Adaptive significance levels in linear regression models

Abstract

The Full Bayesian Significance Test (FBST) for precise hypotheses is presented by Pereira and Stern (1999) as a Bayesian alternative to the traditional significance tests based on p-values. With the FBST the authors introduce the e-value as an evidence index in favor of the null hypothesis (H). An important practical issue for the implementation of the FBST is to establish how small the evidence against H must be in order to decide for its rejection. In this work we present a method to find a cutoff value for the evidence in the FBST by minimizing the linear combination of the averaged type-I and type-II error probabilities for a given sample size and also for a given dimensionality of the parameter space. Furthermore, we compare our methodology with the results obtained from the test proposed by Pereira et al. (2017) and Gannon et al. (2019) which presents the P-value as a decisionmaking evidence measure and includes an adaptive significance level. For that purpose, the scenario of linear regression models under the Bayesian approach is considered.

O método de ponderação bayesiana de modelos para seleção de modelos

“…𝑓 (𝑦|𝛽, 𝜎 2 , 𝑥) = (2𝜋𝜎 2 ) − 𝑛 2 𝑒𝑥𝑝{− 1 2𝜎 2 (𝑦 − 𝑥𝛽) 𝑇 (𝑦 − 𝑥𝛽)} No caso, a variância de 𝜖 é 𝜎 2 desconhecida (Murphy, 2007;Hoyos, 2020). Suponhamos que:…”

“…𝜎 2 ∼ 𝐼 𝐺(𝑎 0 , 𝑏 0 ), isto é, 𝑔(𝜎 2 ) = (𝑏 0 ) 𝑎 0 Γ(𝑎 0 ) (𝜎 2 ) −(𝑎 0 +1) exp{− 𝑏 0 𝜎 2 }𝕀 𝑅 + (𝜎 2 ), e 𝛽|𝜎 2 ∼ 𝑁 𝑝 (𝑚 0 , 𝜎 2 𝑉 0 ), Supondo que 𝑋 𝑇 𝑋 seja uma matriz não singular, temos ainda que 𝑚 * = 𝑉 * (𝑉 −1 0 𝑚 0 + 𝑋 𝑇 𝑋 β), onde β é o vetor de estimativa da 𝛽. Ou seja, a distribuição posteriori de (𝛽, 𝜎 2 ) é Normal Gama Inversa (Hoyos, 2020), isto é, 𝛽, 𝜎 2 |𝑦, 𝑥 ∼ 𝑁 𝑝 𝐼 𝐺(𝑚 * , 𝑉 * , 𝑎1, 𝑏1)…”