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I would like to provide an asymptotic solution, as the sample size $n\rightarrow\infty$. Using the Fenton-Wilkson empirical asymptotic result in [1], we know that the sample sum of i.i.d. lognormals $ …

The question is clarified by Prof.V.Vovk. See his answer below for discussion. Recently, early works of Gammerman, Vanpnik and Vovk[4] are rediscovered by Wasserman et.al[1] and proposed it as a promi …

I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field. I have read (4) and parts of (3). …

Are you trying to construct an infinite-dimensional (Hilbert) manifold of probability measures on a fixed measurable space? Or you simply want to find a Bergman divergence analog in order to generaliz …

@Carlo's answer is very insightful from a physics perspective, thanks a lot for the teaching and learning here. My answer is from a more ML and statistic perspective as @Ed Smith asked. But in the OP, …

I may start answering by pointing out that the term "nonparametrics statistics" is essentially "parametric". The existing methods (e.g. Smoothing splines) in nonparametrics, are somehow all parametriz …

I do not think the conclusion will hold for a general exchangeable sequence. In a more general case, you have to assume that U-statistics itself are not degenerated ($w_i\neq w_j$ for $i\neq j$) and t …

Note that the Laguerre orthogonal polynomials are in form of [1](bearing combinatoric interpretation) and [3] \begin{align} & L_n^\nu(x)=(-1)^n\sum_{m=0}^n \binom n m \prod_{i=1}^m (\nu+2(n-i))(-x)^{ …

It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some n …

I think it is just called (scaled) supremum $L_1$ norm and it is mostly studied in Bayesian nonparametric estimation literature, especially posterior consistency. The following paper investigate condi …

Iosif Pinelis provided a very nice answer, however, I would like to provide a more comprehensive answer to this question. I think the title is a bit misleading because we do not actually need the orde …

$g_\theta(x):=e^{-(\theta x-1)^2/2}=\exp(-\frac{1}{2\frac{1}{\theta^2}}[x-\frac{1}{\theta}]^2)$ is the kernel of $N(\frac{1}{\theta},\frac{1}{\theta^2})$ with density $\frac{1}{\sqrt {2\pi \frac{1}{\t …

Non-Gaussianness is an ambiguous concept. In the continuum of probability distributions such as the uniform, where all events are clustered into a given range and equally likely. On the other s …

$$\mathbb{P}(X-Y<t\mid X>Y) =\frac{\mathbb{P}(X-Y<t,X>Y)}{\mathbb{P}(X>Y)} =\mathbb{P}(X-Y<t)$$ since $X(\omega)>Y(\omega),\ \forall\omega\in\Omega $. $$\mathbb{P}(X-Y<t)=\int_{0}^{\infty}dy\int_{0}^ …

In response to the critical comments below I revised my answer. Hope this is more helpful! (1) Two kinds of metrics are defined on generally different spaces. It is not fair to compare these two met …