几类特殊随机环境下的马氏过程的统计问题
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摘要
作为特殊的Markov过程,生灭过程,分支过程常常被用来模拟生物学,运筹学,人口统计学,工程学,经济学中的随机现象。许多数学家,生物学家,工程师等对此进行了大量的研究,建立了各种各样的模型,得到了很多深刻有趣的结果。然而,还有许多问题有待进一步研究,如转移概率分布的刻画,过程的随机模拟,部分观测样本下的统计推断,随机环境下的生灭,分枝过程等。
本篇论文由相互独立又有联系的四部分组成。在第一部分中,我们主要讨论一类特殊的线性生灭过程—移入-生灭过程。这部分包括第一章和第二章。第一章我们利用矩母函数给出了三个精确而优美的转移概率公式。这些公式对生灭过程的随机模拟与统计推断非常有用。第二章,我们给出两种抽样方式下的最大似然估计—给定区间上的连续抽样与等间隔抽样,并利用随机模拟方法讨论估计的分布性质。
第二部分,我们讨论另一类特殊的具有交互作用的非线性二维生灭过程—随机捕食-被捕食模型。这部分包括第三,四章。在第三章,我们建立了一类新的随机捕食-被捕食模型,并证明了新模型在平均意义上与经典的捕食-被捕食模型的等价性,讨论了其灭绝性,平衡态分布等。在第四章,我们给出了随机捕食-被捕食模型参数的最大似然估计,然后利用随机模拟方法讨论估计的分布性质。
第三部分,即第五章,我们讨论随机环境下的移入-移出-生灭过程。在这一部分,我们将经典的生灭过程理论推广到了随机环境下,证明了对任意给定随机环境移入-移出-生灭密度矩阵q,当生率小于死率时,存在一个唯一的随机环境下的q过程—(?)(θ~*(0);t)和随机环境下的移入-移出-生灭过程(X~*,ζ~*),使得(X~*,ζ~*)严平稳,遍历,其随机转移阵为(?)(θ~*(0);t)。
第四部分,我们讨论随机环境下的分枝链。这部分包括第六章,第七章。在第六章,我们定义了各种条件多元概率母函数,并利用条件多元概率母函数这一强有力工具研究随机环境中r-维分支链的性质,给出了其协方差阵的精确计算公式。在第七章,我们定义了随机环境中r-维分支链的Laplace泛函,讨论了随机环境中r-维分支链的各种矩问题。
A variate of phenomenon in a variate fields, such as biology, operation research, demography, economics, and engineering, can be well modeled by birth-death processes and branching processes. At the same time, the richness of the theory of birth-death process provides standard analytical methods to investigate numerous important quantities in general stochastic process and practice, such as stationary distribution, mean first passage time, etc. Motivated by this, a lot of mathematicians, biologists, engineers, etc., have made great efforts in it. Nevertheless, a lot of issues are still in the way being solved.This thesis consists of four parts. In part Ⅰ (Chapter 1-2), we focuss on investigating a special linear population process—immigration-birth-death process. First, in Chapter 1, we provide some exact, elegant formulae of the transition distribution of immigration-birth-death process by moment generating function. They are very useful in simulation and statistical inference. Second, in Chapter 2, by the exact descriptions we obtained, the maximum likelihood estimation under two sampling schemes: complete observation and equidistant observation, are given and in the end, simulations are carried out to test the bias and efficiency of our estimators.In part Ⅱ (Chapter 3-4), we deal with another special interesting nonlinear population process—prey-predator process. First, in Chapter 3, a two-dimensional interacting birth-death process is established and by means of two-dimensional moment generating function, the equivalence in mean between our model and the classic prey-predator model are proved. Also, in Chapter 4 chapter, using the properties of stochastic prey-predator model, the maximum likelihood inference under two sampling schemes are given and simulations are carried out.In part Ⅲ (Chapter 5 ), the theory of classic birth-death process is generalized to random environment. Our main result is for any immigration-emigration birth and death matrix random environment, q, with birth rate
less than death rate, there are a unique q-process in random environment P(9*(0);t) and a bi-immigration birth and death process in random environment (X*,f) with random transition matrix P(9*{Q);t) such that P(9*(0);t) is ergodic and X* is a strictly stationary process.In part IV (Chapter 6-7), we discuss r-dimension branching chains in random environment. First, in Chapter 6, we define some probability generating functions in random environment and use them to give a exact formula of the the covariance matrix of r-dimension branching chains in random environment. Second, in Chapter 7, we define the Laplace functional of r-dimension branching chains in random environment and then discuss the moments of r-dimension branching chains in random environment.
本篇论文由相互独立又有联系的四部分组成。在第一部分中,我们主要讨论一类特殊的线性生灭过程—移入-生灭过程。这部分包括第一章和第二章。第一章我们利用矩母函数给出了三个精确而优美的转移概率公式。这些公式对生灭过程的随机模拟与统计推断非常有用。第二章,我们给出两种抽样方式下的最大似然估计—给定区间上的连续抽样与等间隔抽样,并利用随机模拟方法讨论估计的分布性质。
第二部分,我们讨论另一类特殊的具有交互作用的非线性二维生灭过程—随机捕食-被捕食模型。这部分包括第三,四章。在第三章,我们建立了一类新的随机捕食-被捕食模型,并证明了新模型在平均意义上与经典的捕食-被捕食模型的等价性,讨论了其灭绝性,平衡态分布等。在第四章,我们给出了随机捕食-被捕食模型参数的最大似然估计,然后利用随机模拟方法讨论估计的分布性质。
第三部分,即第五章,我们讨论随机环境下的移入-移出-生灭过程。在这一部分,我们将经典的生灭过程理论推广到了随机环境下,证明了对任意给定随机环境移入-移出-生灭密度矩阵q,当生率小于死率时,存在一个唯一的随机环境下的q过程—(?)(θ~*(0);t)和随机环境下的移入-移出-生灭过程(X~*,ζ~*),使得(X~*,ζ~*)严平稳,遍历,其随机转移阵为(?)(θ~*(0);t)。
第四部分,我们讨论随机环境下的分枝链。这部分包括第六章,第七章。在第六章,我们定义了各种条件多元概率母函数,并利用条件多元概率母函数这一强有力工具研究随机环境中r-维分支链的性质,给出了其协方差阵的精确计算公式。在第七章,我们定义了随机环境中r-维分支链的Laplace泛函,讨论了随机环境中r-维分支链的各种矩问题。
A variate of phenomenon in a variate fields, such as biology, operation research, demography, economics, and engineering, can be well modeled by birth-death processes and branching processes. At the same time, the richness of the theory of birth-death process provides standard analytical methods to investigate numerous important quantities in general stochastic process and practice, such as stationary distribution, mean first passage time, etc. Motivated by this, a lot of mathematicians, biologists, engineers, etc., have made great efforts in it. Nevertheless, a lot of issues are still in the way being solved.This thesis consists of four parts. In part Ⅰ (Chapter 1-2), we focuss on investigating a special linear population process—immigration-birth-death process. First, in Chapter 1, we provide some exact, elegant formulae of the transition distribution of immigration-birth-death process by moment generating function. They are very useful in simulation and statistical inference. Second, in Chapter 2, by the exact descriptions we obtained, the maximum likelihood estimation under two sampling schemes: complete observation and equidistant observation, are given and in the end, simulations are carried out to test the bias and efficiency of our estimators.In part Ⅱ (Chapter 3-4), we deal with another special interesting nonlinear population process—prey-predator process. First, in Chapter 3, a two-dimensional interacting birth-death process is established and by means of two-dimensional moment generating function, the equivalence in mean between our model and the classic prey-predator model are proved. Also, in Chapter 4 chapter, using the properties of stochastic prey-predator model, the maximum likelihood inference under two sampling schemes are given and simulations are carried out.In part Ⅲ (Chapter 5 ), the theory of classic birth-death process is generalized to random environment. Our main result is for any immigration-emigration birth and death matrix random environment, q, with birth rate
less than death rate, there are a unique q-process in random environment P(9*(0);t) and a bi-immigration birth and death process in random environment (X*,f) with random transition matrix P(9*{Q);t) such that P(9*(0);t) is ergodic and X* is a strictly stationary process.In part IV (Chapter 6-7), we discuss r-dimension branching chains in random environment. First, in Chapter 6, we define some probability generating functions in random environment and use them to give a exact formula of the the covariance matrix of r-dimension branching chains in random environment. Second, in Chapter 7, we define the Laplace functional of r-dimension branching chains in random environment and then discuss the moments of r-dimension branching chains in random environment.
引文
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[3] K. B. Arthreya and S. Karlin, On branching processes with random environment, I Extinction probabilities, Ann.Math.Statist, 1970, 42, 1499-1520.
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[5] N. T. J. Bailey, the mathematical theory of infectious diseases and its applications, 2003, London: Griffin.
[6] R. B. Banks, Growth and diffusion phenomena : mathematical frameworks and applications, 1994, Springer-Verlag.
[7] V. D. Barnett, the Monte-Carlo solution of a competing species problem. Biometrics, 1962, 18 , 76-103.
[8] M. S. Barnett, On the theoretical models for competive and predator biological systems. Biometrika, 1957, 44, 27-42.
[9] N. G. Becker, M. H. Abraham, Estimation in Epidemics with Incomplete Observations, Journal of the Royal Statistical Society, Series B, 1997 , 59 , 415-429.
[10] N. T. Bailey, the mathematical theory of infectious diseases and its applications, 2004, London: Griffin.
[11] N. G. Beker, On Lotka-Volterra predator models, J. Appl. Prob, 1997, 45, 375-381.
[12] N. G. Becker and M.H. Abraham, Esitimation in epidemics with incomplete observations, J. R. Statist. Soc. B, 1997, 59, 415-429.
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