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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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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25 views

Question about an implication relationship

Let $||\cdot||$ be the Euclidean norm. Suppose $X_1,X_2$ are two independent and identically distributed random variables, and $a_N(X_1,X_2)$ is a vector valued function that depends on factor $N$ and ...
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18 views

Assessment of bias by imputation

Lets say i have a large survey dataset which i want to use as a source for income reporting of the population, e.g. parameters of the distribution, poverty and inequality. Due to item nonresponse on ...
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1answer
34 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
35 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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23 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
37 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
36 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
25 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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38 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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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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13 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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14 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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15 views

Creating a transition matrix based on a Markov chain in R

I have four distributions that represent incomes in R. I categorise them by what income group they fall under such as under half the mean, between half the mean and 3/4th of the mean and so on until ...
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5 views

Proving two different expressions of non-centrality parameters are equivalent

I am stuck in proving $$\sum_{i=1}^{K}\xi_i(\mu_i - \bar{\mu})^2 = \sum_{i,j}\xi_i\xi_j(\mu_i - \mu_j)^2,$$ where $\bar{\mu} = \sum_{i=1}^{K}\xi_i\mu_i$ and $\sum_{i=1}^{K}\xi_i = 1$. I am not sure ...
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1answer
37 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
25 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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10 views

Support Vector Machines, finding the Lagrangian multipliers and b

Hi guys, I am trying to use SVM to classify my data samples. You can find my dataset attached above where A B C are negative samples and D E are positive samples. For convenience, I have also attached ...
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7 views

Analysis of large deviations of the empirical survival function

I have been doing some self studying in survival analysis, but I seem to be stuck on this book problem. It's asking us to use the Hoeffding's Inequality for the analysis of large deviations of the ...
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1answer
79 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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63 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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14 views

markov's inequality generalizability

Let $X : \Omega \rightarrow \mathbb{R}$ be a non-negative random variable on probability space $(\Omega, \mathscr{A}, P)$ and let $c > 0$. Then: $$\mathrm{P}[X > c] \leq \frac{\mathbb{E}(X)}{...
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1answer
48 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
50 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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7 views

Sample size for uniform multinomial to control minimum

I'm interested in controlling the minimum for a uniform multinomial distribution. Specifically, What would be the sample size to have the following inequality regarding the minimum? Do you know of ...
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1answer
66 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: $\frac{X_i^2}{\sigma^2}\sim \chi^2_1$ ...
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2answers
116 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 ...
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1answer
39 views

Hoeffding's inequality with a probabilistic bound?

Hoeffding's inequality states that: Let $\mathbf{X}_1, \dots, \mathbf{X}_n$ be independent random variables, such that $\mathbf{X}_i \in [a_i, b_i]$ with probability one. Let $\mathbf{S}_n$ be $\...
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21 views

Expectations, Double Integrals and Jensen's Inequality

Consider two random variables distributed $v\backsim G(.)$ and $c \backsim F(.)$ with pdfs $g(.)$ and $f(.)$. Let the supports of $c$ and $v$ be $[x,y]$. Let $x<a=E(v)<b<y$, so $[a,b]\subset\...
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1answer
25 views

Does this expectation inequality holds?

Let $X\in L_p(P), p>1$. Is the following result true? $$E[\lvert X\rvert I(\lvert X\rvert>C)]\leq C^{1-p}E\lvert X\rvert^p.$$ where $C>0$. It can be found in the proof of Corollary A.1 (...
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2answers
136 views

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 ...
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0answers
20 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?
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28 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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57 views

Some inequality problem, which is larger?

There is a set of n random numbers whose sum equals 1, i.e., $$w_a=\{w_{a1},\; w_{a2},\; \ldots,\; w_{an}\}$$ where $$ w_{a1} + w_{a2}+ \ldots+ w_{an} =1$$ and $$0 < \{ w_{a1}, w_{a2}, \ldots, w_{...
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1answer
85 views

Need help to understand Feller's statement “whenever $r$th moment exists so do all preceding moments”

I am reading the book of Feller called "An Introduction to Probability Theory and Its Applications, Vol I" (third edition, page 227) and am stuck at the moment he explains the notion of variance of a ...
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21 views

Conditioning and linear MSE

Let $\sigma_{X|Y}^2$ denote the linear mean squared error in estimating $X$ from $Y$. Then is it always true that additional conditioning cannot increase the LLSE? In other words, is this true? $$ \...
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1answer
43 views

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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1answer
60 views

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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1answer
152 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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1answer
78 views

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 $Var(Y - \hat Y) = \sigma^2(I-H)$ ...
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2answers
758 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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0answers
37 views

Can I test for inequality in H0 using chi square test?

Let's say I want to test whether an $n$-sided dice is not too unfair. In the standard chi-square test we test the zero-hypothesis $$ H_0\colon (p_1,\dots p_n) = (1/n,\dots,1/n) ,\quad\text{i.e.,}\quad ...
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1answer
36 views

Estimator based on inequality data

$X_i \sim N(\mu, \sigma^2)$ (iid), $i = 1,2,...,N$, I want to estimate $\theta = (\mu, \sigma^2)$. Problem is, I don't observe $x_i$. For each $i$, I only observe $(a_i, b_i)$, and I know that $a_i &...
2
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2answers
47 views

Does an inequality hold as an expectation over a probability distribution?

Suppose I have to functions $f(x)$ and $g(x)$ such that $$ f(x) \leq g(x) \quad \forall x. $$ For a distribution $\pi(x)$ on $x$, is it necessarily true that $$ E_\pi[f(x)] \leq E_\pi[g(x)]? $$ My ...
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1answer
341 views

A different proof for KL divergence non-negativity

KL divergence's non-negativity can be proved in many ways. One could use the inequality $\log x \leq x - 1$ as a main step in the proof, another one could leverage the property of concave of the ...
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3answers
313 views

Cauchy Schwarz inequality proof using discriminant

I know the proof but I'm unclear on one thing. Cauchy-Schwarz inequality: Given X,Y are random variables, the following holds: $$ (E[XY])^2 \le E[X^2]E[Y^2] $$ Proof Let $$ u(t) = E[(tX - Y)^2] $$ ...
4
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0answers
117 views

Proving an inequality for CDF's

I am working on a proof to show that given $x_1, x_2,\ldots,x_k$ random variables with a joint pdf and joint CDF, show that $$ 1-\sum_{i=1}^k \overline{F_i(x_i)} \leq F(x_1,x_2,\ldots,x_k) \leq \min_i ...
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2answers
71 views

Summation of squared x_i if summation of x_i is 1

How to prove "If $\sum_{i=1}^n x_i=1$, then $\sum_{i=1}^n x_i^2>1/n$"? I'm thinking about $Var(x_i)=E(x_i^2)-[E(x_i)]^2=\frac{1}{n}\sum_{i=1}^n x_i^2-1/n^2\ge0$. Is that correct?
1
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1answer
32 views

Help understanding a probability inequality

I'm working throught Wasserman's "All of Statistics" book. When proving convergence of random variables/distributions in chapter 5, he lists the following inequality: $$F_n(x) = \mathbb{P}(X_n\le x)=\...
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1answer
274 views

Prove that $E(X\ln X)\le E X E\ln X$ [closed]

I want to prove it using Jensen inequality, so I need to prove that $g(x)=x\ln x$ is a convex function, which means $$g\left(\frac{a+b}{2}\right)\le \frac{1}{2}\left(g(a)+g(b)\right).$$ How can I ...