 Selection on noisy performance can produce apparent overconfidence

Let $X=A+\epsilon$ 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 $\mathrm{Cov}(S,X)={\sigma }_A^2$ and $\mathrm{Var}(X)={\sigma }_A^2+{\sigma }_{\epsilon }^2$, Gaussian conditional expectation gives $\mathbb{E}[S\mid X=x]=\mu +\lambda (x-\mu )$ with $\lambda ={\sigma }_A^2/({\sigma }_A^2+{\sigma }_{\epsilon }^2)$. 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.