This paper considers the problem of estimating the bounds on the average controlled direct effects （ACDEs） of a treatment variable on an unobserved response variable in the presence of unobserved confounders between...This paper considers the problem of estimating the bounds on the average controlled direct effects （ACDEs） of a treatment variable on an unobserved response variable in the presence of unobserved confounders between an intermediate variable and the response variable. When the response variable is observed, Cai, et al.（2008） derived the formulas for the sharp bounds on the ACDEs. When the response variable is unobserved, the authors propose a graphical criterion for selecting variables affected by the response variable to derive the formulas for the bounds on the ACDEs, which is an extension of the result of Kuroki（2005） to ACDEs. The results enable us not only to judge from the graph structure whether the bounds on the ACDEs can be expressed through observed variables when the response variable is unobserved, but also to provide their formulas when the answer is affirmative.展开更多
In this paper we prove that the distributions of their sojourn time and hitting time at one special state for random walks which are allowed to be finite or infinite and Markov chains on star-graphs with discrete time...In this paper we prove that the distributions of their sojourn time and hitting time at one special state for random walks which are allowed to be finite or infinite and Markov chains on star-graphs with discrete time can uniquely determine the probability distribution of the whole chains. This result also suggests a new statistical method for Markov chains.展开更多
基金This research was partially supported by the National Natural Science Foundation of China under Grant Nos. 10871038, 10926186, and 11025102, the National 973 Key Project of China under Grant No. 2007CB311002, and the Jilin Project （20100401）.
文摘This paper considers the problem of estimating the bounds on the average controlled direct effects （ACDEs） of a treatment variable on an unobserved response variable in the presence of unobserved confounders between an intermediate variable and the response variable. When the response variable is observed, Cai, et al.（2008） derived the formulas for the sharp bounds on the ACDEs. When the response variable is unobserved, the authors propose a graphical criterion for selecting variables affected by the response variable to derive the formulas for the bounds on the ACDEs, which is an extension of the result of Kuroki（2005） to ACDEs. The results enable us not only to judge from the graph structure whether the bounds on the ACDEs can be expressed through observed variables when the response variable is unobserved, but also to provide their formulas when the answer is affirmative.
基金This research is supported by Hunan Province Natural Science Foundation of China(OOJJY2003)Doctoral Program Foundation of Institute of Higher Education of China, and Beijing Natural Science Foundation of China(1042007 ).
文摘In this paper we prove that the distributions of their sojourn time and hitting time at one special state for random walks which are allowed to be finite or infinite and Markov chains on star-graphs with discrete time can uniquely determine the probability distribution of the whole chains. This result also suggests a new statistical method for Markov chains.
