Asymptotics of predictive distributions driven by sample means and variances

Abstract

Let $\hat{I}{\pm }_n(\cdot )=P(X_{n+1}\in \cdot \mid X_1,\dots ,X_n)$ be the predictive distributions of a sequence $(X_1,X_2,\dots )$ of $p$-dimensional random vectors. Suppose $$\hat{I}{\pm }_n={\mathcal{N}}_p(M_n,Q_n)$$ where $M_n=\frac{1}{n}{\sum }_{i=1}^n{X}_i$ and $Q_n=\frac{1}{n}{\sum }_{i=1}^n(X_i-M_n)(X_i-M_n{)}^t$. Then, there is a random probability measure $\hat{I}\pm$ on the Borel subsets of ${\mathbb{R}}^p$ such that $\Vert \hat{I}{\pm }_n-\hat{I}\pm \Vert \stackrel{a.s.}{⟶}0$ where $\Vert \cdot \Vert$ is total variation distance. An explicit expression for $\hat{I}\pm$ is provided and the convergence rate of $\Vert \hat{I}{\pm }_n-\hat{I}\pm \Vert$ is shown to be arbitrarily close to $n^{-1/2}$. Moreover, it is still true that $\Vert \hat{I}{\pm }_n-\hat{I}\pm \Vert \stackrel{a.s.}{⟶}0$ even if $\hat{I}{\pm }_n=\mathcal{L}(M_n,Q_n)$ where $\mathcal{L}$ belongs to a class of distributions much larger than the normal. The predictives $\hat{I}{\pm }_n$ are useful in various frameworks, including Bayesian predictive inference and predictive resampling. Finally, the asymptotic behavior of copula-based predictive distributions (introduced in [13]) is investigated and a numerical experiment is performed.