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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13 views

For which probability distributions is this inequality true?

I have derived the following inequality and now I want to check whether typical probability density functions exist, which fulfill this inequality. My suspicion is that it depends on the slope/...
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54 views

Would this extension of Khintchine's inequality be correct?

This seems trivial, yet I have to make sure it is indeed correct. Referring to Roman Vershynin's High-Dimensional Probability book, the Khintchine's inequality (Exercise 2.6.5, page 27) is defined as ...
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How to compare the size of two l_inf norms?

Suppose $u\in \mathbb{R}^{n}$ is an $n\times 1$ vector of standard Gaussian perturbations - i.e. $u\sim N(0,I)$, $X\in\mathbb{R}^{n\times p}$ is an $n\times p$ fixed matrix such that $p\gg n$, and $L\...
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2answers
146 views

How to calculate lower bound on $P \left[|Y| > \frac{|\lambda|}{2} \right]$?

Let $Y$ be a random variable such that $E[Y] = \lambda$, $\lambda \in \mathbb{R}$ and $E[Y^2]<\infty$. The problem is to find a lower bound on the probability $$ P \left[|Y| > \frac{|\lambda|}{...
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17 views

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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27 views

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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3answers
109 views

Is $P(|X_1|>k)\le P(|X_2|> k)$ when $X_i\sim N(\mu_i,\sigma^2)$ and $|\mu_2| \ge |\mu_1|$?

Suppose $X_1\sim N(\mu_1,\sigma^2)$ and $X_2\sim N(\mu_2,\sigma^2)$ where $\mu_2\ge \mu_1$. Since $\mu_2\ge \mu_1$, based on a characterization of stochastic ordering, we can say that $$P(X_1>c)\le ...
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1answer
34 views

Find the minimum percentage of values within two bounds subject to moment constraints

I am facing the following problem: Variable Z has a mean of 15 and a standard deviation of 2. What is the minimum percentage of Z values that lie between 8 and 17? I have tried the following: Here ...
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41 views

Hoeffding like result for expected number of pulls

Hoeffding's inequality states that if, $X_1, \ldots, X_n$ are independent random variables bounded by the interval $[0, 1]$, i.e. $0 ≤ X_i ≤ 1$. And the empirical mean of these variables is given by, $...
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1answer
114 views

Hoeffding type concentration result for the inverse of a sum of iid random variables

Consider a collection of $n$ i.i.d. Bernoulli random variables $\{ X_i \}_{i=1}^{n}$ with $\mathbb{E}[X_i] = \mu$. Then, if $\hat{\mu}$ is the mean of the $n$ random variables, i.e. if, $$\hat{\mu} = \...
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41 views

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 \begin{equation} \frac{w'X\Delta}{n}\leq \left\lVert\frac{w'X}{n}\right\rVert_{\infty}\lVert\...
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2answers
3k views

Proof that variance is always greater than or equal to zero

It is common knowledge that: $$\begin{equation}\label{3} Var(X) \geq 0 \end{equation}$$ 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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18 views

When does this Deep Neural Network inequality become an equality?

Let $\text{DNN}_k(x)$ be some fully connected DNN with $k$ hidden layers. Let $(x,y)$ be some data points and $\ell$ be a loss function. Then $\forall \ k \in \{1,2,,..\} $, we have that $$\min \ell(\...
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38 views

prove the difference between mean and median is less than the variance [duplicate]

Suppose $X$ is a random variable with finite variance. Let $m$ denote the median of $X$ and $\mu$ the mean of $X$, i.e. $\mu=\mathbb{E}(X)$. Show $$(m-\mu)^2\leq\text{var}(X)$$ Intuitively this is ...
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1answer
148 views

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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20 views

Chebyshev bounds [duplicate]

Can Chebyshev bound be greater than 1? The following question is just in contradiction to the above stated question: a) A rv assume values -1,1,3,5 with respective probabilities, 1/6,1/6,/16,1/2. Find ...
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1answer
107 views

Upper bound for absolute third central moment

Suppose $X\in \mathbb{R}$ is a random variable with expected value $\mathbb{E}X = \mu$. I ran across a proof which uses the inequality $$ \mathbb{E}[|X - \mu|^3] \leq 2^3 \mathbb{E}|X|^3. $$ Can ...
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43 views

cauchy schwarz inequality on sum of squares

Can I and how I can use the Cauchy Schwarz inequality on the amplitude of a imaginary sum of squares? $$Z = X+iY$$ and $$|Z| = \sqrt{X^2 +Y^2}$$ to show that $$|E[Z] | \leq \mathbb{E}[|Z|]$$ where $X,...
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0answers
48 views

How can I weight ordinal observations and reduce them to one statistic? [closed]

It will be a complicated question and I try to briefly explain. I am studying on educational inequalities. The survey I use for analysis incudes ordinal variables and the education degree which ...
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0answers
25 views

How can I test for significant gini coefficient differences?

I calculate the gini coefficients for different countries using the ineq R-package. I want to find out whether the calculated gini coefficients for the respective countries do significantly differ. I ...
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1answer
24 views

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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0answers
122 views

Proof of the multivariate Cramer-Rao inequality

I search a detailed proof of the multivariate Cramer-Rao inequality in the general case where the estimator is not necessarily unbiased. Let $T(X)$ be an estimator of the parameter $\theta\in\mathbb{R}...
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0answers
21 views

How to test that a sequence of variances rank ascendingly?

I am investigating forecast optimality. Diebold (2017, p. 334, list item d) indicates that one of the desirable properties of a good forecast is Optimal forecasts have $h$-step-ahead errors with ...
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1answer
254 views

Chebyshev's inequality for Pareto distribution (3 sigma rule)

According to the Chebyshev's inequality, if we take any distribution, we get >88.8889% of data in +-3 sigma interval. For a normal distribution it is 99.97%. How to calculate the interval for a ...
3
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2answers
88 views

Boundary of $E\left[\frac{\prod_{i=1}^n x_i}{\prod_{i=1}^n x_i+\prod_{i=n+1}^m x_i}\right]$

Suppose $X_i$ are i.i.d. In addition, $X_i>0$ and $E[X_i]>1$. Suppose $E[X_i]$ is known, could we find upper bound or lower bound for the following expectation: $$ E\left[\frac{\prod_{i=1}^n x_i}...
3
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1answer
123 views

When is Jensen's Inequality strict?

For a homework problem, I have to prove that for a random sample $X_1, \ldots, X_n$, drawn from a population with finite variance $\sigma^2$, with sample mean $\bar{x}$ and sample variance $s^2$, that ...
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1answer
57 views

Question about property of 2-increasing, grounded function with margins

The question is about what may very well be an obvious detail in the proof of a lemma from the book An Introduction to Copulas by Roger Nelsen. I will state all the relevant results and definitions ...
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1answer
28 views

Distributive property of probabilistic inequalities involving random variables on both sides

Can I break down $P(h \geq (A + B)$, given all $ A,B,h$ are all random variables. Will the following rule works? $$P[h \geq (A + B)] = P(h\geq A) + P(h\geq B)$$ Actually, in one of my mathematical ...
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40 views

Bounding the norm of the difference between two related probability densities

Suppose we have a continuous random variable $X$ and two continuous functions $f$ and $g$ such that $f(X)$ and $g(X)$ are continuous random variables. Let $p_A$ be the probability density function of ...
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27 views

Generalization of the Payley-Zigmund inequality

The Payley-Zigmund inequality states that for a positive random variable $Z$ the following holds \begin{equation} \operatorname{P}( Z > \theta\operatorname{E}[Z] ) \ge (1-\theta)^2 \frac{\...
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0answers
37 views

Are this simple claim and its proof correct?

Suppose random sequence $\{X_{i}(N)\}_{i=1}^{N}$ is a row-wise i.i.d. triangular array, where $N$ is sample size. This means for any given $N$, $X_{i}(N),\dots,X_{N}(N)$ are i.i.d. following ...
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1answer
321 views

Use Chebyshev's inequality to find a lower bound of a Chi-Square Distribution

I'm trying to solve the following exercise but I'm not sure if what I'm doing is right. "Let $X$ be an r.v. distributed as $\chi_{40}^{2}$. Use Tchebichev’s inequality in order to find a lower ...
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1answer
191 views

Correlation Coefficient Squared is Less Than or Equal to One

Problem Statement: Let $Y_1$ and $Y_2$ be jointly distributed random variables with finite variances. Let $\rho$ denote the correlation coefficient of $Y_1$ and $Y_2.$ Using the inequality $$[E(Y_1Y_2)...
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60 views

How important is triangle inequality for statistical estimators?

(Pearson's) correlation is a measure of co-dependence that does not fulfill certain axioms such non-negativity and triangle inequality. In layman's terms, how would you describe what triangle ...
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1answer
66 views

Upper Bound and Lower Bound on Means when Distributions are bounded?

Suppose we have two different probability distributions $p, q$ defined on input $x \in [0,1]$. We know that for any value of $x$ in the domain, we have $\exp^{-a} \leq \frac{p(x)}{q(x)} \leq \exp^{a} $...
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1answer
42 views

Laplace Inequality

I am trying to prove that if $r_i \sim Lap(0,1/\varepsilon)$ where $\varepsilon >0$ then: $$Pr[r_i \geq 1+r^*] \geq e^{-\varepsilon}Pr[r_i \geq r^{*}]$$. I know that for $r*>0$ it satisfies ...
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1answer
56 views

$var(y)=b^2var(x)+var(e)$

Suposse $y=xb+e$ where $y$ and $x$ are random variable. $e$ is the error of the regression. Since x and e are independent then: $var(y)=b^2var(x)+var(e)$ How can I proof the following double ...
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0answers
43 views

$E(xy)<\infty$ proof

I am reviewing the best linear projection properties proof in Hansen's book on econometrics. Specifically, the proof according to which $E(xy)<\infty$. For this, it is assumed that $E(y^2)<\...
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0answers
16 views

inequality involving mean, median and variance [duplicate]

I'm looking to show $|{\rm med}(x)-\bar{x}|\le{\rm sd}(x)$. I did a bunch of simulations and the statement seems right to me. $$ {\rm Var}(x)=\frac{1}{n}\sum\left(x_i-\bar{x}\right)^2=\frac{1}{n}\sum\...
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0answers
151 views

REINFORCE algorithm, help for the proof of the variance reduction by subtracting a baseline

I'm trying to find a proof or an approximate argument justifying that, in the REINFORCE algorithm, subtracting a baseline to the episode reward reduces the variance. I believe this proof can be done ...
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19 views

How to include information from observations with mathematical inequalities in Ordinary Least Squares regression?

So, I was using Ordinary Least Squares (OLS) linear regression to build a model describing pond water level fluctuations in function of precipitation and potential evapotranspiration (PET) data. The ...
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1answer
68 views

Variance inequality

Why does the following hold? For a random variable X with finite necessary moments, $E(|X|) \leq \sqrt{Var(X)}+|E(X)|$
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1answer
148 views

Expectation of (sum subtract the expectation of sum)

Let's say we have random variables $\mathbf{X}$, and we have $P(\mathbf{X}\in [a, b])=1$, we have $\mathbf{S}_n = \mathbf{X}_1 + \mathbf{X}_2, +\dots + \mathbf{X}_n$. If $\mathbf{X}_1, \mathbf{X}_2, ...
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0answers
56 views

Normal distribution inequality

I came across a theorem that involves inequality in a normal distribution: For $z> 0$, the function $\Phi \left ( z \right )$ satisfies the inequality $$\left ( \frac{1}{z}-\frac{1}{z^{3}} \right )...
5
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1answer
414 views

Difference of two KL-divergence

The Kullback-Leibler (KL) divergence between two distributions $P$ and $Q$ is defined as $$\mbox{KL}(P \| Q) = \mathbb{E}_P\left[\ln \frac{\mbox{d}P}{\mbox{d}Q}\right].$$ My question is that suppose ...
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0answers
90 views

Inequalities on Fisher Information / expected second derivative?

Under some regularity conditions we can compute fisher information as $ - \mathbb{E}_{\theta_0} [\frac{\partial}{\partial \theta^2} \ln f(x;\theta_0)] $ I was wondering if there are some kind of ...
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1answer
118 views

Need mathematical steps for Hoeffding's Inequality applied to Bernoulli Distribution

I am trying to understand Hoeffiding's Inequality in Machine Learning and I am referring to WikiPedia for it. Hoeffding's Inequality is defined as follows: $ P(|\hat{\theta} - \theta)| \ge \epsilon) \...
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1answer
79 views

Strongly convex function evaluated over a mean of n points

Let f(x) be a Strongly-convex function under some m > 0. Given two points x, y it is known that: $$f(\frac{x + y}{2}) \leq \frac{f(x) + f(y)}{2} - \frac{1}{2^3} \cdot m \cdot ||x - y||^2$$ What is ...
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1answer
67 views

Is $E(T) \ge \sigma$ or is $E(T)\le\sigma$?

Let $X_1,\cdots X_n \sim N(0,\sigma^2)$ be i.i.d., and let $T=\sqrt{\frac{1}{n}\Sigma^n_{i=1}(X_i^2)}$. Is $E(T) \ge \sigma$ or is $E(T)\le\sigma$? My work: $\frac{X_i^2}{\sigma^2}\sim \chi^2_1$ ...
5
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2answers
131 views

How can I establish an inequality between $|\frac1n \sum_{i=1}^nX_i|$ and $\frac1n\sum^n_{i=1}|X_i|$ where $X_i \sim N(0,1)$?

Let $X_1, \ldots , X_n$ be a random sample from a $N(0,1)$ population. Define $Y_1=|\frac1n \sum_{i=1}^nX_i|$ and $Y_2=\frac1n\sum^n_{i=1}|X_i|$. Find a relationship between $E(Y_1)$ and $E(Y_2)$. I ...