= Selection on noisy performance can produce apparent overconfidence
{title2=$\mathbb E[S\mid X=x]-x=(1-\lambda)(\mu-x)$}
Let $X=A+\varepsilon$ and $S=A+\eta$, where latent score $A$ has <normal distribution> with mean $\mu$ and variance $\sigma_A^2$, and the centred normal errors are independent of each other and $A$. Since $\operatorname{Cov}(S,X)=\sigma_A^2$ and $\operatorname{Var}(X)=\sigma_A^2+\sigma_\varepsilon^2$, <Gaussian conditional expectation> gives $\mathbb E[S\mid X=x]=\mu+\lambda(x-\mu)$ with $\lambda=\sigma_A^2/(\sigma_A^2+\sigma_\varepsilon^2)$. Subtracting $x$ proves the formula. For $0<\lambda<1$, selecting scores below the mean creates positive expected self-assessment errors relative to those observed scores. This is a mathematical possibility, not an empirical estimate of any particular study's contribution from measurement error.
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