  Given univariate random variables $Y_1, \ldots, Y_n$ with the
$\text{Uniform}(\theta_0 - 1, \theta_0 + 1)$ distribution, the sample midrange
$\frac{Y_{(n)}+Y_{(1)}}{2}$ is the MLE for $\theta_0$ and estimates $\theta_0$
with error of order $1/n$, which is much smaller compared with the $1/\sqrt{n}$
error rate of the usual sample mean estimator. However, the sample midrange
performs poorly when the data has say the Gaussian $N(\theta_0, 1)$
distribution, with an error rate of $1/\sqrt{\log n}$. In this paper, we
propose an estimator of the location $\theta_0$ with a rate of convergence that
can, in many settings, adapt to the underlying distribution which we assume to
be symmetric around $\theta_0$ but is otherwise unknown. When the underlying
distribution is compactly supported, we show that our estimator attains a rate
of convergence of $n^{-\frac{1}{\alpha}}$ up to polylog factors, where the rate
parameter $\alpha$ can take on any value in $(0, 2]$ and depends on the moments
of the underlying distribution. Our estimator is formed by the
$\ell^\gamma$-center of the data, for a $\gamma\geq2$ chosen in a data-driven
way -- by minimizing a criterion motivated by the asymptotic variance. Our
approach can be directly applied to the regression setting where $\theta_0$ is
a function of observed features and motivates the use of $\ell^\gamma$ loss
function for $\gamma > 2$ in certain settings.
