Let X1, , Xn be a sample from the normal distribution with mean and variance

Let X1, ··· , Xn be a sample from the normal distribution with mean θ and variance 1, with cdf denoted by Fθ(·). Let Φ(z) denote the standard normal cdf, so that Fθ(t) = Φ(t − θ). For any two cdfs F and G, let F − G

denote supt |F(t) − G(t)|. Let ˆθn be the estimator of θ minimizing Fˆn − Fθ,

where Fˆn(t) = n−1 n i=1 1(Xi ≤ t) denotes the empirical cdf. In case you are worried about problems of existence or uniqueness, you may assume ˆθn is any estimator satisfying Fˆn − Fθ
ˆn  ≤ inf θ Fˆn − Fθ + n, where n is any sequence of positive constants tending to 0.
(i) Prove ˆθn is a consistent estimator of θ.
(ii) Suppose now the observations come from a cdf F, possibly nonnormal. The problem is to test the null hypothesis that F is normal with variance 1 against the alternative hypothesis that F is not. Consider the test statistic Tn = inf θ Fˆn − Fθ.
Argue, if F is N(θ, 1), then the distribution of Tn does not depend on θ.
(iii) If F is not normal with variance one, argue that Tn tends in probability to the constant γF = infθ F − Fθ, and γF > 0.
(iv) Find a sequence of constants cn so that the test that rejects iff Tn ≥ cn has probability of a Type I error tending to 0, and has power tending to one for any fixed alternative F. Hint: Use the Dvoretzky, Kiefer, Wolfowitz Inequality.
Section 14.3