On efficient prediction and predictive density estimation for spherically symmetric models

Abstract : Let X, U, Y be spherically symmetric distributed having density η d+k/2 f η(x − θ| 2 + u 2 + y − cθ 2) , with unknown parameters θ ∈ R d and η > 0, and with known density f and constant c > 0. Based on observing X = x, U = u, we consider the problem of obtaining a predictive densitŷ q(y; x, u) for Y as measured by the expected Kullback-Leibler loss. A benchmark procedure is the minimum risk equivariant densityq mre , which is Generalized Bayes with respect to the prior π(θ, η) = η −1. For d ≥ 3, we obtain improvements onq mre , and further show that the dominance holds simultaneously for all f subject to finite moments and finite risk conditions. We also obtain that the Bayes predictive density with respect to the harmonic prior π h (θ, η) = η −1 θ 2−d dominatesq mre simultaneously for all scale mixture of normals f. The results hinges on duality with a point prediction problem, as well as posterior representations for (θ, η), which are of interest on their own. Namely, we obtain for d ≥ 3, point predictors δ(X, U) of Y that dominate the benchmark predictor cX simultaneously for all f , and simultaneously for risk functions E f ρ (Y − δ(X, U) 2 + (1 + c 2) U 2) , with ρ increasing and concave on R + , and including the squared error case E f (Y − δ(X, U) 2
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Dominique Fourdrinier, Eric Marchand, William Strawderman. On efficient prediction and predictive density estimation for spherically symmetric models. Journal of Multivariate Analysis, Elsevier, In press. ⟨hal-02335112⟩

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