All Questions
Tagged with upper-lower-bounds statistics
56
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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
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33
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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
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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:
$$Max\quad y'y$$
$$ s.t.\quad 1'y=0,\quad and\quad a'y = 1,\quad and \...
- 1
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27
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Showing upper and lower Bayesian method of survival function
\begin{equation}
\begin{split}
S(t) = \frac{1}{\int_{0}^{1}\prod_{m=1}^M\left( \prod_{i=1}^{n_m} (t_{mi}+yx)^{\beta-1}\right)y^{c-1}(1-y)^{d-1}\frac{ \Gamma(\sum_{m=1}^{M}n_m+a)}{\left[\sum_{m=1}^{...
- 11
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Why is the multivariate normal distribution is $(\Sigma, C)$ sub-gaussian?
The definition of \textit{sub-gaussian} from a book I work with is: $X\in\mathbb{R}^n$ is $(\Sigma,C)$ sub-gaussian if $$\mathbb{P}(\lvert X^\top u\rvert>t)<Ce^{-t^2/(2u^\top\Sigma u)}, \qquad u\...
- 11
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58
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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
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34
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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
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17
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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 ...
- 2,177
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58
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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 $n>0$, $$\mathbb{E}\left[\sum_{t=1}^{n}(1-X_t)\...
- 47
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Upper bound of two binomially distributed random variables
Let be $X_1,X_2$ two i.i.d binomially distributed random variables, where $p$ is the probability of success and $m$ the length of the underlying Bernoulli experiment. In a proof our professor argues
$$...
- 4,564
3
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1
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94
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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 $$P\left( \sum_{i=1}^n X_i \geq m ...
- 153
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48
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Product of random variables greater than dependent sum of random variables
Let $X := {1\dots m}$ be the set of indexes corresponding to the elements of the vector of i.i.d. random variables $w:=(w_1, \dots, w_m)$. Let there be the subsets $Y, Z \subseteq X$ which may or may ...
- 221
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27
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Tight bounds for the expected maximum value of k IID Binomial(n, p) random variables
What is the tightest lower and upper bound for the expected maximum value of k IID Binomial(n, p) random variables
I tried to derive it :
$$Pr[max \leq C] = (\sum_{i = 0}^C {n \choose i}p^i(1 - p)^i)^...
- 21
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1
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368
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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
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91
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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 ...
