Question

Suppose that 01, ..., In are independent realizations from N (u,02). We know that X ~ N(N ). Suppose we are interested in minimizing the objective function h with respect to a where h is defined as h(a) = (Eſa X – ])? + Var(aĚ) Specifically we are interested in finding the global minimizer â, i.e., a = argmin h(a), CER where R+ denote the set of positive real numbers. (a) (7 points) Find Vh(a) and v2h(a). (b) (7 points) Based on Vh(a) find the value ā such that Vh(a) = 0. Prove that ā is the global minimizer by checking the value of v?h(a).

Answer 1

sol - Given That Xv..... Un are Independent redizations forom N(4.6%) that In (mah minimizing the objective function h with respect to a where h as defined Fw1(, Ž) 7 hlas = (t (ax_4])" + var (ax) = (au.ujt mo% (a) Find the phim and so mla) Thia) = 2(0"-"). M+ 2a on 2² hlas = 2M + 25% (6). Thla) value ă such that thlas=o that á is global minimizar value of thla) Thias=0 → zu (a-i) +298/m 20 = a (art ofn) - > a= uta o'hta) = 2 (et + ) The minimizer value aso,

then the condition to automatically satisfied for MER 7 hä) = 2 (M +0/n) so taert U a - global minimizer.