We give a negative answer to a problem posed by James Robins on estimating a constant conditional variance in nonparametric regression under random design. For every $s>1$ and integer $d>4s$, when the regression function is $s$-H\"older, the unknown design density is bounded above and away from zero, and the conditional error laws may depend on the design but have mean zero, a common variance, and uniformly bounded fourth moments, we show that the minimax root-mean-square risk is bounded below by $n^{-\beta}$ with $\beta=\frac{d(3s+1)+8s}{(d+2s)(d+4)}$. Hence the conjectured rate $n^{-4s/(d+4s)}$ is not uniformly attainable. We use a similar argument to establish the minimax rate $n^{-1/2}\vee n^{-4s/(d+4s)}$ when \(s \in (0,1]\).
We study minimax-optimal designs and estimators for estimating the sample average treatment effect in finite population randomized experiments, where both design and estimator are unrestricted. For binary potential outcomes, we show this minimax risk is equivalent to the minimax risk $\rho_n^*$ of an estimation problem...
Timothy Sudijono, Edgar Dobriban, E. Tchetgen· 1 citation
Given $[0,1]$-valued random variables $X_1,\dots,X_n$ such that $\mathbb{E}[X_i | X_1,\dots,X_{i-1}]= \mu$ for all $i$, we propose a new nonasymptotic confidence interval for $\mu$ that is obtained by inverting terminal e-values generated by a novel betting strategy. When the data are iid, its limiting width matches th...
Diego Martinez-Taboada, Aaditya Ramdas· 1 citation
We study expected improvement (EI) for minimizing a deterministic function $f$ in the RKHS $\mathcal H_k$ of a continuous positive-semidefinite kernel $k$ on a nonempty compact set $\mathcal X\subset\mathbb R^d$. Function values are observed exactly, and EI is computed from a fixed zero-mean Gaussian-process model with...
Emmanuel Vazquez, S. Petit· arXiv.org· 0 citations
Consider $n$ independent, non-negative, mean at most one random variables, $X_1,X_2,\ldots$. We show the following bound on the probability of their sum exceeding a threshold $t$: \[ \mathbb{P}\left[\sum_{i=1}^n X_i\ge t\right] \leq 1-\left(1-\frac{1}{t}\right)^n \text{ for all } t\ge 2n+1 \,. \] To prove this, we cons...
Let $X = (X_1, \ldots, X_n)$ be a random vector from any Borel probability law on $\mathbb{R}_+^n$. We revisit the problem of deriving a lower confidence bound (LCB) on a scalar parameter of that law. We recast classical work, beginning with Buehler, in purely probabilistic terms to form a more accessible and extensibl...
G. Bissias, E. Learned-Miller· arXiv.org· 0 citations
A lower bound under the uniform design shows that these separation rates are rate-optimal, which answers a question raised by Mukherjee and Sen (2018).
P. M. Aronow, Patrick Lopatto· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
