In this paper we present a direct and simple approach to obtain bounds on the asymptotic minimax risk for the estimation of constrained binomial and multinomial proportions. Quadratic, normalized quadratic and entropy loss are considered and it is demonstrated that in all cases linear estimators are asymptotically minimax optimal. For the quadratic loss function the asymptotic minimax risk does...

This paper considers simultaneous estimation of multivariate normal mean vector using Zellner's(1994) balanced loss function when 2 is known and unknown. We show that the usual estimator X is minimax and obtain a class of minimax estimators which have uniformly smaller risk than the usual estimator X. Also, we obtain the proper Bayes estimator relative to balanced loss function and nd the minim...

We investigate a state estimation problem for the dynamical system described by uncertain linear operator equation in Hilbert space. The uncertainty is supposed to admit a set-membership description. We present explicit expressions for linear minimax estimation and error provided that any pair of uncertain parameters belongs to the quadratic bounding set. We introduce a new notion of minimax di...

Cai et al. (2010) [4] have studied the minimax optimal estimation of a collection of large bandable covariance matrices whose off-diagonal entries decay to zero at a polynomial rate. They have shown that the minimax optimal procedures are fundamentally different under Frobenius and spectral norms, regardless of the rate of polynomial decay. To gain more insight into this interesting problem, we...

The bounded normal mean problem has important applications in nonparametric function estimation. It is to estimate the mean of a normal distribution whose mean is restricted to a bounded interval. The minimax risk for such a problem is generally unknown. It is shown in Donoho, Liu and MacGibbon(1990) that the linear minimax risk provides a good approximation to the minimax risk. We show in this...

In this paper, we address the problem of regression estimation in the context of a p-dimensional predictor when p is large. We propose a general model in which the regression function is a composite function. Our model consists in a nonlinear extension of the usual sufficient dimension reduction setting. The strategy followed for estimating the regression function is based on the estimation of ...

We provide a complete picture of asymptotically minimax estimation of Lr-norms (for any r ≥ 1) of the mean in Gaussian white noise model over Nikolskii-Besov spaces. In this regard, we complement the work of Lepski, Nemirovski and Spokoiny (1999), who considered the cases of r = 1 (with poly-logarithmic gap between upper and lower bounds) and r even (with asymptotically sharp upper and lower bo...

We investigate the problem of continuous-time causal estimation under a minimax criterion. Let X = {Xt, 0 ≤ t ≤ T} be governed by the probability law Pθ from a class of possible laws indexed by θ ∈ Λ, and Y T be the noise corrupted observations of X available to the estimator. We characterize the estimator minimizing the worst case regret, where regret is the difference between the causal estim...

This paper considers point and interval estimation of the lq loss of an estimator in high-dimensional linear regression with random design. We establish the minimax rate for estimating the lq loss and the minimax expected length of confidence intervals for the lq loss of rate-optimal estimators of the regression vector, including commonly used estimators such as Lasso, scaled Lasso, square-root...