# 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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### Maximum entropy probability distribution with fixed interval and variance?

What is the maximum entropy probability distribution if the support is a fixed interval (e.g. [-1,1]) with an already known variance? If we know the support is a fixed interval, then the maximum ...
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### Reverse Pinsker's inequality for smooth density classes

Suppose we are given a class of probability density functions $\mathcal{F}$ so that for every $f \in \mathcal{F}$ we have $\alpha \leq f \leq \beta$ for some positive $\alpha, \beta \in \mathbb{R}_+$ ...
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### Quotient of estimators

Say $A$ is a set of a finite number of samples, and $\hat{\mu}_A$ and $\hat{\sigma}_A$ are unbiased estimators (computed over $A$) of $\mu$ and $\sigma$ which are some distinct population statistics. ...
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### Concentration result for self-normalized empirical process

In Theorem 1.1 of this paper by Bercu, Gassiat and Rio, a concentration result is derived for the 'self-normalized' empirical process. Specifically, suppose that $(X,X_n)_{n \ge 1}$ is a sequence of i....
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### A property of the distribution related to stochastic ordering

Let $X$ be a random variable with a symmetric support $S\subset[-M,M]$ for some $M>0$. (i.e., if x is a point of increase of CDF $F_X(\cdot)$, so is $-x$.) Has the infimum value of $c$ such that \...
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### Asymptotic stochastic ordering for weighted sum of i.i.d. random variables

Are you aware of any literature focusing on the conditions such that for two i.i.d. sequences of discrete r.v.'s $\{X_n\}$ and $\{Y_n\}$, a_1X_1+a_2X_2+\ldots+a_nX_n\geq_1 a_1Y_1+...
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### Gibbs Priors form a Martingale

I am working on adapting variational inference to the recently developed Martingale posterior distributions. The first case, which reduces the VI framework to Gibbs priors, is proving hard to show as ...
1 vote
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### Joint moments like $\tau(XYXYXY)$ in terms of individual moments of free variables $X,Y$

Terry Tao RMT book has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5 $$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$ ...
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### Evolution of the empirical mean of a list as we remove elements proportional to their value

Consider a list of $N$ integers $k_1,k_2,\dots k_N$, drawn independently from some distribution $P(k)$ with $k_i \geq 1$. We denote its mean with $\langle k\rangle=\sum_{k=1}kP(k)$. The first two ...
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### Points based partial ranking

I want to rank a population $P=\{P_1,\ldots,P_n\}$. I am given a set $R=\{R_1,\ldots,R_k\}$ of partial rankings. The partial rankings may have varying sizes (e.g. the first ranking ranks only 8 ...
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### Minimax optimal multiple hypothesis test

Let us consider the following two-player game between Chooser and Guesser. There is a finite set $\Omega$ and $k$ probability distributions on $\Omega$, denoted by $\mathcal{P} =\{P_1,\ldots,P_k\}$. ...
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### Sum of arrival times of Chinese Restaurant Process (CRP)

Suppose that a random sample $X_1, X_2, \ldots$ is drawn from a continuous spectrum of colors, or species, following a Chinese Restaurant Process distribution with parameter $|\alpha|$ (or ...
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### Continuous-time Wold decomposition

I'm looking for a reference for the Wold–Zasukhin decomposition in continuous time for stationary random processes on the real line. I am aware of the classic result in the book from Rozanov, which ...
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1 vote
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### Approximation of continuous function by multilayer Relu neural network

For continuous/holder function $f$ defined on a compact set K， a fix $L$ and $m_1,m_2,\dots,m_L$, can we find a multilayer Relu fully connected network g with depth $L$ and each $i$-th layer has width ...
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1 vote
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### conjecture for general form of minimax estimator

I had previously posed an overly ambitious version of this conjecture here, Form of minimax estimator, which was quickly shot down by Václav Voráček (on twitter) and Iosif Pinelis (MO answer in the ...
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This question is inspired by this paper. For those who are interested in more details and applications about record index and record values, you can find them in the paper. Let $X=\{0, 1, 2, \cdots, N\... 1 vote 1 answer 325 views ### Form of minimax estimator Let$\Delta$be the set of all probability distributions over$\mathbb{N}=\{1,2,\ldots\}$and fix some$\mathcal{P}\subseteq\Delta$. Suppose additionally that$\Delta$is endowed with some norm$||\...
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Let $(\Omega, \mathcal{F}, \mathbb{P})$ denote a probability space supporting a standard Brownian motion $B$. Let $\Pi=\{\pi_n : n \ge 0\}$ denote the sequence of dyadic uniform partitions of the ...