Abstract: Based on the theory for M-estimation of the location parameters. the influence functions of the L^p estimation (one dimension and multi-dimension) have been derived and the robustness of L^p estimation has been analyzed while the observational errors obey the contaminated distribution which the data include the outliers disturbed. and the asymptotic variance and efficiency of L^p estimation have been determined too. The properties of L^p estimation have been discussed in the both robustness and efficiency. The observations with outliers disturbed can be processed using L^p estimation which the parameter p should be between 1.2 and 1.5.