A surveyor measures an angle with two instruments; call them A and B. Both instr

Question

A surveyor measures an angle with two instruments; call them A and B. Both instruments produced unbiased measurements, but the standard deviation of measurements made with instrument A is twice that of B. Due to environmental effects (wind, thermal expansion, etc) simultaneous measurements made with the two instruments have a correlation of r=0.5.

        Let X and Y be random variables representing simultaneous measurements of a bearing made with instruments A and B, respectively.  You are going to use a linear combination of the two measurements as your estimator (call it S) of the true bearing (call it s):

S=aX+bY

Where a and b are predetermined weights.

(a)What conditions must be placed on the weights a and b so that E[S]=s

That is, so that S is an unbiased estimator of s?

(b)What values of a and b will minimize the variance of S? The answer to this question will take the from a equals a specific number and b equals a specific number.

Explanation / Answer

1) For E(s) to be unbiased estimator of s, a+b = 1

2) If SD of B is d, SD of A is 2D

V(a) = 4d^2 , V(b) = d^2

V(s) = a^2*V(a) + b^2*V(b) + 2*a*b*2d*d*0.5 = d^2 * (4a^2 + b^2 + 2ab)

Putting b = 1-a, V(s) becomes d^2 * (3a^2 + 1)

For minimum variance, dV(s)/da = 0, which gives a = 0.

Therefore a=0 and b=1 for minimum variance.

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