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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.

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0answers
18 views

Standard normal variable times an independent random variable [on hold]

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 [on hold]

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
167 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
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0answers
18 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
503 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 ...
1
vote
0answers
32 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 ...
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 ...
-2
votes
0answers
23 views

Kolmogorov–Smirnov test in logistic regression [on hold]

When applying KS-test (as goodness-of-fit test) on logistic regression (class: 0,1), where x-axis = probability of being classified as class 1, sorting ascendingly. Here are the 2 questions: Why are ...
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
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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
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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 ...
0
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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:...
2
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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 \...
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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(...
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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 $y \sim \mathcal{N}(0,1)$.
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0answers
27 views

Measurement theory and sample size calculation for multivariate testing

Let $\mathbf{Y}$ be a vector of independent, normally-distributed random variables. Let $S_1$, $S_2$ and $S_3$ be three non-overlapping samples of sizes $N_1$, $N_2$ and $N_3$, respectively. Let $M_A$ ...
0
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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 ...
1
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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 ...
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 (...
1
vote
1answer
86 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 ...
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
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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
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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 ...
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 ...
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). ...
0
votes
1answer
95 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
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 ...
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 ...
3
votes
1answer
104 views

Why the Fisher information matrix is equal to the Hessian matrix of the Kullback–Leibler distance at the true parameter?

I'm reading 《Algebraic geometry and statistical learning theory》.My problem is why the Fisher information matrix is equal to the Hessian matrix of the Kullback–Leibler distance at the true parameter?...
2
votes
1answer
70 views

upper bound on power of neyman-pearson hypothesis test

Let $H_0$ and $H_1$ be two distributions. The Neyman-Pearson lemma says that of all rejection regions $R$ with fixed probability $\alpha$ under $H_0$, the one with maximal probability under $H_1$ is ...
1
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0answers
33 views

Choice of residual function for least squares error minimization

Good morning, I have the a set of data $(\sigma,D,\alpha_0)_i$, $i=1...n$ data. I want to determine two parameters $K_{IC}$, $C_f$ in the basic equation given as $K_{IC} = \sigma \sqrt{D} k_0(\...
3
votes
1answer
108 views

Largest eigenvalues of a (random) correlation matrix?

I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some ...
0
votes
0answers
67 views

Asymptotic distribution of $n\mathbb E_{\hat{P}_n}[g(Z;\theta)]^T\operatorname{Cov}_{\hat{P}_n}[g(Z;\theta)]^{-1}\mathbb E_{\hat{P}_n}[g(Z;\theta)]$

Setup This question is a followup on this question. I'm interested in the asymptotic distribution of certain quadratic forms. So, let $Z$ be a $p$-dimensional random vector with (unknown) ...
2
votes
1answer
66 views

How does a statistical divergence change under a Lipschitz push-forward map?

Let $\mu, \nu$ be two probability measures on a space $X$ (assume Polish space). $T: X \rightarrow Y$ is a Lipschitz-map that acts as a push-forward on these measures; let $\mu^\prime = T_{\#\mu}$ and ...
4
votes
0answers
189 views

What happens in the martingale CLT if I norm by the conditional variance instead?

TLDR: I'm a statistician (bear with me!) trying to use the martingale CLT but I only can estimate the conditional variance instead of the unconditional one. Can I do anything to get a CLT with norming ...
1
vote
1answer
51 views

References for Hellinger distance/affinity involving mixture distributions

For two continuous probability distributions $F,G$ and their densities, $f,g$, the (squared) Hellinger distance/affinity is given by $d^2_H(F,G)=1-\int_{\mathbb{R}} \sqrt{fg}~dx$. Suppose that $f,g$ ...
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, ...
2
votes
0answers
76 views

Open problems in Monte Carlo Simulation [closed]

I want to know some open problems in Monte Carlo Simulation, which is being studied or in a stalemate. Could you please give me some advice? Thanks a alot
3
votes
2answers
228 views

Effect of perturbing the atoms of a measure on the Wasserstein distance

Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
1
vote
1answer
141 views

Asymptotic distribution of $\mathbb E_{\hat{P}_n}[Z]^T\operatorname{Cov}_{\hat{P}_n}[Z]^{-1}\mathbb E_{\hat{P}_n}[Z]$

Under very general conditions on the random $p$-dimensional vector $Z$, what can be said about the asymptotic distribution of the (random) scalar quantity $R_n := \mathbb E_{\hat{P}_n}[Z]^T\...
1
vote
0answers
16 views

Large Deviations Rate of Convergence and Robbins Monro

I am looking for a result/paper (if there is any) on the large deviations rate of convergence of the Robbins-Monro (RM) algorithm. Specifically, given $X_k \rightarrow X$ a.s. in the RM algorithm, I ...
1
vote
2answers
122 views

lower bound the probability of at least L collisions

Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$. If we now ask ...
2
votes
1answer
232 views

Why the VC dimension of triangles in 2D space is not greater than 7?

I understand that there are sets of 7 points on a circle that can be fully shattered using triangles.But, it is not clear to me why it cannot shatter 8 points. Is there any intuitive way of arriving ...
0
votes
1answer
115 views

Is the normal product distribution sub-gaussian?

Consider the normal product distribution, which is the distribution of the product of two or more independent normal variables. Particulary, focus in the case where the multiplied normal variables are ...
3
votes
1answer
143 views

Why does the assumption $|U_t| \le \frac1{p_{\min}}$ work in this paper?

I am reading a 2009 paper right now "Importance Weighted Active Learning" and on page 5, there is a theorem that uses the inequality $|U_t| \leq \frac{1}{p_{\min}}$. I am not sure how the paper found ...