Shrinking a noisy return estimate toward zero

Question

You estimate a signal's true mean \mu with a noisy measurement x of variance \sigma^2 = 16. Absent evidence, your anchor is zero, so you shrink toward b = 0: \hat\mu = w\,x + (1-w)\cdot 0 = w\,x. Suppose the true mean is \mu = 6. What weight w minimizes the mean squared error?

Accepted Answer

The idea. Multiplying your estimate by w < 1 pulls it toward zero. That shrinks the noise but introduces a bias, because the truth is not actually zero. Balance the two. Write the MSE. With anchor b = 0, the distance is d = \mu - 0 = 6. The estimate w\,x has variance w^2\sigma^2 and bias (1-w)\,d, so 	ext{MSE}(w) = w^2 \sigma^2 + (1-w)^2 d^2. Minimize. w^\star = \frac{d^2}{d^2 + \sigma^2} = \frac{36}{36 + 16} = \frac{36}{52} = \frac{9}{13} \approx 0.692. Keep about 69 percent of your raw estimate and let the rest decay toward zero. Check the payoff. The optimal MSE is \frac{\sigma^2 d^2}{\si…

Shrinking a noisy return estimate toward zero

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