In this question, we will prove the formula for SE(log(0)) = + + + 1 n 21 n22 using delta method. The delta method is a useful method to derive the asymp- totic variance of a test statistic. Suppose that 0 = Pup22, where puj is defined as below. Column 1 2 Row 1 P11 P12 P1+ 2 P21 P22 P2+ P+1 P+2 N= n++ We want to derive the variance of log(0). The multivariate version of the delta method is Var(0) ~ Vf(PIl, P12, P21, P22) Cov(P11, P12, P21, P22) V f(P11, P12, P21, P22) Where V is the gradient vector. That is of Of Vf(P11, P12, P21, P22) = ap11 ap.22 We assume the multinomial sampling since the total number of observations is fixed. I try to : SE ( logo ) = J Var ( logo ) = JVf