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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
0answers
52 views

How sensitive are probability distributions to noise?

Let $\sigma: \mathbb{R}^d \rightarrow (0,1)^d$ be the softmax operator which maps $x$ to the manifold of the probability distributions on $d$ variables defined in the following way: $$\sigma(x)_i = \...
2
votes
1answer
33 views

Mutual information between continuous and discrete variables from numerical data

I am looking for references/measures to estimate the mutual information between a continuous (C) and discrete (D) variable, given a real-world (i.e. finite sample) data set. C is uniformly distributed ...
-1
votes
0answers
32 views

PCA on conditional heteroscedastic stochastic process data

What is the correct method of application of Principal Component Analysis (PCA) on time series data? Since the time series may exhibit conditional heteroscedasticity, application of normal PCA might ...
-4
votes
0answers
41 views

Let X∈x^2(2,1). Find the pdf of Y=sqrt(X) [on hold]

How is this problem suposed to be done, I'm confused about the request? Let X∈x^2(2,1). Find the pdf of Y=sqrt(X)
0
votes
0answers
22 views

P-value in Likelihood Ratio Test definition

According to Williams, D.: Weighing the Odds the p-value of observed data in the likelihood ratio setting is defined as $$\mathrm{p_{val}}(y^{obs}) := \mathrm{sup}_{\theta \in B_0} \mathbb{P}\big(\...
2
votes
1answer
207 views

How to simulate random paths of a non-homogeneous continuous-time Markov process with discrete state space for a given infinitesimal generator matrix?

Let $X=(X_{t},\,t \in T)$ be a non-homogeneous, continuous time Markov process with a finite state space $S=\{1,...,K\}$. Let $\alpha_{i,j}(t)$ be the hazard rates of some $\varGamma$-distributed ...
-1
votes
0answers
19 views

Standard normal variable times an independent random variable [closed]

Hello I am unsure about the distribution of a variable. If X~N(0,1) and P[V=1]=P[V=-1]=0.5 Show that the distribution of VX is also standard normal. I understand that E[V]=0 and Var[V]=1 but I'm ...
-2
votes
0answers
16 views

Standardising a score of values help [closed]

I am working on developing on a standardised score from a number of scores. The values will be ranging from large to small of each score and I am trying to develop a way of proportionally measuring ...
2
votes
1answer
172 views

Are Linear Maps resistant to Noise?

Let's assume I have a $m \times m$ matrix $M$ with Frobenius norm $1$ and a unit vector $x \in S^{m-1}$. I also have a second $m \times m$ matrix $M^*$ which is obtained from the first one plus some ...
0
votes
0answers
19 views

Parametric statistics: how to estimate the supremum of a set of parameters from a random sample

I would like to ask a question on how to estimate the supremum norm of a set of parameters in the following setting. I appreciate any pointer or suggestion. Thanks. Question: Suppose we have $m$ ...
8
votes
2answers
509 views

Random Walks on high dimensional spaces

I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in arbitrary directions (uniformely on the unit sphere $S^1$, not left-right-up-down), ...
5
votes
1answer
165 views

Relation between the two possible KL divergences of two distributions

Given that I know $$D\left(P\parallel Q\right)<\alpha,$$ can I say anything about $D\left(Q\parallel P\right)$ in terms of an upper bound on it? Also, given this upper bound on $D\left(P\parallel ...
3
votes
0answers
72 views

Upper bounding the start of a distribution's CDF, given bounds on first moments

Take nonnegative random variables $X$ whose first $K$ moments have bounds: $\mu^k\leq E[X^k]\leq c\mu^k$ for each $k=1,\dots,K$. In this case what is an upper bound for $P(X\leq O(\mu))$? I am ...
1
vote
2answers
43 views

Approximate the variance of multiple normal distributions with the same standard deviation

Given a number of normal distributions $N(\mu_1, \sigma^2), N(\mu_2, \sigma^2), ..., N(\mu_n, \sigma^2)$ with fixed variance $\sigma^2$, but not necessary equal means. My question is how to ...
1
vote
0answers
35 views

Hypergraph partitioning and bipartite graph partitioning

Are hypergraph partitioning, and bipartite graph partitioning related, or equivalent, given that hypergraphs can be represented as bipartite graphs? In the first case, we want to partition the set of ...
12
votes
4answers
695 views

Good introduction to statistics from a algebraic point of view?

There are already lots of questions on this subject like Is there an introduction to probability theory from a structuralist/categorical perspective? Is there a combinatorial/topological treatment ...
0
votes
1answer
43 views

Anti-concentration: upper bound for $P(\sup_{a \in \mathbb S_{n-1}}\sum_{i=1}^na_i^2Z_i^2 \ge \epsilon)$

Let $\mathbb S_{n-1}$ be the unit sphere in $\mathbb R^n$ and $z_1,\ldots,z_n$ be a i.i.d sample from $\mathcal N(0, 1)$. Question Given $\epsilon > 0$ (may be assumed to be very small), what is ...
0
votes
0answers
19 views

Anti concentration of $\frac{1}{n}zz^T + \text{diag}(z_1^2,\ldots,z_n^2)$ for sub-gaussian i.i.d $z_1,\ldots,z_n$ and $z:=(z_1,\ldots,z_n)$

Let $z_1,\ldots,z_n$ be an i.i.d sample from a sub-gaussian distribution. Define the $n$-by-$n$ p.s.d matrix $C_n := \frac{1}{n}zz^T + \text{diag}(z_1^2,\ldots,z_n^2)$, where $z:=(z_1,\ldots,z_n)$. ...
0
votes
0answers
17 views

Concentration of $X^T\eta\eta^TX \in \mathbb R^d$ for i.i.d $(x_i,\eta_i)$ and sub-gaussian $\eta_i$

Suppose $(x_1,\eta_1),\ldots,(x_n,\eta_n)$ are $n$ i.i.d points in $\mathbb R^{d+1}$ such that $\eta_1,\ldots,\eta_n$ are $\sigma$-subgaussian. Let $X \in \mathbb R^{n \times d}$ be the vertical ...
3
votes
1answer
73 views

Regression with correlation structure

I have a theoretical question about regression models. Let's say I measured multiple responses from $n$ subjects and these responses are correlated with each other. For example, let's say I measured ...
0
votes
0answers
56 views

Does the linear combination of the quantile $\alpha F^{-1}(\tau)+\beta G^{-1}(\tau)$ still a quantile

$F(x)$ and $G(y)$ are distribution functions. Define the $\tau$th quantile for cdf $F(x)$, $G(y)$ as $$\xi_\tau\equiv F^{-1}(\tau)=\inf\{x:F(x)\ge \tau\}$$ and $$\eta_\tau\equiv G^{-1}(\tau)=\inf\{y:...
0
votes
1answer
42 views

Cumulative Order Statistics of Independent Non-Identical Distributions

I understand that the p.d.f of order statistics for Independent Non-Identical Distributions are given by the Bapat-Beg theorem as previously explained in another question. As explained in the article, ...
4
votes
1answer
395 views

Backwards random codebook generation

$$X \longrightarrow \fbox{$\phantom{\int}P_{Y|X}\phantom{\int}$}\longrightarrow Y$$ The information capacity of this channel is $C=\max_{P_X} I(X;Y)$, and it can be achieved by associating each ...
2
votes
0answers
39 views

Enumerating lattice points in a product of balls, in limit with dimension

Fix $(L_n)_n$ to be a sequence of lattices, each $L_n\subset \mathbb{R}^n$, where both the effective-inradius and effective-outradius go to 1 (i.e. the Voronoi region of the lattices approach a ball ...
0
votes
0answers
41 views

Limiting a sequence of moment generating functions [migrated]

I was trying to solve the following problem: Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables with the probability mass function $P\{X_n = \pm1 \} = \frac{1}{2}$, $n \in \...
29
votes
4answers
6k views

Applications of algebraic geometry to machine learning

I am interested in applications of algebraic geometry to machine learning. I have found some papers and books, mainly by Bernd Sturmfels on algebraic statistics and machine learning. However, all this ...
-2
votes
0answers
47 views

Any ideas how to proceed with interpreting this expected loss?

For this post, $\Gamma$ is an expectation operator, and we have two distributions $S$ and $T$. $(x_S,y_S)\sim S$ and $(x_t,y_t)\sim T$. Define $\Gamma^*_f = \arg \min_{\Gamma} \Gamma l(f(x_t),y_s)$ ...
1
vote
1answer
64 views

Sharp tail bounds for the maximum of an iid sample of a random variable supported on $[0, 1]$

Let $X_1,\ldots,X_n$ be an iid sample from a distribution supported on $[0, 1]$. Question What are some sharp concentration inequalities (i.e tail bounds) empirical statistic defined by $Z_n := \max(...
-2
votes
1answer
74 views

Existence or impossibility of Gaussian factory

Gaussian factory problem: given an iid sequence $x_i \sim \mathcal{N}(\mu,\sigma^2)$, $i=1,2,\dots$, with $\mu$ and $\sigma^2$ both unknown, construct a realization $y \sim \mathcal{N}(0,1)$.
1
vote
0answers
34 views

Equivalent of linearity for the Fréchet mean under translations

Following Pennec (2006), the Fréchet mean for a set of draws $\{x_i\}_{i=1,\dots,n}$ from a metric space $M$ with distance function $d$ is defined as the minimiser of the variance of the draws, $$ E[\...
100
votes
14answers
22k views

Statistics for mathematicians

I'm looking for an overview of statistics suitable for the mathematically mature reader: someone familiar with measure theoretic probability at say Billingsley level, but almost completely ignorant of ...
-1
votes
2answers
120 views

What is the likehood function in the noise free observation case

In the nonlinear Bayesian Tracking problem, if we consider the noise exists only in the state equation : x[k] = f(x[k-1],v[k-1]) where vk-1 here is an iid process noise sequence And we suppose that ...
6
votes
1answer
1k views

Proof of Karlin-Rubin's theorem

I asked this question on Math Exchange, but as I did not receive a successful answer, maybe you could help me. Karlin-Rubin's theorem states conditions under which we can find a uniformly most ...
5
votes
1answer
557 views

Earth mover/Wasserstein distance between a pdf and an empirical distribution

This question is inspired by this much older question: Convergence of an empirical distribution w.r.t. the Hellinger distance Let $P$ be a continuous probability distribution on a compact subset of $...
8
votes
3answers
9k views

What's an efficient way to calculate covariance for a large data set?

What is the best algorithm for computing covariance that would be accurate for a large number of values like 100,000 or more?
0
votes
0answers
16 views

Non-asymptotic tail-bounds for Hotelling $T^2$ statistic

Let $X_1,\ldots,X_n$ be an i.i.d sample from a distribution on $\mathbb R^p$ with mean $\mu = 0 \in \mathbb R^p$ and $p$-by-$p$ covariance matrix $\Sigma$ of rank $r \le p$. Consider the centered ...
0
votes
1answer
100 views

A problem related to the comparison of two integer-valued random variables

Consider an urn containing red, blue and green balls (the situation is the same illustrated in this post). Let $X$ be the non-negative, integer-valued random variable defined as the number of trials (...
4
votes
2answers
456 views

Practical bounds for the Wasserstein distance in 2 dimensions

Let $X_1,\dots,X_n$ be a set of independent samples of a distribution $\mu$ on the unit square, let $\hat\mu_n$ be the empirical distribution on the points $X_1,\dots,X_n$, and let $W_1(\mu,\hat\mu_n)$...
4
votes
3answers
111 views

Expected distance of nearest matching pair in the game of pairs

Recently I was playing several rounds of the game of pairs with my children. I was surprised that almost every time, one matching pair was adjacent (either next to each other in a row, or vertically). ...
1
vote
0answers
19 views

Practical statistics for queueing networks

There is a theory for queueing networks where we postulate some nicely behaving base distributions of arrival processes and service processes and then calculate the behaviour of the system. Now, in ...
1
vote
1answer
87 views

Minimum number of support vectors? [closed]

I'm learning SVM and its written everywhere that the minimal number of support vectors is 2? But I couldn't find any formal proofs of that. Why cant there be less than 2 support vectors? Can somebody ...
0
votes
0answers
44 views

Does lattice mod preserve direction?

For high enough dimension $n$ there are lattices $L_n$ in $\mathbb{R}^n$ whose Voronoi partition's base regions encompass all but a negligible proportion of a $(1-\varepsilon)$-ball, and also nearly ...
2
votes
0answers
69 views

How to formally connect the log-Euclidean and Riemannian metrics for Symmetric Positive Definite matrices?

I'm a statistician working on a research project dealing with metrics on SPD matrices, specifically the log-Euclidean $d_{LE}(X, Y) = \|\log(X) - \log(Y)\|$ and the Riemannian metric $d_{R}(X, Y) = \|\...
2
votes
0answers
54 views

Proof of a technical fact in the book of Schapire and Freund on boosting

Disclaimer: I asked this question on math.stackexchange.com two weeks ago but it has not been answered yet so I figured that I might as well try to also post it here. I am currently looking at ...
0
votes
1answer
60 views

Reconstructing Euclidian space from distance matrix

The setup. Let's say that we have a set of objects $O_i$ for which we have a dissimilarity measure $M(O_1,O_2)$. With this we can build a distance matrix $D_{ij}$. Let's also assume that we have NO ...
0
votes
1answer
96 views

Expected values of two non-negative, integer-valued random variables related to an urn problem

Consider an urn containing $c$ distinguishable balls, $\alpha$ of which are red, $\beta$ of which are blue, and $\gamma$ of which are green, and $\alpha+\beta+\gamma=c$. We assume $\alpha,\beta,\gamma&...
-1
votes
1answer
40 views

Characterisation of a superset of the simplex

Does there exist a nice description of the following set: \begin{equation} A:=\left\lbrace x\in\mathbb{R}^{n}\ \colon\ 0< x_{i}-\bar{x}+\frac{1}{n}< 1\ \text{for} \ i=1,\dots,n\right\rbrace, \...
3
votes
1answer
82 views

Non-asymptotic tail bounds for $D_{\text{Hellinger}}(P\|\hat{P}_N)$

Let P be a distribution on a finite set of size $k$ and let $\hat{P}_N=(N_1/N,\ldots,N_k/N)$ be the empirical distribution (frequencies) from a samples of size $N$. Consider the Hellinger distance ...
4
votes
1answer
319 views

Minimize the variance of a Boltzmann distribution

N.B.: Sorry for cross-posting from https://stats.stackexchange.com/posts/347804/edit (I realized it was the wrong venue for the question, but couldn't find an easy way to transfer the question here). ...
1
vote
0answers
44 views

Gaussian as a product of two independent random variables [duplicate]

Ideally what I am looking for two random variables, $X$ and $Y$ (if one is positive then that's even better) such that $Z=X\cdot Y\sim\mathcal{N}(0,1)$ where $X,Y$ are some distributions I can ...