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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 [tag:diversity].

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A question involving directional derivatives and differential inequalities

This is a follow-up question to A question about copulas and directional derivatives. Since no answer was given, I am going to precise the definition of copula. I am interested in proving (or ...
17 views

Extreme Value Theory - Determining the positive normalising constant in the Extremal Types Theorem

I am working through the following question and cannot seem to work out how the final result is obtained from the last inequality involving $a_n$. Can someone shed some light?
26 views

How do I solve a linear inequality system ($X\beta+b<0$)?

Given a low-dimension linear regression problem $\mathbf{y}=\mathbf{X}\beta + \epsilon$, we can easily estimate $\beta$ with $(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^Ty$. However, the problem seems ...
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Moment inequality: $E\mid X_1 X_2 X_3\mid \leq (E(\mid X_1\mid^3)+E(\mid X_2\mid^3)+E(\mid X_3\mid^3))/3$ for zero-mean r.v.'s?

Let $X_1, X_2, X_3$ be zero mean random variables and assume $E(\mid X_i \mid ^{4+\delta})\leq C, i=1,2,3$ where $C$ is a constant and $\delta>0$ some positive small constant. How can I show that ...
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Unusual Markov inequality for normal distribution

I'm trying to answer the following question from Larry Wassermans book on statistical inference. My question is how did they arrive at the Markov bound, it does not seem like the normal form of the ...
68 views

Variance and covariance inequality

Given a real-valued random variable $X$, is $$2\mathbb E[X] \mathrm{Var}(X) \geq \mathrm{Cov}(X, X^2)$$ true? Any pointers for how to tackle this problem would be immensely helpful.
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Bounding residual variance with distance from mean

For a linear regression $Y = X\beta + \varepsilon$ with $\varepsilon \sim \mathcal N(0,\sigma^2 I)$, we have $\hat Y = H Y$ for $H = X(X^TX)^{-1}X^T$. This means that $Var(Y - \hat Y) = \sigma^2(I-H)$ ...
251 views

Proving efficiency of OLS over GLS

I'm trying to prove the efficiency of OLS over GLS when the covariance matrix of the error $\varepsilon$ is mistakenly assumed to be $\sigma^2\Sigma$ instead of $\sigma^2 I$. After deriving the ...
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Oracle Inequality : In basic terms

I'm going through a paper that uses oracle inequality to prove something but I'm unable to understand what it is even trying to do. When I searched online about 'Oracle Inequality', some sources ...
84 views

Inequality on two random variables

This seems like a really straightforward question but I think maybe I lack the vocabulary to search for it correctly. Given two random variables $X$ and $Y$ with known probability distribution ...
495 views

Zero mean unit variance random variables bound on probability

Let $X_1, X_2$ be zero-mean, unit variance Random Variables with correlation coefficient $\rho$ then $$P(|X_1|\le\epsilon,|X_2|\le\epsilon) \ge 1-\epsilon^{-2}(1+\sqrt{1-\rho^2})$$ I tried to used ...
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Lower Bound on $E[\frac{1}{X}]$ for positive symmetric distribution

Let $X$ be positive random variable and its distribution is symmetric about its mean value $m$. Then $$E\left[\frac{1}{X}\right] \geq \frac{1}{m} + \frac{\sigma^2}{m^3},$$ where $\sigma^2$ is ...
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Is there a method to fit a bound to the plot of an linear inequality?

I have a physical dataset that is bounded by several different processes, and thus the plot takes the form of a linear inequality: I'm specifically interested in studying the upper bound. Is there a ...
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Inequality on variance of sum

I want to prove that $$\operatorname{Var}\left(\sum\limits_{i=1}^m{X_i}\right) \leq m\sum\limits_{i=1}^m{\operatorname{Var}(X_i)} \,. \>$$ A too complicated proof is to write  a_{ij}=\sqrt {Cov(...
In Cover & Thomas, Elements of Information Theory, at the section on Conditional Limit Theorem (11.6), it is proved that the KL divergence bounds the $\cal{L}_1$-norm from above, \$\frac{1}{2\ln2}\...