Asymptotics of predictive distributions driven by sample means and variances

Abstract

Let

be the predictive distributions of a sequence

left parenthesis upper X 1 comma upper X 2 comma ellipsis right parenthesis ( X 1 , X 2 , … ) $(X_1,X_2,\ldots)$

of p -dimensional random vectors. Suppose

alpha Subscript n Baseline equals script upper N left parenthesis upper M Subscript n Baseline comma upper Q Subscript n Baseline right parenthesis α n = N ( M n , Q n ) $\alpha_n=\mathcal{N}(M_n,Q_n)$

, where

upper M Subscript n Baseline equals left parenthesis 1 divided by n right parenthesis sigma summation Underscript i equals 1 Overscript n Endscripts upper X Subscript i M n = ( 1 / n ) ∑ i = 1 n X i $M_n=({1}/{n})\sum_{i=1}^nX_i$

and

upper Q Subscript n Baseline equals left parenthesis 1 divided by n right parenthesis sigma summation Underscript i equals 1 Overscript n Endscripts left parenthesis upper X Subscript i Baseline minus upper M Subscript n Baseline right parenthesis left parenthesis upper X Subscript i Baseline minus upper M Subscript n Baseline right parenthesis Superscript down tack Q n = ( 1 / n ) ∑ i = 1 n ( X i − M n ) ( X i − M n ) ⊤ $Q_n=({1}/{n})\sum_{i=1}^n(X_i-M_n)(X_i-M_n)^{\top}$

. Then there is a random probability measure

alpha α $\alpha$

on the Borel subsets of

double struck upper R Superscript p R p $\mathbb{R}^p$

such that

StartMetric alpha Subscript n Baseline minus alpha EndMetric long right arrow Overscript normal a period normal s period Endscripts 0 ‖ α n − α ‖ ⟶ a . s . 0 $\lVert\alpha_n-\alpha\rVert\overset{\mathrm{a.s.}}\longrightarrow 0$

, where

StartMetric dot EndMetric ‖ ⋅ ‖ $\lVert{\cdot}\rVert$

is the total variation distance. An explicit expression for

alpha α $\alpha$

is provided and the convergence rate of

StartMetric alpha Subscript n Baseline minus alpha EndMetric ‖ α n − α ‖ $\lVert\alpha_n-\alpha\rVert$

is shown to be arbitrarily close to

n Superscript negative 1 divided by 2 n − 1 / 2 $n^{-1/2}$

. Moreover, it is still true that

StartMetric alpha Subscript n Baseline minus alpha EndMetric long right arrow Overscript normal a period normal s period Endscripts 0 ‖ α n − α ‖ ⟶ a . s . 0 $\lVert\alpha_n-\alpha\rVert\overset{\mathrm{a.s.}}\longrightarrow 0$

even if

alpha Subscript n Baseline equals script upper L left parenthesis upper M Subscript n Baseline comma upper Q Subscript n Baseline right parenthesis α n = L ( M n , Q n ) $\alpha_n=\mathcal{L}(M_n,Q_n)$

, where

script upper L L $\mathcal{L}$

belongs to a class of distributions much larger than the normal. The predictives

alpha Subscript n α n $\alpha_n$

are useful in various frameworks, including Bayesian predictive inference and predictive resampling. Finally, the asymptotic behavior of copula-based predictive distributions is investigated and a numerical experiment is performed.