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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 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 ...
Voyager's user avatar
  • 305
2 votes
1 answer
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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-...
Don P.'s user avatar
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2 votes
1 answer
48 views

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 $u$ \begin{equation} \...
Don P.'s user avatar
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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 ...
Stephen Jiang's user avatar
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0 answers
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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, ...
larry77's user avatar
  • 203
7 votes
2 answers
434 views

Advantage of GLMs over transformation models

According to the book I am currently reading, we should prefer GLMs over a simple transformation model (e.g. $\log(y) = x_i^T \beta + u_i$ ) The argument is derived by Jensen's inequality: $$ E(\log(\...
Marlon Brando's user avatar
1 vote
0 answers
28 views

Mutual Information decay

Consider $m$ channels indexed by $i$ with $1 \leq i \leq m$. The input alphabets are from the same finite set $\mathcal{X}$. Let $\pi$ denote a probability distribution on $\mathcal{X}$. Define the ...
Sushant Vijayan's user avatar
1 vote
1 answer
39 views

If event $A$ is a union of elements of $S(A)$, then $\min_{Z\in S(A):Z\subset A}P(B\mid Z)\leq P(B\mid A)$ for any event $B$

Let $A$ and $B$ be events in a probability space and $S(A)$ a collection of events such that $A$ is a union of some elements in $S(A)$. How could I then conclude that the conditional probabilities ...
Cartesian Bear's user avatar
1 vote
0 answers
16 views

Effect on entropy when we scale Bernoulli plus Gaussian

Question: Given $X\sim\text{Bernoulli}(\alpha)$, $Y\sim\mathcal{N}(0,1)$, and non-random positive constants $C,\epsilon>0$. Let $H(\cdot)$ be the differential entropy. Is it true that $$ H((C+\...
Resu's user avatar
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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: $$\mathbb{V}\left[\frac1n\sum_{i=1}^nX_i\right]$$...
Tristan Nemoz's user avatar
0 votes
1 answer
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 ...
ornitorrinco's user avatar
1 vote
1 answer
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 ...
Galen's user avatar
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1 answer
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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 $X^{(1)}$...
jack of all woes's user avatar
3 votes
1 answer
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 ...
AvidLearner's user avatar
1 vote
0 answers
29 views

Derivation of the target upper-bound with Jensen's inequality in 'Divide-and-Conquer RL'

I am currently stuck in one line in the paper named "Divide-and-Conquer Reinforcement Learning"(Ghosh et. al., 2018 ICLR). It is the equation (1) in the page 4 which is like below. $$E_\pi[...
KCLEE's user avatar
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0 answers
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Is there any theory pertaining to an assumption to would allow the following inequality to hold (constrained LASSO)

Suppose we observe the vector-matrix pair $(y,\mathbf{X})\in\mathbb{R}^n\times\mathbb{R}^{n\times d} $ which is linked by the observation model: \begin{equation} y=\mathbf{X}\theta^*+\epsilon \end{...
Carl's user avatar
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3 votes
2 answers
163 views

Is it true that $\langle X^4\rangle \ge 3 \langle X^2\rangle^2$?

Consider a real random variable $X$ with zero mean. Does the following inequality hold in general? $$\langle X^4\rangle \ge 3 \langle X^2\rangle^2$$ I'm not sure how to prove this or if a counter-...
a06e's user avatar
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3 votes
1 answer
127 views

Sign of Correlation between $X$ and $f(X)$ for strictly monotonic $f$

This question is a follow up to this question. Suppose $f$ is strictly increasing. Can we say $$\text{Cov}(X,f(X))\geq 0?$$ Ben's answer on the aforementioned linked post can be extended to show the ...
Golden_Ratio's user avatar
3 votes
1 answer
165 views

Expectation of the Inverse of the Sample Mean vs. Inverse of Expectation of the Sample Mean

For an iid sample of $n$ realizations of random variable $X$ with $\mathbb{E}\left[X\right] = \mu > 0$, let $\bar{X}_n$ denote its sample mean. Is there a distribution for $X$ such that $\mathbb{E}\...
Jakob J's user avatar
  • 43
0 votes
0 answers
146 views

On the difference between two independent beta distribution

This is a follow up to this question. Consider again two independent rvs $X\sim Beta(a_1+1,b_1+1)$ and $Y \sim Beta(a_2+1,b_2+1)$. Here $a_1,b_1,a_2,b_2$ are in $\mathbb{N}$. Is it true that $\mathbb{...
Sushant Vijayan's user avatar
7 votes
1 answer
209 views

Property of two independent Beta distribution

I have been working with beta-bernoulli posteriors recently. Is it true that if $X,Y$ are independent rvs with $X \sim Beta(a_1+1,b_1+1)$ and $Y \sim Beta(a_2+1,b_2+1)$ then $\mathbb{P}(X>Y)>0.5$...
Sushant Vijayan's user avatar
0 votes
0 answers
37 views

Probability of mixture distribution

Let \begin{equation} p_1=\Pr(0.8z_1^2+0.2z_2^2 > c), \qquad p_2=\Pr(0.2z_1^2+0.2z_2^2 > c), \end{equation} where $z_1,z_2 \sim N(0,1)$. Then, can I have $p_1 >p_2$ since $0.8z_1^2+0.2z_2^2 &...
user0131's user avatar
  • 357
3 votes
1 answer
324 views

Testing a Null Hypothesis Nested by the Alternative Hypothesis

Consider a parameter of interest $\beta \in \mathcal{B}$, and two hypotheses $$H_0:\;\beta\in\mathcal{B_0}\quad versus \quad H_1:\;\beta\in\mathcal{B}_1$$ where $\mathcal{B}_0\cup\mathcal{B_1}=\...
M.C. Park's user avatar
  • 935
1 vote
0 answers
57 views

Testing a special inequality hypothesis

Suppose that we are facing a hypothesis: $$H_0: \mu\geq 0 \quad vs \quad H_1: \mu\geq c$$ where $\mu$ is the parameter of interest and $c$ is a unknown paraneter. Here, we only have the distribution ...
M.C. Park's user avatar
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0 answers
23 views

Probability of higher hourly earnings by gender from UK data [duplicate]

I'm trying to find the probability that in the UK, in a male, female pair picked at random, the female will have the higher annual income. I have following data from https://www.ons.gov.uk/...
Robin Andrews's user avatar
1 vote
1 answer
479 views

What does it mean that the decomposition is based on the linear systematic component? And how can I interpret my result?

I'm using the oaxaca package to implement a Blinder-Oaxaca decomposition on a logistic model with binary outcome. The vignette says that: Note that, if a non-linear function such as glm() is chosen, ...
robertspierre's user avatar
1 vote
2 answers
88 views

Sufficient condition for $ \sigma_{X}^{2} \leq \sigma_{Y}^{2}$

Suppose $X$ and $Y$ are random variables whose expected values are $\mu_X$ and $\mu_Y$, and variances are $\sigma_{X}^{2}$ and $\sigma_{Y}^{2}$, respectively. Also, we suppose $F_x$ and $F_Y$ are the ...
user12518177's user avatar
5 votes
1 answer
533 views

Taylor expansion in Hoeffding's Lemma proof

Hoeffding's Lemma proof uses Taylor expansion with this statement: From Taylor's theorem, for some $ 0\leq \theta \leq 1$ $ L(h) = L(0) + h L'(0) + \frac{1}{2} h^2 L''(h\theta) \leq \frac{1}{8}h^2 $ ...
Tavakoli's user avatar
  • 151
0 votes
0 answers
364 views

What is the difference between MVB UMVUE and MVUE.?

Cramer Rao inequality gives MVB and if MVB exist it is MLE. Rao Blackwell gives UMVUE, but isn’t when we have MVB estimator for unbiased it is UMVUE? Then what is MVUE? MVB minimum variance bound ...
User0405's user avatar
2 votes
0 answers
40 views

For $Y \geq 0$, prove that $Pr(Y \geq k) \leq E(Y)/k$

Let $Y$ be a non-negative random variable, $k$ be any positive constant, show that $Pr(Y \geq k) \leq E(Y)/k$. My attempt (using integration by parts): \begin{align} \int_0^k y \,dF(y) &\leq E(Y) \...
Dayne's user avatar
  • 2,661
4 votes
1 answer
87 views

Sum of values with different probabilities

Suppose I have the following linear expression: $S = x_1 + x_2+ \dots + x_n$, in which each $x_i$ can only assume the following values: -2, -1, 0, 1, 2 whose probabilities are 0.1, 0.2, 0.2, 0.25, 0....
donut's user avatar
  • 263
2 votes
1 answer
432 views

Probability Conditioned on Inequality

Assume that $A \sim \mathcal{N}(0, 1)$, $B \sim \mathcal{N}(0, 1)$. I am trying to calculate $P(A \,|\, A < B)$. For the sake of this problem, we can assume that $A \perp B$, but (for obvious ...
zen_of_python's user avatar
1 vote
0 answers
112 views

Khintchine inequality for the linear combination of sparse Bernoulli random variables

Let $\{\epsilon_{n}\}_{n=1}^{N}$ be i.i.d. random variables with $P(\epsilon_{n} = \pm 1) = 1/2$ for $n=1,2, \ldots, N$ i.e. a sequence of Rademacher distribution. Let $0<p<\infty$ and let $x_{1}...
Bhisham's user avatar
  • 319
8 votes
2 answers
543 views

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 ...
sayda's user avatar
  • 309
1 vote
1 answer
26 views

Conditionalizing events on more than one event

I am currently working on a question which seems to have an obvious answer, but it it seems just impossible for me to find a stringent proof of this relation (if it is true). Imagine the following ...
Zito's user avatar
  • 11
0 votes
1 answer
867 views

Why is "Jensen's Inequality" Important in Probability?

Why is Jensen's Inequality Important in Probability and Statistics? I was reading the Wikipedia page on "Convex Functions" (https://en.wikipedia.org/wiki/Convex_function), and came across ...
stats_noob's user avatar
6 votes
2 answers
218 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|}{...
Bhisham's user avatar
  • 319
0 votes
0 answers
29 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?
M.C. Park's user avatar
  • 935
2 votes
0 answers
277 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 \...
Carl's user avatar
  • 1,226
6 votes
3 answers
139 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 ...
StubbornAtom's user avatar
  • 11.5k
2 votes
1 answer
49 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 ...
shan's user avatar
  • 131
5 votes
1 answer
489 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} = \...
ijuneja's user avatar
  • 185
1 vote
0 answers
232 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\...
Carl's user avatar
  • 1,226
10 votes
2 answers
5k 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 ...
OliverVD's user avatar
  • 139
1 vote
0 answers
54 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(\...
MathIsLife12's user avatar
2 votes
0 answers
47 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 ...
Tan's user avatar
  • 1,499
9 votes
2 answers
737 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 ...
Martund's user avatar
  • 545
5 votes
1 answer
735 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 ...
safelyanonymous's user avatar
0 votes
0 answers
115 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,...
user avatar
1 vote
0 answers
379 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 ...
Maxpayne's user avatar