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Asymptotics of predictive distributions driven by sample means and variances

Abstract

Let αn()=P(Xn+1X1,,Xn)\alpha_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr) be the predictive distributions of a sequence (X1,X2,)(X_1,X_2,\ldots) of pp-variate random variables. Suppose \alpha_n=\mathcal{N}_p(M_n,Q_n) where Mn=1ni=1nXiM_n=\frac{1}{n}\sum_{i=1}^nX_i and Qn=1ni=1n(XiMn)(XiMn)tQ_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\alpha on Rp\mathbb{R}^p such that αnα\alpha_n\rightarrow\alpha weakly a.s. If p{1,2}p\in\{1,2\}, one also obtains αnαa.s.0\lVert\alpha_n-\alpha\rVert\overset{a.s.}\longrightarrow 0 where \lVert\cdot\rVert is total variation distance. Moreover, the convergence rate of αnα\lVert\alpha_n-\alpha\rVert is arbitrarily close to n1/2n^{-1/2}. These results (apart from the one regarding the convergence rate) still apply even if αn=Lp(Mn,Qn)\alpha_n=\mathcal{L}_p(M_n,Q_n), where Lp\mathcal{L}_p belongs to a class of distributions much larger than the normal. Finally, the asymptotic behavior of copula-based predictive distributions (introduced in [13]) is investigated and a numerical experiment is performed.

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