Affine Equivariant Estimator Research Articles

Improved estimation of a covariance matrix under quadratic loss

For the quadratic loss function, it is shown that the best affine equivariant estimator of the normal covariance matrix is improved on by Stein-type estimators.

Entropy loss and risk of improved estimators for the generalized variance and precision

Let the distributions of X(p×r) and S(p×p) be N(ζ, Σ⊗I r) and W p(n, Σ) respectively and let them be independent. The risk of the improved estimator for |Σ| or {ei329-1} based on X and S under entropy loss (=d/|Σ| −log(d/|Σ|)−1 or d|Σ|−log(d|Σ|)−1) is evaluated in terms of incomplete beta function of matrix argument and its derivative. Numerical comparison for the reduction of risk over the best affine equivariant estimator is given.

Decision-theoretic estimation of generalized variance and generalized precision

Consider a random data matrix X=(X1,...,Xk):pXk with independent columns [sathik] and an independent p X p Wishart matrix [sathik]. Estimators dominating the best affine equivariant estimators of [sathik] are obtained under four types of loss functions. Improved estimators (Testimators) of generalized variance and generalized precision are also considered under convex entropy loss (CEL).

On improved estimators of the generalized variance

Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix S: p × p ∼ W p ( n, Σ) and an independent normal random matrix X: p × k ∼ N( ξ, Σ ⊗ I k ) with ξ( p × k) unknown. Denote the columns of X by X (1) ,…, X ( k) and set ψ (0)(S, X) = { (n − p + 2)! (n + 2)! } | S | , ψ (i)(X, X) = min[ψ (i−1)(S, X), { (n − p + i + 2)! (n + i + 2)! } | S + X (1) X′ (1) + ⋯ + X (i) X′ (i) |] and Ψ ( i) ( S, X) = min[ ψ (0)( S, X), { (n − p + i + 2)! (n + i + 2)! }| S + X (1) X′ (1) + ⋯ + X (i) X′ (i) |] , i = 1,…, k. Our result is that the minimax, best affine equivariant estimator ψ (0)( S, X) is dominated by each of Ψ ( i) ( S, X), i = 1,…, k and for every i, ψ ( i) ( S, X) is better than ψ ( i−1) ( S, X). In particular, ψ (k)(S, X) = min[{ (n − p + 2)! (n + 2)! } | S |, { (n − p + 2)! (n + 2)! } | S + X (1)X′ (1)|,…,| { (n − p + k + 2)! (n + k + 2)! } | S + X (1)X′ (1) + ⋯ + X (k)X′ (k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result ( Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance.

An Improved Estimator of the Generalized Variance

A multivariate extension is made of Stein's result (1964) on the estimation of the normal variance. Here the generalized variance $|\Sigma|$ is being estimated from a Wishart random matrix $S: p \times p \sim W(n, \Sigma)$ and an independent normal random matrix $X: p \times k \sim N(\xi, \Sigma \otimes 1_k)$ with $\xi$ unknown. The main result is that the minimax, best affine equivariant estimator $((n + 2 - p)!/(n + 2)!)|S|$ is dominated by $\min\{((n + 2 - p)!/(n + 2)!)|S|, ((n + k + 2 - p)!/(n + k + 2)!)|S + XX'|\}$. It is obtained by a variant of Stein's method which exploits zonal polynomials.

Estimating the Scale Parameter of the Exponential Distribution with Unknown Location

Let $X_1,\cdots, X_n$ be i.i.d. random variables each with a density which is $a^{-1} \exp (b - x)$ or 0 according as $x \geqq b$ or $x < b$ where $-\infty < b < \infty, a > 0$ are unknown constants. Let $\bar{X} = n^{-1} \sum X_i$ and $M = \min X_i$. The maximum likelihood estimator of $a$ is $\bar{X} - M$ and is, if quadratic loss is assumed, the best affine equivariant estimator of $a$. It is shown that if loss is measured by any member of a large class of "bowl-shaped" functions which includes quadratic loss, the best affine equivariant estimator is inadmissible. The proof entails an examination of the conditional expected loss given the maximal invariant under the scale group. It is carried out by exhibiting a superior alternative. In the case of quadratic loss, for example, the result is as follows. Given any estimator $u = (\bar{X} - M)T\lbrack M(\bar{X} - M)^{-1}\rbrack$, let $T^\ast(y) = T(y)$ if $y < 0$ and $T^\ast(y) = \min \{T(y), n(n + 1)^{-1}(1 + y)\}$ if $y > 0$. If $T^\ast \neq T$ with positive probability, then the estimator, obtained from $u$ by replacing $T$ by $T^\ast$, has uniformly smaller risk than $u$. Using a generalization of the author's conditions for admissibility [Ann. Math. Statist. 41 (1970) 446-457] a class, $B$, of generalized Bayes estimators within $D$, the class of scale equivariant estimators, are obtained with each member of $B$ admissible in $D$. The improper measures determining members of $B$ have densities on the orbit space $R$, created in the parameter space by the action of the group of scale changes. These prior densities, $g$, satisfy $\int^\infty_1 (t^2g(t))^{-1} dt = \int^{-1}_{-\infty} (t^2g(t))^{-1} dt = \infty$.