Asymptotic behaviour of the posterior predictive p-value

Abstract

The posterior predictive p-value is a Bayesian-motivated alternative to the classical concept of p-value. This paper is devoted to study its asymptotic behaviour. Under mild assumptions, it is proved that:

a)When θ^*, the true value of the parameter, belongs to the null hypothesis, the distribution of the posterior predictive p-value converges to the uniform distribution over the interval (0,1).

b)When θ^* does not belong to the null hypothesis, the posterior predictive p-value converges almost surely to zero

Keywords:

• p-value
• posterior predictive p-value
• discrepancy variable
• asymptotic behaviour
• goodness of fit