# 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.

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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

**0**answers

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**

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**1**answer

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**

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**1**answer

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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**2**answers

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 ...

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**1**answer

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 $w_i^\top X, \ldots, w_k^\top X$ ...

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**3**answers

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-...

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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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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 ...

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**0**answers

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 ...

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**1**answer

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 ...

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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} \...

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**1**answer

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 ...

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**0**answers

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 = \...

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**1**answer

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 $y_i$...

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**1**answer

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)\...

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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 ...

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**0**answers

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**

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**1**answer

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 ...

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**2**answers

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 $\text{Med}[y]$...

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**1**answer

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 ...

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**0**answers

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 ...

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**1**answer

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 ...

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**0**answers

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 ...

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**1**answer

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 ...

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**3**answers

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**

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**1**answer

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 ...

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**0**answers

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 ...

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**1**answer

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)-...

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**1**answer

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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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 ...

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**1**answer

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

**1**answer

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\...

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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}\...

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**1**answer

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 ...

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**1**answer

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 ...

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votes

**1**answer

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(\...

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**2**answers

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 $\delta$...

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**1**answer

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, ...

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**1**answer

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**

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**1**answer

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 ...

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**1**answer

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

**1**answer

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**

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**0**answers

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 ...

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**1**answer

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 ...

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**0**answers

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 $E$ ...

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**2**answers

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

**1**answer

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 \...

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**0**answers

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**

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**1**answer

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 ...