Asymptotics of predictive distributions driven by sample means and variances
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| Format: | Preprint |
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2024
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| _version_ | 1866917775372451840 |
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| author | Garelli, Samuele Leisen, Fabrizio Pratelli, Luca Rigo, Pietro |
| author_facet | Garelli, Samuele Leisen, Fabrizio Pratelli, Luca Rigo, Pietro |
| contents | Let $α_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)$ be the predictive distributions of a sequence $(X_1,X_2,\ldots)$ of $p$-dimensional random vectors. Suppose $$α_n= \mathcal{N} _p (M_n,Q_n)$$ where $M_n=\frac{1}{n}\sum_{i=1}^nX_i$ 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 $α$ on the Borel subsets of $\mathbb{R}^p$ such that $\lVertα_n-α\rVert\overset{a.s.}\longrightarrow 0$ where $\lVert\cdot\rVert$ is total variation distance. An explicit expression for $α$ is provided and the convergence rate of $\lVertα_n-α\rVert$ is shown to be arbitrarily close to $n^{-1/2}$. Moreover, it is still true that $\lVertα_n-α\rVert\overset{a.s.}\longrightarrow 0$ even if $α_n=\mathcal{L}(M_n,Q_n)$ where $\mathcal{L}$ belongs to a class of distributions much larger than the normal. The predictives $α_n$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16828 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotics of predictive distributions driven by sample means and variances Garelli, Samuele Leisen, Fabrizio Pratelli, Luca Rigo, Pietro Statistics Theory Methodology Let $α_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)$ be the predictive distributions of a sequence $(X_1,X_2,\ldots)$ of $p$-dimensional random vectors. Suppose $$α_n= \mathcal{N} _p (M_n,Q_n)$$ where $M_n=\frac{1}{n}\sum_{i=1}^nX_i$ 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 $α$ on the Borel subsets of $\mathbb{R}^p$ such that $\lVertα_n-α\rVert\overset{a.s.}\longrightarrow 0$ where $\lVert\cdot\rVert$ is total variation distance. An explicit expression for $α$ is provided and the convergence rate of $\lVertα_n-α\rVert$ is shown to be arbitrarily close to $n^{-1/2}$. Moreover, it is still true that $\lVertα_n-α\rVert\overset{a.s.}\longrightarrow 0$ even if $α_n=\mathcal{L}(M_n,Q_n)$ where $\mathcal{L}$ belongs to a class of distributions much larger than the normal. The predictives $α_n$ 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. |
| title | Asymptotics of predictive distributions driven by sample means and variances |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2403.16828 |