Propriedades lógicas de classes de testes de hipóteses

Abstract

When performing simultaneous hypotheses testing is expected that the decisions obtained therein are logically consistent with each other. In this work, we find restrictions under which simultaneous Bayes tests meet logical conditions separately or jointly. It is shown that the conditions for the simultaneous tests meet these conditions alone are quite intuitive. However, when trying to obey the conditions jointly, we lose optimality. Furthermore, we evaluate the relationship between these tests and simultaneous Bayes tests generated by estimators, ie, we show that, under some conditions, to choose an estimator based on Bayes decision is equivalent to choosing a decision based on a Bayes test. Finally, we show that if we take a decision based on Maximum Likelihood Estimators, then that decision should be equal to taking a Bayes test and concluded that these decisions are admissible and obey the Likelihood Principle.

A Bayesian Decision-Theoretic Approach to Logically-Consistent Hypothesis Testing

“…and µ I x pθq ¡ 0, for all θ Θ, together with the family of 0-1 loss functions, one will obtain a Bayesian TS that coincides with ϕ, as well (see [14] for the details).…”

“…This example is further discussed in [8,13]. Several other (in)coherent testing schemes are explored by [8,14]. Coherence is by far the most emphasized logical requisite for simultaneous test procedures in the literature.…”

“…It is easy to verify that B T pAq tTpα 1 , α 2 , α 3 q : pα 1 , α 2 , α 3 q Au is a convex set. Indeed, B is a triangle (see Figure 2) with vertices P 1 T p1, 0, 0q p∆ A 1 pθ 1 q, ∆ A 2 pθ 1 q, ∆ A 3 pθ 1 qq, P 2 T p0, 1, 0q p∆ A 1 pθ 2 q, ∆ A 2 pθ 2 q, ∆ A 3 pθ 2 qq and P 3 T p0, 0, 1q p∆ A 1 pθ 3 q, ∆ A 2 pθ 3 q, ∆ A 3 pθ 3 qq (these points are not aligned owing to the restrictions on the quantities ∆ A i pθ j q, [14]). Now, we turn to the main argument of the proof.…”

“…that is, µ x is the probability measure degenerate at the point δpxq [14]. Furthermore, let µ 0 be any probability measure defined on PpX q.…”