Majorization Under Incomplete Information And Application

Based on the Schur-convex function,theory of majorization,Shannon entropy and inequality measures and their relevant advances,this thesis focuses on studying the sufficient condition for the establishment of the relationship of majorization(SCM)between two variable vectors under incomplete information,and discusses its applications.The main work of this thesis is as follows:1.Based on the Schur-convex function and under the conditions of a few known variable values in a variable vector,the thesis proposes a method to determine the upper and lower bounds of majorization for the discrete finite variable vectors.According to this method,the thesis gives an algorithm to determine the upper and lower bounds of majorization for any variable vectors and the SCM between two variable vectors under incomplete information.2.The thesis applies the SCM to the Shannon entropy H.Under the conditions of a few known discrete probability values of two discrete probability distributions P and Q,based on two intervals generated by the upper and lower bounds of majorization for P and Q respectively,this thesis analyzes three cases with the location of these two intervals,which is non-overlapping,non-overlapping regions with overlapping and overlapping regions with overlapping,and gives the sufficient condition for the claim that H(P)? H(Q)under incomplete information,respectively.At last,the thesis gives the necessary and sufficient conditions for the conclusion that H(P)? H(Q)with the three-dimensional discrete probability distributions P and Q.3.The thesis applies the SCM to the inequality measure.Based on the data of the number of employed persons in urban and rural areas in China Statistical Yearbook in 2016,under some assumptions,this thesis mainly analyzes whether the distribution of urban employees in 2014 is more uniform than that in 2015 with 10 different economic types.With the complete data,a model is established and it is concluded that the distribution of urban employees in 2015 is more uneven than the distribution in 2014.With the incomplete data,using the related proposed theorems and obtained corollaries in the thesis,the conclusion is consistent with the former.The latter also points out that urban residents' attitudes to the 10 different economic types is unequal,and it results in uneven distribution of urban employees.Therefore,it is worth to study the SCM between two variable vectors under incomplete information.