In size-n random sampling from a bivariate population f(x, y), suppose that the

Question

In size-n random sampling from a bivariate population f(x, y), suppose that the objective is to estimate the parameter Ax For example, the population may consist of married couples, with Y = husband's earnings and X - wife's earnings. The sample statistics X, Y, s2, s , and SXY are available. (a) Propose a statistic T that is an unbiased estimator of . Show that it is unbiased. (b) Find its variance V(T) in terms of the population variances and covariance of X and Y. (c) For the practical case, in which those population variances and covariance are unknown, propose an unbiased estimator of V(T). Show that it is unbiased. (d) What statistic would you report in practice as a standard erroir for T?

Explanation / Answer

Part (a)

Given = µY - µX, let T = Ybar – Xbar. Then T is an unbiased estimator of . ANSWER

Proof: E(T) = E(Ybar – Xbar) = E(Ybar) – E(Xbar) = µY - µX.

=> (Ybar – Xbar) is an unbiased estimator of . DONE

Part (b)

V(T) = V(Ybar – Xbar) = V(Ybar) + V(Xbar) – 2Cov(Ybar, Xbar) = (1/n)(12 + 22 - 212),

Where 12 , 22 and 12 are respectively variance of X, Y and Cov(X, Y).

Thus, V(T) = (12 + 22 - 212)/n ANSWER

Part (c)

E{(n - 1)SX2} = 12 , E{(n - 1)SY2} = 22 , and E{(n - 1)SXY} = 12 . So,

E[(n - 1)(SX2 + SY2 - 2SXY)] = (12 + 22 - 212)

=> E[{(n - 1)/n}(SX2 + SY2 - 2SXY)] = (12 + 22 - 2 12)/n

=> [{(n - 1)/n}(SX2 + SY2 - 2SXY)] is an unbiased estimator of V(T) ANSWER

Part (d)

From Part (b), V(T) = (1/n)(12 + 22 - 2 12)

=> SE(T) = sqrt{(1/n)(12 + 22 - 212)} ANSWER

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