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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5
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1answer
436 views

Square root of normal distribution

Let $X$ and $Y$ be independent random variates with the same probability distribution, $P(x)$. Assuming that the product $Z=XY$ is a random variate with normal distribution, say $$f_Z(x) = ...
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0answers
114 views

How to decide a value of learning rate for Stochastic Gradient Descent?

I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation, $w_{i+1} = w_i + -\eta \nabla ...
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0answers
45 views

Maximum likelihood estimation with several distributions

My question concerns using Maximum likelihood to estimate unknown parameters used by several (poisson) distributions. The parameters are the pairs $(a_1,b_1),\dots,(a_N,b_N)$, and for each pair ...
6
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1answer
267 views

Reference on (discrete) log-concave probability distributions

A discrete distribution $p$ over $\mathbb{N}$ is said to be log-concave if it satisfies the following conditions: The support of $p$ is a contiguous interval, i.e. $\exists a \leq b$ s.t. $p_i > ...
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99 views

Upper bound for chi-square divergence in terms of KL divergence

In my research I need an upper bound for chi-square divergence in terms KL divergence which works for general alphabets. To make this precise, note that for two probability measures $P$ and $Q$ ...
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73 views

How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...
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44 views

Taking power of the integrand in a Riemann-Stieltjie Integral

This is a problem I am trying to solve as part of a calculation for Value-at-Risk. Given that $P(X<x)=F(x)=\int_{\theta}F(x|\theta)dG(\theta)=1-\alpha$, where $F$ and $G$ are CDF's, is there a ...
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200 views

Inequality with CDF of order statistics

here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go: Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...
5
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2answers
398 views

Applications of cohomology to probability and statistics

Are there interesting/useful applications of cohomology (and homological algebra in general) to probability and statistics, or information theory? By "interesting/useful", I mean "not merely ...
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42 views

Is a conditional copula invariant under strictly increasing transformations?

currently I am working on conditional copulas and I have a theoretical question. In "An Introduction to Copulas", Nelsen (2006) there is a theorem (2.4.3) which says: Let $X$ and $Y$ be continuous ...
2
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2answers
121 views

What is the sum capacity of a scalar gaussian broadcast channel?

"On the Achievable Throughput of a Multiantenna Gaussian Broadcast Channel" by Giuseppe Carie and Shlomo Shamai talks, in part, about the following type of link (paraphrasing): A transmitter with ...
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32 views

Is it possible to distinguish between to edge orientation while learning a network structure?

I'm considering the case of learning bayesian network structure using a dataset $\mathcal{D}$ with scoring methods : $$\mathcal{G}^*=\max_{\mathcal{G}}\text{Score}(\mathcal{G}, \mathcal{D})$$ I'm ...
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1answer
54 views

assumptions on local rademacher complexities

A lot of the work on Local Rademacher complexities of Koltchinskii, and Bartlett for fast rates of convergence is based on Bousquet's version of Talagrand's inequality [1] (Theorem 2.11). However the ...
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105 views

What's the variance in the Six Degrees model?

Recall the six degrees of Kevin Bacon game. You can even play the game at The Oracle of Bacon, and their search works via Breadth First Search. I interpret the punchline as saying that if I start ...
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301 views

An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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72 views

Continuous self-information

Let $I(X,Y)$ be the mutual information between two continuous random variables $X$ and $Y$. We have $I(X,Y) = H(X)-H(X|Y)$, and setting $X=Y$ leads to $I(X,X) = H(X)-H(X|X)$. If $X$ was discrete, ...
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1answer
43 views

distances-based dispersion measuring approach

Is there any known approach or method to measure the dispersion of a set depending on the distances between its points (i.e.: without calculating the average or the mean) ? thanks.
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135 views

Bound the expectation of trace norm of random Hermitian matrix

Suppose $H_i$ are traceless $d\times d$ Hermitians, $X_i$ are Standard normal distribution for $1\leq i\leq d^2$. We would like to bound the following expectation on the trace norm ...
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1answer
161 views

1-wasserstein distance v.s. total variation distance

Suppose that $\mu_1$ and $\mu_2$ are two distributions defined on $\mathbb{R}^n$ and $\gamma$ is a symmetric distribution (around $0$) on $\mathbb{R}^n$ with compact support. Let $\gamma_x$ denote the ...
2
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1answer
304 views

Statistical distance between discrete and continuous distributions

Are there any statistical distance functions that are capable of comparing a continuous and a discrete distribution? From reading this list http://en.wikipedia.org/wiki/Statistical_distance the only ...
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62 views

Integral over conditioning variable of a Gaussian

The marginal of a multivariate Gaussian can be computed in closed form, i.e., $p(x) = \int_y \mathcal{N}((x,y);\mu,\Sigma)\ dy$ is simple. But what I need is $L(x) = \int_y \mathcal{N}((x\mid y); ...
3
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1answer
140 views

Does bounding moments make distributions close in total variation distance?

Let $W\sim\mathcal{N}(0,\sigma^2)$ be a "reference" Gaussian random variable. Suppose I have a set of distributions, $\mathcal{W}$, where $W_a\in\mathcal{W}$ if it satisfies the following criteria: ...
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70 views

Lower bound on difference between polynomials at moderate distance

Fix $r > 0$ and $k, n \in \mathbb{N}$. Also consider a function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$. Let $x_{1},\ldots, x_{n+1}$ be points chosen uniformly from $[-r,r]^{d}$. For $1 \leq i ...
3
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2answers
207 views

PDF of the product of normal and Cauchy distributions

I am having trouble in finding out the resulting PDF of the product of normal and Cauchy distributions. It turns out that we have a general formula for calculating the PDF of product of two random ...
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2answers
116 views

What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them? Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all ...
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4answers
754 views

What does it mean when we say we have computed a number to a certain accuracy using a probabilistic algorithm?

My intention is to ask a general question about probabilistic (Monte Carlo) algorithms. But to keep things simple, I will focus on a few specific examples. Let me start the discussion with ...
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0answers
79 views

limit distribution of multinomial distribution with increasing categories

If $\bf{X} \sim \text{multi}(n,p)$ with $k$ categories, we know $$ \sqrt{n}\left( \frac{\bf{X}}{n} - \bf{p} \right) \rightarrow^D N(0,\Sigma),$$ where $\bf{X}=(X_1,\ldots,X_k)^T$ and ...
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35 views

Inverse of the covariance of the estimate of a covariance

I have a covariance matrix, $V_{ij}$, which (for reasons that aren't important) I'm going to call the visibilities. I have an estimator for the visibilities $\hat V_{ij}$, and I've derived that the ...
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0answers
44 views

Kaplan-Meier estimator for mixtures

Let $\mathbf F=(F_1,\ldots ,F_m)^\top$ be a vector of continuous CDFs and $W$ is a matrix of weight coefficients, such that: $W\in \operatorname{Mat}_{n\times m}([0,1])$ $\forall i\in [n]:\sum ...
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28 views

Coordinates Poisson Cluster parent point

Is there any method to know the position of parent point in 'Poisson Cluster Process'? For information I use data with poisson distribution. data consist of (longitude, latitude, date). I want ...
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107 views

Sum of the entries of the inverse covariance matrix

Let $T \in\left(0,1\right)$, $n\in\mathbb{N}$ and $e_n = [1,\ldots,1]\in\mathbb{R}^n$. Consider the covariance matrix $\mathfrak{A}_n = ...
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31 views

How to sample from the ratio between two distributions?

I want to sample a lot of $\theta$s from the density function below: $$ r(\theta) = \frac{prior(\theta)}{Z}\frac{\int p(\theta,z_1)dz_1}{\int q(\theta,z_2)dz_2} $$ where $Z$ is the constant for ...
0
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1answer
224 views

Continuity of a Functional

A certain functional $T$ is defined as: $$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$ where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$. The result that above functional is ...
3
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0answers
150 views

Donsker's Theorem for triangular arrays

I should mention that I already posed this question on Math Stack Exchange, but didn't receive much feedback. Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given ...
8
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1answer
701 views

Table with the most seated customers in Chinese restaurant process

Suppose we have some initial configuration of people seated at some tables. We start taking new customers and seat them following Chinese restaurant process. Is there some known work on finding the ...
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0answers
97 views

In what sense is the Bayesian posterior mean a “convex combination”?

I asked this on math.stackexchange with no response, I'm hoping someone here might have something. Suppose I want to estimate $x \in \mathbb{R}^n$ from two signals with zero mean, normally ...
3
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1answer
109 views

Characterizing space that preserves positive-definiteness property

Given a symmetric positive-definite matrix $\Sigma$, consider the space $\mathcal{D}$ of diagonal matrices such that $\forall D\in\mathcal{D}$, the matrix $\Sigma-D\Sigma^{-1}D$ is positive definite. ...
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48 views

Finding a general form of the density function when we have a four dimensional random variable

Consider a subject having time of the specific event $T_i$, which is a single sample from a distribution $F_i$ with density $f_i$ and support $[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...
1
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1answer
102 views

Rademacher complexity of a Lipschitz class: Are the boundedness constraints necessary?

Consider the following function class: $F={f:R^d\rightarrow [a,b], f(x)=\sigma(w^Tx)}$ where $\sigma(.)$ is Lipschitz, and $w\in R^d$ is a parameter vector. The problem I'm working on is a machine ...
0
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1answer
92 views

Expectation of exp(-1/(ax^2)) when x is a standard normal variable and a>0 is a parameter [closed]

I would like to know if the mean value of $\exp(-1/(ax^2)) $ when $x \sim N(0,1)$ and $a>0$ is a parameter is known.
4
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1answer
128 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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2answers
67 views

Is it possible to find an asymptotic distribution for the LRT without the ML estimators being consistent?

I'm reading a comment(last page) to a paper, and the author states that sometimes, even though the estimators (found by ML or maximum quasilikelihood) may not be consistent, the test may be ...
3
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2answers
320 views

Consistent price index

This question came out of a discussion with a colleague from economics about price indices. Here is MattF's formulation of the question which differs somehow from the original problem. Let ...
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0answers
62 views

Is there an efficient algorithm for sampling from the negative hypergeometric distribution? [closed]

I'm writing a small statistics library currently. One of the algorithms I'm implementing has two variants: one that samples the hypergeometric distribution and one that samples the negative ...
2
votes
1answer
155 views

Proof for power-law tail of Poisson-Dirichlet distribution (Pitman-Yor process & Zipf's law)

I'm trying to understand the motivation of using Pitman-Yor (PY) processes in language modeling, in particular Teh's hierarchical LM based on PY processes. A motivation frequently stated in research ...
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2answers
138 views

Is a function of complete statistics again complete?

suppose $T$ is a complete stats for a parameter $\theta$. Is any function $f(T)$ again complete? It sounds weird but the definition seems to confirm that $f(T)$ is indeed complete..
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36 views

Bounds on Product of CDF or Beta function

I have functions of the form \begin{align} I_i = \int_0^\infty F_0(x)^aF_1(x)^b(1-F_0(x))^c(1-F_1(x))^ddF_i(x)~~~~i = 0,1 \end{align} $F_0(x)$ and $F_1(x)$ are CDFs corresponding to the random ...
2
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0answers
78 views

Mean and variance of a general multivariate skew normal distribution

I have a problem about a general multivariate skew normal distribution. There is a $p\times 1$ vector, $\mathbf{y}=(\mathbf{y}_1',\mathbf{y}_2',\ldots,\mathbf{y}_n')',p>n$, which has the density as ...
6
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1answer
166 views

Closure of random rotations

Are matrix Fisher random variables closed under multiplication? For those unfamiliar with the jargon, let me unpack the terms above and repose my question. This is a question about probability ...
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3answers
321 views

A. Markov's papers?

A. Markov published several papers on his chains, starting in 1906, so it is written, in the journal: (1) Извѣстія Физико-математического общества при Казанском университете I am surprised by the ...