In various frameworks, to assess the joint distribution of a $k$-dimensional random vector $X=(X_1,\ldots,X_k)$, one selects some putative conditional distributions $Q_1,\ldots,Q_k$. Each $Q_i$ is regarded as a possible (or putative) conditional distribution for $X_i$ given $(X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_k)$. The $Q_i$ are compatible if there is a joint distribution $P$ for $X$ with conditionals $Q_1,\ldots,Q_k$. Three types of compatibility results are given in this paper. First, the $X_i$ are assumed to take values in compact subsets of $\mathbb{R}$. Second, the $Q_i$ are supposed to have densities with respect to reference measures. Third, a stronger form of compatibility is investigated. The law $P$ with conditionals $Q_1,\ldots,Q_k$ is requested to belong to some given class $\mathcal{P}_0$ of distributions. Two choices for $\mathcal{P}_0$ are considered, that is, $\mathcal{P}_0=\{$exchangeable laws$\}$ and $\mathcal{P}_0=\{$laws with identical univariate marginals$\}$.

Compatibility results for conditional distributions

RIGO, PIETRO
2014-01-01

Abstract

In various frameworks, to assess the joint distribution of a $k$-dimensional random vector $X=(X_1,\ldots,X_k)$, one selects some putative conditional distributions $Q_1,\ldots,Q_k$. Each $Q_i$ is regarded as a possible (or putative) conditional distribution for $X_i$ given $(X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_k)$. The $Q_i$ are compatible if there is a joint distribution $P$ for $X$ with conditionals $Q_1,\ldots,Q_k$. Three types of compatibility results are given in this paper. First, the $X_i$ are assumed to take values in compact subsets of $\mathbb{R}$. Second, the $Q_i$ are supposed to have densities with respect to reference measures. Third, a stronger form of compatibility is investigated. The law $P$ with conditionals $Q_1,\ldots,Q_k$ is requested to belong to some given class $\mathcal{P}_0$ of distributions. Two choices for $\mathcal{P}_0$ are considered, that is, $\mathcal{P}_0=\{$exchangeable laws$\}$ and $\mathcal{P}_0=\{$laws with identical univariate marginals$\}$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11571/784231
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