All Questions
Tagged with lower-bounds st.statistics
11 questions
1
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1
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105
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How to lower bound the probability that one Gaussian exceeds many others
I am interested in upper and lower bounding probability that one component of a multivariate Gaussian exceeds all others. For instance, say we have a multivariate Gaussian RV $X \sim N(\mu, \Sigma)$ ...
- 283
0
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0
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52
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Classifier-specific lower bounds on the misclassification rate in binary classification
Consider a binary classification problem for $(X,Y)$, and let $\hat{f}$ be a proposed classifier. We wish to bound the misclassification rate $P(\hat{f}(X)\ne Y)$. There are many known lower bounds on ...
- 13
2
votes
1
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349
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Lower bound on sum of independent heavy-tailed random variables
I have a sum of $n$ i.i.d random variables $X_i$ such that $E[X_i] = 0$,$\mathrm{E}[|X_i|^{1 + \delta}]$ exists for some $0 < \delta < 1$ but $\mathrm{E}[|X_i|^{1 + \delta+ \epsilon}]$ does not ...
- 23
4
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1
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168
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Existence of copula bound pointwise strictly smaller than the Fréchet-Hoeffding upper bound
Consider bivariate copulas $C_1$ and $C_2$ with $\max\{C_1(u,v), C_2(u,v)\}< M_2(u,v)$ for all $u,v \in(0,1)$, where $M_2(u,v) := \min\{u,v\}$ is the Fréchet-Hoeffding upper bound.
Is there a ...
- 143
2
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0
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222
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When does Le Cam's method give tight lower bounds for distribution testing?
In the context of statistical estimation or distribution testing, Le Cam's method is a way to prove lower bounds on the minimax sample complexity ([1,2,3,4], further details below). My question is: ...
- 33
0
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0
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330
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Lower-bound smallest eigenvalue of covariance matrix of $y = f(Ax)$, for $x$ uniform on unit-sphere
Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a ...
- 6,853
2
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1
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1k
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Finding the expectation $\mathrm{E} (1/ X)$ for a negative binomial random variable $X$
Suppose a random variable $X$ is distributed as $\operatorname{NB}(\mu, \theta)$, and its mass is as follows
$$ \mathrm{P}(X = y) = \binom{y + \theta - 1}{y} \left(\frac{\mu}{\mu + \theta}\right)^{y}\...
- 331
3
votes
1
answer
87
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If $X \sim N(0,I_m)$, what is a necessary and sufficient condition on $u_m > 0$ such that $\lim\sup_{m\to \infty} P(\|X\|^2 \ge u_m|X_1|) = 1$
Let $m$ be a large positive integer and $X=(X_1,\ldots,X_m) \sim N(0,I_m)$. I wish to show that the squared norm of $X$ is much much bigger than the absolute value of any of the $X_j$'s. For example, ...
- 6,853
3
votes
2
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233
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Lower bounding decoding error in a noisy adversarial channel
Problem description
Suppose we have a finite alphabet $\mathcal{X}$, where each letter $X \in \mathcal{X}$ indexes into some fixed set of distributions, $\{P_{1},\ldots,P_{|\mathcal{X}|}\}$. For ...
- 181
3
votes
1
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165
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Non-asymptotic tail bounds for $D_{\text{Hellinger}}(P\|\hat{P}_N)$
Let P be a distribution on a finite set of size $k$ and let $\hat{P}_N=(N_1/N,\ldots,N_k/N)$ be the empirical distribution (frequencies) from a samples of size $N$. Consider the Hellinger distance ...
- 6,853
11
votes
0
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536
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Bounding the probability that a random variable is maximal
Question: Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_N$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$.
I am looking ...
- 119