Statistical Inference for Discrete Patterns

我们研究了随机变量序列和随机有向树等随机结构中的运行等待时间分布和模式。得到如下结果：1。我们已经得到了在{0,1}值的马尔可夫树上指定长度的“1”运行数的确切分布。该结果可用于有向树上连续系统的可靠度计算。得到了连续系统在有向树上的精确寿命分布。我们已经研究了马尔可夫相关双变量试验中运行等待时间早晚的精确分布。我们已经给出了关于等待时间的条件概率生成函数的线性方程组并解出了它们。该结果可应用于线性连通- (r, s) -out- (r+1, n): F格系的可靠度计算。引入了多项式型马尔可夫链可嵌套向量和可返回型马尔可夫链可嵌套变量，并讨论了它们的一些性质。利用这一结果，我们推导出了一系列多状态试验中指定长度的运行次数的分布。我们为运行引入了一个统一的计数方案，称为1-重叠计数。在马尔可夫链和高阶马尔可夫链等相关随机序列中，给出了指定长度的1-重叠1-运行数的精确概率生成函数。我们引入了一种新的依赖序列，称为（k, r）阶二进制序列，并推导了该序列中成功和失败运行的早晚等待时间的确切分布。

We have studied waiting time distributions of runs and patterns in random structures such as sequences of random variables and random directed trees. The following results are derived.1. We have obtained the exact distribution of the number of "1" -runs of a specified length on {0, 1} -valued Markov trees. This result can be applied to calculate the reliability of a consecutive system on a directed tree. We also obtained the exact distribution of the life time of the consecutive system on the directed tree.2. We have studied exact distributions of sooner and later waiting times for runs in Markov dependent bivariate trials. We have given systems of linear equations with respect to conditional probability generating functions of the waiting times and have solved them. This result can be applied to calculate the reliability of the linear connected- (r, s) -out-of- (r+1, n) : F lattice system.3. We have introduced a Markov chain imbeddable vector of multinomial type and a Markov chain imbeddable variables of returnable type and have discussed some of their properties. By using the result we have derived the distribution of numbers of occurrences of runs of specified lengths in a sequence of multi-state trials.4. We have introduced a unified counting scheme for runs called 1-overlapping counting. We have given exact probability generating function of the number of 1-overlapping 1-runs of a specified length in some dependent random sequences such as a Markov chain and a heigher order Markov chain.5. We have introduced a new type of dependent sequence called a binary sequence of order (k, r) and have derived the exact distributions of sooner and later waiting times for success and failure runs in the sequence.