All Questions
Tagged with upper-lower-bounds statistics
56
questions
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
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
3
votes
1
answer
885
views
Lower bounds on sum of squared sub-gaussians
Letting $\left\{X_{i}\right\}_{i=1}^{n}$ be an i.i.d. sequence of zero-mean sub-Gaussian variables with parameter $\sigma,$ define
$Z_{n} :=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2} .$ Prove that
$$
\...
- 73
3
votes
1
answer
94
views
Minimum number of Bernoulli trials until sum reaches threshold with high probability
Let $X_1, X_2, \dots$ be i.i.d. $Bern(p)$ with $p\in (0, 1)$. Let $\delta \in (0,1)$ and $m \in \mathbb{N}$. What is the smallest integer $n \in \mathbb{N}$ such that $$P\left( \sum_{i=1}^n X_i \geq m ...
- 153
3
votes
1
answer
2k
views
Cramer-Rao Casella Berger 7.38 for exponential family
The question states ''let $X_{1}, \dots, X_{n}$ be random sample from $f(x \mid \theta) = \theta\cdot x^{\theta-1}$ for $0 < x< 1 ; \theta > 0$. Is there a function of $\theta, g(\theta)$ ...
- 771
3
votes
1
answer
81
views
The expected weight-ratio between weighted and un-weighted balls when picked from a bin without replacement
The Problem
The problem, I believe, can be stated in the following way: Given $K$ white balls all with without weight (one can say that the weight is $0$) and $N - K$ red balls with individual ...
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
2
votes
1
answer
2k
views
Log det of covariance and entropy
I understand log of determinant of covariance matrix bounds entropy for gaussian distributed data. Is this the case for non gaussian data as well and if so, why?
What does Determinant of Covariance ...
- 211
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
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 ...
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
2
votes
0
answers
70
views
Show that $\operatorname{Pr}(Z-X \geq 0)$ converges to one
Suppose that $V_i$, for $i \in \mathbb{N}$, are i.i.d. standard normal random variables and $Y_i = \sum_{k=1}^i V_k$ for $i \in \mathbb{N}$ with $Y_0 = 0$. Let $X_n = (\sum_{i=1}^n V_i Y_{i-1})^2 Y_n^...
- 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
1
vote
2
answers
33
views
How to calculate the width of a variance
Short version
I have a series of results that sit within clear upper and lower bounds relative to the starting value. I do not know how to find those bounds (and thus the width of the band). I would ...
1
vote
1
answer
47
views
Variance of sum of deviations
Suppose I have an i.i.d. sample $\{X_i\}_{i=1}^M$ for some positive integer $M$, and suppose that $X_i \sim X$ for some random variable $X$ with finite variance. Then, denote by
$$
E_M = \frac1M\sum_{...
- 1,450