6.13 For the situation of Example 6.9, consider as another family of distributions, the contaminated normal mixture family suggested by Tukey (1960) as a model for observations which usually follow a normal distribution but where occasionally something goes wrong with the experiment or its recording, so that the resulting observation is a gross error. Under the Tukey model, the distribution function takes the form Fτ,V (t) = (1 − V)X(t) + VX  t

τ



.

That is, in the gross error cases, the observations are assumed to be normally distributed with the same mean θ but a different (larger) variance τ 2. 9

(a) Show that if the Xi’s have distribution Fτ,V (x − θ), the limiting distribution of δ2n is unchanged.

(b) Show that the limiting distribution of δ1n is normal with mean zero and variance n

n−1



φ

1* n n−1 (a − θ)

22

(1 − V + Vτ 2).

(c) Compare the asymptotic relative efficiency of δ1n and δ2n.