Let Xl, X2, ..., Xn be an i.i.d. sample and Fn(x) the empirical distribution function: statistics Home Work MU, NZ

Problem 3

Let Xl, X2, ..., Xn be an i.i.d. sample and Fn(x) the empirical distribution function. We know from the central limit theorem that- 4 MCO, 02) with 02 — Fx(x)) depending on x. Hence, for large n the distribution of Fn(x) will be close to a normal distribution.

(a) (2 marks) Find an upper bound for the variance of Fn(x) that depends on n but does not depend on x. The upper bound should be as small as possible.

(b) (10 marks) Assume that Fn(x) is normally distributed for each x. Compute the width of a 90% asymptotic point-wise confidence band for Fx (x) using the empirical
distribution function and the upper bound on the variance from (a). In the end the width of the confidence band may depend on n but not on x.

Remember For a normal random variable Y with mean E(Y) and variance Var(Y) 02, P (—1.6450 Y — Lt 1.6450) 0.9.

(b) (5 marks) Compare the width of the point-wise confidence band to the width of a 90% uniform confidence band constructed with the DKW inequality. What do you
observe?