All Questions
Tagged with upper-lower-bounds statistics
56
questions
0
votes
1
answer
31
views
strange bound on correlation for symmetric pdf
I am puzzled by a rather simple fact:
The correlation of a symmetric multivariate pdf seems to be bound from below (increasingly strong with the number of dimensions). That seems unlikely to me.
But I ...
- 120
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
0
answers
64
views
How to find vertices of intersection of two hyperplanes?
According to Shapiro and Wilk(1965) in lemma 3,
$W$ has lower bound: $na_1^2/(n-1).$
To find this value, they solve the problem:
$$Max\quad y'y$$
$$ s.t.\quad 1'y=0,\quad and\quad a'y = 1,\quad and \...
- 1
1
vote
0
answers
27
views
Showing upper and lower Bayesian method of survival function
\begin{equation}
\begin{split}
S(t) = \frac{1}{\int_{0}^{1}\prod_{m=1}^M\left( \prod_{i=1}^{n_m} (t_{mi}+yx)^{\beta-1}\right)y^{c-1}(1-y)^{d-1}\frac{ \Gamma(\sum_{m=1}^{M}n_m+a)}{\left[\sum_{m=1}^{...
- 11
0
votes
0
answers
54
views
Why is the multivariate normal distribution is $(\Sigma, C)$ sub-gaussian?
The definition of \textit{sub-gaussian} from a book I work with is: $X\in\mathbb{R}^n$ is $(\Sigma,C)$ sub-gaussian if $$\mathbb{P}(\lvert X^\top u\rvert>t)<Ce^{-t^2/(2u^\top\Sigma u)}, \qquad u\...
- 11
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
0
votes
0
answers
34
views
2-Stirling number simplification with polynomial
I am looking for a way to either 1) simplify the following equation or 2) provide a reasonably tight upper bound to the following. Note that $\epsilon < 1$, and reasonably also $\epsilon \ll 1$.
\...
1
vote
0
answers
17
views
Calculation of lower range confidence interval
Question
We observe $x$, the maximum of $n$ values in a random sample from the uniform distribution between $0$ and $c$, where $c > 0$. Find an exact lower range $100(1 - \alpha)\%$ confidence ...
- 2,187
0
votes
1
answer
58
views
Bound on the expected time of first success in a series of Bernoulli RVs
Given an infinite series of Bernoulli RVs $X_1,X_2,...$ (which may be differently distributed and mutually dependent), we are given that for every $n>0$, $$\mathbb{E}\left[\sum_{t=1}^{n}(1-X_t)\...
- 47
0
votes
0
answers
30
views
Upper bound of two binomially distributed random variables
Let be $X_1,X_2$ two i.i.d binomially distributed random variables, where $p$ is the probability of success and $m$ the length of the underlying Bernoulli experiment. In a proof our professor argues
$$...
- 4,564
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 $$P\left( \sum_{i=1}^n X_i \geq m ...
- 153
1
vote
0
answers
48
views
Product of random variables greater than dependent sum of random variables
Let $X := {1\dots m}$ be the set of indexes corresponding to the elements of the vector of i.i.d. random variables $w:=(w_1, \dots, w_m)$. Let there be the subsets $Y, Z \subseteq X$ which may or may ...
- 221
1
vote
0
answers
27
views
Tight bounds for the expected maximum value of k IID Binomial(n, p) random variables
What is the tightest lower and upper bound for the expected maximum value of k IID Binomial(n, p) random variables
I tried to derive it :
$$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
90
views
Proving the set where probability density function becomes infinite is bounded
For a continuous random variable $X$, with probability density function $p_X(x)$, it is known that there exists a $p_{min} > 0$ such that $p_X(x) \geq p_{min} \forall x \in X$. Also, I know that $X$...
- 11
0
votes
3
answers
230
views
Lower bound for $\sqrt{x} - \sqrt{x-1} $ [closed]
Does anyone know any useful lower bound for $\sqrt{x} - \sqrt{x-1} $ for $x>1$. I have a problem where I want to find a lower bound for $$\sqrt{C \log(n)} - \sqrt{C \log(n)-1} $$ for a positive ...
- 3
1
vote
0
answers
48
views
Upper Bound for Moments for Product of Sample Means
I have a question about the upper bound of the following moment.
Suppose that $(A_1, e_1),\ldots, (A_n, e_n)$ are i.i.d. with $E(e_i)=0$. I am wondering if we have the bound
$$E\bigg(\bigg\|\frac{\...
- 53
1
vote
0
answers
102
views
Concentration bound for the distribution of the difference of two random variables
If we use $\Rightarrow$ to represent convergence in distribution and suppose that $X_n \Rightarrow N(0,\sigma_1)$ and $Y_n \Rightarrow N(0,\sigma_2)$, and $X_n$ and $Y_n$ are independent, then we all ...
- 11
2
votes
2
answers
118
views
Lower bound on the $\Phi$-entropy of a Gaussian variable
I am trying to prove that for $X$ a centered Gaussian variable,
$$\limsup_{n\in\mathbb{N}}\,\mathbb{E}\left[(X+n)^2\log\left(\frac{(X+n)^2}{1+n^2}\right)\right]=2.$$
I already know by the Gaussian ...
- 652
0
votes
1
answer
31
views
Lower bound on empirical Gaussian probablity
Consider $\{a_1,a_2,...,a_n\}$ n points in $\mathbb{R}$. Assume their mean is $0$, their standard deviation is then given by $\sigma=\sqrt{\dfrac{1}{n}\sum_ia_i^2}$. Let $p(x)=\dfrac{1}{\sqrt{2\pi}\...
- 700
1
vote
1
answer
94
views
Probabilistic Bound on Random Walk with Drift
For Gaussian random variable $\xi_t$ with mean $\mu$ and standard deviation $\sigma$, consider the random walk with initial condition $P_0=100$, such that
\begin{equation}
P_t=P_{t-1}(1+\xi_t).
\end{...
- 175
0
votes
0
answers
31
views
Maximum possible correlation for $N$ random variables sharing idential correlation coefficient
Let $X_1, X_2, \ldots, X_n$ be random variables. For every $i\ne j$, assume $corr(X_i,X_j)=\rho$, where $\rho$ is some constant.
What is the maximum possible value of $\rho$?
- 840
1
vote
2
answers
33
views
How to calculate the width of a variance
Short version
I have a series of results that sit within clear upper and lower bounds relative to the starting value. I do not know how to find those bounds (and thus the width of the band). I would ...
0
votes
0
answers
60
views
Asymptotic propagation of error
Let $\tilde{s}_n$ and $\tilde{p}_n$ be estimators of the quantities $s$ and $p$, respectively ($\mathbb{E}[\tilde{s}_n]=s$ and $\mathbb{E}[\tilde{p}_n]=p$). Imagine we have obtained asymptotic bounds ...
- 984
1
vote
1
answer
47
views
Variance of sum of deviations
Suppose I have an i.i.d. sample $\{X_i\}_{i=1}^M$ for some positive integer $M$, and suppose that $X_i \sim X$ for some random variable $X$ with finite variance. Then, denote by
$$
E_M = \frac1M\sum_{...
- 1,450
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
20
views
Use the statistical process control method to find the indicated control limits.
Question : The table gives $10$ samples of three measurements, made during a production run. Use the statistical process control method to find the indicated control limits.
Using $k_2=2.568$ and $...
1
vote
0
answers
46
views
Generalization Bounds
Given the loss function $L(\hat{y},y)$
the generalization error is defined as
$$R(h) = \underset{(x,y)\sim D}{\mathrm{E}}[L(h(x),y)]$$
the empirical error is defined as
$$\hat R(h) = \frac{1}{m}\...
1
vote
0
answers
52
views
Does logistic regression not fulfill an inequality required for Wilks' Theorem or am I missing something?
The required inequality:
Wilks' Theorem is given in the source below as Theorem 12.4.2, p. 515. Before stating the inequality, some definitions are needed:
Let $Z_1, \dots, Z_n$ be i.i.d. according to ...
- 47
2
votes
0
answers
212
views
Hoeffding's Inequality Assumptions
I'm looking for the assumptions of the Hoeffding's inequality to check it is applicable to my problem. So far the only assumptions I can find are the variables $Z_i$ are IID and bounded. However, Im ...
- 21
1
vote
0
answers
20
views
On the difference between the main effect in a one-factor and a two-factor regression
This question was asked on Cross Validated where it received little attention and no comments or answers, but as it is purely mathematically oriented it may well be more suitable here.
Consider a ...
- 27.4k
0
votes
1
answer
110
views
Markov Inequality question confusion bound problem
"If a fair die is rolled 200 times, count the number of 1’s. Give an upper bound for the probability that the count of 1’s stays below 8. "
So here for Markov inequality,
P(X>=8)<=E[X]/8.
So here,...
- 1
-1
votes
1
answer
3k
views
How to find upper bound of a probability?
We roll a fair die 50 times and count the number of 2’s. Give an upper bound for the probability that the count of 2’s stays below 7.
How can I approach this problem?
- 1
1
vote
0
answers
131
views
High probability upper bound for linear combination of Gaussian random variables
Suppose that $x_1, \dots, x_n$ are i.i.d. with $x_i \sim N(0,I_k)$. Let $A_1, \dots, A_n$ be matrices with dimension $k \times k$ and $\|A_i\|_2 \leq 1$. Consider the following random vector
$$y = \...
- 1,180
1
vote
0
answers
41
views
Nice bounds for 3rd folded central moment in terms of variance?
Say we have some real numbers $x_1, \ldots, x_n$, and let $\mu$ be their mean and $\sigma^2$ be the variance. Are there some nice bounds for $\frac 1n \sum_{i=1}^n |x_i - \mu|^3$ based on $\sigma^2$? ...
- 8,945
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
0
answers
55
views
Showing that the following ratio expectation is bounded by $O(1/n)$
Suppose that $V_i$, $i \in \mathbb{N}$ are i.i.d. standard normal random variables and let $Y_i = \sum_{k=1}^i V_i$ for $i\geq 1$ and $Y_0 = 0$. Consider the following ratio random variable
$$R = \...
- 1,180
0
votes
1
answer
662
views
VC dimension of a circle
I have a circle where all the points inside a circle are labelled positively and all the points outside the circle are labelled negatively. The center of the circle is h and radius is r. Let H+ be ...
- 5
0
votes
1
answer
84
views
Upper bound for variance of kernel averages
I'm reading Hansen's (2008, p. 729) Theorem 1 where he bounds the variance of averages of the form
$$\hat\Psi(x)=\frac1{Th}\sum\limits_{t=1}^T Y_t K\bigg(\frac{x-X_t}h\bigg)$$
given that $\{(Y_t,X_t)\...
- 2,689
1
vote
0
answers
26
views
How concentrated is the $t^{th}$ smallest discrete uniform order statistic?
Let $n,z,t$ be positive integers and let $X_1,\ldots,X_{z\cdot t}$ be i.i.d. random variables that are uniformly distributed over $\{0,\ldots,n\}$.
Let $X_{(t)}$ denote the $t^{th}$ smallest ...
- 2,436
-1
votes
1
answer
74
views
Probability. Define Upper bound and Lower bound for double integral. [closed]
I am finding trouble of finding bounds for double integral for statistic.
Here the question looks like:
given joint pdf = $\frac{3x}4$, $0<x<1,\;0<y<4x$.
I have to find pdf of $R=XY$.
...
- 13
0
votes
1
answer
165
views
Upper bound for Variance of linear combination of random variables: $\operatorname{Var}\left(x^Ta\right) \leq \frac{\|a\|^2}{4}. $
I found this while reading a paper where they used it as a casual fact.
Say, you have a vector $x = (x_1, x_2, \dots, x_n)$ where $x_i \in [0,1]$ are independent random variables. Consider linear ...
- 691
1
vote
3
answers
906
views
Is there a way to bound expected value with limited information of the CDF?
Suppose I want to evaluate $E[X]$, where $X$ is a univariate random variable and takes values in $\mathcal{X}$, where the smallest element of $\mathcal{X}$ is 0 and the largest element of $\mathcal{X}$...
- 403
0
votes
2
answers
2k
views
upper bound on cross entropy or relative entropy
Am looking for upper bounds for relative-entropy or cross-entropy for non-gaussian case . All I am aware of is bounds for entropy based on determinant of covariance matrix.
What about for relative ...
- 211
3
votes
1
answer
885
views
Lower bounds on sum of squared sub-gaussians
Letting $\left\{X_{i}\right\}_{i=1}^{n}$ be an i.i.d. sequence of zero-mean sub-Gaussian variables with parameter $\sigma,$ define
$Z_{n} :=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2} .$ Prove that
$$
\...
- 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