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