# Questions tagged [inequality]

Use this tag if you question involves the use of an inequality. The inequality may have probabilistic origins or be a purely mathematical inequality. Do not use for measures of inequality, for instance income inequality. For that use the [diversity] tag.

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### Is $f(a) = EX^{1+a}EX^{-(1+a)}$ non-decreasing?

$X$ is a non-negative random variable and $a$ is a non-negative real number. Define $$f(a)= EX^{1+a}EX^{-(1+a)}.$$ Is $f(a)$ non-decreasing with $a$? Original problem: when I read a paper, I encounter ...
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### Expectation under convex order by multiplying

I am trying to understand if the following statement is true, or the conditions under it is satisfied. Let $M,N$ and $X>0$ be random variables. If the following inequality holds for any concave non-...
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### Expectation under convex order

I am trying to understand if the following statement is true. Let $M,N$ and $X$ be random variables. If the following inequality holds for any concave non-decreasing function $u$ \...
• 157
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### Does there exist a matrix Bernstein type inequality for weakly dependent random variables?

When reviewing the literature, I managed to find the following papers, which deduce Bernstein type inequalities for weakly dependent random variables (where weak dependence is defined in some special ...
13 views

### Monitoring the Evolution of Inequality in Time-Dependent Distributions

I created a self-contained example in R that you find at the end of the post. I would like to be reassured that what I do makes sense. I have a dataset with 2 variables: year is a time variable, ...
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### Misunderstanding on the use of Popoviciu and von Szokefalvi Nagy's inequalities on the variance of a unbiased estimator

Let $X_1,\cdots,X_n$ be (discrete in my case) i.i.d. and bounded between $m$ and $M$. I'm interested in bounding the variance of an unbiased estimator: $$\mathbb{V}\left[\frac1n\sum_{i=1}^nX_i\right]$$...
86 views

### Direction of inequality sign in alternative hypothesis -- determining from word problems

SUPER basic stats 101 type question here, sorry. My teacher likes to set problems that call for a one-tailed Z- or T-test. We've been instructed to always use the equals sign in the H0. Sometimes I ...
1 vote
60 views

### Adjustment needed for multivariate Dvoretzky–Kiefer–Wolfowitz inequality on MCMC samples?

I was thinking about studying bounds on the multivariate empirical cumulative distribution function for samples from an MCMC chains. The multivariate Dvoretzky–Kiefer–Wolfowitz inequality would seem ...
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### When a non-subgaussian vector has subgaussian components

The following remark is from Asymptotically Normal and Efficient Estimation of Covariate-Adjusted Gaussian Graphical Model by Chen et al. 2016: Here, $X$ is a design matrix with iid rows and $X^{(1)}$...
201 views

### Expectation of first of moment of symmetric r.v. in terms of variance

Let $X$ be a symmetric random variable with bounded moments and standard deviation $\sigma$. I want to lower-bound $\mathbb E[|X|]$ in terms of $\sigma$. Here is the formal conjecture; I wonder if ...
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1 vote
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### Statistical Test for Some Inequality Condition

Suppose that we have three random variables, named $V_1, V_2$ and $V_3$. Here, I want to test the following inequality: $V_1 \geq V_2, V_3$ Is there any reference dealing with this topic?
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### Problem understanding the intuition behind Slepian's inequality

Slepian's inequality is defined as follows: Let $X\in\mathbb{R}^n$ and $Y\in\mathbb{R}^n$ be centered Gaussian random vectors such that \begin{align} \mathbb{E}X_iX_j&\geq \mathbb{E}Y_iY_j,\quad \...
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### Holder's inequality in the case of $L_1$ and $L_{\infty}$ norm

I am referring to Wainwright's High-Dimensional Statistics book, where at some point it is deduced that \frac{w'X\Delta}{n}\leq \left\lVert\frac{w'X}{n}\right\rVert_{\infty}\lVert\...
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### Proof that variance is always greater than or equal to zero

It is common knowledge that: $$$$\label{3} Var(X) \geq 0$$$$ for every random variable $X$. Despite this, I do not remember seeing a formal proof of this. Is there a proof ...
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1 vote