Calibrate thresholds for a misspecification test based on non-degenerate beliefs under rule (ER)

Determine finite-sample calibration of the total variation threshold ε and the observation count n for the hypothesis test that rejects correct specification of the candidate family of processes 𝒫 when min_{θ∈Θ} ||q_n − δ_θ||_{TV} > ε, where q_n is the belief produced by the conservative updating rule (ER). The calibration must explicitly balance Type I and Type II error probabilities under the identification condition that p_θ ≠ p_θ′ for θ ≠ θ′.

Background

The paper proposes using the long-run behavior of the belief process under the conservative updating rule (ER) as a diagnostic for model misspecification. If the belief does not converge to a point mass on a single parameter, the model must be misspecified (assuming identification). This motivates a test that rejects correct specification when the belief remains sufficiently far from any degenerate distribution.

The authors suggest operationalizing this idea with a threshold on total variation distance, rejecting when min_{θ∈Θ} ||q_n − δθ||{TV} > ε for a given sample size n. They note that specifying ε and n requires trading off Type I and II errors and explicitly state that this calibration is an open question.