# 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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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

**1**answer

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

**0**answers

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

**0**answers

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)

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**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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 ...

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**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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 ...

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vote

**2**answers

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

**0**answers

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

**4**answers

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 ...

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votes

**1**answer

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

**0**answers

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 $z:=(z_1,\ldots,z_n)$.
...

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**0**answers

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

**1**answer

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 ...

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votes

**0**answers

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:...

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votes

**1**answer

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

**1**answer

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

**0**answers

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 ...

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votes

**0**answers

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 \...

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votes

**4**answers

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 ...

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**0**answers

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

**1**answer

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(...

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votes

**1**answer

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 $y \sim \mathcal{N}(0,1)$.

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vote

**0**answers

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[\...

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votes

**14**answers

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 ...

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votes

**2**answers

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 ...

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votes

**1**answer

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

**1**answer

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 $...

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votes

**3**answers

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?

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votes

**0**answers

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

**1**answer

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 (...

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votes

**2**answers

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 $W_1(\mu,\hat\mu_n)$...

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votes

**3**answers

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). ...

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vote

**0**answers

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 ...

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vote

**1**answer

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 ...

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votes

**0**answers

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 ...

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votes

**0**answers

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) = \|\...

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**0**answers

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

**1**answer

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

**1**answer

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&...

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votes

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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 ...