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Questions tagged [st.statistics]

Applied and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments.

1
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2answers
162 views

Weak convergence for discrete-time processes using characteristic functions

I am looking for a good reference about the analogues of the Bochner Theorem and the Lévy Continuity Theorem for probability measures on $\mathbb{R}^{\mathbb{N}}$ with the product topology. ...
4
votes
0answers
73 views

What is the entropy of binomial decay?

Let's play a game. I start with $N$ indistinguishable tokens, and I wait $T$ turns. Every turn, each token has probability $p$ of disappearing. I want an analytic formula for the entropy of this ...
3
votes
1answer
247 views

Poisson process with stochastic intensity correlated with a Brownian Motion

I am currently confused with the moment of non-homogeneous compound Poisson process and a Brownian Motion. I know that generally Poisson Process and Brownian Motion are independent if they are adapted ...
5
votes
1answer
587 views

Does MCMC overcome the curse of dimensionality?

I want to compute an integral like this $$\frac{\int_y g(y) e^{-\beta f(y)} \text{d} y } {\int_y e^{-\beta f(y)} \text{d} y}$$ where $f(y)$ is not necessarily convex and the dimension $d$ of $y$ is ...
28
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2answers
1k views

Manifold of probability measures: connections between two types of metrics

The space of probability measures could be viewed as an infinite-dimensional manifold, equipped with two possible types of metrics — (1) Wasserstein and (2) Fisher-Rao. Metric (1) is connected with ...
7
votes
1answer
163 views

Expectation involving maximum of Gaussian variables

Let $X\sim N(0, I_d)$ be a $d$-dimensional Gaussian random vector. Let $W_1, \ldots, W_k \in \mathbb{R}^d$ be $k$ fixed vectors in general positions. It is clear that $w_i^\top X, \ldots, w_k^\top X$ ...
4
votes
3answers
290 views

Determinant of correlation matrix of autoregressive model

I wonder if there is a paper that can point out how to compute the determinant of a $d \times d$ autoregressive correlation matrix of the form $$R = \begin{pmatrix} 1 & r & \cdots & r^{d-...
2
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0answers
212 views

Matrix optimization of a random quadratic form

I am interested in maximizing a quadratic form which looks like $$f(\Sigma) = E(\operatorname{trace}(SJ)) = E(1^{\top} S 1)$$ where $J$ is a matrix of $1$'s, $S= \Sigma_{mm} - \Sigma_{mo} \Sigma_{oo}...
4
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0answers
164 views

Distributions over permutation groups $\mathcal{S}_n$

Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...
2
votes
0answers
88 views

Relationship between the Hurst exponent and the alpha parameter

I have a question about the relationship between the Hurst exponent $H$ and the $\alpha$ parameter in the autocorrelation function when long memory is present. As we know in this case the decay of the ...
3
votes
1answer
144 views

Wishart type matrix

Assume a positive semi-definite $M\times M$ matrix $A$, not with full rank, and an $M\times N$ matrix $X$, where $M>N$. The elements of $X$ are independent, zero-mean complex Gaussian with variance ...
2
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0answers
100 views

Min Max Equality in Information Theory

Let $\mathcal{Y}$ and $\mathcal{X}$ be finite sets and let $Q_Y$ be a fixed probability mass function on $\mathcal{Y}$. Also, let $P_{X | Y}$ be some fixed conditional distribution on $\mathcal{X} \...
5
votes
1answer
393 views

Strong duality for a particular moment problem

Reading the paper in this Link (see pag 13) with the objective of understanding a topic related to stochastic optimization I came across a problem in demonstrating one of the theorems. The situation ...
2
votes
0answers
82 views

Algorithm for optimal grouping for canonical correlation analysis?

In Canonical Correlation Analysis (CCA), we have two sets of column vectors $X = \{x_1, x_2 ... x_n \}$ and $Y = \{y_1, y_2 ... y_n \}$ and find the linear combinations of each set, says $a = \...
1
vote
1answer
79 views

Accounting for unobserved events in baysian learning

I wanted to use Bayes theorem to help me automate the task of deciding if I should ignore events, but I am not sure how to update the posterior if I do The simple story goes like this: An event $y_i$...
3
votes
1answer
145 views

Convex lower bound for probability that a random subset of [n] has cardinality at most k

For $n\in\mathbb{N}$, the probability that a random subset of $[n]=\{1,\cdots n\}$ has cardinality at most $k$ is $f_k(n)=2^{-n}\sum\limits_{i=0}^k{n\choose i}$. I'm looking for a lower bound $g_k(x)\...
6
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2answers
373 views

Need help with a model, Whatsapp data analysis

This is not actually a research question. It is more an exercise which I posed myself in mathematical/statistical modelling. I have some Whatsapp data of a chat with someone. I want to find a ...
1
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0answers
59 views

Hedges' estimator of $\tau$ in the random effects model ( meta analysis)

In the random effects model we observe the $y_i$ with the standard errors $s_i^2$ where it is assumed that $y_i = \psi + a_i + e_i$ where $a_i$ is normally distributed with mean $0$ and standard ...
2
votes
1answer
72 views

Is there a general theory supporting the construction of conditional confidence intervals?

Conditional confidence intervals are intervals whose confidence statements apply even after considering the actual data collected (i.e., conditional on the data actually observed, not averaged over ...
5
votes
2answers
595 views

Median and mean of the sample mean of i.i.d. log-normal

Let $y:=\frac1n\sum_{i=1}^n x_i$, where $\{x_i\}_{i=1}^n$ is a set of i.i.d. random variables, and every $x_i$ has a lognormal distribution $x_i \sim\text{Lognormal}(\mu,\sigma^2)$. Let $\text{Med}[y]$...
1
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1answer
115 views

Variance bound of a functional

$X_1,\ldots,X_n$ are i.i.d standard normal random variables. $a_1,\ldots, a_n$ are constants with $a_i \in [\kappa_1, \kappa_2]$ for all $i$ and $\kappa_1>0$. $\hat c_n$ is given as the solution ...
2
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0answers
39 views

Legendre expansion of $r(x) = f(x)/g(x)$ using a finite number of samples from $f(x)$ and $g(x)$

I have two finite sets of events $\{x_1, ..., x_N\}$ and $\{y_1, ..., y_N\}$ that are sampled from the PDFs $f(x)$ and $g(x)$, respectively, where $x \in [-1,+1]$. I want to estimate the Legendre ...
3
votes
1answer
81 views

Constructive approximation of Hölder functions using kernel functions

Suppose I have a function $f \in \mathcal C^{\alpha, L}([0,1])$, where $\mathcal C^{\alpha, L}([0,1])$ is the space of $\alpha$-smooth Hölder functions with norm $L$. I am interested in efficiently ...
1
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0answers
139 views

Relation between pseudo-dimension and Rademacher complexity

With techniques of Dudley's entropy bound and Haussler's upper bound one can show that there exists a constant $C$ such that any class of $\{0,1\}$ indicator functions with Vapnik-Chervonenkis ...
5
votes
1answer
281 views

What are some of results in low dimensional statistics that do not hold in high dimensions?

This question is partially inspired by the following MO post: What are some of the surprising results of finite sample statistical estimation? and current heated research front of high dimensional ...
5
votes
3answers
383 views

Why a random variable is better described by its cumulants than by its characteristic funtion?

It is a classical and well known problem that a random variable $X$ is not uniquely determined by its moments $\mathbb{E}(X_n)$. The moment problem is the problem of determining the probability ...
3
votes
1answer
146 views

Bound for expectation of function of 3 normal distributions

Let $X,Y,Z$ be three standard normal distribution. Let $\rho_{XY},\rho_{YZ},\rho_{XZ}$ be the correlation between those random variables. Let $f()$ be a monotone, odd, bounded, and differentiable ...
3
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0answers
72 views

Asymptotic results on statistical graph models

This post is partly inspired by this post. Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix While it is well-known that two basic ...
3
votes
1answer
170 views

Is this generalization bound proof wrong?

This is an ICML02 paper by Garg, Har-Peled & Roth: http://sarielhp.org/p/01/bounds/bounds.pdf The equation after eq. (3) is the well-known symmetrization trick for $\sup_{h\in {\mathcal H}} |E(h)-...
2
votes
1answer
150 views

Order of independent random variables

Let $(p_\pi)_{\pi\in S_3}$ be given nonnegative reals such that $\sum_{\pi \in S_3} p_\pi = 1$. What are necessary and sufficient conditions for there to exist independent random variables $X_1,X_2,...
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0answers
141 views

Strict monotonicity of conditional variances

Let $K \geq 2$ be a positive integer and $C$ be any $K \times K$ non-singular matrix (if necessary, can assume that all $K$ rows of $C$ are needed to span the coordinate row vector $e_1'$). For ...
4
votes
1answer
493 views

Minimizing KL divergence: the asymmetry, when will the solution be the same?

The KL divergence between two distribution $p$ and $q$ is defined as $$ D( q \| p)\int q(x)\log \frac{q(x)}{p(x)} dx $$ and is known to be asymmetry: $D(q\|p)\neq D(p\|q)$. If we fix $p$ and try to ...
3
votes
1answer
134 views

Concentration inequality of joint event over time of a submartingale

Consider a discrete time submartingale $X_n$ with bounded difference $|X_n-X_{n-1}|\leq c$. With Azuma inequality we have the concentration of a single time event as $$ P(X_t-X_0 \leq -t) \leq exp\...
4
votes
0answers
85 views

Is there an example that both Berry-Essen bound and DKW bound are attained?

The Berry-Essen bound stated that $$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$ where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\...
1
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1answer
54 views

Clarification on margin bound uniform w.r.t. the margin parameter

Theorem 4.5. in the book "Foundations of Machine Learning" by Mohri et al: http://prlab.tudelft.nl/sites/default/files/Foundations_of_Machine_Learning.pdf derives a generalization bound to hold ...
-1
votes
1answer
54 views

Minimax solution but game has no value

Fix convex sets $\Delta,\Pi$ and let $r: \Pi \times \Delta \in [0,\infty]$ be linear (i.e., concave and convex) in its first parameter for every fixed second parameter. I'm looking for a situation ...
2
votes
1answer
85 views

Does maximizing $D_u$ imply stochastic ordering?

Let $\mathscr P _0$ and $\mathscr P _1$ be two non-overlapping sets of probability distributions defined on $(\Omega,\mathcal{A})$. Consider the distance defined as $$D_u(P_0,P_1)=\int_\Omega \left(\...
2
votes
2answers
301 views

Lower bound on number of samples for an epsilon delta approximation matching the Chernoff bound

So we have two biased coins, one comes out head w.p. $1/2+\epsilon$ and the other w.p. $1/2-\epsilon$. How many times should we flip these two coins to be able to tell them apart w.p. at least $\delta$...
-3
votes
1answer
118 views

Non-random movements [closed]

I know that the hedge fund Renaissance Technologies use computer-based models to predict price changes in financial instruments. These models are bases on analyzing as much data as can be gathered, ...
2
votes
1answer
383 views

Rademacher complexity of composition of functions

I am looking for a bound on the empirical Rademacher complexity of the following class: $G=\left\{x \rightarrow \frac{h^T f(x)}{\|h\|_2 \cdot \|f(x)\|_2} : h\in R^d, f()=(f_1(),\ldots,f_d()), f_j \in ...
4
votes
1answer
647 views

Rate of convergence of uniform order statistics to their expectations

This is a problem that I encountered in my research and have no clues to fully resolve it. Basically, I need large (or moderate) deviation bounds on the difference between an order statistic of ...
10
votes
1answer
942 views

How is the “conformal prediction” conformal?

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 ...
2
votes
1answer
231 views

Extension of Talagrand contraction lemma (on empirical Rademacher complexity)

Is the following true? Let $(x_1,...,x_N)$ be a set of points on the unit sphere $S^{d-1}$. Let $\ell_x: [-1,1]\rightarrow [0,1]$ be a family of Lipschitz functions indexed by $x\in S^{d-1}$, with ...
3
votes
0answers
117 views

Derivative of rank $r$ approximation of matrix

Let $Y \in \mathbb R^{n \times c}$ and $r$ be an integer with $1 \le r \le \operatorname{rank}(Y)$. Consider the problem $$\text{minimize} \|Y-X\|_{\text{Fro}}^2\text{ over }X \in \mathbb R^{n \times ...
2
votes
1answer
217 views

Weak convergence of sum of log normal random variables

Let $S_t$ be the Geometric Brownian Motion, we know that $$dS_t=rS_tdt+\sigma S_tdW_t, t\in [0,T], S_0>0, r>0,\sigma>0$$ and the distribution of $S_t$ is known explicitly. Please see the ...
5
votes
0answers
350 views

Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance

Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
5
votes
2answers
190 views

Polynomial related to lognormal moments

Consider the polynomial: $$p(x) = \sum_{k=0}^{r}(-1)^{r-k} {r \choose k} x^{k(k-1) / 2}$$ I want to show that $$p(x) = (x - 1)^{\lceil r/2 \rceil} \, q(x)$$ That is, $(x - 1)^{\lceil r/2 \rceil}$ ...
2
votes
1answer
145 views

Distribution-free statistics on compact Lie groups

(Cross-listed from the math stackexchange) Let $(X_i)_{i=1}^n$ be iid random variables with joint cdf $F$. Recall that the empirical distribution function is: $$ F_n(x) = \frac{1}{n} \sum_{i=1}^n \...
1
vote
0answers
243 views

Generalized non-central chi distribution: pdf and cdf

I am looking for the closed-form expression of the CDF of the product of two independent generalized non-central chi distributions (not chi-squared) each with k=2 degrees of freedom. A generalized non-...
1
vote
1answer
50 views

Realization property implies expectation property

Is there a theorem that says if every realization $X(t)$ of a random process $X_{\omega}(t)$ satisfies some property, then the expectation $\mathbb{E}X(t)$ also satisfies the same property? What ...