1. Let (X1,X2, . . . ,Xn) be jointly normal with EXi = , var(Xi) = 2,...

1. Let (X1,X2, . . . ,Xn) be jointly normal with EXi = μ, var(Xi) = σ2, and cov(Xi,Xj) =

ρσ2 if |i−j| = 1, i = j, and = 0 otherwise.

(a) Show that var(X) =

σ2 n

1+2ρ



1− 1 n



and E(S2) = σ2



1− 2ρ

n



.

(b) Show that the t-statistic

√

n(X−μ)/S is asymptotically normally distributed with mean 0 and variance 1+2ρ. Conclude that the significance of t is overestimated for positive values of ρ and underestimated for ρ < 0 in large samples.

(c) For finite n, consider the statistic T2 =

n(X−μ)2 S2 .

Compare the expected values of the numerator and the denominator of T2 and study the effect of ρ = 0 to interpret significant t values (Scheffé [101, p. 338].)