The value of a population mean increases linearly through time: µ(t) = a + ßt while the variance…

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The value of a population mean increases linearly through time: μ(t) = α + βt while the variance remains constant. Independent simple random samples of size n are taken at times t = 1, 2, and 3.

a. Find conditions on w1, w2, and w3 such that

is an unbiased estimate of the rate of change, β. Here Xi denotes the sample mean at time ti .

b. What values of the  minimize the variance subject to the constraint that the estimate is unbiased?