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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.

2
votes
2answers
72 views

Why the autoregressive process to generate random time series?

in order to test some forecasting methods, I desire to generate random time series. I'm about to use the AR(1) model: $X_k=\alpha X_{k-1}+ \epsilon_k$ With eventually: $\alpha>1$ How can I ...
1
vote
0answers
46 views

Empirically random, quickly multiplicable matrices

I have encountered a need for fast computation of a transformation $Ax$ where $A\in \mathbb{C}^{K\times N},\ K\sim 10^7,\ N\sim 10^3$ is designed, and $x\in \mathbb{C}^N$ has iid $\mathcal{CN}(0,1)$ ...
1
vote
0answers
91 views

Rate of convergence of probabilities under Wasserstein convergence of measures?

Suppose $\mu=\sum_{i=1}^k p_i\delta_{x_i}$ and $\mu'=\sum_{i=1}^k p_i'\delta_{x_i'}$ are two atomic probability measures. Let $W_r(\mu,\mu')$ be the Wasserstein distance between $\mu$ and $\mu'$. Are ...
3
votes
1answer
81 views

Concentration inequalities specialized for log-likelihood / log-density functions

Let $P$ be a probability measure and $f$ be some probability density function (not necessarily related to $P$). Consider the function $$ L(X_1,\ldots,X_n) =\frac1n\sum_{i=1}^n\log f(X_i), \quad X_i\...
2
votes
0answers
80 views

Full-rank factorization property of integer-valued matrices

$\newcommand{\al}{\alpha} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\...
4
votes
1answer
74 views

Improved estimates of $n$ quantities via $n$ measurements

$\newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\epsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\si}{\sigma} \newcommand{\Si}{\...
3
votes
1answer
70 views

Optimal linear measurement operator

Let $x\in R^n$ be an unknown vector. Suppose I am allowed to choose any $A\in R^{m\times n}$, under the constraint that each row of $A$ has $\ell_2$ norm at most $1$. Then I carry out a "measurement", ...
4
votes
1answer
452 views

Mean and Variance of maximum of random variables

Given a set of random variables $x_1,x_2,...,x_n$, and we know their means and variances $(\mu_1,\sigma_1),(\mu_2,\sigma_2),...,(\mu_n,\sigma_n)$. How to compute mean and variance of the maximum ...
3
votes
1answer
75 views

Distribution of largest entry in a random vector

If we have a random unit vector on $\mathbb{C}^n$, drawn from the Fubini-Study metric, the marginal distribution of the squared absolute values of each of the coefficients in the vector is given by a ...
3
votes
0answers
35 views

Control the variance of some coincidence statistic

Let $X = X_1, \cdots, X_m \sim D_d$ a sample of size $m$ from a $[d]$-supported distribution. Let $$K_1 = K_1(X_1, \cdots, X_m) = \sum_{i = 1}^{d} \unicode{x1D7D9}\{B_i = 1\}$$ with $$B_i = B_i(X_1, ...
0
votes
1answer
105 views

Confusion about homogeneity

This might sound trivial or a simple misunderstanding, but please bear with me as I'm not a Math major. I want to investigate some aspects of PCA in homogeneous directions and needed simple ...
0
votes
0answers
28 views

Are there any non-asymptotic bounds for the minimum empirical risk vs theoretical risk?

I'm trying to see if there's any bounds on the difference between $f_{ERM}$ and $f^{*}$. For now, define $\mathcal{F}$ to be a function class. Let $P$ be a probability measure and $\hat{P_n}$ be the ...
3
votes
0answers
140 views

Partitioning the coupons collected in the classical coupon collector's problem

Suppose that there is an urn containing $n$ different coupons, from which $m$ coupons are being collected, equally likely, with replacement. Let $C(m)$ be the whole set of the $m$ collected coupons. ...
4
votes
1answer
47 views

Coordinates of Dirichlet Distribution Negatively Associated?

Suppose $X$ is sampled from a symmetric Dirichlet distribution with arbitrary shape and $n$ dimensions. Equivalently we can independently sample $z_i \sim \text{Gamma}(\alpha, 1)$ and then set $x_i=\...
0
votes
1answer
87 views

1D functional equation: solve for function with given expected value w.r.t normal density

Given scalars $c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz) $\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation $$ \begin{split} \mathbb ...
2
votes
3answers
112 views

Statistical test for boundedness of Expectation

I have $n$ i.i.d samples from a unknown distribution. I want to prove or disprove that the mean is finite. Are there any statistical test for this hypothesis ?
2
votes
1answer
201 views

What is the extreme value distribution for the Kolmogorov-Smirnov D statistic?

I occasionally find that I want to apply a K-S test in the context of unit-testing software that involves random behaviors. Unit testing with sampling statistics is a bit tricky because you want to ...
1
vote
1answer
95 views

Ratio of perfectly correlated gaussian distributions

Let $M$ be a positive definite matrix and let $w \in S^{d-1}$ be a unit vector uniformly distributed over the sphere. I want to understand the distribution of the quadratic form $\frac{w^T M^3 w}{w^T ...
-1
votes
1answer
77 views

How can one quantify the convergence of relative frequency to probability?

I have already asked this on stackexchange but did not get any answer. Say I run a simple Bernoulli trial a number of times and compute the relative frequency for success. Clearly the relative ...
3
votes
1answer
138 views

Concentration inequalities on the supremum of average after time $n$

Let $R_1, R_2, \cdots$ be i.i.d. Rademacher random variables (taking values $-1,+1$ w.p. $0.5$). At time $k$, their average is $\frac{1}{k}\sum_{i=1}^k R_i$. One can imagine after $k\geq n$ for some $...
0
votes
0answers
33 views

In paper “Neighbourhood Components Analysis”, how to determine the optimal number of neighbours (K)?

Recently, I am reading the paper "Neighbourhood Components Analysis". At the last paragraph on the second page of the paper, "Notice that by learning the overall scale of A as well as the relative ...
0
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0answers
77 views

How to calculate the exact probability $p$ at which maxima occurs in the curves in an infinite system?

I'm writing with respect to this paper: Khatun, Dutta, and Tarafdar - "Islands in Sea" and "Lakes in Mainland" phases and related transitions simulated on a square lattice Here's a link to the PDF ...
1
vote
0answers
52 views

Non-diagonalizable matrix in a discretized Ornstein-Uhlenbeck process

I am attempting to implement a pairs trading algorithm for two securities by approximating a discretized version of the Ornstein-Uhlenbeck process: \begin{equation*} d\mathbf{S}_t = \mathbf{\kappa}(\...
2
votes
1answer
259 views

Naviers Stokes equation and machine learning

I am looking for a reference explaining how to solve Navier-Stokes numerically using Machine learning algorithms . Thank you in advance for your help .
0
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0answers
46 views

Using empirical expectation of orthogonal polynomials for density estimation

Estimating the probability density function using empirical moments is quite popular. Is there any advantage to using a different polynomial basis than the usual $1$,$x$, $x^2$ ... etc? For the moment ...
2
votes
1answer
89 views

Minimising the f-divergence to a conditional probability constraint

Let $P$ be a probability distribution and let $A$ and $B$ be some events, and suppose that we want to minimise an $f$-divergence between $P$ and the set of all distributions $Q$ that satisfy that ...
1
vote
1answer
68 views

Markov processes: Construction of the state variables

I have asked this question on stats.se.com but I did not receive an answer. Given is the description of a probabilistic finite state machine and I want to 'translate this' into a Markov process 'on it'...
0
votes
0answers
126 views

Continuity of the conditional expectation

Consider the conditional expectation of $x$ given $y$, $$ \mathbb{E}(x | y) $$ where $x \in X$ and $y \in Y$ where $X, Y$ are Hilbert spaces (possibly infinite dimensional). Question : I am looking ...
8
votes
3answers
202 views

Regularized linear vs. RKHS-regression

I'm studying the difference between regularization in RKHS regression and linear regression, but I have a hard time grasping the crucial difference between the two. Given input-output pairs $(x_i,y_i)...
3
votes
1answer
311 views

Kullback–Leibler divergence of product distributions

Say the KL divergence between two distributions $A$ and $B$ is $\varepsilon$. Can we give bounds, or a precise computation, of the KL divergence between $A^k$ and $B^k$ (the product distributions)?
0
votes
1answer
99 views

How to compute the entropy of a random variable with values in a metric space? [closed]

I have a cloud of points, and I want to compute its 'diversity'. Variance is not appropriate, because a cloud clustering around few points can still have a large variance. To that end, I see the ...
2
votes
1answer
184 views

Does variants of Bernstein and Freedman concentration inequalities exist with NO uniform bound on the range of RV or martingale differences

A classic formulation of the Bernstein inequality (from Wikipedia) is as follow: Let $X_1, \ldots, X_n$ be independent zero-mean random variables. Suppose that $|X_i|\leq M$ almost surely, for all $i$...
0
votes
0answers
34 views

probabilistic transfer lipschitzness

https://arxiv.org/pdf/1705.08848.pdf In this paper, there's mention of probabilistic transfer lipschitzness. Currently, I'm trying to relax this assumption but I'm not familiar with other ...
4
votes
1answer
111 views

Probability of ruin in the Sparre Andersen renewal risk model

In 1957, Erik Sparre Andersen proposed using a renewal risk model to describe the behavior of the insurers surplus $$U(t)=u+ct-\sum\limits_{i=1}^{\Theta(t)}Z_i, \quad t \geqslant 0$$ where: $u \...
0
votes
0answers
38 views

Radial Basis Function Centers

Given $n$ inputs $x_1,...x_n$ and $n$ responses $f(x_1),...,f(x_n)$, a commonly used approach of estimating $f(\cdot)$, is by using a positive definite radial basis function $\phi(\cdot)$: \begin{...
1
vote
0answers
25 views

Finding most Representative Sample in “Pair Statistics”

By "Pair Statistics" I understand statics that are based on values $\varphi:\mathcal{P}\times\mathcal{P}\ni(p,q)\mapsto y\in\mathbb{R}$ that can be observed for every pair $(p,q)$ of individuals of a ...
2
votes
1answer
75 views

Quantifying an increasing spacing between data points

Is there a measure or statistic that could quantify a steady increase in the spacing between data points in a time series? For instance, in the figure, the points are clustered and dense near 0, but ...
4
votes
0answers
106 views

Total Variation distance of polynomials of Bernoulli R.V.s

Let $X_i, Y_i$ be i.i.d Bernoulli $0/1$ random variables with $\mathbb{E}[X_i] = p$ and $\mathbb{E}[Y_i] = q$. Let \begin{align*} X &= X_1 X_2 + Χ_2 Χ_3 + \ldots +X_{n-2} X_{n-1}+ X_{n-1} X_n\\...
3
votes
0answers
116 views

Finding a mixture of 1st and 0'th order Markov models that is closest to an empirical distribution

I am interested in finding the distribution "$p^*$" closest to an empirical distribution $\hat{p}$ where $p^*$ is a mixture of first and zeroth order Markov models. That is, I want to find $$ p^* = \...
3
votes
2answers
162 views

Largest eigenvalue of the adjacency matrix of weighted random graph

I find the theorem for largest eigenvalue of the adjacency matrix of ER random graph in here https://arxiv.org/pdf/math/0106066.pdf. The adjacency matrix is a symmetric random matrix s.t. diagonal ...
3
votes
0answers
82 views

Largest eigenvalue divided by $n$

Let $X$ be an $n\times n$ symmetric random matrix whose diagonal is fixed as $1$, and every element in the upper triangle (excluding the diagonal) is drawn from Bernoulli($p$). The elements in the ...
1
vote
0answers
28 views

Probability of increase in $\frac{\lambda}{n}$ of correlation matrix (principal score) after random sample the original matrix

Suppose $X$ is the original matrix of $n$ columns, and $P$ is the $n \times n$ correlation matrix of $X$. $P$ is symmetric positive semi-definite. Denote the largest eigenvalue of $P$ is $\lambda$, ...
0
votes
0answers
44 views

For 1-NN (Next Neighbor), what is the expectation of the largest probability of being the nearest neighbor?

Suppose we sample $n$ points $X_1,X_2,...,X_n$ independently from a distribution $P_X$ on $[0,1]^d$. For a new point $X$ independently from $P_X$, we find its nearest neighbor in $X_1,X_2,...,X_n$. ...
6
votes
1answer
144 views

Martingale version of Bernstein-type inequality for (slightly) heavy-tailed distributions?

It is known that for sub-exponentially distributed martingale difference sequence, the following Bernstein-type inequality holds: $$ ℙ\left(\left| \sum_{i=1}^N a_i X_i \right| \ge t \right) \le 2\...
2
votes
1answer
210 views

Distribution of ratio between complex Gaussian and Chi-square R.V.s

What would be the distribution (p.d.f.) of the following ratio? $z = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$ where $x_{i} \sim \mathcal{CN}(0,a), \forall i$ and $a > 1$. As can be ...
1
vote
0answers
65 views

Is there a complete countable axiomatization of conditional independence? (Graphoids)

Note: A pointer to a reference, or a yes/no answer with a 1-2 sentence incomplete/non-rigorous justification would suffice for answers. I am just curious about whether the result is true; it is fairly ...
2
votes
0answers
92 views

Stochastic Approximation in Reproducing Kernel Hilbert Space

Consider an iterative algorithm with incremental updates \begin{align} x_{t+1} = x_t + \alpha_t \cdot [ h(x_t) + M_{t+1}], \end{align} where $\{x_t \}_{t \geq 0}$ is in a reproducing kernel Hilbert ...
2
votes
1answer
105 views

Understanding some Hoeffding-type martingale inequality

Would anyone know how to prove the following, coming from the proof of theorem 2 in this paper (https://arxiv.org/pdf/1605.08671.pdf)? Consider i.i.d. Sub Gaussian random variables $(X_t)_{t\geq 1}$ ...
1
vote
1answer
82 views

Inequality satisfied for $t=1/2$ and Measurability implies Log-Concavity

Say $f:\mathbb{R}\to(0,\infty)$ is measurable, and that $$\forall a,b\in\mathbb{R}~~a<b\implies\log f(a)+\log f(b)\leq2\log f(\frac{a+b}{2}).$$ Why must $f$ be log-concave? (That is, why must $$\...
3
votes
1answer
139 views

Bounding the “spikiness” of a probability distribution

Are there any well-known conditions that guarantee that a probability distribution isn't too "spiky"? I ask this question because I am interested in the families of probability distributions $f(x)$ ...