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.

Filter by
Sorted by
Tagged with
1 vote
1 answer
633 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 ...
1 vote
1 answer
38 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 ...
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+\...
7 votes
1 answer
200 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 $\mathbb{P}(X>Y)>0.5$...
0 votes
0 answers
34 views

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: $$\mathbb{V}\left[\frac1n\sum_{i=1}^nX_i\right]$$...
0 votes
1 answer
48 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 ...
3 votes
1 answer
124 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 ...
1 vote
1 answer
55 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 ...
4 votes
1 answer
86 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....
0 votes
1 answer
22 views

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 $X^{(1)}$...
3 votes
1 answer
195 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 ...
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[...
0 votes
0 answers
18 views

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{...
3 votes
2 answers
155 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-...
3 votes
1 answer
144 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}\...
0 votes
0 answers
113 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{...
17 votes
1 answer
4k views

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 ...
3 votes
1 answer
2k 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}...
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 &...
3 votes
1 answer
317 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}=\...
1 vote
0 answers
56 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 ...
1 vote
1 answer
413 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, ...
0 votes
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/...
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 ...
5 votes
1 answer
513 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 $ ...
0 votes
0 answers
344 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 ...
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) \...
2 votes
1 answer
386 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 ...
1 vote
0 answers
110 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}...
8 votes
2 answers
530 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 ...
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 ...
0 votes
1 answer
780 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 ...
0 votes
0 answers
95 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 )...
2 votes
2 answers
414 views

Detecting outliers in binary data using Mahalanobis distance

I have a binary vector $X_i$, $i=1...N$ of independent Bernoulli variables with parameters $p_i, \mu_i = p_i, \sigma_i^2 = p_i(1-p_i)$ (which is known) and I'm looking for some sort of tail bound to ...
6 votes
2 answers
217 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|}{...
0 votes
0 answers
28 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?
2 votes
0 answers
257 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 \...
6 votes
3 answers
137 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 ...
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 ...
5 votes
1 answer
444 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} = \...
1 vote
0 answers
222 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\...
5 votes
1 answer
1k 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 ...
10 votes
2 answers
4k 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 ...
1 vote
0 answers
53 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(\...
2 votes
0 answers
46 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 ...
9 votes
2 answers
644 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 ...
5 votes
1 answer
681 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 ...
1 vote
1 answer
743 views

Inequality of two independent random variables

My question is related to this one but more specific. Inequality on two random variables We have two continuous random variables, $X$ and $Y$. We know that the expected value of both is 0 (or more ...
0 votes
0 answers
110 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,...
3 votes
1 answer
592 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 ...