As in Section 14.3.2, consider the Chi-squared test for testing uniformity on (0, 1) based on k + 1 cells; call if n,k. Fix

As in Section 14.3.2, consider the Chi-squared test for testing uniformity on (0, 1) based on k + 1 cells; call if φ∗

n,k. Fix any B < ∞ and  > 0.

Let UB be the set of u with u = 0 and u2 ≤ B. For alternative sequences of the form (14.25) with bn = n−1/2 , show that, if k is large enough (but fixed), then lim sup n sup u:u∈UB Efn (φ∗
n,k) ≤ α +  .