Combining two noisy estimates of fair value

Question

Two independent, unbiased estimates of an asset's fair value land on your desk. Estimate $X_1$ has a standard deviation of $3$ ; estimate $X_2$ has a standard deviation of $4$ .

What single number best combines them, and how good is the combined estimate?

Accepted Answer

For independent unbiased estimates, the minimum-variance (and, under Gaussian noise, maximum-likelihood) combination weights each one by its precision, the reciprocal of its variance: \hat\mu = \frac{X1/\sigma1^2 + X2/\sigma2^2}{1/\sigma1^2 + 1/\sigma2^2}. Plug in \sigma1^2 = 9,\ \sigma2^2 = 16: \frac{1}{9} = 0.1111, \qquad \frac{1}{16} = 0.0625, \qquad 	ext{sum} = 0.1736. Weights: w1 = \frac{0.1111}{0.1736} = 0.64, \qquad w2 = \frac{0.0625}{0.1736} = 0.36. So \hat\mu = 0.64\,X1 + 0.36\,X2, the sharper estimate gets nearly twice the weight of the noisier one, even though \sigma1 and \sigma2 …

Combining two noisy estimates of fair value

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