Math 281A Homework 5
Due: Nov 14, in class
1. Let {xi}n
i=1be i.i.d. sample from a strictly positive density that is symmetric about θ, show that the
Huber M-estimator for location is consistent for θ.
2. Let {xi}n
i=1be i.i.d. sample from a strictly positive density. Define
ψ(x)=2
1+e−x−1,
and ˆ
θnbe the solution of n
∑
i=1
ψ(Xi−θ)=0.
(a) Show that ˆ
θn
P
Ð→ θ0for some θ0, and express θ0in the density of observations;
(b) Show that √n(ˆ
θn−θ0)converges in distribution and find the limit variance.
3. Let {xi}n
i=1be i.i.d. sample from Uniform(0,1), determine the relative efficiency of the sample median
and the sample mean.
4. Let {xi}n
i=1be i.i.d. sample from N(θ, 1), find the relative efficiency of the Huber estimator and the
sample mean.