TY - JOUR

T1 - Truncated linear estimation of a bounded multivariate normal mean

AU - Kortbi, Othmane

AU - Marchand, Éric

N1 - Funding Information:
We are grateful to two anonymous referees for several useful comments and suggestions. The research work of Éric Marchand is partially supported by NSERC of Canada . During Othmane Kortbi's Ph.D. studies at the Université de Sherbrooke, he benefited from financial support from several sources but he wishes to thank especially the ISM (Institut de sciences mathématiques) and the CRM (Centre de recherches mathématiques).

PY - 2012/9

Y1 - 2012/9

N2 - We consider the problem of estimating the mean θ of an N p(θ, I p) distribution with squared error loss ∥δ-θ∥ 2 and under the constraint ∥θ∥≤m, for some constant m>0. Using Stein's identity to obtain unbiased estimates of risk, Karlin's sign change arguments, and conditional risk analysis, we compare the risk performance of truncated linear estimators with that of the maximum likelihood estimator δ mle. We obtain for fixed (m, p) sufficient conditions for dominance. An asymptotic framework is developed, where we demonstrate that the truncated linear minimax estimator dominates δ mle, and where we obtain simple and accurate measures of relative improvement in risk. Numerical evaluations illustrate the effectiveness of the asymptotic framework for approximating the risks for moderate or large values of p.

AB - We consider the problem of estimating the mean θ of an N p(θ, I p) distribution with squared error loss ∥δ-θ∥ 2 and under the constraint ∥θ∥≤m, for some constant m>0. Using Stein's identity to obtain unbiased estimates of risk, Karlin's sign change arguments, and conditional risk analysis, we compare the risk performance of truncated linear estimators with that of the maximum likelihood estimator δ mle. We obtain for fixed (m, p) sufficient conditions for dominance. An asymptotic framework is developed, where we demonstrate that the truncated linear minimax estimator dominates δ mle, and where we obtain simple and accurate measures of relative improvement in risk. Numerical evaluations illustrate the effectiveness of the asymptotic framework for approximating the risks for moderate or large values of p.

KW - Asymptotic analysis

KW - Dominance

KW - Maximum likelihood

KW - Multivariate normal

KW - Point estimation

KW - Restricted parameters

KW - Squared error loss

KW - Truncated linear estimators

KW - Truncated linear minimax

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U2 - 10.1016/j.jspi.2012.03.022

DO - 10.1016/j.jspi.2012.03.022

M3 - Article

AN - SCOPUS:84860884342

VL - 142

SP - 2607

EP - 2618

JO - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

SN - 0378-3758

IS - 9

ER -