Suppose that, to assess the joint distribution of a random vector \$(X_1,\ldots,X_n)\$, one selects the kernels \$Q_1,\ldots,Q_n\$ with \$Q_i\$ to be regarded as a possible conditional distribution for \$X_i\$ given \$(X_j:j\ne i)\$; \$Q_1,\ldots,Q_n\$ are compatible if there exists a joint distribution for \$(X_1,\ldots,X_n)\$ with conditionals \$Q_1,\ldots,Q_n\$. Similarly, \$Q_1,\ldots,Q_n\$ are improperly compatible if they can be obtained, according to the usual rule, with an improper distribution in place of a probability distribution. In this paper, compatibility and improper compatibility of \$Q_1,\ldots,Q_n\$ are characterized under some assumptions on their functional form. The characterization applies, in particular, if each \$Q_i\$ belongs to a one parameter exponential family. Special attention is paid to Gaussian conditional autoregressive models.

### A note on compatibility of conditional autoregressive models

#### Abstract

Suppose that, to assess the joint distribution of a random vector \$(X_1,\ldots,X_n)\$, one selects the kernels \$Q_1,\ldots,Q_n\$ with \$Q_i\$ to be regarded as a possible conditional distribution for \$X_i\$ given \$(X_j:j\ne i)\$; \$Q_1,\ldots,Q_n\$ are compatible if there exists a joint distribution for \$(X_1,\ldots,X_n)\$ with conditionals \$Q_1,\ldots,Q_n\$. Similarly, \$Q_1,\ldots,Q_n\$ are improperly compatible if they can be obtained, according to the usual rule, with an improper distribution in place of a probability distribution. In this paper, compatibility and improper compatibility of \$Q_1,\ldots,Q_n\$ are characterized under some assumptions on their functional form. The characterization applies, in particular, if each \$Q_i\$ belongs to a one parameter exponential family. Special attention is paid to Gaussian conditional autoregressive models.
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2017
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11571/1172727`
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