由概似函數建構離散型分配的信賴區間

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(1)國立交通大學統計學研究所碩士論文. 由概似函數建構離散型分配的信賴區間 Likelihood-Based Confidence Intervals for Discrete Distributions. 研究生：施婉菁指導教授：陳鄰安博士. 中華民國九十四年六月.

(2) 由概似函數建構離散型分配的信賴區間 Likelihood-Based Confidence Intervals for Discrete Distributions 研究生：施婉菁. Student：Wan-Ching Shih. 指導教授：陳鄰安博士. Advisor：Dr. Lin-An Chen. 國立交通大學理學院統計研究所碩士論文. A thesis Submitted to Institute of Statistics College of Science National Chiao Tung University in partial Fulfillment of the Requirements for the Degree of Master in Statistics June 2005 Hsinchu, Taiwan, Republic of China. 中華民國九十四年六月.

(3) 由概似函數建構離散型分配的信賴區間研究生：施婉菁指導教授：陳鄰安博士. 國立交通大學統計研究所. 中文摘要延伸 Chen(2004,處理連續行分配的問題)的觀念，我們利用概似函數的頂端區域來建構離散型分配參數的信賴區間。有幾個地方值得我們注意。首先，這個區間顯示在具有相同信賴係數的區間中所包含 x 的點數最少。因為期望長度是一般用來比較信賴區間的準則，而實際上在建構平均長度一致最短的信賴區間目前並未有令人滿意的結果。利用樣本數量的方式來衡量離散型分配的信賴區間不失為一個好方法。第二，傳統區間估計方法得到的信賴區間最被批評的一點是：它可能不包含欲估之參數(即最大概似估計量)。而由概似函數所建構的信賴區間則具有此一性質。另外，現有的信賴區間的建構並未利用到所有參數的資訊。此種區間具有最大概似估計量的性質，如不變性和充分性。由於貝氏型態的事後分配信賴區間也具有最短長度之最佳性，但貝氏過於依賴先驗分配，且並未具有不變性的特性。 i.

(4) Likelihood-Based Confidence Intervals for Discrete Distributions Student: Wan-Ching Shih Adviser: Dr. Lin-An Chen Institute of Statistics National Chiao Tung University Hsinchu, Taiwan Abstract Generalizing from Chen (2004) of dealing for continuous distributions, we introduce a likelihood-based confidence set for the discrete distributions. Besides some properties extended from maximum likelihood estimator, four additional properties are of special interest. First, this set is shown to have volume for the sample falling in the confidence set the smallest among classes of 100(1 − α)% confidence sets. With the fact that expected-length is a popularly used criteria for comparing confidence intervals and, actually, there is no satisfactory results for constructing the optimal one, minimizing the volume, in terms of sample x, is a good criterion for evaluating confidence interval for discrete distributions. Second, the likelihood-based confidence sets include maximum likelihood estimate, the most plausible parameter value after an observation has been made, whereas the traditional frequentist approaches of confidence set are criticized for that may not include it. Third, the construction of these confidence sets are based on Fisher’s likelihood principle which ask that any statistical procedure should depend upon the likelihood function whereas the existed confidence sets do not fulfill this desirability. Fourth, properties behaving for maximum likelihood estimator such as invariance and sufficiency have been carried over to these approaches. The property of invariance is interesting for the fact that the Bayesian highest posterior density confidence set has also an optimality of shortest width, however, this Bayesian interval is criticized for depending on the prior density and for not having the desired property of invariance. Key words: Confidence interval; confidence set; likelihood based confidence interval. 1.

(5) 致謝由衷感謝指導老師陳鄰安老師的教導，論文方能完成，並感謝老師在待人處世生活上，對我的許多建議與協助，在此僅致上我最深的祝福與謝意。還要感謝黃榮臣老師、許文郁老師及洪慧念老師三位口試委員，在百忙之中撥冗審稿與校正，並在口試時給予我建議與指導。謝謝所有所上的老師在這兩年間對於我的指導，讓我獲益匪淺。另外也要謝謝我的研究室同學們，揚波、秀仁、如美、淑儀、怡娟、翊琳及雅靜，一路的互相扶持。能在統計所這樣溫馨的大家庭中成長，是件很幸福的事情。最後要感謝我的家人，在就學期間給予我的支持與鼓勵，讓我能無後顧之憂地專心向學。. 施婉菁 2005.06. iii.

(6) Contents 中文摘要. i. 英文摘要. ii. 誌謝. iii. Contents. iv. 1. Introduction. 1. 2. Likelihood-Based Confidence Set and MLE Related Properties. 3. 3. Existence and Optimality of the Likelihood-Based Confidence Set. 6. 4. Derivation of Likelihood-Based Confidence Intervals. 8. 5. An Example of Exact UHLCI with Fixed Confidence Coefficient. 11. 6. General formulation of UHLCI for Binomial Distribution. 12. 7. Appendix. 15. Reference. 19. Table 1. 21. iv.

(7) 1. Introduction One concern in statistical practice is to divide possible parameters into a plausible set and implausible set. In the late 1920s, following the success of his work with Pearson on hypothesis testing, Neyman began to develop a frequentist approach to estimation of plausible set, called the confidence set. For decades, the application of this technique has been received considerable theoretical and practical value at least. Although with great attention received for confidence set construction, however, serious deficiencies has long been considered for these existed techniques. Silvey (1975) and Lindsey (1996) both pointed out that much of frequentist theory and techniques for confidence sets appear ad hoc because it need not be model-based, not relying on the likelihood function, nor any other single unifying principle, not even having the requirement that all of the information in the data and model must be used. This leads to the first deficiency that these existed techniques are less convincing in the interpretation of generating a subset of plausible values in the parameter space after an observation X = x has been taken. For the second, it is desired that, on the fact that there is nothing unique about a confidence set of given a confidence level, there is a suitable criterion in some sense “smallest” for deciding to choose the best one. Have we ever set a proper criterion for optimal interval estimation? Effort has been made on minimizing the expected length of confidence interval. Pratt (1961) showed that the interval inverted from a two sided UMP test has the property of uniformly smallest expected-length. Unfortunately it is rare in literature to have two sided UMP test. Moreover, Zacks (1971, Section 10.3) considered an example of a sample X of size one drawn from normal distribution N (θ, 1) and showed that there is a 100(1 − α)% confidence interval for θ which has shorter expected length than the popularly used interval (X −zα/2 , X +zα/2 ) when θ is zero. This provides an example that this familiar one can’t be the best in terms of expected length. Therefore, an appropriate criterion that the optimal ones may be obtainable is desired to be introduced for confidence set. For solving these two deficiencies, Chen (2004) extended the concept of maximum likelihood estimate to define a likelihood-based confidence interval. The method of maximum likelihood is appealing because in some sense a maximum likelihood estimate is the most plausible parameter value after an observation of the random sample has been made, that is, it is that parameter value which gives greatest probability. With this appealing, this point estimation technique gains the efficiency 1.

(8) 2. of attaining Cramer-Rao lower bound asymptotically. For interval estimation, why not to use the most plausible parameter set, i.e., the highest part of the likelihood function, is appealing in set estimation if its random version does cover θ with the desired confidence coefficient. This consideration by Chen (2004), unlike the traditional frequentist approaches, does use the likelihood function for interval construction. Apparently this new approach avoids the first deficiency that the traditional freqentist approaches are less convincing in the interpretation of generating a subset of plausible values in the parameter space after an observation X = x has been taken. This approach also has been shown to carry over several properties of maximum likelihood estimate to the interval estimate that it is not guaranteed for the traditional techniques of frequentist confidence intervals. Moreover, a criterion minimizing the quantity of the set {x : θ ∈ C(x)}, C(X) is a confidence set, among the class of 100(1 − α)% confidence set has been introduced where optimal one does often obtainable. Then, for Zacks’ example, this new confidence interval not only defeat the interval (X − zα/2 , X + zα/2 ) at the point θ = 0 but also defeat the whole class of 100(1 − α)% confidence intervals at whole parameter space when we measure interval against this criterion. Unfortunately, the design of the optimality and theory in Chen (2004) are only appropriate for a sample drawn from a continuous distribution. Our concern in this paper is that can we extend this approach of likelihood-based confidence set to a sample with discrete distributions. There are reasons that is worth to develop likelihood-based confidence set for discrete random variables. First, the confidence interval or set in Chen (2004) asking for that it covers parameter θ with a fixed probability, say 1 − α, uniformly in θ is, in general, not attainable for discrete distributions. The difficulty of constructing an interval with this need for discrete distribution has been pointed out by several authors that discrete distributions are not generally possible to construct a confidence interval for their unknown parameters with an exactly specified confidence coefficient using only a set of observations (see for examples, Welsh (1996, page 146) and Kotz and Johnson (1982)). Second, the confidence set for discrete distributions has been received with little attention in literature where one exception is the credible region of Bayesian approach and fiducial interval where they, actually, are not frequentist approaches. The Bayesian approach regards the parameter as a random variable with a prior distribution where the major controversy about this approach is the determi-.

(9) 3. nation of the prior distribution. On the other hand, the fiducial interval was proposed by Fisher (1930, 1933, 1935, 1939) through a series of examples and without any formal structure or theory, the fiducial inferences. Essentially, he proposed using the distribution function of F (t|θ) of a sufficient statistic T in order to make conditional probability statement about θ given T = t, thus somehow transferring the probability measure from space of T to space of θ. However, no formal justification was offered for this controversial “transfer” (please see one special “transfer” by Clopper and Pearson (1934), see also Welsh (1996) and Kotz and Johnson (1982)). Therefore, alternative approach, especially the one relying on likelihood function, is desirable. Extending from the approach in Chen (2004), we introduce a likelihood-based confidence set for the discrete distributions. Several interesting results are also discovered. First, a new version of optimality of smallest volume uniformly in θ has been established and, as displayed in our examples, the optimal ones are often obtainable. This optimality criteria seems to be reasonable to replace the one based on expected length for the interval estimation for discrete distributions. Second, some properties behaving for maximum likelihood estimator such as invariance and sufficiency have been carried over to some of these approaches. The property of invariance is interesting for the fact that the Bayesian highest posterior density confidence set has also an optimality of shortest width, restricted to a prior distribution , however, this Bayesian interval is criticized for depending on the prior density and for not having the desired property of invariance. Third, the intervals defined in this paper, if they exist, always include the maximum likelihood estimate whereas the traditional frequentist approaches are criticized for that do not always happen. 2. Likelihood-Based Confidence Set and MLE Related Properties Let X be a random variable or vector with likelihood function L(θ, x) with parameter space Ω. With this likelihood function and regarding that x is fixed, we consider that parameter θ1 is more plausible than parameter θ2 if L(θ1 , x) > L(θ2 , x). The likelihood-based 100(1 − α)% confidence set for θ introduced by Chen (2004) is a random set S(X) with S(x) = {θ ∈ Ω : L(θ, x) ≥ a(x)} and 1 − α = Pθ (θ ∈ S(X)) for θ ∈ Ω,. (2.1). where a(x) may be set as a positive function or a positive constant. Basically the parameter subset S(x) does being relatively plausible after observation x has been.

(10) 4. observed and then the confidence set S(X) is more convincing than the classical ones. In consideration of discrete distribution, its discreteness makes asking for that coverage probability in (2.1) exactly equal to 1 − α is generally impossible (see this point in Angus and Schafer (1984) and Welsh (1996)). A reasonable modification is one has confidence at least 100(1 − α)% that the true parameter is covered by the random set. Two types of confidence set are available to introduce. Definition 2.1. Suppose that we have a random set C(X) that satisfies (a) For X = x ∈ Rn , L(θ, x) ≥ L(θ0 , x) if θ ∈ C(x) and θ0 6∈ C(x), (b) infθ∈Ω Pθ (θ ∈ C(X)) = 1 − α. Then we call C(X) a 100(1 − α)% highest likelihood confidence set (HLCS) for parameter θ. If C(X) = (a(X), b(X)), a random interval, then it is called a 100(1 − α)% highest likelihood confidence interval (HLCI). The method of highest likelihood has a strong intuitive appeal and according to it, we estimate the true parameter θ by any parameter set which is located on top of the picture of the likelihood function; such a set belongs to the set relatively more plausible after we have observed x. In general, the picture in shape of the likelihood function L(θ, x) in terms of θ varies in x. Then, condition (a) provides us the benefit that we may choose a(x) varying in x for guaranteeing that, for all x, C(x) is not an empty set. In the following, we revise condition (a) so that the confidence sets C(x) are with equal lowest level plausibilities. Definition 2.2. If there is `α ∈ R such that the set C(x) = {θ : L(θ, x) ≥ `α } satisfying that its corresponding random set C(X) satisfies infθ∈Ω Pθ (θ ∈ C(X)) = 1 − α, then C(X) is called a 100(1 − α)% uniformly highest likelihood confidence set (UHLCS). If C(X) = (a(X), b(X)), a random interval, then it is called a 100(1 − α)% uniformly highest likelihood confidence interval (UHLCI). We state three properties for the HLCS and the UHLCS that are inherited from the maximum likelihood estimator. First, we show that HLCS contains all information regarding θ which is contained in the sample X. Theorem 2.3. Let T = t(X1 , ..., Xn ) be a sufficient statistic for θ. Then if HLCS exists and is unique, it is a function of T . Proof. The factorization theorem implies the following representation of the likelihood.

(11) 5. function: L(θ, x) = g(t(x1 , ..., xn ), θ)h(x1 , ..., xn ), where g and h are nonnegative and h is independent of θ. Therefore, for given x1 , ..., xn , L(θ, x) ≥ L(θ0 , x) if and only if g(t(x1 , ..., xn ), θ) ≥ g(t(x1 , ..., xn ), θ0 ). Thus, if a unique HLCS exists, it will have to be a function of T .. ¤. The Bayesian highest posterior density confidence intervals, although have the optimality of smallest volume in terms of prior distribution, are not parameterization invariant, i.e., the highest posterior density confidence interval for parameter θ cannot simply be transformed to yield the highest posterior density confidence interval for, say, 1/θ, although, it will have the same confidence level. The HLCS and UHLCS do being parameterization invariant. Theorem 2.4. (Invariance Property of Highest Likelihood Confidence Interval) Let C(X) be a 100(1 − α)% HLCS for parameter θ and φ be a one-to-one function defined on Ω onto Ω∗ . Then φ(C(X)) = {φ(θ) : θ ∈ C(X)} is a 100(1 − α)% HLCS for parameter φ(θ). On the other hand, if C(X) is a UHLCS for θ, then φ(C(X)) is a UHLCS for φ(θ). Proof. Consider that C(X) is a HLCS. Set θ∗ = φ(θ). Then, for given x ∈ Rn , L(θ, x) = L(φ−1 (θ∗ ), x), call it L∗ (θ∗ , x). It follows that L∗ (θ1∗ , x) ≥ L∗ (θ2∗ , x) if and only if L(θ1 , x) ≥ L(θ2 , x). (6.1). with θ1 = φ−1 (θ1∗ ) and θ2 = φ−1 (θ2∗ ). By assuming the existence of a HLCS for θ, clearly the HLCI for θ∗ exists and is φ(C(X)). On the other hand, if C(X) is a UHLCS, then the proof is analogous to the above except that (6.1) is replaced by, for x1 , x2 ∈ Rn , L∗ (θ1∗ , x1 ) ≥ L∗ (θ2∗ , x2 ) if and only if L(θ1 , x1 ) ≥ L(θ2 , x2 ).. ¤. The traditional frequentist approaches of confidence set are often criticized for that they do not, in general, include the most plausible parameter value, i.e., the maximum.

(12) 6. likelihood estimate. However, this new frequentist approach does has this property, simply followed from its definition. Theorem 2.5. The HLCS and UHLCS both include the maximum likelihood estimate. 3. Existence and Optimality of the Likelihood-Based Confidence Set As mentioned earlier, the discreteness of the considered random variable does not guarantee to have confidence set at any confidence level 1 − α. We then ask if there is some value 1 − α such that the UHLCS exists. This is answered in the following theorem. Theorem 3.1. If random vector X has a sample space of finite elements, there exists a UHLCS for a confidence level 1 − α which is not the trivial 0 or 1. Proof. Let’s denote by γa = inf{Pθ (L(θ, X) ≥ a) : θ ∈ Ω}. It is seen that γ0 = 1 and, by boundness of density function, there is positive constant au such that γau = 0. These are the trivial cases. Without loss of generality, let the sample space of X be denoted by Rx and with number k of elements. Let a be a positive value with 0 < a ≤ k1 . For every θ ∈ Ω, P with the property 1 = x∈Rx L(θ, x), there is xθ ≥ a such that L(θ, xθ ) ≥ k1 . That indicates Pθ (L(θ, x) ≥ a) ≥ L(θ, x) ≥ It is proved by letting 1 − α = k1 .. 1 k. for θ ∈ Ω. Then infθ∈Ω Pθ (L(θ, X) ≥ a) ≥ k1 .. ¤. In point estimation and statistical hypothesis, there are satisfactory criteria for developing optimal procedures. For examples, minimum variance from the class of unbiased estimators for point estimation and maximizing powers from the class of level α tests for hypothesis testing are satisfactory criteria since their corresponding rules are often obtainable. In set estimation, the popular criterion is defined through the set’s expected length. However, as proved by Pratt (1961) a confidence set that has uniformly smallest expected length has to be an inversion of a two sided UMP test which is rare existed in literature. This indicates that expected length might not be appropriate as a criterion for defining an optimal confidence set. We, extending from the optimal confidence set for continuous distribution in Chen (2004), introduce an appropriate criterion in terms of quantity as a rule for developing optimal confidence set for discrete distributions..

(13) 7. Definition 3.2. A random set C(X) is called the uniformly smallest super quantity random set if, for given θ, the following holds quantity({x : θ ∈ C(x)}) ≤ quantity({x : θ ∈ D(x)}). (3.1). for any random set D(X) with Pθ (θ ∈ C(X)) ≤ Pθ (θ ∈ D(X)) and this statement holds for every θ ∈ Ω. If this uniformly smallest super quantity random set is used as a confidence set with confidence coefficient 1 − α, we call it a 100(1 − α)% uniformly smallest super quantity (USSQ) confidence set. Furthermore, if this 100(1−α)% USSQ confidence set is an interval, then we call it the 100(1 − α)% USSQ confidence interval. For continuous distributions, we may ask for a UHLCS with coverage probability 1 − α uniformly in θ ∈ Ω that, then, is available to have an optimal one that satisfies (3.1) for every D(X) with Pθ (θ ∈ D(X)) ≥ 1 − α (see this point in Chen (2004)). In discrete distributions, since a random set that covers parameter θ with probability 1 − α uniformly is usually not available, we then have to relax its definition of a USSQ confidence set. The following theorem is extended from an optimality property for the highest posterior density credible region of a Bayesian approach (see Garthwaite etc (2002)) that provides a desirable property of smallest quantity for the UHLCS. Theorem 3.3. A 100(1 − α)% UHLCS is a 100(1 − α)% USSQ confidence set. Proof. The proof is similar to the proof of the Neyman-Pearson lemma. Let C(X) be the 100(1 − α)% UHLCS. For a fixed value θ, let D(X) be a confidence set that satisfies. X {x:θ∈C(x)}. L(θ, x) ≤. X. L(θ, x).. {x:θ∈D(x)}. Deleting the subset common to {x : θ ∈ C(x)} and {x : θ ∈ D(x)} yields X X L(θ, x). L(θ, x) ≤ {x:θ∈C(x)}∩{x:θ∈D(x)}c. (3.2). {x:θ∈C(x)}c ∩{x:θ∈D(x)}. The region on two integrals of (3.2) cover the true parameter θ with unequal probability. Now, the definition of UHLCS also indicates that L(θ, x1 ) ≥ L(θ, x2 ) for x1 with θ ∈ C(x1 ) and x2 with θ 6∈ C(x2 ), i.e., the likelihood function defined on region in left integral is greater or equal to it in the right integral. Thus, quantity({x : θ ∈ C(x)}∩{x : θ ∈ D(x)}c ) ≤ quantity({x : θ ∈ C(x)}c ∩{x : θ ∈ D(x)}), (3.3).

(14) 8. so, adding the quantity of {x : θ ∈ C(x)} ∩ {x : θ ∈ D(x)} to both sides of (3.3), quantity({x : θ ∈ C(x)} ≤ quantity({x : θ ∈ D(x)}).. ¤. There is nothing unique about a confidence set of given confidence level. The USSQ seems to be appropriate playing the criterion of defining an optimal set since Theorems 3.1 and 3.2 indicate that optimal ones are obtainable. 4. Derivation of Likelihood-Based Confidence Intervals Classical interval estimators and the approaches of HLCI and UHLCI for continuous distributions in Chen (2004) are formulated based on pivotal quantity. However, it is rare that a discrete distribution involves a pivotal quantity. In this section, we develop explicit forms of HLCI and UHLCI for several cases of two distributions. The proofs of the theorems in this section are quite technical and then are listed in Appendix. Let X be a random variable with binomial distribution b(n, p). Our interest is the HLCI and UHLCI for parameter p. Theorem 4.1. Suppose that X has binomial distribution with n = 1. (a). Then the HLCI is of the form ½ C(x) =. (a, 1) if x = 1 (0, b) if x = 0. (4.1). for some a, b with 0 < a < 1, 0 b . min{b, 1 − a} if a < b. If we consider a 100(1 − α)% HLCI for p, then a and b are values satisfy 1 − α = min{b, 1 − a}. That is, besides a < b, we need to let either a = α or b = 1 − α. (b). The UHLCI has the form ½ C(x) =. (a, 1) if x = 1 (0, 1 − a) if x = 0. for some 0 < a < 1 where its confidence coefficient is ½ 0 if a > 0.5 γ= . 1 − a if a ≤ 0.5. (4.2).

(15) 9. Suppose that we want a 100(1 − α)% UHLCI, we need to set a = α. Theorem 4.2. Suppose that X has binomial distribution with n = 2. (a). Then a HLCI has the form  . (0, b) C(x) = (0.5 − c, 0.5 + c)  (a, 1). if x = 0 if x = 1 if x = 2. (4.3). for some a, b, c with 0 < a, b < 1 and 0 < c < 0.5. The coverage probability function of this HLCI is πC (p) = p2 I(p > a) + 2p(1 − p)I(0.5 − c < p < 0.5 + c) + (1 − p)2 I(p < b).. (4.4). (b). The UHLCI has the form    C(x) =.  . q (0.5 −. 1 4. (0, 1 − a) q 2 − a2 , 0.5 + 41 − (a, 1). if x = 0. a2 2. 1 ) if x = 1 for some a ∈ (0, √ ), 2 if x = 2. indicating that there is no confidence interval when x = 1 and a > coefficient of the UHLCI is q  2   21 − a2 + 41 − γ= 2a(1 − a)   0 Since. 1 2. 2. − a2 +. q 1 4. −. a2 2. a2 2. √1 . 2. (4.5). The confidence. if 0 < a < 0.5 if 0.5 < a < if 32 < a. 2 3. .. is approximately 0.7285 when a = 0.5, a UHLCI of confidence. coefficient 1 − α with 1 − α > 0.75 is the C(X) of (4.5) with a satisfying α = q a2 1 4 − 2 .. 1 2. 2. − a2 +. Deriving the HLCS and UHLCS for binomial distribution b(n, p) with n ≥ 3 may be analogously developed, however, its complexity will increases as n goes larger. We will provide some studies of UHLCS for general case of this distribution in next section. Let D(X) be any random set. Then, for each p ∈ (0, 1), {x : p ∈ D(x)} is a set of finite points since it is a subset of the finite sample space {0, 1, 2, ..., n}. Let’s denote #A as the number of elements in set A if it is finite. We next state a result induced from theorem 3.3..

(16) 10. Corollary 4.3. Let X have a binomial distribution b(n, p) and C(X) be a UHLCS for p. Then #{x : p ∈ C(x)} ≤ #{x : p ∈ D(x)} for any set D(X) with Pp (p ∈ C(X)) ≤ Pp (p ∈ D(X)) and this holds uniformly in p ∈ (0, 1). Consider another example of discrete distribution related to an experiment applied in industry. Let Z be the number of independent trials of some component until it fails, where 1−p is the probability of failure on each trial. We record the exact number of trials, X = Z, if Z ≤ r; otherwise we record X = r + 1, where r is a fixed positive integer. It may be seen that the pdf of X is ½ f (x, p) =. (1 − p)px−1 pr. if x = 1, ..., r . if x = r + 1. (4.6). Theorem 4.4. Suppose that X has a distribution with pdf of (4.6) with r = 2. (a). Then a HLCI has the form   C(x) =. (0, b) (0.5 − c, 0.5 + c)  (a, 1). if x = 1 if x = 2 if x = 3. (4.7). for some a, b, c with 0 < a, b < 1 and 0 < c < 0.5. (b). The UHLCI C(X) is the empty set, for a > 1, and has the form, for 0 < a < 1,  . (0, 1√ − a2 ) if x = 1 √ C(x) = (0.5 − 0.25 − a2 , 0.5 + 0.25 − a2 )I(0 < a < 0.5) if x = 2  (a, 1) if x = 3 with the confidence coefficient as  (1 − a2 )2    0.5 + √0.25 − a2 γ= 1−a    0. (4.8). if 0 < a < 0.495 if 0.495 ≤ a < 0.5 √ if 0.5 ≤ a < −1+2 5 √ if a ≥ −1+2 5. where 0.495 is an approximated value solving (1 − a2 )2 = 0.5 +. √ 0.25 − a2 . Since. (1 − a2 )2 is about 0.56 when a is 0.495, a 100(1 − α)% UHLCS for p when 1 − α > 0.6 p is the one in (4.8) with a = 1 − (1 − α)1/2 ..

(17) 11. 5. An Example of Exact UHLCI with Fixed Confidence Coefficient Consider a random variable X that has a distribution with pdf ½ (1 − p)px−1 if x = 1, ..., ` L(p, x) = p` if x = ` + 1. (5.1). This likelihood function has the following properties: (b1) The picture of L(p, 1) = 1 − p is a decreasing straight line with range (0, 1) and it of L(p, `) = p` is monotone increasing with range (0, 1) too. (b2) L(p, 2) > L(p, 3) > ... > L(p, `), p ∈ (0, 1). (b3) L(p, 2) achieves maximum at 0.5 with L(0.5, 2) = 0.25. An UHLCI is setting by defining L(p, x) ≥ a for some a > 0. Lemma 5.1. If a > 0.25, the confidence coefficient γ ≤ 0.5. Proof. From (b2) and (b3), the UHLCI has the form  3 if x = 1  (0, 4 ) −1/` C(x) = (4 , 1) if x = ` + 1 ,  φ if x = 2, 3, ..., ` where 4−1/` < 43 , with coverage probability function  if 0 0.5 for ` > 2, γ = inf0<p<1 πC (p) ≤ inf0<p<4−1/` (1 − p) < 0.5.. ¤. Theorem 5.2. If the constant ` is ≥ 3, then, for γ ≥ 0.5, a 100γ% UHLCI is the one with a = 1 − γ 1/` . Proof. If we put all curves for likelihood functions L(p, x), x = 1, ..., ` + 1 in terms of p on a plane associated with the constant line g(p) = a, we may see the following facts: (c1) g(p) = a meets the line L(p, 1) at p = 1 − a and the curve L(p, 2) = p(1 − p) at √ p = 0.5 − 0.25 − a. (c2) For x, 2 < x ≤ ` + 1, any intersecting point px , if it exists, of figures of g(p) and L(p, x) must satisfies 0.5 −. √ 0.25 − a < px < 1 − a.. (5.2).

(18) 12. The statement of (4.2) for x = 3, ..., ` is followed from (b2) in this section. It holds for x = ` + 1 due the fact that L(p, ` + 1) < L(p, 2) for 0 L(p, 2) for p0 < p < 1. 6. General formulation of UHLCI for Binomial Distribution The interest in this section is to develop the confidence interval for general binomial distribution. The likelihood function of a binomial distribution b(n, p) is L(p, x) = µ ¶ n px (1 − p)n−x where 0 < p < 1 and x = 0, 1, ..., n. We consider the confidence x interval is a function of a ∈ R+ as C(x, a) = {p : L(p, x) ≥ a}, a ≥ 0. Since the likelihood function L(p, x) is never greater than value 1 and never smaller than value 0, than we have. ½ C(X, a) = ½. with πC (a) = inf0<p<1 =. φ if a > 1 (0, 1) if a ≤ 0. 0 if a > 1 1 if a ≤ 0. Lemma 6.1. Let L(p, x) be the likelihood function of the binomial random variable. Then, if n = 2k − 1, x x (a) L( 2k−1 , x) is symmetric on {0, 1, 2, ..., k −1, k, ..., 2k −1} in sense that L( 2k−1 , x) =. L( 2k−1−x 2k−1 , 2k − 1 − x), for x = 0, 1, 2, ..., k − 1, x (b) L( 2k−1 , x) is strictly monotone decreasing on {0, 1, 2, ..., k − 1}.. On the other hand, if n = 2k, then x x (a) L( 2k , x) is symmetric on {0, 1, 2, ..., 2k} in sense that L( 2k , x) = L( 2k−x 2k , 2k − x),. for x = 0, 1, 2, ..., k − 1, x , x) is strictly monotone decreasing on {0, 1, 2, ..., k − 1}. (b) L( 2k µ ¶ ¢n−x n ¡ x ¢x ¡ x . The symmetry 1 − nx Proof. The likelihood at X = x with p = n is n x property is induced from the fact that. ¶n−(n−x) ¶n−x µ µ ¶µ n−x n−x n−x n 1− , n − x) = L( n−x n n n µ ¶³ ´ ´ ³ x n−x x x n = 1− x n n x = L( , x) n if x = 0, 1, , 2, ..., k − 1 for either n = 2k − 1 or 2k. Now, with several arrangements,.

(19) 13. we have the likelihood ratio between x + 1 and x as L( x+1 n , x + 1) = L( nx , x). µ. x+1 x. ¶x µ. n − (x + 1) n−x. ¶n−(x+1) .. (6.1). Consider the case that n = 2k − 1. To show the monotone property, from (5.1), we need only to show the following `(k) =. L( k−j+1 2k−1 , k − j + 1) k−j L( 2k−1 , k − j). µ =. k−j+1 k−j. ¶k−j µ. k+j−2 k+j−1. ¶k+j−2 < 0,. (6.2). for j = 1, 2, ..., k. It is easy to check that, for a fixed j, `(k) converges to to 1 as k → ∞. Then to show (5.2) needs only to show that `(k) is a monotone increasing function of k for each j = 1, 2, ..., that is equivalent to show that the derivative of the logarithm of `(k) is positive when we treat k as positive real-value. With some P∞ m simplications and by the series ln(1 − x) = − m=1 xn , |x| < 1, it may be seen the following ∂ln`(k) 2(j − 1) 2(j − 1) = −ln(1 − )− ∂k (k − j + 1)(k + j − 1) (k − j + 1)(k + j − 1) =. 2j−1 m ∞ ( X (k−j+1)(k+j−1) ) m=2. m. which obviously is greater than zero.. ¤. Theorem 6.2. For n = 2k − 1, C(x, a) =  (0, 1 − a1/n )I(x = 0), (a1/n , 1)I(x = n) if L( n1 , 1) < a ≤ 1     ..    .   1/n m   (0, 1 − a )I(x = 0), ∪ (L(p, j − 1) ≥ a)I(x = j − 1),  j=2   2k−1   ∪j=2k−m+1 (L(p, j − 1) ≥ a)I(x = j − 1), (a1/n , 1)I(x = n), m−1 φI(x = m, m + 1, ..., 2k − m − 1) if L( m n , m) < a ≤ L( n , m − 1)    for m, 2 ≤ m ≤ k − 1    .   ..     2k−1  (0, 1 − a1/n )I(x = 0), ∪j=2 (L(p, j − 1) ≥ a)I(x = j − 1),    1/n (a , 1)I(x = n) if a ≤ L( k−1 n , k − 1) For n = 2k, C(x, a) =.

(20) 14.  (0, 1 − a1/n )I(x = 0), (a1/n , 1)I(x = n) if L( n1 , 1) < a ≤ 1     ..    .    1/n m  (0, 1 − a )I(x = 0), ∪j=2 (L(p, j − 1) ≥ a)I(x = j − 1),    1/n  , 1)I(x = n),  ∪2k j=2k−m+2 (L(p, j − 1) ≥ a)I(x = j − 1), (a m−1 φI(x = m, m + 1, ..., 2k − m) if L( m n , m) < a ≤ L( n , m − 1)    for m, 2 ≤ m ≤ k    .   ..     1/n 2k  (0, 1 − a )I(x = 0), ∪j=2 (L(p, j − 1) ≥ a)I(x = j − 1),    (a1/n , 1)I(x = n) if 0 < a ≤ L( nk , k) Lemma 6.3 The pictures of L(p, x) and L(p, x + 1) meets at the point p =. x+1 n+1. x+1 for x = 0, 1, ..., n − 1. Moreover, the function L( n+1 , x) is monotone decreasing on. {0, 1, ..., k − 1} when n = 2k − 1 and on {0, 1, ..., k − 1} when n = 2k. Proof. First, solving the equation L(p, x) = L(p, x + 1), we may have the answer p=. x+1 n+1 .. Consider the case that n = 2k − 1. By letting µ ¶µ ¶x µ ¶2k−1−x x+1 x+1 2k − 1 − x 2k − 1 s(x) = L( , x) = . x 2k 2k 2k By some simplifications, we may see that µ ¶x+1 µ ¶2k−x−2 s(x + 1) x+2 2k − x − 2 0 ` (x) = = . s(x) x+1 2k − x − 1. (6.3). The same argument in the proof of Lemma 5.1 showing that (5.3) is less than 1 for x = 0, 1, ..., k − 2 which proves the lemma.. ¤. Consider the case that n = 2k − 1. By letting p0 =. (k−1)+1 (2k−1)+1. = 21 , the point of p of. smallest joining value between two neighboring likelihood curve, for any p0 6= p0 , there exists a > L( 21 , k − 1) and at least one x ∈ {0, 1, ..., 2k − 1} such that p is a point in the interval {p : L(p, x) ≥ a} which indicates that πC (p) > 0. This property holds for p0 =. (k−1)+1 2k+1. =. k 2k+1. when n = 2k. We state this implication in the next theorem.. 1 Theorem µ ¶ 6.4. If n = 2k − 1, the confidence coefficient is zero when a > L( 2 , k − 1) = 2k − 1 ¡ 1 ¢2k−1 . On the other hand, if n = 2k, the confidence coefficient is zero 2 k−1 µ ¶³ ´k+1 ´k−1 ³ 2k k+1 k k . when a > L( 2k+1 , k − 1) = 2k+1 2k+1 k−1.

(21) 15. Lemma 6.5. If a ≤ ( 21 )n , the straight line connecting the points (a1/n , L(a1/n , n)) and (1 − a1/n , L(1 − a1/n , 0)) is under the curves of L(p, x) for all x = 0, 1, ..., n on the region (a1/n , 1 − a1/n ). Proof. The result is induced from the following three facts:. µ ¶ n ¡ 1 ¢n (a). For s = 1, ..., k − 1, L(p, s) and L(p, n − s) meets at = which 2 s ¡ 1 ¢n is greater than 2 , the intersection point between L( 21 , 0) and L( 21 , n).. L( 21 , s). (b). For s = 1, ..., n, the likelihood function at x = 0 L(p, 0) = (1 − p)n is smaller µ ¶ n than ps (1 − p)n−s = L(s, p) on the region ( 21 , 1) and, for s = 0, 1, ..., n − 1, the s likelihood function at x = n L(p, n) = pn is smaller than L(s, p) on the region (0, 21 ). (c). The straight line connecting the points (a1/n , a) and ( 21 , a) is under the curve of L(p, n) on (a1/n , 21 ) and the straight line connecting the points ( 21 , a) and (1 − a1/n , a) is under the curve of L(p, 0) on ( 21 ), 1 − a1/n ). Theorem 6.6. If a ≤ ( 21 )n , πC (p) = 1 for p ∈ (a1/n , 1 − a1/n ). We have computed the exact confidence intervals for various sample size n. Some examples are attached after the references. 7. Appendix Proof of Theorem 4.1. The likelihood function of X is ½ 1 − p if x = 0 L(p, x) = . p if x = 1 The HLCI is the random set with 1 − p ≥ `1 if x = 0 and p ≥ `2 if x = 1 for some `1 , `2 ≥ 0 which implies (4.1) with b = 1 − `1 and a = `2 . The coverage probability function of C(X) in (4.1) is πC (p) = Pp (p ∈ C(X)) =. 1 X. px (1 − p)1−x. x=0,p∈C(x). = pI(p ∈ (a, 1)) + (1 − p)I(p ∈ (0, b)). Now, if a > b, we further have  if p a.

(22) 16. that indicates γ = 0. On the other hand, if a < b, we have  if p > b  p πC (p) = 1 if a b), 1I(a < p < b), (1 − p)I(p < a)} = inf{b, 1 − a}. To have a UHLCI, we need to ensure L(p, x) ≥ a for some 0 < a < 1 which implies that b = 1 − a in (4.1). Then a UHLCI C(X) has the form of (4.2). The coverage probability function of UHLCI, from (4.2), is πC (p) = pI(p ∈ (a, 1)) + (1 − p)I(p ∈ (0, 1 − a)). In case that a ≤ 0.5,  if p < a 1 − p πC (p) = 1 if a ≤ p ≤ 1 − a  p if p > 1 − a which implies that the confidence level is inf{(1 − p)I(p ≤ a), 1I(a ≤ p ≤ 1 − a), pI(p > 1 − a)} = 1 − a. On the other hand, if a > 0.5,  if p < 1 − a 1 − p πC (p) = 0 if 1 − a ≤ p ≤ a  p if p > a which implies that the confidence level is 0.. ¤. Poof of Theorem 4.2. The likelihood function   (1 − p)2 L(p, x) = 2p(1 − p)  p2. is if x = 0 if x = 1 . if x = 2. Then (4.3) is implies by the fact that p(1 − p) for 0 < p < 1 is symmetric about 0.5 and (4.4) is obviously implied from the likelihood function.. On the other hand, let’s consider the UHCI which requires L(p, x) attaining the same constant. Since p(1 − p) is symmetric at 0.5, a UHCI C(X) has the form  if x = 0   q (0, 1 − a) q 1 2 2 C(x) = (0.5 − 41 − a2 , 0.5 + 41 − a2 ) if x = 1 for some a ∈ (0, √ ).  2  (a, 1) if x = 2.

(23) 17. That implies that if a > this UHLCI is. √1 , 2. there is no UHCI. The coverage probability function of r. r 1 a2 1 a2 πC (p) = p I(p > a)+2p(1−p)I(0.5− − ≤ p ≤ 0.5+ − )+(1−p)2 I(p < 1−a). 4 2 4 2 For making further region dividing, we have, if 0 < a < 0.5, r 1 a2 πC (p) = (1 − p)2 I(0 < p < 0.5 − − ) + ((1 − p)2 + 2p(1 − p))I(0.5 4 2 r 1 a2 − − ≤ p ≤ a) + I(a ≤ p ≤ 1 − a) + (2p(1 − p) + p2 )I(1 − a ≤ p ≤ 0.5 4 2 r r 1 a2 1 a2 + − ) + p2 I(0.5 + − ≤ p ≤ 1), (6.1) 4 2 4 2 We need to find the confidence coefficient of the UHLCI for this case. The picture of 2. the coverage probability function πC in (6.1) is symmetric about p = 0.5. By letting g1 (p) = (1 − p)2 and g2 (p) = (1 − p)2 + 2p(1 − p) = 1 − p2 . They are both decreasing functions on their corresponding domains and the central part determining by the indicator function I(a ≤ p ≤ 1 − a) has value 1. Then the infimum of πCq (p) achieves 2 at minimums of right ends of g1 and g2 . Now the ends are g1 (0.5 − 41 − a2 ) = q 1 a2 1 a2 2 2 − 2 + 4 − 2 and g2 (a) = 1 − a . Then we have r 1 a2 1 a2 1 a2 (g1 (0.5 − − ) − ( − ))2 = − and 4 2 2 2 4 2 1 a2 1 a2 a4 (g2 (a) − ( − ))2 = − + 2 2 4 2 4 which implies r 2 1 a2 a 1 (6.2) − + 1 − a2 > − 2 4 2 2 for 0 < a < 1. Then, in this case, the confidence coefficient is r r 1 a2 1 a2 1 a2 − . + − )= − inf0<p<1 πC (p) = g1 (0.5 − 2 4 2 2 2 4 2 From (6.1), if 0.5 < a < 3 , r 1 a2 − ) + (1 − p2 )I(0.5 πC (p) = (1 − p)2 I(0 < p < 0.5 − 2 4 r 1 a2 ≤ p ≤ 1 − a) + 2p(1 − p)I(1 − a ≤ p ≤ a) − − 2 4 r r 2 1 a2 a 1 ≤ p ≤ 1). − − ) + p2 I(0.5 + + (2p(1 − p) + p2 )I(a ≤ p ≤ 0.5 + 2 4 2 4 (6.3).

(24) 18. Now, (6.3) is symmetric at p = 0.5. By letting g1 (p) = (1 − p)2 , g2 (p) = (1q− p2 and 2 g3 (p) = 2p(1 − p), they are monotone decreasing, respectively, on (0, 0.5 − 41 − a2 ), q 2 (0.5 − 41 − a2 , 1 − a) and (0.5, a) where g3 is symmetric at 0.5 on (1 − a, a) with r. 1 a2 a2 g1 (0.5 − − ) = 0.5 − + 4 2 2 g2 (1 − a) = a(2 − a),. r. 1 a2 − , 4 2. (6.4) (6.5). g3 (a) = 2a(1 − a).. (6.6). The confidence coefficient of the UHLCI for case that 0.5 < a <. 2 3. is the minimum. of the values in (6.4)-(6.6). From (4.7) andqthe fact that a(2 − a) = 2a − a2 > 2 2 1 − a2 which implies a(2 − a) > 0.5 − a2 + 41 − a2 . We now compare (6.4) and (6.6) and, actually, we want to show It is equivalent q that (6.6) is the2 minimum. q a2 1 a2 a 1 a2 to show that h1 (a) = (0.5 − 2 + 4 − 2 ) − (0.5 − 2 ) = 4 − 2 is larger than h3 (a) = (2a(1 − a) − (0.5 − h1 ( 32 ) on. = h3 ( 32 ) (0.5, 32 ).. a2 2 ). = 2a − 0.5 −. 3a2 2 .. This is true since h1 (0.5) > h3 (0.5),. and the fact that h1 and h3 are both decreasing and concave downward. From (4.6), if. 2 3. <a<. √1 , 2. r. 1 a2 πC (p) = (1 − p) I(0 < p < 1 − a) + 2p(1 − p)I(0.5 − − ≤ p ≤ 0.5 4 2 r 1 a2 + − ) + p2 I(a ≤ p ≤ 1) 4 2 q q 2 a2 1 which is zero on (1−a, 0.5− 4 − 2 ) and (0.5+ 41 − a2 , a) indicating inf 0<p<1 πC (p) = 2. 0.. ¤. Proof of Theorem 4.4. The likelihood function is   (1 − p) if x = 1 L(p, x) = (1 − p)p if x = 2  p2 if x = 3 which has picture with several properties: (a). L(p, 2) has mode point L(0.5, 2) = 0.25 and its picture is totally behind it of L(p, 1) by the fact that L(p, 2) < L(p, 1), 0 0. The UHLCI C(X) is empty for a > 1 which is obvious since L(p, x) ≤ 1. The facts that L(p, 2) ≤ 0.25 and L(p, 1) and L(p, 3) intersect at p0 > 0.25 we may consider C(x) in three categories, in terms of a, as p0 ≤ a ≤ 1, 0.5 ≤ a < p0 and 0 < a < 0.5. These facts indicate that C(X) = φ when p 6∈ (0, 1−a2 ) or 6∈ (a, 1) ∪ (p0 , 1) and then γ = 0 for any a ∈ (p0 , 1). For a ∈ (0.5, p0 ), the coverage probability function is πC (p) = p2 I(a, 1)+(1−p)I(0 < p < 1 − a2 ) which indicates that γ = min{1 − a, (1 − a2 )2 } = 1 − a for this region of a. On the other hand, for a ∈ (0, 0.5), the coverage probability function is p p πC (p) = (1 − p)I(0 < p < 1 − a2 ) + p(1 − p)I(0.5 − 0.25 − a2 ≤ a ≤ 0.5 + 0.25 − a2 ) + p2 I(a ≤ p ≤ 1) Note that L(p, 2) and L(p, 3) intersects at p = 0.5 such that the picture of L(p, 3) is behind it of L(p, 2) and we also know that the picture of L(p, 2) is also behind it of L(p, 1). Then, it is seen that the confidence coefficient for a in this region is √ min{0.5 + 0.25 − a2 , (1 − a2 )2 } that indicates the display of γ for a ∈ (0, 0.5) by √ noticing that 0.5 + 0.25 − a2 > (1 − a2 )2 when 0 < a < 0.495 and < holds when 0.495 < a < 0.5. ¤ References Angus, J. E. and Schafer, R. E. (1984). Improved confidence statements for the binomial parameter. American Statistician, 38, 189-191. Chen, L. A. (2004). Likelihood-Based Intervals.Unpublished manuscript. Fisher, R. A. (1921). On the probable error of a coefficient of correlation deduced from a small sample. Metron, 1, 3-32. Fisher, R. A. (1930). Inverse probability. Proc. Camb. Phil. Soc., 26, 528-535. Fisher, R. A. (1933). The concepts of inverse probability and fiducial probability referring to unknown parameters. Proc. Roy. Soc. A, 139, 343-348. Fisher, R. A. (1935). The fiducial argument in statistical inference. Ann. Eugenics. 6, 391-398. Fisher, R. A. (1939). A note on fiducial inference. Annals of Statistics. 10, 383-388. Garthwaite, P. H., Jolliffe, I. H. and Jones, B. (2002). Statistical Inference. Oxford University Press Inc.: New York..

(26) 20. Kotz, S. and Johnson, N. L. (1982). Encyclopedia of Statistical Sciences. Wiley: New York. Lindsey, J. K. (1996). Parametric Statistical Inference. Oxford Science Publication: New York. Pratt,J. W. (1961). Length of confidence interval. Journal of the American Statistical Association. 56, 549-567. Silvey, S. D. (1975). Statistical Inference. Chapman and Hall: London. Welsh, A. H. (1996). Aspects of Statistical Inference. Wiley: New York. Zacks, S. (1971). The Theory of Statistical Inference. Wiley: New York..

(27) Table 1. Confidence interval for n = 2 x. γ = 0.8000. γ = 0.8500. γ = 0.9000. γ = 0.9500. γ = 0.9900. 0 1 2. (0, 0.5654) (0.1055,0.8944) (0.4345,1). (0, 0.6206) (0.0780,0.9219) (0.3793,1). (0, 0.6879) (0.0513,0.9486) (0.3120,1). (0, 0.7778) (0.0253,0.9746) (0.2221,1). (0, 0.9001) (0.0050,0.9949) (0.0999,1). n=3 x. γ = 0.7500. γ = 0.8623. γ = 0.9000. γ = 0.9500. γ = 0.9900. 0 1 2 3. (0, 0.4999) (0.0457,0.7668) (0.2331,0.9542) (0.5000,1). (0, 0.5000) (0.0457,0.7669) (0.2330,0.9542) (0.5000,1). (0, 0.5477) (0.0329,0.8041) (0.1958,0.9670) (0.4522,1). (0, 0.6377) (0.0163,0.8646) (0.1353,0.9836) (0.3622,1). (0, 0.7860) (0.0032,0.9410) (0.0589,0.9967) (0.2140,1). n=4 x. γ = 0.8349. γ = 0.8500. γ = 0.8750. γ = 0.9500. γ = 0.9900. 0 1 2 3 4. (0, 0.3865) (0.0400,0.6135) (0.1896,0.8103) (0.3864,0.9599) (0.6135,1). (0, 0.4089) (0.0338,0.6368) (0.1723,0.8276) (0.3631,0.9661) (0.5910,1). (0, 0.4999) (0.0164,0.7212) (0.1153,0.8846) (0.2787,0.9835) (0.5000,1). (0, 0.5364) (0.0119,0.7513) (0.0971,0.9028) (0.2486,0.9880) (0.4635,1). (0, 0.6869) (0.0024,0.8591) (0.0417,0.9582) (0.1408,0.9975) (0.3130,1). n=5 x. γ = 0.8000. γ = 0.8351. γ = 0.9100. γ = 0.9375. γ = 0.9900. 0 1 2 3 4 5. (0, 0.3287) (0.0309,0.5219) (0.1486,0.6957) (0.3042,0.8513) (0.4780,0.9690) (0.6712,1). (0, 0.4007) (0.0165,0.5992) (0.1035,0.7632) (0.2367,0.8964) (0.4007,0.9834) (0.5992,1). (0, 0.4007) (0.0165,0.5992) (0.1035,0.7632) (0.2367,0.8964) (0.4007,0.9834) (0.5992,1). (0, 0.4999) (0.0064,0.6916) (0.0614,0.8351) (0.1648,0.9385) (0.3083,0.9935) (0.5000,1). (0, 0.6065) (0.0019,0.7781) (0.0322,0.8943) (0.1056,0.9677) (0.2218,0.9980) (0.3934,1). n=6 x. γ = 0.8078. γ = 0.8835. γ = 0.9000. γ = 0.9519. γ = 0.9900. 0 1 2 3 4 5 6. (0, 0.3260) (0.0170,0.4999) (0.0969,0.6517) (0.2125,0.7874) (0.3482,0.9030) (0.5000,0.9829) (0.6740,1). (0, 0.3369) (0.0153,0.5120) (0.0911,0.6630) (0.2033,0.7966) (0.3369,0.9088) (0.4879,0.9846) (0.6630,1). (0, 0.3652) (0.0115,0.5422) (0.0775,0.6908) (0.1812,0.8187) (0.3091,0.9224) (0.4577,0.9884) (0.6347,1). (0, 0.4113) (0.0071,0.5886) (0.0595,0.7317) (0.1502,0.8497) (0.2682,0.9404) (0.4113,0.9928) (0.5886,1). (0, 0.5422) (0.0015,0.7067) (0.0261,0.8269) (0.0843,0.9156) (0.1730,0.9738) (0.2932,0.9984) (0.4577,1). 21.

(28) n=7 x. γ = 0.7952. γ = 0.8601. γ = 0.8950. γ = 0.9500. γ = 0.9914. 0 1 2 3 4 5 6 7. (0, 0.2913) (0.0139,0.4466) (0.0806,0.5833) (0.1776,0.7086) (0.2913,0.8223) (0.4166,0.9193) (0.5533,0.9860) (0.7086,1). (0, 0.2913) (0.0139,0.4466) (0.0806,0.5833) (0.1776,0.7086) (0.2913,0.8223) (0.4166,0.9193) (0.5533,0.9860) (0.7086,1). (0, 0.3523) (0.0071,0.5131) (0.0549,0.6476) (0.1345,0.7649) (0.2350,0.8654) (0.3523,0.9450) (0.4868,0.9928) (0.6477,1). (0, 0.3927) (0.0044,0.5542) (0.0424,0.6852) (0.1118,0.7962) (0.2037,0.8881) (0.3147,0.9575) (0.4457,0.9955) (0.6072,1). (0, 0.5000) (0.0011,0.6543) (0.0203,0.7713) (0.0664,0.8635) (0.1364,0.9335) (0.2286,0.9796) (0.3456,0.9988) (0.4999,1). n=8 x. γ = 0.7834. γ = 0.8500. γ = 0.8828. γ = 0.9422. γ = 0.9900. 0 1 2 3 4 5 6 7 8. (0, 0.2569) (0.0127,0.3959) (0.0721,0.5195) (0.1575,0.6348) (0.2569,0.7430) (0.3651,0.8424) (0.4804,0.9278) (0.6040,0.9872) (0.7431,1). (0, 0.2903) (0.0085,0.4344) (0.0571,0.5590) (0.1328,0.6722) (0.2245,0.7754) (0.3277,0.8671) (0.4409,0.9428) (0.5655,0.9914) (0.7096,1). (0, 0.3089) (0.0068,0.4549) (0.0503,0.5794) (0.1209,0.6910) (0.2086,0.7913) (0.3089,0.8790) (0.4205,0.9496) (0.5450,0.9931) (0.6910,1). (0, 0.3646) (0.0034,0.5132) (0.0341,0.6353) (0.0915,0.7409) (0.1677,0.8322) (0.2590,0.9084) (0.3646,0.9658) (0.4867,0.9965) (0.6354,1). (0, 0.4506) (0.0010,0.5965) (0.0181,0.7106) (0.0585,0.8046) (0.1183,0.8816) (0.1953,0.9414) (0.2893,0.9818) (0.4034,0.9989) (0.5493,1). n=9 x. γ = 0.8121. γ = 0.8660. γ = 0.9040. γ = 0.9556. γ = 0.9871. 0 1 2 3 4 5 6 7 8 9. (0, 0.2298) (0.0116,0.3554) (0.0651,0.4677) (0.1415,0.5735) (0.2298,0.6744) (0.3255,0.7701) (0.4264,0.8584) (0.5322,0.9348) (0.6445,0.9883) (0.7701,1). (0, 0.2592) (0.0079,0.3901) (0.0520,0.5045) (0.1198,0.6098) (0.2012,0.7080) (0.2919,0.7987) (0.3901,0.8801) (0.4954,0.9479) (0.6098,0.9920) (0.7407,1). (0, 0.2754) (0.0064,0.4085) (0.0461,0.5234) (0.1095,0.6281) (0.1873,0.7245) (0.2754,0.8126) (0.3718,0.8904) (0.4765,0.9538) (0.5914,0.9935) (0.7245,1). (0, 0.3233) (0.0034,0.4605) (0.0322,0.5751) (0.0843,0.6766) (0.1524,0.7674) (0.2325,0.8475) (0.3233,0.9156) (0.4248,0.9677) (0.5394,0.9965) (0.6766,1). (0, 0.4317) (0.0006,0.5681) (0.0137,0.6758) (0.0460,0.7661) (0.0947,0.8423) (0.1576,0.9052) (0.2338,0.9539) (0.3241,0.9862) (0.4318,0.9993) (0.5682,1). n = 10 x. γ = 0.7984. γ = 0.8571. γ = 0.8991. γ = 0.9495. γ = 0.9900. 0 1 2 3 4 5 6 7 8 9 10. (0, 0.2277) (0.0081,0.3463) (0.0503,0.4511) (0.1134,0.5488) (0.1882,0.6415) (0.2705,0.7294) (0.3584,0.8117) (0.4511,0.8865) (0.5488,0.9496) (0.6536,0.9918) (0.7722,1). (0, 0.2327) (0.0075,0.3523) (0.0483,0.4574) (0.1100,0.5552) (0.1836,0.6476) (0.2650,0.7349) (0.3523,0.8163) (0.4447,0.8899) (0.5425,0.9516) (0.6476,0.9924) (0.7672,1). (0, 0.2727) (0.0043,0.3978) (0.0349,0.5049) (0.0866,0.6020) (0.1516,0.6915) (0.2261,0.7738) (0.3084,0.8483) (0.3979,0.9133) (0.4950,0.9650) (0.6021,0.9956) (0.7272,1). (0, 0.3179) (0.0022,0.4464) (0.0242,0.5534) (0.0664,0.6482) (0.1228,0.7334) (0.1900,0.8099) (0.2665,0.8771) (0.3517,0.9335) (0.4465,0.9757) (0.5535,0.9977) (0.6820,1). (0, 0.3832) (0.0008,0.5125) (0.0140,0.6167) (0.0450,0.7063) (0.0904,0.7845) (0.1476,0.8523) (0.2154,0.9095) (0.2936,0.9549) (0.3832,0.9859) (0.4874,0.9991) (0.6167,1). 22.

(29) n = 11 x. γ = 0.8053. γ = 0.8500. γ = 0.8885. γ = 0.9541. γ = 0.9904. 0 1 2 3 4 5 6 7 8 9 10 11. (0, 0.2060) (0.0077,0.3148) (0.0470,0.4113) (0.1052,0.5019) (0.1736,0.5886) (0.2486,0.6718) (0.3281,0.7513) (0.4113,0.8263) (0.4980,0.8947) (0.5886,0.9529) (0.6851,0.9922) (0.7939,1). (0, 0.2116) (0.0071,0.3214) (0.0448,0.4186) (0.1013,0.5094) (0.1685,0.5960) (0.2424,0.6787) (0.3212,0.7575) (0.4039,0.8314) (0.4905,0.8986) (0.5813,0.9551) (0.6785,0.9928) (0.7883,1). (0, 0.2464) (0.0042,0.3619) (0.0331,0.4616) (0.0809,0.5529) (0.1404,0.6380) (0.2081,0.7177) (0.2822,0.7918) (0.3619,0.8595) (0.4470,0.9190) (0.5383,0.9668) (0.6380,0.9957) (0.7536,1). (0, 0.2850) (0.0023,0.4044) (0.0237,0.5051) (0.0634,0.5955) (0.1156,0.6781) (0.1770,0.7539) (0.2460,0.8229) (0.3218,0.8843) (0.4044,0.9365) (0.4948,0.9762) (0.5955,0.9976) (0.7149,1). (0, 0.3782) (0.0004,0.5000) (0.0103,0.5982) (0.0351,0.6829) (0.0724,0.7574) (0.1201,0.8229) (0.1770,0.8798) (0.2425,0.9275) (0.3170,0.9648) (0.4017,0.9896) (0.4999,0.9995) (0.6217,1). n = 12 x. γ = 0.7990. γ = 0.8500. γ = 0.9083. γ = 0.9500. γ = 0.9886. 0 1 2 3 4 5 6 7 8 9 10 11 12. (0, 0.1939) (0.0067,0.2955) (0.0417,0.3855) (0.0940,0.4701) (0.1557,0.5511) (0.2235,0.6292) (0.2955,0.7044) (0.3707,0.7764) (0.4488,0.8442) (0.5298,0.9059) (0.6144,0.9582) (0.7044,0.9932) (0.8060,1). (0, 0.2007) (0.0060,0.3037) (0.0391,0.3946) (0.0895,0.4796) (0.1496,0.5606) (0.2161,0.6383) (0.2870,0.7129) (0.3616,0.7838) (0.4393,0.8503) (0.5203,0.9104) (0.6053,0.9608) (0.6962,0.9939) (0.7992,1). (0, 0.2249) (0.0040,0.3321) (0.0312,0.4252) (0.0756,0.5109) (0.1305,0.5915) (0.1926,0.6678) (0.2601,0.7398) (0.3321,0.8073) (0.4084,0.8694) (0.4890,0.9243) (0.5747,0.9687) (0.6678,0.9959) (0.7750,1). (0, 0.2613) (0.0022,0.3729) (0.0223,0.4677) (0.0592,0.5533) (0.1071,0.6324) (0.1632,0.7060) (0.2257,0.7742) (0.2939,0.8367) (0.3675,0.8928) (0.4466,0.9407) (0.5322,0.9776) (0.6270,0.9977) (0.7386,1). (0, 0.3544) (0.0004,0.4698) (0.0093,0.5637) (0.0316,0.6454) (0.0652,0.7181) (0.1080,0.7833) (0.1587,0.8412) (0.2167,0.8919) (0.2818,0.9347) (0.3545,0.9683) (0.4362,0.9906) (0.5301,0.9995) (0.6455,1). n = 13 x. γ = 0.7916. γ = 0.8491. γ = 0.9024. γ = 0.9500. γ = 0.9901. 0 1 2 3 4 5 6 7 8 9 10 11 12 13. (0, 0.1791) (0.0063,0.2735) (0.0390,0.3574) (0.0875,0.4364) (0.1447,0.5123) (0.2072,0.5859) (0.2735,0.6573) (0.3426,0.7264) (0.4140,0.7927) (0.4876,0.8552) (0.5635,0.9124) (0.6425,0.9609) (0.7264,0.9936) (0.8208,1). (0, 0.2001) (0.0044,0.2988) (0.0316,0.3852) (0.0747,0.4655) (0.1270,0.5417) (0.1856,0.6147) (0.2486,0.6845) (0.3154,0.7513) (0.3852,0.8143) (0.4582,0.8729) (0.5344,0.9252) (0.6147,0.9683) (0.7011,0.9955) (0.7998,1). (0, 0.2071) (0.0039,0.3070) (0.0295,0.3941) (0.0709,0.4747) (0.1218,0.5509) (0.1790,0.6235) (0.2411,0.6929) (0.3070,0.7588) (0.3764,0.8209) (0.4490,0.8781) (0.5252,0.9290) (0.6058,0.9704) (0.6929,0.9960) (0.7928,1). (0, 0.2523) (0.0017,0.3580) (0.0190,0.4476) (0.0513,0.5288) (0.0938,0.6039) (0.1437,0.6740) (0.1995,0.7395) (0.2604,0.8004) (0.3259,0.8562) (0.3960,0.9061) (0.4711,0.9486) (0.5523,0.9809) (0.6419,0.9982) (0.7476,1). (0, 0.3251) (0.0004,0.4346) (0.0092,0.5245) (0.0306,0.6036) (0.0622,0.6748) (0.1020,0.7396) (0.1487,0.7983) (0.2016,0.8512) (0.2603,0.8979) (0.3251,0.9377) (0.3963,0.9693) (0.4754,0.9907) (0.5653,0.9995) (0.6748,1). 23.

(30) n = 14 x. γ = 0.7855. γ = 0.8399. γ = 0.9030. γ = 0.9538. γ = 0.9915. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14. (0, 0.1665) (0.0060,0.2546) (0.0366,0.3331) (0.0818,0.4072) (0.1350,0.4785) (0.1932,0.5478) (0.2547,0.6154) (0.3186,0.6813) (0.3845,0.7452) (0.4521,0.8067) (0.5214,0.8649) (0.5927,0.9181) (0.6668,0.9633) (0.7453,0.9939) (0.8334,1). (0, 0.1850) (0.0043,0.2772) (0.0300,0.3582) (0.0704,0.4338) (0.1193,0.5057) (0.1738,0.5749) (0.2323,0.6416) (0.2939,0.7060) (0.3583,0.7676) (0.4250,0.8261) (0.4942,0.8806) (0.5661,0.9295) (0.6417,0.9699) (0.7227,0.9956) (0.8149,1). (0, 0.2080) (0.0028,0.3041) (0.0236,0.3873) (0.0589,0.4639) (0.1031,0.5360) (0.1536,0.6046) (0.2087,0.6700) (0.2676,0.7323) (0.3299,0.7912) (0.3953,0.8463) (0.4639,0.8968) (0.5360,0.9410) (0.6126,0.9763) (0.6958,0.9971) (0.7919,1). (0, 0.2405) (0.0015,0.3405) (0.0169,0.4256) (0.0461,0.5027) (0.0846,0.5743) (0.1300,0.6415) (0.1807,0.7047) (0.2359,0.7640) (0.2952,0.8192) (0.3584,0.8699) (0.4256,0.9153) (0.4972,0.9538) (0.5743,0.9830) (0.6594,0.9984) (0.7595,1). (0, 0.3149) (0.0003,0.4193) (0.0077,0.5051) (0.0264,0.5807) (0.0543,0.6492) (0.0897,0.7118) (0.1314,0.7691) (0.1785,0.8214) (0.2308,0.8685) (0.2881,0.9102) (0.3507,0.9456) (0.4192,0.9735) (0.4948,0.9922) (0.5806,0.9996) (0.6850,1). n = 15 x. γ = 0.8002. γ = 0.8572. γ = 0.8963. γ = 0.9500. γ = 0.9900. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15. (0, 0.1556) (0.0057,0.2382) (0.0344,0.3119) (0.0768,0.3815) (0.1265,0.4487) (0.1809,0.5141) (0.2382,0.5781) (0.2978,0.6408) (0.3591,0.7021) (0.4218,0.7617) (0.4858,0.8190) (0.5512,0.8733) (0.6184,0.9231) (0.6880,0.9655) (0.7617,0.9942) (0.8443,1). (0, 0.1722) (0.0041,0.2587) (0.0285,0.3349) (0.0665,0.4061) (0.1123,0.4741) (0.1633,0.5397) (0.2179,0.6033) (0.2752,0.6650) (0.3349,0.7247) (0.3966,0.7821) (0.4602,0.8366) (0.5258,0.8876) (0.5938,0.9334) (0.6650,0.9714) (0.7412,0.9958) (0.8277,1). (0, 0.1959) (0.0026,0.2866) (0.0219,0.3653) (0.0547,0.4379) (0.0957,0.5065) (0.1425,0.5719) (0.1935,0.6345) (0.2479,0.6946) (0.3053,0.7520) (0.3654,0.8064) (0.4280,0.8574) (0.4934,0.9042) (0.5620,0.9452) (0.6346,0.9780) (0.7133,0.9973) (0.8040,1). (0, 0.2248) (0.0014,0.3194) (0.0160,0.4000) (0.0434,0.4734) (0.0794,0.5419) (0.1217,0.6065) (0.1687,0.6676) (0.2198,0.7255) (0.2744,0.7801) (0.3323,0.8312) (0.3934,0.8782) (0.4580,0.9205) (0.5265,0.9565) (0.5999,0.9839) (0.6805,0.9985) (0.7751,1). (0, 0.2901) (0.0004,0.3893) (0.0078,0.4714) (0.0260,0.5444) (0.0528,0.6110) (0.0864,0.6725) (0.1255,0.7294) (0.1695,0.7820) (0.2179,0.8304) (0.2705,0.8744) (0.3274,0.9135) (0.3889,0.9471) (0.4555,0.9739) (0.5285,0.9921) (0.6106,0.9995) (0.7098,1). 24.

(31) n = 16 x. γ = 0.8044. γ = 0.8522. γ = 0.9081. γ = 0.9476. γ = 0.9900. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16. (0, 0.1562) (0.0044,0.2365) (0.0287,0.3076) (0.0658,0.3744) (0.1101,0.4385) (0.1589,0.5007) (0.2109,0.5614) (0.2653,0.6206) (0.3215,0.6784) (0.3793,0.7346) (0.4385,0.7890) (0.4992,0.8410) (0.5614,0.8898) (0.6255,0.9341) (0.6923,0.9712) (0.7634,0.9955) (0.8437,1). (0, 0.1612) (0.0039,0.2426) (0.0271,0.3144) (0.0629,0.3817) (0.1061,0.4461) (0.1539,0.5084) (0.2051,0.5690) (0.2588,0.6280) (0.3145,0.6854) (0.3719,0.7411) (0.4309,0.7948) (0.4915,0.8460) (0.5538,0.8938) (0.6182,0.9370) (0.6855,0.9728) (0.7573,0.9960) (0.8388,1). (0, 0.1828) (0.0025,0.2683) (0.0210,0.3427) (0.0521,0.4116) (0.0908,0.4768) (0.1348,0.5392) (0.1826,0.5993) (0.2335,0.6572) (0.2869,0.7130) (0.3427,0.7664) (0.4006,0.8173) (0.4607,0.8651) (0.5231,0.9091) (0.5883,0.9478) (0.6572,0.9789) (0.7316,0.9974) (0.8172,1). (0, 0.2153) (0.0013,0.3054) (0.0145,0.3822) (0.0396,0.4523) (0.0728,0.5178) (0.1117,0.5797) (0.1550,0.6385) (0.2020,0.6945) (0.2523,0.7476) (0.3054,0.7979) (0.3614,0.8449) (0.4202,0.8882) (0.4821,0.9271) (0.5476,0.9603) (0.6177,0.9854) (0.6945,0.9986) (0.7846,1). (0, 0.2823) (0.0003,0.3773) (0.0067,0.4560) (0.0228,0.5261) (0.0469,0.5902) (0.0772,0.6495) (0.1126,0.7046) (0.1525,0.7559) (0.1963,0.8036) (0.2440,0.8474) (0.2953,0.8873) (0.3504,0.9227) (0.4097,0.9530) (0.4738,0.9771) (0.5439,0.9932) (0.6226,0.9996) (0.7176,1). n = 17 x. γ = 0.7971. γ = 0.8500. γ = 0.9015. γ = 0.9502. γ = 0.9904. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17. (0, 0.1482) (0.0041,0.2243) (0.0269,0.2918) (0.0616,0.3552) (0.1031,0.4161) (0.1488,0.4752) (0.1976,0.5329) (0.2485,0.5894) (0.3011,0.6447) (0.3552,0.6988) (0.4105,0.7514) (0.4670,0.8023) (0.5247,0.8511) (0.5838,0.8968) (0.6447,0.9383) (0.7081,0.9730) (0.7756,0.9958) (0.8518,1). (0, 0.1539) (0.0036,0.2314) (0.0250,0.2997) (0.0584,0.3637) (0.0986,0.4249) (0.1432,0.4842) (0.1909,0.5420) (0.2410,0.5983) (0.2930,0.6533) (0.3466,0.7069) (0.4016,0.7589) (0.4579,0.8090) (0.5157,0.8567) (0.5750,0.9013) (0.6362,0.9415) (0.7002,0.9749) (0.7685,0.9963) (0.8460,1). (0, 0.1713) (0.0025,0.2522) (0.0202,0.3228) (0.0497,0.3882) (0.0863,0.4503) (0.1278,0.5099) (0.1728,0.5675) (0.2206,0.6232) (0.2707,0.6771) (0.3228,0.7292) (0.3767,0.7793) (0.4324,0.8271) (0.4900,0.8721) (0.5496,0.9136) (0.6117,0.9502) (0.6771,0.9797) (0.7477,0.9974) (0.8286,1). (0, 0.2019) (0.0013,0.2873) (0.0140,0.3605) (0.0379,0.4274) (0.0693,0.4901) (0.1060,0.5496) (0.1468,0.6064) (0.1909,0.6607) (0.2378,0.7126) (0.2873,0.7621) (0.3392,0.8090) (0.3935,0.8531) (0.4503,0.8939) (0.5098,0.9306) (0.5725,0.9620) (0.6394,0.9859) (0.7126,0.9986) (0.7980,1). (0, 0.2714) (0.0002,0.3624) (0.0060,0.4380) (0.0208,0.5055) (0.0429,0.5673) (0.0709,0.6247) (0.1035,0.6784) (0.1403,0.7286) (0.1806,0.7756) (0.2243,0.8193) (0.2713,0.8596) (0.3215,0.8964) (0.3752,0.9290) (0.4326,0.9570) (0.4944,0.9791) (0.5619,0.9939) (0.6375,0.9997) (0.7285,1). 25.

(32) n = 18 x. γ = 0.7901. γ = 0.8505. γ = 0.9000. γ = 0.9477. γ = 0.9894. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18. (0, 0.1397) (0.0039,0.2118) (0.0256,0.2758) (0.0586,0.3359) (0.0979,0.3938) (0.1412,0.4500) (0.1873,0.5050) (0.2354,0.5590) (0.2850,0.6119) (0.3359,0.6640) (0.3880,0.7149) (0.4409,0.7645) (0.4949,0.8126) (0.5499,0.8587) (0.6061,0.9020) (0.6640,0.9413) (0.7241,0.9743) (0.7881,0.9960) (0.8602,1). (0, 0.1538) (0.0028,0.2290) (0.0213,0.2951) (0.0510,0.3567) (0.0872,0.4156) (0.1278,0.4724) (0.1715,0.5275) (0.2176,0.5813) (0.2656,0.6337) (0.3152,0.6847) (0.3662,0.7343) (0.4186,0.7823) (0.4724,0.8284) (0.5275,0.8721) (0.5843,0.9127) (0.6432,0.9489) (0.7048,0.9786) (0.7709,0.9971) (0.8461,1). (0, 0.1627) (0.0023,0.2396) (0.0190,0.3068) (0.0468,0.3691) (0.0813,0.4284) (0.1204,0.4855) (0.1627,0.5406) (0.2076,0.5941) (0.2546,0.6461) (0.3034,0.6965) (0.3538,0.7453) (0.4058,0.7923) (0.4593,0.8372) (0.5144,0.8795) (0.5715,0.9186) (0.6308,0.9531) (0.6931,0.9809) (0.7603,0.9976) (0.8373,1). (0, 0.1974) (0.0010,0.2795) (0.0123,0.3498) (0.0340,0.4140) (0.0626,0.4742) (0.0963,0.5313) (0.1339,0.5859) (0.1746,0.6381) (0.2180,0.6882) (0.2637,0.7362) (0.3117,0.7819) (0.3618,0.8253) (0.4140,0.8660) (0.4686,0.9036) (0.5257,0.9373) (0.5859,0.9659) (0.6501,0.9876) (0.7204,0.9989) (0.8025,1). (0, 0.2599) (0.0002,0.3472) (0.0056,0.4199) (0.0193,0.4849) (0.0400,0.5447) (0.0660,0.6003) (0.0964,0.6526) (0.1306,0.7017) (0.1681,0.7479) (0.2086,0.7913) (0.2520,0.8318) (0.2982,0.8693) (0.3473,0.9035) (0.3996,0.9339) (0.4552,0.9599) (0.5150,0.9806) (0.5800,0.9943) (0.6527,0.9997) (0.7400,1). n = 19 x. γ = 0.8073. γ = 0.8549. γ = 0.8989. γ = 0.9504. γ = 0.9902. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19. (0, 0.1322) (0.0038,0.2007) (0.0245,0.2615) (0.0559,0.3187) (0.0932,0.3738) (0.1343,0.4273) (0.1780,0.4798) (0.2235,0.5314) (0.2705,0.5822) (0.3187,0.6321) (0.3678,0.6812) (0.4178,0.7294) (0.4685,0.7763) (0.5201,0.8219) (0.5726,0.8656) (0.6261,0.9067) (0.6812,0.9440) (0.7384,0.9754) (0.7992,0.9961) (0.8677,1). (0, 0.1459) (0.0027,0.2176) (0.0203,0.2805) (0.0485,0.3393) (0.0828,0.3954) (0.1212,0.4497) (0.1625,0.5025) (0.2060,0.5541) (0.2513,0.6045) (0.2981,0.6537) (0.3462,0.7018) (0.3954,0.7486) (0.4458,0.7939) (0.4974,0.8374) (0.5502,0.8787) (0.6045,0.9171) (0.6606,0.9514) (0.7194,0.9796) (0.7823,0.9972) (0.8540,1). (0, 0.1632) (0.0018,0.2381) (0.0161,0.3032) (0.0408,0.3634) (0.0719,0.4205) (0.1073,0.4752) (0.1460,0.5281) (0.1872,0.5794) (0.2306,0.6292) (0.2757,0.6774) (0.3225,0.7242) (0.3707,0.7693) (0.4205,0.8127) (0.4718,0.8539) (0.5247,0.8926) (0.5794,0.9281) (0.6365,0.9591) (0.6967,0.9838) (0.7618,0.9981) (0.8367,1). (0, 0.1853) (0.0010,0.2634) (0.0121,0.3305) (0.0330,0.3919) (0.0605,0.4497) (0.0926,0.5047) (0.1283,0.5573) (0.1669,0.6080) (0.2079,0.6568) (0.2511,0.7037) (0.2962,0.7488) (0.3431,0.7920) (0.3919,0.8330) (0.4426,0.8716) (0.4952,0.9073) (0.5502,0.9394) (0.6080,0.9669) (0.6694,0.9878) (0.7365,0.9989) (0.8146,1). (0, 0.2445) (0.0002,0.3281) (0.0055,0.3980) (0.0189,0.4607) (0.0388,0.5185) (0.0636,0.5726) (0.0926,0.6236) (0.1250,0.6718) (0.1604,0.7175) (0.1985,0.7607) (0.2392,0.8014) (0.2824,0.8395) (0.3281,0.8749) (0.3763,0.9073) (0.4273,0.9363) (0.4814,0.9611) (0.5392,0.9810) (0.6019,0.9944) (0.6718,0.9997) (0.7554,1). 26.

(33) n = 20 x. γ = 0.8031. γ = 0.8500. γ = 0.9018. γ = 0.9486. γ = 0.9902. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20. (0, 0.1255) (0.0036,0.1907) (0.0234,0.2486) (0.0534,0.3031) (0.0889,0.3557) (0.1280,0.4068) (0.1695,0.4570) (0.2128,0.5063) (0.2575,0.5549) (0.3031,0.6029) (0.3497,0.6502) (0.3970,0.6968) (0.4450,0.7424) (0.4936,0.7871) (0.5429,0.8304) (0.5931,0.8719) (0.6442,0.9110) (0.6968,0.9465) (0.7513,0.9765) (0.8092,0.9963) (0.8744,1). (0, 0.1391) (0.0026,0.2075) (0.0193,0.2676) (0.0460,0.3238) (0.0786,0.3775) (0.1150,0.4295) (0.1542,0.4801) (0.1954,0.5296) (0.2383,0.5780) (0.2825,0.6255) (0.3279,0.6720) (0.3744,0.7174) (0.4219,0.7616) (0.4703,0.8045) (0.5198,0.8457) (0.5704,0.8849) (0.6224,0.9213) (0.6761,0.9539) (0.7323,0.9806) (0.7924,0.9973) (0.8608,1). (0, 0.1540) (0.0018,0.2253) (0.0157,0.2873) (0.0394,0.3449) (0.0691,0.3995) (0.1030,0.4521) (0.1399,0.5029) (0.1791,0.5523) (0.2202,0.6004) (0.2630,0.6472) (0.3072,0.6927) (0.3527,0.7369) (0.3995,0.7797) (0.4476,0.8208) (0.4970,0.8600) (0.5478,0.8969) (0.6004,0.9308) (0.6550,0.9605) (0.7126,0.9842) (0.7746,0.9981) (0.8459,1). (0, 0.1801) (0.0009,0.2554) (0.0109,0.3199) (0.0303,0.3791) (0.0558,0.4347) (0.0858,0.4876) (0.1192,0.5384) (0.1552,0.5873) (0.1936,0.6344) (0.2339,0.6800) (0.2761,0.7238) (0.3199,0.7660) (0.3655,0.8063) (0.4126,0.8447) (0.4615,0.8807) (0.5123,0.9141) (0.5652,0.9441) (0.6208,0.9696) (0.6800,0.9890) (0.7445,0.9990) (0.8198,1). (0, 0.2374) (0.0002,0.3179) (0.0050,0.3853) (0.0173,0.4459) (0.0357,0.5018) (0.0588,0.5542) (0.0858,0.6037) (0.1160,0.6506) (0.1490,0.6951) (0.1845,0.7374) (0.2223,0.7776) (0.2625,0.8154) (0.3048,0.8509) (0.3493,0.8839) (0.3962,0.9141) (0.4457,0.9411) (0.4981,0.9642) (0.5540,0.9826) (0.6146,0.9949) (0.6820,0.9997) (0.7625,1). n = 21 x. γ = 0.8000. γ = 0.8459. γ = 0.8983. γ = 0.9524. γ = 0.9892. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21. (0, 0.1201) (0.0034,0.1825) (0.0222,0.2379) (0.0507,0.2901) (0.0845,0.3404) (0.1216,0.3893) (0.1610,0.4374) (0.2022,0.4847) (0.2446,0.5313) (0.2880,0.5774) (0.3322,0.6228) (0.3771,0.6677) (0.4225,0.7119) (0.4686,0.7553) (0.5152,0.7977) (0.5625,0.8389) (0.6106,0.8783) (0.6595,0.9154) (0.7098,0.9492) (0.7620,0.9777) (0.8174,0.9965) (0.8798,1). (0, 0.1335) (0.0024,0.1991) (0.0182,0.2567) (0.0435,0.3106) (0.0744,0.3621) (0.1089,0.4120) (0.1460,0.4606) (0.1851,0.5081) (0.2257,0.5547) (0.2676,0.6005) (0.3106,0.6453) (0.3546,0.6893) (0.3994,0.7323) (0.4452,0.7742) (0.4918,0.8148) (0.5393,0.8539) (0.5879,0.8910) (0.6378,0.9255) (0.6893,0.9564) (0.7432,0.9817) (0.8008,0.9975) (0.8664,1). (0, 0.1488) (0.0016,0.2173) (0.0146,0.2770) (0.0368,0.3323) (0.0649,0.3848) (0.0968,0.4354) (0.1315,0.4843) (0.1685,0.5319) (0.2073,0.5783) (0.2477,0.6235) (0.2894,0.6676) (0.3323,0.7105) (0.3764,0.7522) (0.4216,0.7926) (0.4680,0.8314) (0.5156,0.8684) (0.5645,0.9031) (0.6151,0.9350) (0.6676,0.9631) (0.7229,0.9853) (0.7826,0.9983) (0.8511,1). (0, 0.1708) (0.0009,0.2428) (0.0106,0.3046) (0.0292,0.3614) (0.0537,0.4149) (0.0824,0.4659) (0.1142,0.5150) (0.1485,0.5623) (0.1850,0.6081) (0.2233,0.6524) (0.2632,0.6953) (0.3046,0.7367) (0.3475,0.7766) (0.3918,0.8149) (0.4376,0.8514) (0.4849,0.8857) (0.5340,0.9175) (0.5850,0.9462) (0.6385,0.9707) (0.6953,0.9893) (0.7571,0.9990) (0.8291,1). (0, 0.2286) (0.0002,0.3062) (0.0047,0.3712) (0.0163,0.4297) (0.0336,0.4839) (0.0554,0.5347) (0.0808,0.5828) (0.1093,0.6285) (0.1403,0.6721) (0.1737,0.7136) (0.2092,0.7531) (0.2468,0.7907) (0.2863,0.8262) (0.3278,0.8596) (0.3714,0.8906) (0.4171,0.9191) (0.4652,0.9445) (0.5160,0.9663) (0.5702,0.9836) (0.6287,0.9952) (0.6937,0.9997) (0.7713,1). 27.

(34) n = 22 x. γ = 0.8000. γ = 0.8417. γ = 0.8944. γ = 0.9486. γ = 0.9907. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22. (0, 0.1193) (0.0029,0.1802) (0.0198,0.2341) (0.0459,0.2847) (0.0771,0.3334) (0.1116,0.3807) (0.1484,0.4270) (0.1869,0.4725) (0.2267,0.5174) (0.2675,0.5616) (0.3092,0.6053) (0.3516,0.6483) (0.3946,0.6907) (0.4383,0.7324) (0.4825,0.7732) (0.5274,0.8130) (0.5729,0.8515) (0.6192,0.8883) (0.6665,0.9228) (0.7152,0.9540) (0.7658,0.9801) (0.8197,0.9970) (0.8806,1). (0, 0.1272) (0.0023,0.1899) (0.0175,0.2451) (0.0418,0.2967) (0.0714,0.3461) (0.1044,0.3940) (0.1398,0.4407) (0.1771,0.4864) (0.2158,0.5313) (0.2558,0.5754) (0.2967,0.6188) (0.3385,0.6614) (0.3811,0.7032) (0.4245,0.7441) (0.4686,0.7841) (0.5135,0.8228) (0.5592,0.8601) (0.6059,0.8955) (0.6538,0.9285) (0.7032,0.9581) (0.7548,0.9824) (0.8100,0.9976) (0.8727,1). (0, 0.1415) (0.0016,0.2071) (0.0141,0.2642) (0.0355,0.3173) (0.0624,0.3678) (0.0929,0.4164) (0.1262,0.4635) (0.1615,0.5094) (0.1985,0.5542) (0.2369,0.5980) (0.2766,0.6408) (0.3173,0.6826) (0.3591,0.7233) (0.4019,0.7630) (0.4457,0.8014) (0.4905,0.8384) (0.5364,0.8737) (0.5835,0.9070) (0.6321,0.9375) (0.6826,0.9644) (0.7357,0.9858) (0.7928,0.9983) (0.8584,1). (0, 0.1679) (0.0008,0.2376) (0.0095,0.2975) (0.0266,0.3524) (0.0494,0.4040) (0.0761,0.4534) (0.1058,0.5007) (0.1379,0.5465) (0.1721,0.5908) (0.2080,0.6337) (0.2455,0.6753) (0.2844,0.7155) (0.3246,0.7544) (0.3662,0.7919) (0.4091,0.8278) (0.4534,0.8620) (0.4992,0.8941) (0.5465,0.9238) (0.5959,0.9505) (0.6475,0.9733) (0.7024,0.9904) (0.7623,0.9991) (0.8320,1). (0, 0.2164) (0.0002,0.2910) (0.0047,0.3537) (0.0160,0.4102) (0.0328,0.4626) (0.0538,0.5120) (0.0783,0.5589) (0.1055,0.6035) (0.1351,0.6462) (0.1669,0.6871) (0.2007,0.7262) (0.2363,0.7636) (0.2737,0.7992) (0.3128,0.8330) (0.3537,0.8648) (0.3964,0.8944) (0.4410,0.9216) (0.4879,0.9461) (0.5373,0.9671) (0.5897,0.9839) (0.6462,0.9952) (0.7089,0.9997) (0.7835,1). n = 23 x. γ = 0.7971. γ = 0.8546. γ = 0.9050. γ = 0.9485. γ = 0.9898. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23. (0, 0.1167) (0.0026,0.1757) (0.0182,0.2279) (0.0426,0.2768) (0.0719,0.3238) (0.1044,0.3694) (0.1392,0.4141) (0.1756,0.4580) (0.2132,0.5011) (0.2519,0.5438) (0.2915,0.5858) (0.3317,0.6273) (0.3726,0.6682) (0.4141,0.7084) (0.4561,0.7480) (0.4988,0.7867) (0.5419,0.8243) (0.5858,0.8607) (0.6305,0.8955) (0.6761,0.9280) (0.7231,0.9573) (0.7720,0.9817) (0.8242,0.9973) (0.8832,1). (0, 0.1215) (0.0023,0.1816) (0.0169,0.2345) (0.0402,0.2840) (0.0685,0.3315) (0.1002,0.3775) (0.1341,0.4224) (0.1698,0.4664) (0.2068,0.5096) (0.2449,0.5522) (0.2840,0.5941) (0.3239,0.6354) (0.3645,0.6760) (0.4058,0.7159) (0.4477,0.7550) (0.4903,0.7931) (0.5335,0.8301) (0.5775,0.8658) (0.6224,0.8997) (0.6684,0.9314) (0.7159,0.9597) (0.7654,0.9830) (0.8183,0.9976) (0.8784,1). (0, 0.1349) (0.0016,0.1978) (0.0137,0.2527) (0.0343,0.3037) (0.0601,0.3522) (0.0893,0.3990) (0.1212,0.4444) (0.1550,0.4887) (0.1903,0.5319) (0.2270,0.5743) (0.2648,0.6158) (0.3037,0.6565) (0.3434,0.6962) (0.3841,0.7351) (0.4256,0.7729) (0.4680,0.8096) (0.5112,0.8449) (0.5555,0.8787) (0.6009,0.9106) (0.6477,0.9398) (0.6963,0.9656) (0.7472,0.9862) (0.8021,0.9983) (0.8650,1). (0, 0.1612) (0.0007,0.2283) (0.0091,0.2859) (0.0255,0.3388) (0.0472,0.3887) (0.0726,0.4363) (0.1010,0.4822) (0.1316,0.5265) (0.1641,0.5694) (0.1982,0.6112) (0.2338,0.6517) (0.2707,0.6910) (0.3089,0.7292) (0.3482,0.7661) (0.3887,0.8017) (0.4305,0.8358) (0.4734,0.8683) (0.5177,0.8989) (0.5636,0.9273) (0.6112,0.9527) (0.6611,0.9744) (0.7140,0.9908) (0.7716,0.9992) (0.8387,1). (0, 0.2143) (0.0001,0.2867) (0.0040,0.3476) (0.0143,0.4024) (0.0297,0.4532) (0.0491,0.5011) (0.0717,0.5466) (0.0970,0.5899) (0.1247,0.6315) (0.1543,0.6712) (0.1858,0.7094) (0.2191,0.7459) (0.2540,0.7808) (0.2905,0.8141) (0.3287,0.8456) (0.3684,0.8752) (0.4100,0.9029) (0.4533,0.9282) (0.4988,0.9508) (0.5467,0.9702) (0.5975,0.9856) (0.6523,0.9959) (0.7132,0.9998) (0.7856,1). 28.

(35) n = 24 x. γ = 0.8003. γ = 0.8499. γ = 0.8998. γ = 0.9515. γ = 0.9898. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24. (0, 0.1116) (0.0025,0.1682) (0.0177,0.2182) (0.0411,0.2652) (0.0693,0.3103) (0.1005,0.3542) (0.1339,0.3971) (0.1688,0.4394) (0.2049,0.4810) (0.2420,0.5221) (0.2799,0.5626) (0.3184,0.6027) (0.3575,0.6424) (0.3972,0.6815) (0.4373,0.7200) (0.4778,0.7579) (0.5189,0.7950) (0.5605,0.8311) (0.6028,0.8660) (0.6457,0.8994) (0.6896,0.9306) (0.7347,0.9588) (0.7817,0.9822) (0.8317,0.9974) (0.8883,1). (0, 0.1223) (0.0019,0.1814) (0.0148,0.2331) (0.0359,0.2815) (0.0619,0.3277) (0.0912,0.3724) (0.1227,0.4159) (0.1560,0.4585) (0.1907,0.5003) (0.2265,0.5414) (0.2633,0.5818) (0.3009,0.6215) (0.3393,0.6606) (0.3784,0.6990) (0.4181,0.7366) (0.4585,0.7734) (0.4996,0.8092) (0.5414,0.8439) (0.5840,0.8772) (0.6275,0.9087) (0.6722,0.9380) (0.7184,0.9640) (0.7668,0.9851) (0.8185,0.9980) (0.8776,1). (0, 0.1350) (0.0013,0.1965) (0.0120,0.2500) (0.0308,0.2996) (0.0545,0.3468) (0.0817,0.3922) (0.1114,0.4362) (0.1430,0.4790) (0.1761,0.5209) (0.2106,0.5618) (0.2462,0.6019) (0.2828,0.6411) (0.3203,0.6796) (0.3588,0.7171) (0.3980,0.7537) (0.4381,0.7893) (0.4790,0.8238) (0.5209,0.8569) (0.5637,0.8885) (0.6077,0.9182) (0.6531,0.9454) (0.7003,0.9691) (0.7499,0.9879) (0.8034,0.9986) (0.8649,1). (0, 0.1534) (0.0007,0.2178) (0.0090,0.2733) (0.0248,0.3243) (0.0458,0.3724) (0.0703,0.4184) (0.0976,0.4628) (0.1270,0.5057) (0.1582,0.5474) (0.1908,0.5880) (0.2249,0.6275) (0.2601,0.6660) (0.2965,0.7034) (0.3339,0.7398) (0.3724,0.7750) (0.4119,0.8091) (0.4525,0.8417) (0.4942,0.8729) (0.5371,0.9023) (0.5815,0.9296) (0.6275,0.9541) (0.6756,0.9751) (0.7266,0.9909) (0.7821,0.9992) (0.8465,1). (0, 0.2065) (0.0001,0.2765) (0.0039,0.3353) (0.0136,0.3884) (0.0283,0.4378) (0.0469,0.4843) (0.0685,0.5285) (0.0926,0.5708) (0.1189,0.6114) (0.1471,0.6503) (0.1770,0.6878) (0.2086,0.7238) (0.2416,0.7583) (0.2761,0.7913) (0.3121,0.8229) (0.3496,0.8528) (0.3885,0.8810) (0.4291,0.9073) (0.4714,0.9314) (0.5156,0.9530) (0.5621,0.9716) (0.6115,0.9863) (0.6646,0.9960) (0.7234,0.9998) (0.7934,1). 29.