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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0answers
148 views

Maximization of a total variation distance subject to another total variation distance in Markov chain

Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...
8
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7answers
3k views

Lower bound for sum of binomial coefficients?

Hi! I'm new here. It would be awesome if someone knows a good answer. Is there a good lower bound for the tail of sums of binomial coefficients? I'm particularly interested in the simplest case ...
2
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1answer
254 views

Asymptotic behavior of max of chi-squared distribution

Suppose $X_{\max}$ is the maximum in a sequence $X_1,X_2,\ldots,X_n$ where each $X_i\sim\chi^2_k$ is an i.i.d. chi-squared random variable with $k$ degrees of freedom. Since chi squared distribution ...
21
votes
2answers
868 views

Drawing natural numbers without replacement.

Suppose we start with an initial probability distribution on $\mathbb{N}$ that gives positive probability to each $n$. Let's call this random variable $X_1$ so we have $P(X_1=n)=p_{1,n}>0$ for all ...
13
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7answers
2k views

Correlation and Causation. When can we believe correlation (reasonably, at least) imply causation

We always hear, when reading on correlation, that "correlation does not imply causation." Still, I have never seen any source that tries to answer the question of when can we reasonably conclude a ...
14
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2answers
539 views

Persistent homology of Gaussian Fields in Euclidean space

If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be ...
7
votes
3answers
1k views

randomness in nature [closed]

What is the explanation of the apparent randomness of high-level phenomena in nature? For example the distribution of females vs. males in a population (I am referring to randomness in terms of the ...
12
votes
1answer
636 views

Error to sum of Euler phi-functions

The number theory identity $\phi(1) + \phi(2) + \dots + \phi(n) \approx \frac{3n^2}{\pi^2}$ can be interpreted as counting relatively prime pairs of numbers $0 \leq \{ x,y \} \leq n$ . Has anyone ...
2
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1answer
3k views

Conjugate prior of the Dirichlet distribution?

What is the conjugate prior distribution of the Dirichlet distribution?
4
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3answers
409 views

Incremental entropy computation

After a quick internet search I found no method for incremental entropy computation. Question 1 Let $\{x_i\}_{i=1}^n$ and $\{x_i\}_{i=1+n}^{n+m}$ be two samples and let $S_i^j:=\sum_{k=i}^j x_k$. ...
11
votes
2answers
694 views

Estimate rate of real correct/wrong from 4 answers quiz.

I recently read that one in ten students think the first man on the moon was Buzz Lightyear, a Toy story cartoon. I'm not here to discuss the data in itself, rather, this reading got me into a problem ...
7
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4answers
954 views

averages of Euler-phi function and similar

What are the odds two numbers are relatively prime? This is known to be $\frac{6}{\pi^2}$. The proof involves calculating averages of the Euler phi function. \[ \phi(1) + \phi(2) + \dots + \phi(n) ...
4
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2answers
1k views

Elo Rating System Help with the Maths around number of matches

I'm creating a system that will allow people to rate images. My idea is to use an Elo Rating system (http://en.wikipedia.org/wiki/Elo_rating_system) for each image and then use crowdsourcing to have ...
1
vote
3answers
89 views

How to estimate the entropy of a distribution on a power set?

Given a probability distribution $(X,p)$, its entropy is defined as $H=-\sum_{x\in X} p(x)\log p(x)$. Given a sample of observations $x_n,n=1..N$, one can estimate $p(x)=\frac{\#\{i:x_i=x\}}{N}$ and ...
0
votes
1answer
914 views

Unbiased estimate of the variance of an *unnormalised* weighted mean

I have a follow-up question to this one: unbiased estimate of the variance of a weighted mean Specifically, how do I generalise the result given here (and on Wikipedia) for the unbiased sample ...
5
votes
3answers
252 views

Concentration inequalities for the maximum of the rescaled/normalized sum of iid random variables

I am interested in concentration inequalities for the maximum of the rescaled/normalized sum of iid random variables. Let $X_1,..., X_n$ be i.i.d random variables, $S_n$ their centered sum and $M_n$ ...
5
votes
2answers
381 views

Is the Binomial Expectation of Convex Function Convex in p?

Suppose $X$ has a binomial distribution with success probability $p$ and $n$ trials and let $h(\cdot)$ be a positive convex real-valued function. Is the function $g(p)=\mathbb{E}[h(X)\ |\ p]$ convex ...
3
votes
2answers
745 views

expected values over binomial distributions

In some works of economics/risk analysis etc., I have seen situations where people take the expected value of a function (such as a utility function/cost function) over a binomial distribution: ...
2
votes
1answer
141 views

Empirical estimator fot the total variation distance on a finite space

I have two probability measures $p$ and $p'$ on a finite set $X$ which I do not know precisely, but which I can sample from. I would like to estimate their total variation (omitting multiplier $2$): ...
2
votes
0answers
473 views

A combinatorial bound involving Stirling numbers of the second type

My previous question was solved in a very elegant way, hopefully this (seemingly more complicated) case is also easy for experts. I need the inequality ...
2
votes
1answer
375 views

Multinomial transformation for matrices

Suppose we have a vector of probabilities $\mathbf{p}=(p_1,...,p_n)$, where $p_i>0$ for $i=1,...n$ and $\sum p_i=1$. Define new vector $\mathbf{r}=(r_1,...,r_{n-1})$ in a following way: ...
1
vote
1answer
65 views

What is known about the distribution of the errors in empirical approximation of a CDF?

Let $X_1,X_2,\ldots,X_n$ be i.i.d. random variables in $\mathbb{R}$ with common cumulative distribution function (CDF) $F(x)$. The empirical approximation to $F(x)$ is defined as follows: ...
1
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0answers
105 views

How far away is the maximum of $n$ i.i.d. chi-squared random variables from the rest of the sequence as $n$ gets large?

Suppose that I have a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom $X_1, X_2, \ldots, X_n$, and denote $X_{\max}=\max(X_1, X_2, \ldots, X_n)$. Let $k$ be increasing ...
0
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1answer
55 views

Estimating the variance of error in empirical approximation to a distribution

Let $X_1,X_2,\ldots,X_n$ be i.i.d. random variables in $\mathbb{R}$ with common cumulative distribution function (CDF) $F(x)$. The empirical approximation to $F(x)$ is defined as follows: ...
0
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0answers
102 views

Behavior of the sum of the exponents of chi-squared random variables normalized by their maximum

Let $X_1,X_2,\ldots,X_n$ be a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom, and denote by $X_\max$ the maximum of this sequence. Furthermore, let $k=\omega(1)$ ...