Let (Y₁, W₁), . . . , (Y_n, W_n) be an i.i.d. sample of random vectors with finite covariance matrix

Let Y̅ and W̅ be the sample averages. Let g(y, w) be a function with continuous partial derivatives g₁ and g₂ with respect to y and w, respectively. Let Z = g(Y, W). The two dimensional Taylor expansion of g around a point (y₀, w₀) is

g(y, w) = g(y₀, w₀) + g₁(y₀, w₀)(y − y₀)

                                                + g₂(y₀, w₀)(w − w₀),                                        (12.2.7)

plus an error term that we shall ignore here. Let (y, w) = (Y̅, W̅) and (y₀, w₀) = (E(Y), E(W)) in Eq. (12.2.7).To the level of approximation of Eq. (12.2.7), prove that

 Var(Z) = g₁(E(Y), E(W))²σ_yy + 2g₁(E(Y), E(W))g₂(E(Y ), E(W))σ_yw + g₂(E(Y), E(W))²σ_ww.

Use the formula for the variance of a linear combination of random variables derived in Sec. 4.6.

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