Predictive distributions driven by sample means, variances and clustering

Several statistical procedures are based on predictive distributions. Some examples are Bayesian predictive inference, machine learning, species sampling sequences, the prequential approach, causal inference and predictive resampling. This thesis introduces a novel idea for building predictive distributions, where the new datum depends on past observations only through their shape, location and dispersion. A sequence of predictives (beta_n) is proposed, where beta_n depends only on the mean and covariance matrix of past data. Then, beta_n is generalised to another predictive gamma_n, which has a mixture structure based on the clustering of past data. The updating rule of gamma_n is inspired by the Pòlya urns scheme. In addition, gamma_n is used to construct a sequence (r_n) of predictives, which are suitable for regression problems in a Bayesian framework. The beta_n are shown to converge a.s. in total variation to a random law known in closed form. When beta_n is Gaussian, the convergence rate is arbitrarily close to n^{-1/2}. The gamma_n converge in total variation too. Their limit is known explicitly and it has an interesting statistical interpretation. Similar results hold for r_n. Furthermore, the asymptotic behaviour of the copula-based predictive distributions (Hahn et al. 2018) is investigated, as they are a popular predictive model. Sufficient conditions are provided for them to converge in total variation. Various experiments of Bayesian inference via predictive resampling are carried out. It turns out beta_n estimates well population mean and variance, while gamma_n captures also moments of higher order and quantiles. Moreover, gamma_n is compared to two natural competitors: the copula-based predictives and the predictives of Cui and Walker (2024). They all provide precise posterior estimates, but predictive resampling is faster with gamma_n. Finally, r_n is employed for the estimation of regression coefficients. Satisfying outcomes are obtained for different types of data, including non-linear and heteroscedastic ones.