一元Laplace分布的L_1-范估计的无偏性

The Unbiasedness of L_1 Estimation of Monistic Laplace Distribution

根据Laplace分布的概率密度函数公式 ,推导了中位数的概率密度 。

L 1 estimation is often used to process surveying data containing gross errors or abnormal values,it is a method of robust estimation.It is proved that L 1 estimation can resist disturbances of gross errors and its parameter MLE value is median of observed values. To the problem of unbiasedness of L 1 estimation,basing on uniqueness of solution,Zhou Shijiang proved it according to dual theorem of linear programming; and Wang Zhizhong proved it according to probability statistics theorem by using the method from special to general; also,basing on error distribution theorem and probability statistics theorem,the authors proved it.First,we deprived probability density of median closely according to probability density formula of Laplace distribution,from general to special; then we proved the unbiasedness of L 1 estimation according to probability density of median. When n is odd number,our reasoning thoughts are: (1) we rearrange observed values,big or small. (2) we deprive probability density of subsample according to probability density function of Laplace distribution. (3) we deprive probability density of median according to probability density of subsample. (4) we proved the unbiasedness of L 1 estimation according to probability density of median. Finally,we draw the conclusions: (1) L 1 estimation is unbiased estimation,and this conclusion shows L 1 estimation has good statistical characteristic. (2) the conclusion drawn by this method is same as that deprived from dual theorem of linear programming in reference [2] ,but,this conclusion is deprived from probability statistical theorem,it is simple and accepted easily. (3) this reasoning method can enlighten us to prove the unbiasedness of L p estimation.