Error Density’s Estimates Based on M-Estimates

For linear model Y_i=x'_iβ十e_i,i=1,2,..., the error sequence e_{ii= 1}^∞i.i.d., e₁ has an unknown density f（x）. In this paper, we estimate f（x） by \widehatf_n(x)=\frac12 n a_n \sum_i=1^n I\left(x-a_n \leq \widehate_i \leq x+a_n\right) and obtain the strong convergence rate, the strong uniform convergence rate, L₁-norm consistency asymptotic normality, law of iterated logarithm.181