Premium
MINIMAX ESTIMATION OF A MULTIVARIATE NORMAL MEAN UNDER A CONVEX LOSS FUNCTION
Author(s) -
Lin PiErh,
Mousa Amany
Publication year - 1983
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1983.tb01218.x
Subject(s) - combinatorics , minimax , mathematics , regular polygon , function (biology) , convex function , prime (order theory) , mathematical economics , geometry , evolutionary biology , biology
Summary Let X = (X 1 , ‐ X p )prime; ˜ N p (μ, Σ) where μ= (μ 1 , ‐, μ p )′ and Σ= diag (Σ 2 1 , ‐, Σ 2 p ) are both unknown and p3. Let (n i ‐ 2) w i /Σ 2 i ! X 2 ni , independent. of w i (I ≠ j = 1, ‐, p). Assume that (w 1 , ‐, w p ) and X are independent. Define W = diag (w 1 , ‐, w p ) and ¶ X ¶ 2 w = X′W ‐1 Q ‐1 W ‐1 X where Q = diag (q 1 , ‐,n q p ), q i > 0, i = 1, ‐, p. In this paper, the minimax estimator of Berger & Bock (1976), given by δ (X, W) = [I p ‐ r(X, W) ¶ X ¶ ‐2 w Q ‐1 W ‐1 ] X, is shown to be minimax relative to the convex loss (δ ‐ μ)′[αQ + (1 ‐ α) Σ ‐1 ] δ ‐ μ)/C, where C =α tr (Σ) + (1 ‐ α)p and 0 α 1, under certain conditions on r(X, W). This generalizes the above mentioned result of Berger & Bock.