Question

for (b): Use a Lagrange multiplier. To get a formula for the variance of Xc, use the bilinearity property of covariance and the formula for Cov(Xi , Xj ) (i 6= j) derived in class. When you have the sum of all cj such that j 6= k, you might find it useful to replace that sum by 1 − ck. (Explain.)

ForarandomsampleofsizenfromapopulationofsizeN,considerthefollowing as an estimate of μ: n Xc = wheretheci arefixednumbersandX1,...,Xn isthesample. a. Find a condition on the ci such that the estimate is unbiased. b. Showthatthechoiceofci thatminimizesthevariancesoftheestimatesubject to this condition is ci = 1/n, where i = 1,...,n.

Answer 1

Find a condition on the ci such that the estimate is unbiased

b)

Show that the choice of i that minimizes the variances of the estimate subject to this condition

hence we have to minimize sum c_i^2 with constraint that sum c_i = 1

which happens when c_i are equal and values are 1/n