We present new M-estimators of the mean and variance of real valued random
variables, based on PAC-Bayes bounds. We analyze the non-asymptotic minimax
properties of the deviations of those estimators for sample distributions
having either a bounded variance or a bounded variance and a bounded kurtosis.
Under those weak hypotheses, allowing for heavy-tailed distributions, we show
that the worst case deviations of the empirical mean are suboptimal. We prove
indeed that for any confidence level, there is some M-estimator whose
deviations are of the same order as the deviations of the empirical mean of a
Gaussian statistical sample, even when the statistical sample is instead
heavy-tailed. Experiments reveal that these new estimators perform even better
than predicted by our bounds, showing deviation quantile functions uniformly
lower at all probability levels than the empirical mean for non Gaussian sample
distributions as simple as the mixture of two Gaussian measures.

基于 PAC-Bayes bounds 的新 M-estimators 方法在随机变量具有有界方差和有界峰度或者仅有界方差的情况下具有非渐进极小方差偏差特性，在分布为重尾分布的情况下，其对某个置信度水平的偏差与统计样本的经验均值的偏差的顺序相同。

挑战经验均值和经验方差：偏差研究

Challenging the empirical mean and empirical variance: a deviation study

We calculate the cosmological evolution of the 1-point probability
distribution function (PDF), using an analytic approximation that combines
gravitational perturbation theory with the Edgeworth expansion of the PDF. Our
method applies directly to a smoothed mass density field or to the divergence
of a smoothed peculiar velocity field, provided that rms fluctuations are small
compared to unity on the smoothing scale, and that the primordial fluctuations
that seed the growth of structure are Gaussian. We use this `Edgeworth
approximation' to compute the evolution of $<\delta|\delta|>$ and $<|delta|>$;
these measures are similar to the skewness and kurtosis of the density field,
but they are less sensitive to tails of the probability distribution, so they
may be more accurately estimated from surveys of limited volume. We compare our
analytic calculations to cosmological N-body simulations in order to assess
their range of validity. When $\sigma \ll 1$, the numerical simulations and
perturbation theory agree precisely, demonstrating that the N-body method can
yield accurate results in the regime of weakly non-linear clustering. We show
analytically that `biased' galaxy formation preserves the relation $<\delta^3>
\propto <\delta^2>^2$ predicted by second-order perturbation theory, provided
that the galaxy density is a local function of the underlying mass density.

本研究尝试通过结合重力扰动理论和 PDF 的 Edgeworth 扩展的分析逼近方法，计算 1 点概率分布函数的宇宙演化，并使用类似于密度场偏斜和峭度的措辞计算其演化情况，同时将分析计算与 N - 体模拟进行比较以验证其准确性。