Consider the regression model Yi = xn,i + i , i = 1,..., n , where

Consider the regression model Yi = βxn,i + i , i = 1,..., n , where β ∈ R is unknown, the xn,i are fixed and known, and the i are i.i.d. standard normal. Consider testing the null hypothesis β = 0 versus β > 0 using the UMP level-α test.

(i) Let xn,i = i/n. For what constant p is the limiting local power function against alternatives of the form h/n p not degenerate, i.e., strictly between 0 and 1. Calculate the optimal limiting local power function against such a sequence.

(ii) Repeat (i) in the case xn,i = i.

Section 15.4