## Abstract

Assume X = (X_{1}, ..., X_{p})′ is a normal mixture distribution with density w.r.t. Lebesgue measure, f(x)=∫ 1 (2π)^{p/2}δ{divides}σ{divides}^{1/2}e ^{-(x-θ)′(σ-1(x-θ))/2σ 2} dF(σ), where Σ is a known positive definite matrix and F is any known c.d.f. on (0, ∞). Estimation of the mean vector under an arbitrary known quadratic loss function L_{Q}(θ, a) = (a - θ)′ Q(a - θ), Q a positive definite matrix, is considered. An unbiased estimator of risk is obatined for an arbitrary estimator, and a sufficient condition for estimators to be minimax is then achieved. The result is applied to modifying all the Stein estimators for the means of independent normal random variables to be minimax estimators for the problem considered here. In particular the results apply to the Stein class of limited translation estimators.

Original language | English (US) |
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Pages (from-to) | 141-150 |

Number of pages | 10 |

Journal | Journal of Multivariate Analysis |

Volume | 35 |

Issue number | 2 |

DOIs | |

State | Published - Nov 1990 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty

## Keywords

- minimax estimator
- multivariate normal mixture mean
- quadratic loss