Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies.

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-2
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0answers
21 views

AQA A Level Normal Distribution [on hold]

The question goes like: A wholesaler decides to grade such oranges by weight. He decided that the smallest 30% should be graded as small, largest 20% as large and in between as medium. The ...
-6
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0answers
25 views

Trouble with an assignment [on hold]

Can anyone please be kind enough to help me with this. On one shelf there are 5 hardcover books and 6 paperbacks and on the other shelf there are 7 hardcover and 4 paperback. From the first shelf ...
3
votes
1answer
63 views

Stochastic Covering Number of a Convex Set

Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^2$ ...
0
votes
0answers
45 views

continuous vs discrete random walk [on hold]

For 1D random walk in discrete case the probability $P_N(X)$ of finding walker at position $X$ after $N$ steps has a binomial distribution, moreover when $N+X$ is odd then probability is 0. Let's ...
0
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0answers
34 views

Maximal Correlation with Weak Gaussian Perturbation

Let a pair of random variables $(X,Y)$ be continuous random variables (i.e., they both have density with respect to Lebesgue measure) with joint distribution $P_{XY}$. The maximal correlation ...
2
votes
1answer
65 views

Covariance matrix as optimization problem solution?

I have seen the expectation of a random vector expressed as the solution to the optimization problem: \begin{equation} \mathbb{E}[X]=argmin_{v \in \mathbb{R}^n}\mathbb{E}[\|X-v\|_{l^2}^2](:= ...
5
votes
2answers
103 views

Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian

This is a question related to the statistical model behind independent component analysis (ICA). We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...
1
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0answers
46 views

Simulate a graph from a certain distribution

I am wondering if anyone can indicate whether the following is a solved problem. I don't care about time of the algorithm currently. Consider a general probability distribution F on simple graphs ...
3
votes
1answer
149 views

A lottery on coins in a convex set

You play the following game. You get $4n$ gold coins and have to arrange them in the unit square in general position (no two coins have the same x or the same y coordinate). Call this set of coins ...
4
votes
1answer
104 views

Hellinger integral for the Student/Cauchy family

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral is $H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$. Let now $p$ be ...
5
votes
0answers
81 views

Maximal Correlation versus Correlation Coefficient When one RV is Gaussian

Let a pair of random variables $(X,Y)$ be continuous random variables (i.e., they both have density with respect to Lebesgue measure) with joint distribution $P_{XY}$. The maximal correlation ...
4
votes
1answer
79 views

Negative population variable importance

I asked this question on stats.stackexchange and even elsewhere, but it never received an answer. I just state the probabilistic problem here. It is about the optimality of the conditional ...
1
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0answers
17 views

How to find a subset of a matrix that has minimum condition number? [duplicate]

Suppose matrix $A$ is consist of M column vectors, how can we find a subset $B$, consisting of N column of $A$ (N<M), that has minimum condition number (the ratio of maximum singular value by minimum ...
1
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0answers
89 views

Monotonicity of the Hellinger integral/distance

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are $H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and ...
1
vote
1answer
58 views

“Convergence speed” results for the Langevin process

The Langevin process is defined by the following stochastic differential equation: $$ \dot X = - \nabla \phi + \sqrt 2 dW_t $$ Its equilibrium distribution is the following: $$ p_\infty (x) \propto ...
8
votes
2answers
180 views

Constructing an independent uniform random variable from two independent ones

Does there exist a continuous (differentiable) function $h:[0,1]\times [0,1] \to [0,1]$ such that if $\alpha,\beta\in [0,1]$ are independent and uniformly distributed on $[0,1]$, the random variable ...
2
votes
0answers
64 views

Laplace transform of a integral function of CIR/CEV process

The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...
17
votes
3answers
681 views

Deceptively simple inequality involving expectations of products of functions of just one variable

For a proof to go through in a paper I am writing, I need to prove the following deceptively simple inequality: $$(*)\qquad E(X^a) E(X^{a+1}\log X) > E(X^{a+1})E(X^a\log X) $$ where $X>e$ has ...
0
votes
0answers
90 views

Comparison of Parameter estimation using maximum likelihood and Maximum entropy

I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
4
votes
1answer
157 views

Random walk with continuously distributed steps on [-1,1]

A simple random walk $S_n = X_1 +\cdots +X_n$, where $P(X_i = 1) = p \not = 0.5$ and $P(X_i=-1)= q \triangleq 1-p$, admits the following probability $$P(S_n \textrm{ reaches } a \textrm{ before} -b) ...
0
votes
0answers
34 views

Calculate the KL divergence between two transition matrices

I want to calculate how different two markov transition matrices are. For example: $\begin{pmatrix} .2 & .8 \\ .1 & .9 \end{pmatrix}$ and $\begin{pmatrix} .3 & .7 \\ .1 & .9 ...
0
votes
2answers
51 views

A way to possibly calculate one Binomial CDF function from another closely related one?

Let $y < z$ be two numbers between $0$ and $1$, is there a way to relate the CDF functions $F_{n,y}(s)$ and $F_{n,z}(s)$... or approximate one from another, without just saying $F_{n,z}(s) \le ...
1
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0answers
53 views

A variance-preserving Boolean function [closed]

Let a random variable $X$ be given with $P_X$ supported over $\mathcal{X}$. What are the necessary conditions for the existence of a boolean function $f:\mathcal{X}\to \{0,1\}$ such that ...
4
votes
1answer
174 views

How are the real-space RG transformations defined?

I'm reading Shang-keng Ma's book Modern theory of critical phenomena, and I'm a bit confused as to how the real-space RG transformations are defined. Ma basically says that these transformations are ...
5
votes
1answer
278 views

Hypothesis test beyond simple hypotheses (mathematical statistics)

In mathematical statistics, the following problem (simple hypothesis test) is considered: given a data sample, test the hypothesis $H_0$ stating that all sampled values are values of a random variable ...
1
vote
0answers
36 views

Relation between Aitchison Distance on a Simplex and Geodesic distance on the multinomial manifold [closed]

I am trying to understand the difference/relation between the Aitchison distance on a simplex $$\left[ \sum^D_{k=1} (\log{\frac{x_{ik}}{g(\mathbf{x}_i)}} - \log{\frac{x_{jk}}{g(\mathbf{x}_j)}})^2 ...
0
votes
0answers
76 views

Hoeffding's lemma for unbounded r.v with bounded exponential map

Let $X$ be a real r.v with $E[e^{\lambda X}] < \infty $ for all $\lambda \in [-c,c]$. Is it possible to get an Hoeffding's lemma like bound on $E[e^{\lambda(X-EX)}]$. That is, an upper bound: ...
1
vote
0answers
98 views

Chain Rule for Maximal Correlation

Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...
1
vote
1answer
74 views

An inequality for Maximal Correlation over a Markov Chain

Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...
1
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0answers
50 views

Monte Carlo Simulation - efficient simulation of tail outcomes [closed]

When running Monte Carlo type simulations in situations where you're only interested in tail outcomes, do you know of a way to only simulate those outcomes, so that you can come up with more reliable ...
1
vote
1answer
109 views

Independence of two random variable

Let $W$ and $S$ are two positive valued continuous random variable. Suppose $g: [0,\infty)\rightarrow [0,\infty)$ is a convex function with a constraint that $g$ can't be of the form $g(x)=cx$, $c$ ...
10
votes
1answer
152 views

Probability distribution derived from gamma function - does it have a name?

Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$. Now, let's fix $\sigma$ and let t vary. Then consider the following expression: $$|\Gamma(\sigma+it)|^2$$ For any choice of ...
3
votes
0answers
30 views

Bound on principal angle of uniform random subspaces of different dimensions?

This paper derives the distribution of the largest principal angle between two subspaces sampled (independently) uniformly from the Grassmanian manifold of $p$-dimensional subspaces in $\mathbb{R}^d$, ...
5
votes
1answer
269 views

An Inequality of KL Divergence

Given two probability distributions $P$ and $Q$ defined over a finite set $\mathcal{X}$, one can define the KL divergence between $P$ and $Q$ as $$D(P||Q):=\sum_{x\in ...
0
votes
0answers
37 views

How to prove a CAN estimator is also root n-consistent?

Given a consistent asymptotically normal estimator $\hat{\theta_n}$ of an unknown parameter $\theta$, how to prove that $\hat{\theta_n}$ is also a $\sqrt{n}$-consistent estimator of $\theta$? ...
2
votes
1answer
89 views

Convergence of a test statistic

I'm reading a paper of Shao and Zhang: Testing for Change Points in Time series. In this paper they claim the following: The are testing whether there is a change in the mean of a time series. So ...
4
votes
2answers
195 views

Prediction with positive weights?

Consider a covariance function (positive definite function) on $\mathbb{Z}$: $$ \gamma(k)=(1+|k|)^{-\alpha},\quad \alpha>0. $$ It is guaranteed to be positive definite by Polya's criterion ...
1
vote
1answer
116 views

Large deviations for sums of random variables whose correlation function decays exponentially

Let $X_1, \ldots, X_N$ be a string random variables taking values $X_i \in [-1,1]$ and jointly distributed according $P(X_1, X_2, \ldots, X_{N-1}, X_N)$, which is invariant under cyclic permutations ...
1
vote
0answers
59 views

Characterization of certain families of functions

For $R$ equal $\mathbb{R}$ or $\mathbb{Z}$, let $D^+_R:=\{(x,y)\in R^2\colon x<y\}$. For each natural $n$, let $F_{n,R}$ denote the set of all Borel-measurable functions $f\colon ...
9
votes
1answer
210 views

Normal approximation of tail probability in binomial distribution

My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...
5
votes
2answers
108 views

Gaps between descending order statistics

Let $\{X_{1},X_{2},\cdots,X_{n}\}$ be a random sample of size $n$. Denote $(X_{(1)},X_{(2)},\cdots,X_{(n)})$ to be its descending order statistics. Define gap $g_{i}(n)$ to be ...
1
vote
0answers
77 views

Finding an error estimation for the De Moivre–Laplace theorem with Stirling's formula

Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De ...
3
votes
1answer
102 views

$\int_0^t f(s)\,dB_s$ normally distributed, mean and variance

Suppose that $f(t)$ is a (non-random) continuous function on $[0, \infty)$. Let$$Z_t = \int_0^t f(s)\,dB_s.$$ How do I see that $Z_t$ is normally distributed? What is the mean and variance? I need ...
1
vote
1answer
83 views

KL divergence Inequality

I am trying to find a proof for the following inequality, but I did not get anywhere following the references from the paper I was reading. Consider two probability measures $P$ and $Q$ both ...
1
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0answers
35 views

Adding weights to the Brier score

Fix $n > 0$, and consider the space $\cal P$ of probability functions defined over the Boolean closure of a fixed $\cal S = \{ s_1, \ldots, s_n \}$. The Brier score of $P \in \cal P$ at $s_i \in ...
2
votes
0answers
112 views

Convergence rate of Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...
3
votes
1answer
123 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 ...
5
votes
2answers
142 views

Reference to iterated logarithm law and Smirnov law of empirical CDF

I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws. Let ...
0
votes
0answers
32 views

Consistency of M-estimators when the constraint set also has to be estimated

Let $K \subset \mathbb R^n$ compact and convex. Also let $H$, $G_i, \; i \in \{1,\dotsc,m\} $: $K \to \mathbb R$ be convex functions. Assume we have the following convex optimization problem: $$ ...
2
votes
1answer
103 views

Is there any parameter space of Cramér–Rao_bound

It is known that Cramér–Rao_bound is the lower bound of variance of a parameter. A useful link is https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound There is also a term called ...