Statistical inference for discrete patterns in dependent sequences

相关序列中离散模式的统计推断

基本信息

批准号：
14540114
负责人：
AKI Sigeo
金额：
$ 2.3万
依托单位：
Kansai University (2004-2005)Osaka University (2002-2003)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540114/
关键词：
discrete patterns discrete distribution theory probability generating function Markov chain urn models system reliability waiting time problem conditional expectation 条件付き確率離散確率分布起動試験工学的システムの信頼性 start-up demonstration test 幾何分

项目摘要

We have studied the distributions of numbers of occurrences of discrete patterns and waiting time distributions for them in various dependent sequences. The following results have been derived.1. By using a combinatorial method it is shown that for every finite pattern, the distribution of the waiting time for the reversed pattern coincides with that of the waiting time for the original pattern in a multi-state dependent sequence with a certain type of exchangeability. The corresponding results for the waiting time for the r-th occurrence of the pattern, and for the number of occurrences of a specified pattern in n trials are also studied. Illustrative examples based on urn models are also given.2. Several waiting time random variables for a duplication within a memory window of size k in a sequence of {1,2,...,m}-valued random variables are investigated. The exact distributions of the waiting time random variables are derived by the method of conditional probability generating functio … More ns. In particular, the exact distribution of the waiting time for the first k-match is obtained when the underlying sequence is generated by higher order Markov dependent trials. Examples for numerical calculations are also given.3. We consider waiting time problems for two-dimensional pattern in a sequence of i.i.d. random vectors each of whose entries is 0 or 1. A general method for obtaining the exact distribution of the waiting time for the first occurrence of the pattern in the sequence is presented. The method is an extension of the method of conditional probability generating functions and it is very suitable for computations with computer algebra systems as well as usual numerical computations. Computational results applied to computation of exact system reliability are also given.4. Let k and m be positive integers with k>m. The probability generating function of the waiting time for the first occurrence of consecutive k successes in a sequence of m-th order Markov dependent trials is given as a function of the conditional probability generating functions of the waiting time for the first occurrence of consecutive m successes. This provides an efficient algorithm for obtaining the probability generating function when k is large. In particular, in the case of independent trials a simple relationship between the geometric distribution of order k and the geometric distribution of order k-1 is obtained.5. Various new distributions related to runs and patterns in dependent sequences are obtained by using the stepwise smoothing formula for conditional expectations. Less

我们已经研究了离散模式出现数量的分布和以各种相关序列的等待时间分布的分布。得出以下结果1。通过使用组合方法，可以证明，对于每个有限模式，相反图案的等待时间的分布与多态依赖性序列中原始模式的等待时间与某种类型的交换性相一致。第rth发生模式的等待时间的相应结果以及在N试验中指定模式的发生数量也是研究的。还给出了基于urn模型的插图示例2。研究了几个等待时间的随机变量，以{1,2，...，m}值为值的随机变量的顺序在大小k的内存窗口中进行重复。等待时间随机变量的确切分布是通过条件概率生成函数的方法得出的……更多ns。特别地，当基础序列由高阶马尔可夫依赖性试验产生时，获得了第一个k匹配的等待时间的确切分布。还给出了数值计算的示例3。我们考虑以I.I.D的顺序等待二维模式的等待时间问题。随机向量的每个条目为0或1。出现了一种用于在序列中首次出现该模式的等待时间的确切分布的一般方法。该方法是有条件概率生成函数方法的扩展，它非常适合计算机代数系统以及通常的数值计算计算。还给出了用于计算精确系统可靠性的计算结果4。令K和M为k> m的正整数。第一次出现连续k成功的等待时间的概率生成函数以M-ther阶依赖性试验的序列，作为条件概率生成等待时间的函数的函数。这提供了一种有效的算法，用于当k较大时获得概率生成函数。特别是，在独立试验的情况下，获得了阶k的几何分布与阶数K-1的几何分布之间的简单关系。5。通过使用逐步平滑公式来获得条件期望，可以获得与依赖序列中的运行和模式相关的各种新分布。较少的