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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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 ...
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Bounds on Cov(X, Y) given Var(X), Var(Y)?

I'm generating random multivariate normal data using the rmultnorm() function in R, which allows users to specify a vector of $k$ population means and a $k \times k$...
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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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Prove that Kurtosis is at least one more than the square of the skewness

Wikipedia claims it, and on reading the paper that it linked I found that the proof that was written there was quite difficult. Is there a simple proof possible for this identity? The proof given in ...
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What is the correlation between a random variable and its probability integral transform?

Are there known bounds on the $\operatorname{cor}(X,F(X))$? $X$ is a random variable with CDF $F(X)$. Let $X$ have a fixed variance, for example $\operatorname{var}(X)=1$. What $X$ can maximize or ...
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L2 SVM (squared hinge) theory

The linear L2 SVM can be intuitively understood as $$\text{minimize } f(\boldsymbol{w}) = \frac{1}{2} \Vert\boldsymbol{w}\Vert^2_2 + C \sum_{i=1}^m \xi_i^2 \tag{1}$$ ...
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Implications of zero limiting variance

Assume that I have a sequence of random variables $X_1, X_2, \dots$ with means $\mu_1, \mu_2, \dots$ such that $\lim_{n \to \infty} \operatorname{Var}(X_n) = 0$. Can I claim that for large enough $n$ ...
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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 ...
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