# 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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### Continuity of Nash equilibrium for a family of games

The question may be too vague, but ultimately in search of various (counter)examples or theorems to exhibit the following: Do continuous families $t\mapsto G_t$ of "games" (say each $G_t$ is ...
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### Existence of disintegrations for improper priors on locally-compact groups

In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically ...
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### VC-based risk bounds for classifiers on finite set

Let $X$ be a finite set and let $\emptyset\neq \mathcal{H}\subseteq \{ 0,1 \}^{\mathcal{X}}$. Let $\{(X_n,L_n)\}_{n=1}^N$ be i.i.d. random variables on $X\times \{0,1\}$ with law $\mathbb{P}$. ...
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### Asymptotic expansion on the following integral of exponential function

I wish to obtain the asymptotic for the following integral $$\int_{r: \|r\|\le 1} \exp(M\cdot a^Tr) \, dr,$$ where $a$ is a given vector in $\mathbb{R}^d$, $\|\cdot\|$ is a general norm function and ...
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### Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?

This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian $$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$ are ...
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### Precise asymptotics for moments of order statistics of normal distribution

Let $X_1, \cdots, X_n \sim N(0,1)$ be i.i.d. normal random variates. I am interested in understanding the first two moments of the quasi-range $X_{(n)}-X_{(n-1)}$ (i.e., the maximum value minus the ...
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### Probability bounds of some ranked version of Dirichlet distribution

Recently I have come across a distribution defined on the open ranked simplex $\nabla^{n-1}_+ = \{\vec x \in \mathbb{R}^n:\sum_{k=1}^n x_k =1, x_1 \geq x_2 \geq \cdots \geq x_n > 0\}$, whose ...
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### Popular mistakes in probability

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...
First, let us give the setting. Let $(\Omega, \Sigma, \mathbf{P})$ be a probability space, let $T$ be some interval of time, and let $X: T \times \Omega \rightarrow S$ be a stochastic process. By Mean ...
*I am not sure if this question is better-suited for here or the MSE. According to A Tutorial on Fisher Information (pg. 40), for so-called regular parametric models we have  \sqrt n(\tilde\theta-\...