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.
1,928 questions
3
votes
1
answer
114
views
Stochastic order of generalized chi-square distributions (Extended)
I seek a generalization of the idea and answer found in
Stochastic order of generalized chi-square distributions.
Given that $χ^2_a=∑^n_{i=1}a_iZ^2_i$ where $Z_i∼N(0,1)$ are i.i.d. $\forall i$. The ...
0
votes
0
answers
75
views
Quadratic constrained optimization problem with solution linked to $m$-dimensional Delauney triangulation [closed]
I am wondering if the statement in my screenshot is correct, regarding "the optimum is reached when all the weights $\omega^*_k(x)$ are zero except for the nodes $k$ where $\beta_k$ is one of the ...
- 3,188
5
votes
1
answer
282
views
Stochastic order of generalized chi-square distributions
For any $a\in \mathbb{R}^n_{>0}$ define the random variable $\chi^2_a=\sum_{i=1}^n a_i Z_i^2
$
where the $Z_i\sim \mathcal{N}(0,1)$ are i.i.d. Given $a,b\in \mathbb{R}^n_{>0}$ I am looking for ...
- 105
1
vote
1
answer
241
views
Alternating projections between e-flat and m-flat manifolds
Consider an exponential family $E = \{ p_\theta : \forall x \in \mathcal{X},~ p_{\theta}(x) = \exp(S(x)^\top \theta - F(\theta)), A \theta = b \}$ affinely constrained in its natural parameter and let ...
- 247
6
votes
0
answers
242
views
QR code problem
Motivation. Today, I saw a QR code with an unusually large black square (a largish group of “pixels” coloured black and forming a square). This inspired the following problem.
Problem. Fix $n\in\...
- 54.6k
4
votes
1
answer
146
views
Best known exponent for Wasserstein Law of Large Numbers on a compact domain on $\mathbb R^d$
for $p \geq 1$ define the Wasserstein-$p$ distance to be $$ W_p^p(\mu, \nu) = \inf_{\gamma\in \Gamma_{\mu,\nu}} \int_{\mathbb R^d} |x - y|^p\, \mathrm d \gamma$$ where $\Gamma_{\mu,\nu}$ is the set of ...
- 131
2
votes
2
answers
238
views
Is there a fully one-sided version of Etemadi's inequality?
$\def\P#1{\mathbf{P}\!\left[#1\right]}\def\E#1{\mathbf{E}\!\left[#1\right]}$Let $X_1, \dots, X_n$ be independent real random variables (but not necessarily i.i.d.), and let $S_t = \sum_{i = 1}^t X_i$ (...
- 740
4
votes
0
answers
204
views
Uniformly picking numbers without memory [duplicate]
Motivation. For my older son's spelling bee contest, he gave me a list of $n$ difficult words to read to him, so he could write them down. As I was filling the dishwasher at the same time, I didn't ...
- 54.6k
0
votes
1
answer
86
views
How to uniformly sample from a nonlinearly transformed simplex?
Let $S = \{ w \in \mathbb{R}^3 : w_1 + w_2 + w_3 = 1,; w_i \ge 0 \}$ be the standard 2-simplex. Consider the transformation
$$
T(w_1, w_2, w_3) = (w_1, w_2^p, w_3),
$$
not followed by a ...
0
votes
0
answers
38
views
Characterizing exponential families with elementary normalizing transformations
Let $f_\theta(x) = \exp(\theta T(x) - K(\theta))$ be a one-parameter exponential family of probability density functions with respect to the Lebesgue measure on $\mathbb{R}$, for $\theta$ in an open ...
- 763
0
votes
1
answer
90
views
Donsker-Varadhan duality in conditional sense?
A coherent risk measure named Entropic Value-at-Risk was introduced as follows: Let $(\Omega,\mathcal{F},\mathbf{P})$ be a probability space, $X$ be a random variable and $\beta$ be a positive ...
- 111
0
votes
1
answer
81
views
Convergence of Kaplan-Meier estimator for pooled sample
I am new to survival analysis. Recently I have been thinking about the Kaplan-Meier estimator for pooled sample. Suppose we have two group of samples, group 1 has $n_1$ samples from the survival ...
- 35
6
votes
1
answer
436
views
Sampling from two distributions to determine which is which
Peter Winkler wrote the following in his book "Mathematical Puzzles (revised edition)":
As it turns out, it’s a theorem that in trying to determine which is which of two known probability ...
- 17.8k
3
votes
1
answer
225
views
Matching marginal distributions based on two conditional distributions
Suppose we have two Markov kernels $(A, t) \mapsto K(X \in A \mid T=t)$ and $(A, t) \mapsto H(X \in A \mid T = t)$ that represent conditional distributions of $X$ given $T=t$. We obtain the marginal ...
- 208
2
votes
0
answers
162
views
Partial order on a probability space denoting "more typical"
$\DeclareMathOperator\supp{supp}\newcommand\teq{\underset t=}\newcommand\tlt{\underset t<}$Let $(X, \mathcal{B}(X),\mu)$ be a probability space, where $\mathcal{B}(X)$ is the Borel $\sigma$-algebra ...
- 518