Abstract:
We study the problem of estimating a high-dimensional vector $(\mu_1 , \ldots, \mu_n)$ based on independent $Y_i \sim N(\mu_i , 1)$, $i = 1, \ldots, n$, under a square loss. The idea is to try to imitate an `oracle' that knows the ordered vector $(\mu_{(1)}, \ldots, \mu_{(n)})$. We derive a novel method. The ratio of its risk with the risk obtained by the above oracle approaches 1 as $n$ approaches infinity. This is under conditions where the oracle does not have extreme advantage. Such an extreme advantage is, e.g., when $\mu_i = 0$, $i = 1, \ldots, n$, and when knowing the ordered vector is equivalent to knowing the vector itself. When the vector $(\mu_1 , \ldots, \mu_n)$ is not extremely sparse, the performance of our method in simulations is very good in comparison to existing methods.