑎). Then 𝑍₁,𝑍₂,…,𝑍_𝑛 are independent, identically distributed symmetric random variables over the interval [0,1]. Let 𝑐^′=𝑐/(𝑏−𝑎). Then selecting a 𝑐-alternating subsequence from the random sample 𝑋₁,𝑋₂,…,𝑋_𝑛 is exactly the same to select a 𝑐^′-alternating subsequence from the random sample 𝑍₁,𝑍₂,…,𝑍_𝑛. So the asymptotic selection rate is still the same. Or we can find 𝜆(1) directly by solving the following equation: ₍₁₎^{𝑏−𝑎−𝜆(1)}_{𝑎+𝜆+𝑐}𝐺()𝑔(𝑥)((1((1).𝑥−𝑐𝑑𝑥=𝐺𝑎+𝜆))[1−𝐺𝑎+𝜆+𝑐)] (24) For 0≤𝑐≤(𝑏−𝑎), 𝑎≤𝑦≤(𝑏−𝑐), and 𝑎≤𝜆<𝑎+(𝑏−𝑎−𝑐)/2 the threshold function 𝑓^∗ is defined by 𝑓^∗(𝑦)=max(𝜆,𝑦)+𝑐 for all 𝑎≤𝑦≤(𝑏−𝑐). This time, we recursively define random variables {𝑌_𝑖∶𝑖=1,2,…} by setting 𝑌₀=𝑦 and taking 𝑌_𝑖=𝑌_𝑖−1 if 𝑋_𝑖<𝑓^∗(𝑌_𝑖−1), 𝑌_𝑖=𝑏−𝑋_𝑖 if 𝑋_𝑖≥𝑓^∗(𝑌_𝑖−1). Here 𝑎≤𝑦≤(𝑏−𝑐), 𝐺₁(𝑥)=()12𝑥1−𝑥if0≤𝑥≤2,1−2𝑥+2𝑥²1if2𝐺<𝑥≤1,₂(𝑥)=2𝜋sin⁻¹√𝑥,𝐺₃(𝑥)𝐺=𝑥,₄(𝑥)=3𝑥²−2𝑥³,𝐺₅(𝑥)=2𝑥²1if0≤𝑥≤2,−2𝑥²1+4𝑥−1if2<𝑥≤1. (25)

From Table 1, it seems that when the distribution has higher chances on the tails, then the asymptotic optimal selection rate is higher. On the other hand, when the distribution has higher chance near the center, then the asymptotic optimal selection rate is lower. However, we do not have a proof for this statement.

Table 1 

The asymptotic optimal selection rates for various distributions and 𝑐.

We are now considering the case when 𝑋₁,𝑋₂,…,𝑋_𝑛 have an arbitrary distribution. We have made some progress, but it is still in the premature state. We hope to be able to find the asymptotic optimal selection rate in a forth coming paper.