# 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.

309 questions
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### Research situation in the field of Information Geometry

I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field. I have read (4) and parts of (3). ...
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### Random Distance Matrices

My question is motivated by the following recent paper: http://arxiv.org/abs/1110.6333 Assume you have a metric space $(X,d)$ equipped with a Borel probability measure $\mu$. We can further assume ...
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### Pettis Integrability and Laws of Large Numbers

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...
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### Given that a conditional measure is Gaussian, how bad can the original measure be?

Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
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### Taylor approximation of a function of a random variable

Suppose we have a random variable $X$ and a smooth function $g$. We want to calculate the expectation value $\mathbb{E}(g(X))$. To be able to write down at least an approximate solution, we perform a ...
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### Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance

Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
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### Hierarchical Random Walk (also known as Hierarchical Hidden Markov Model)

Let us consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (https://en.wikipedia.org/wiki/...
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### 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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### Inverse moment of the number of inversions of a permutation

Let $\pi$ be a permutation of $\{1,2,...,n\}$. A pair of elements ($\pi_i$,$\pi_j$) is called an inversion if $i$ $>$ $j$ and $\pi_i$ $<$ $\pi_j$. The total number of inversions in $\pi$ is ...
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### Positive estimator

Suppose that one knows how to generate (independent) random samples $X_1, X_2, \ldots$ distributed as the random varable $X$ with $\mathbb{E}[X]=\mu \in \mathbb{R}$. It is then easy to construct an ...
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### Another generalized coupon collector's problem

Suppose there are $L$ types of coupons, the probabilities that they appear are $a_1,a_2,\ldots,a_L$ respectively, $\sum_i^La_i=1$. Each of them is associated with a constrain number $d_i,i=1,\ldots,L$....
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### Multidimensional Berry-Esseen for probability density functions

This is a follow up to this recent question: Berry Esseen type result for probability density functions There exists a multidimensional version of the usual Berry-Esseen theorem (for cumulative ...
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### Missing mass estimate

Let $S$ be a finite set with probability distribution $P$. Define the random variable $m_i$ to be the "missing mass" after seeing $i$ iid samples from $S$ under $P$. That is, $m_i$ is the total mass ...
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### 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 ...
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### Total Variation distance of polynomials of Bernoulli R.V.s

Let $X_i, Y_i$ be i.i.d Bernoulli $0/1$ random variables with $\mathbb{E}[X_i] = p$ and $\mathbb{E}[Y_i] = q$. Let \begin{align*} X &= X_1 X_2 + Χ_2 Χ_3 + \ldots +X_{n-2} X_{n-1}+ X_{n-1} X_n\\...
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### What is the entropy of binomial decay?

Let's play a game. I start with $N$ indistinguishable tokens, and I wait $T$ turns. Every turn, each token has probability $p$ of disappearing. I want an analytic formula for the entropy of this ...
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### Distributions over permutation groups $\mathcal{S}_n$

Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...
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### Probability distribution function for singular value sum of Gaussian random matrix

Let $\mathbf{X}$ be an $N \times N$ random matrix with IID Gaussian entries. They can be standard normal, but $N$ is not large: that is $N$ $<$ 6, typically. Call its singular value decomposition (...
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### Sum of non-identical categorical random variables

Is there a named distribution for the sum of non-identical categorical random variables? When the categorical variables are i.i.d., the sum is a multinomial distribution. When the categorical ...
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### Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
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### Concentration of sum of independent random variables

Let $X_1, ..., X_n$ be i.i.d. sub-Gaussian random variables with mean $0$ and variance $1$. That is, we have $\Pr[|X_i| > t] \leq \exp(1-t^2/K^2)$ for all $t>0$ and a parameter $K$. Then we can ...
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### Bochner's theorem, in stages

Bochner's theorem (for the real line version) asserts an infinite tower of inequalities, as a positivity condition. Taking each one, what do they mean, in an elementary fashion (at least at the start)?...
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### Expected operator norm of inverse Wishart matrix

Let $W\sim W_p(n,I)$ be a white $p\times p$ Wishart matrix, and assume $n>p+1$, which ensures that $W$ is invertible almost surely. Let $\|W^{-1}\|_{\text{op}}$ be the operator norm (maximum ...
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### Has anyone used reflection in bootstrapping methods for one parameter hypothesis tests?

Here's my idea for a bootstrapping method for testing hypotheses about one parameter. Please tell me if you have seen this somewhere before. If not, I'd appreciate pointers for direction of further ...