论文检索

密度积分泛函的估计AI 解读

Let $\varphi$ be a smooth function of $k + 2$ variables. We shall investigate in this paper the rates of convergence of estimators of $T(f) = \int\varphi(f(x), f'(x), \ldots, f^{(k)}(x), x) dx$ when $f$ belongs to some class of densities of smoothness $s$. We prove that, when $s \geq 2k + \frac{1}{4}$, one can define an estimator $\hat{T}_n$ of $T(f)$, based on $n$ i.i.d. observations of density $f$ on the real line, which converges at the semiparametric rate $1/ \sqrt n$.…

In this paper we shall investigate some nonasymptotic properties of the Grenander estimator of a decreasing density $f$. This estimator is defined as the slope of the smallest concave majorant of the empirical c.d.f. It will be proved that its risk, measured with $\mathbb{L}^1$-loss, is bounded by some functional depending on $f$ and the number $n$ of observations. For classes of uniformly bounded densities with a common compact support, upper bounds for the functional are…

序约束下的密度估计：非渐近极小极大风险AI 解读

Let us consider the class of all unimodal densities defined on some interval of length $L$ and bounded by $H$; we shall study the minimax risk over this class, when we estimate using $n$ i.i.d. observations, the loss being measured by the $\mathbb{L}^1$ distance between the estimator and the true density. We shall prove that if $S = \operatorname{Log}(HL + 1)$, upper and lower bounds for the risk are of the form $C(S/n)^{1/3}$ and the ratio between those bounds is smaller…