All Questions
Tagged with upper-lower-bounds statistics
56
questions
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
$$
\...
- 73
1
vote
1
answer
118
views
Inequality involving log-sum-exp, variance, and mean
Fix $z_1,\ldots,z_n \in \mathbb R$. Let $\mu_n:= mean(z_1,\ldots,z_n):=\frac{1}{n}\sum_{i=1}^n z_i$, $lse_n(z_1,\ldots,z_n)=\log(\sum_{i=1}^n e^{z_i})$, and
$\sigma^2_n := variance(z_1,\ldots,z_n):=\...
- 9,575
3
votes
1
answer
81
views
The expected weight-ratio between weighted and un-weighted balls when picked from a bin without replacement
The Problem
The problem, I believe, can be stated in the following way: Given $K$ white balls all with without weight (one can say that the weight is $0$) and $N - K$ red balls with individual ...
0
votes
0
answers
45
views
Using Markov's Inequality to Derive a Conclusion about random variable
I'm wondering whether I can use Markov's inequality to reach the following statement:
Given Markov's inequality on a non-negative random variable X:
$ P[X\geq a] \leq \frac{E[X]}{a}$
We can do the ...
- 548
2
votes
1
answer
2k
views
Log det of covariance and entropy
I understand log of determinant of covariance matrix bounds entropy for gaussian distributed data. Is this the case for non gaussian data as well and if so, why?
What does Determinant of Covariance ...
- 211
1
vote
1
answer
354
views
Rademacher Complexity Result
I was looking at one of the Rademacher Generalisation bound proofs, which says:
If $G$ is a family of functions mapping from $Z$ to $[0, 1]$ and $\mathcal{R_m}(G)$ denotes the Rademacher Complexity ...
- 127
1
vote
1
answer
359
views
What is the motivation for Gaussian Tail Bounds?
Perhaps I am missing something here, but I'm not seeing the value of having an upper tail bound for a Gaussian random variable. In my statistics class, we motivate tail inequalities for situations in ...
- 17
1
vote
0
answers
396
views
Tight upper and lower bounds of the CDF of a summation of random variables
I have this random variable
$$Y = \sum_{k=1}^KX_k$$
where $X_k$ are i.i.d. random variables with CDF and PDF $F_X(x)$ and $f_X(x)$, respectively.
In my application, the CDF of $Y$ denoted by $F_Y(...
- 390
0
votes
1
answer
49
views
$\max_{\{X_N\}}{\frac{\mu^2(X_N)}{\mu^2(X_N)+\sigma^2(X_N)}}$ for $X_N=\{x_1,...,x_n\},x_i\in \mathbb{N}^+,\exists(i,j):x_i\neq x_j$
Short version of the question
Consider
\begin{equation}
g(X_n)=\frac{\mu^2(X_N)}{\mu^2(X_N)+\sigma^2(X_N)}\\
X_N=\{x_1,x_2,...,x_n\},n>1,\forall i:x_i\in \mathbb{N}^+,\exists(i,j):x_i\neq x_j
\end{...
- 401
0
votes
1
answer
199
views
Lower bound for conditional probability
I have $X_1,X_2$ both identically and independently distributed $\text{Bin}(n,\theta)$. For some $\theta_0\in(0,1)$, and integers (depending on $n$) $a_n$ and $b_n$ satisfying
$$
n\theta_0\leq a_n<...
- 408
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