All Questions
16
questions
2
votes
0
answers
33
views
Lower bound of $\frac{\|(\mathbf X \otimes \mathbf X^\top)\theta\|_2^2}{np}$
According to Theorem 7.16 of High-Dimensional Statistics: A Non-Asymptotic Viewpoint (M. Wainwright, 2019), we know that for $\mathbf X\in\mathbb R^{n\times p}, X_{ij}\overset{iid}{\sim}N(0,1),$ there ...
- 63
0
votes
1
answer
58
views
Bounds on the ratio between second raw moment and expected of absolute value squared
I'm interested in bounds for the ratio $E[X^2]/E[|X|]^2$. The best lower bound is $1$ since $E[|X|] \geq E[X]$ and $E[X^2] - E[X]^2 = Var(X) \geq 0$. On the other hand, I would like to know if there ...
- 63
3
votes
1
answer
94
views
Minimum number of Bernoulli trials until sum reaches threshold with high probability
Let $X_1, X_2, \dots$ be i.i.d. $Bern(p)$ with $p\in (0, 1)$. Let $\delta \in (0,1)$ and $m \in \mathbb{N}$. What is the smallest integer $n \in \mathbb{N}$ such that $$P\left( \sum_{i=1}^n X_i \geq m ...
- 153
1
vote
0
answers
27
views
Tight bounds for the expected maximum value of k IID Binomial(n, p) random variables
What is the tightest lower and upper bound for the expected maximum value of k IID Binomial(n, p) random variables
I tried to derive it :
$$Pr[max \leq C] = (\sum_{i = 0}^C {n \choose i}p^i(1 - p)^i)^...
- 21
4
votes
1
answer
372
views
Exponential bound for tail of standard normal distributed random variable
Let $X\sim N(0,1)$ and $a\geq 0$. I have to show that $$\mathbb{P}(X\geq a)\leq\frac{\exp(\frac{-a^2}{2})}{1+a}$$
I have no problem showing that $\mathbb{P}(X\geq a)\leq \frac{\exp(\frac{-a^2}{2})}{a\...
- 103
2
votes
0
answers
92
views
Probabilistic bound on difference of Lipschitz random function
I am currently facing the following problem :
Let $(X_1,Z_1),\ldots,(X_n,Z_n)$ be $n$ i.i.d. sample points from some distribution $p$ supported on $\mathcal X\times\{-1,1\}$ where $\mathcal X\subseteq ...
1
vote
0
answers
102
views
Concentration bound for the distribution of the difference of two random variables
If we use $\Rightarrow$ to represent convergence in distribution and suppose that $X_n \Rightarrow N(0,\sigma_1)$ and $Y_n \Rightarrow N(0,\sigma_2)$, and $X_n$ and $Y_n$ are independent, then we all ...
- 11
2
votes
2
answers
118
views
Lower bound on the $\Phi$-entropy of a Gaussian variable
I am trying to prove that for $X$ a centered Gaussian variable,
$$\limsup_{n\in\mathbb{N}}\,\mathbb{E}\left[(X+n)^2\log\left(\frac{(X+n)^2}{1+n^2}\right)\right]=2.$$
I already know by the Gaussian ...
- 652
15
votes
1
answer
204
views
Show that $\mathbb{E}\left|\hat{f_n}-f \right| \leq \frac{2}{n^{1/3}}$ where $\hat{f_n}$ is a density estimator for $f$
Question
Suppose we have a continuous probability density $f : \mathbb{R} \to [0,\infty)$ such that $\text{sup}_{x \in \mathbb{R}}(\left|f(x)\right| + \left|f'(x)\right|) \leq 1. \;$ Define the ...
- 153
1
vote
0
answers
52
views
Does logistic regression not fulfill an inequality required for Wilks' Theorem or am I missing something?
The required inequality:
Wilks' Theorem is given in the source below as Theorem 12.4.2, p. 515. Before stating the inequality, some definitions are needed:
Let $Z_1, \dots, Z_n$ be i.i.d. according to ...
- 47
2
votes
0
answers
212
views
Hoeffding's Inequality Assumptions
I'm looking for the assumptions of the Hoeffding's inequality to check it is applicable to my problem. So far the only assumptions I can find are the variables $Z_i$ are IID and bounded. However, Im ...
- 21
1
vote
0
answers
131
views
High probability upper bound for linear combination of Gaussian random variables
Suppose that $x_1, \dots, x_n$ are i.i.d. with $x_i \sim N(0,I_k)$. Let $A_1, \dots, A_n$ be matrices with dimension $k \times k$ and $\|A_i\|_2 \leq 1$. Consider the following random vector
$$y = \...
- 1,180
1
vote
3
answers
906
views
Is there a way to bound expected value with limited information of the CDF?
Suppose I want to evaluate $E[X]$, where $X$ is a univariate random variable and takes values in $\mathcal{X}$, where the smallest element of $\mathcal{X}$ is 0 and the largest element of $\mathcal{X}$...
- 403
0
votes
0
answers
45
views
Using Markov's Inequality to Derive a Conclusion about random variable
I'm wondering whether I can use Markov's inequality to reach the following statement:
Given Markov's inequality on a non-negative random variable X:
$ P[X\geq a] \leq \frac{E[X]}{a}$
We can do the ...
- 548
1
vote
1
answer
354
views
Rademacher Complexity Result
I was looking at one of the Rademacher Generalisation bound proofs, which says:
If $G$ is a family of functions mapping from $Z$ to $[0, 1]$ and $\mathcal{R_m}(G)$ denotes the Rademacher Complexity ...
- 127