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.
136
questions
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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 ...
- 6,988
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votes
1
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18
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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 $X^{(1)}$...
3
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1
answer
154
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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 ...
- 238
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0
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26
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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.
$$E_\pi[...
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0
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18
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Is there any theory pertaining to an assumption to would allow the following inequality to hold (constrained LASSO)
Suppose we observe the vector-matrix pair $(y,\mathbf{X})\in\mathbb{R}^n\times\mathbb{R}^{n\times d} $ which is linked by the observation model:
\begin{equation}
y=\mathbf{X}\theta^*+\epsilon
\end{...
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2
votes
2
answers
139
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Is it true that $\langle X^4\rangle \ge 3 \langle X^2\rangle^2$?
Consider a real random variable $X$ with zero mean. Does the following inequality hold in general?
$$\langle X^4\rangle \ge 3 \langle X^2\rangle^2$$
I'm not sure how to prove this or if a counter-...
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1
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91
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Sign of Correlation between $X$ and $f(X)$ for strictly monotonic $f$
This question is a follow up to this question.
Suppose $f$ is strictly increasing. Can we say
$$\text{Cov}(X,f(X))\geq 0?$$
Ben's answer on the aforementioned linked post can be extended to show the ...
- 523
3
votes
1
answer
93
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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}\...
- 43
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15
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Lower bound on gradient of the restricted function
Let $A$ be an $N\times n$ matrix where $N < n$ and $b$ be a vector in $\mathbb{R}^N$. Let $f: \mathbb{R}^n \to \mathbb{R}$ be a quadratic function $f(x)=\frac{1}{N}\|Ax-b\|^2$.
Define two sets $J \...
- 111
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0
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82
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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{...
- 145
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votes
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156
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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$...
- 145
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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 &...
- 345
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292
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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$$
where $\mathcal{B}_0\cup\mathcal{B_1}=\...
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0
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Testing a special inequality hypothesis
Suppose that we are facing a hypothesis:
$$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 ...
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0
votes
0
answers
23
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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/...
- 265
1
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1
answer
319
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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, ...
- 2,040
1
vote
2
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88
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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 ...
- 15
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450
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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 $
...
- 151
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269
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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
...
- 13
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38
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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) \...
- 2,361
4
votes
1
answer
74
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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....
- 253
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1
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294
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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 ...
- 115
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0
answers
90
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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}...
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481
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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 ...
- 309
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vote
1
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26
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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 ...
- 11
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582
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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 ...
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2
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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|}{...
- 199
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votes
0
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26
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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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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 \...
- 1,144
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3
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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
$$P(X_1>c)\le ...
- 10.6k
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1
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49
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Find the minimum percentage of values within two bounds subject to moment constraints
I am facing the following problem:
Variable Z has a mean of 15 and a standard deviation of 2. What is the minimum percentage of Z values that lie between 8 and 17?
I have tried the following: Here ...
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4
votes
1
answer
370
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Hoeffding type concentration result for the inverse of a sum of iid random variables
Consider a collection of $n$ i.i.d. Bernoulli random variables $\{ X_i \}_{i=1}^{n}$ with $\mathbb{E}[X_i] = \mu$.
Then, if $\hat{\mu}$ is the mean of the $n$ random variables, i.e. if,
$$\hat{\mu} = \...
- 155
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0
answers
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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\...
- 1,144
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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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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
$$\min \ell(\...
- 121
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0
answers
46
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prove the difference between mean and median is less than the variance [duplicate]
Suppose $X$ is a random variable with finite variance. Let $m$ denote
the median of $X$ and $\mu$ the mean of $X$, i.e. $\mu=\mathbb{E}(X)$.
Show $$(m-\mu)^2\leq\text{var}(X)$$
Intuitively this is ...
- 1,459
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votes
2
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467
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Prove that Kurtosis is at least one more than the square of the skewness
Wikipedia claims it, and on reading the paper that it linked I found that the proof that was written there was quite difficult. Is there a simple proof possible for this identity?
The proof given in ...
- 525
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1
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558
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Upper bound for absolute third central moment
Suppose $X\in \mathbb{R}$ is a random variable with expected value $\mathbb{E}X = \mu$. I ran across a proof which uses the inequality
$$
\mathbb{E}[|X - \mu|^3] \leq 2^3 \mathbb{E}|X|^3.
$$
Can ...
- 170
0
votes
0
answers
99
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cauchy schwarz inequality on sum of squares
Can I and how I can use the Cauchy Schwarz inequality on the amplitude of a imaginary sum of squares?
$$Z = X+iY$$ and $$|Z| = \sqrt{X^2 +Y^2}$$
to show that $$|E[Z] | \leq \mathbb{E}[|Z|]$$
where $X,...
user310375
1
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0
answers
284
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How can I weight ordinal observations and reduce them to one statistic? [closed]
It will be a complicated question and I try to briefly explain. I am studying on educational inequalities. The survey I use for analysis incudes ordinal variables and the education degree which ...
- 39
3
votes
1
answer
49
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Implications of zero limiting variance
Assume that I have a sequence of random variables $X_1, X_2, \dots$ with means $\mu_1, \mu_2, \dots$ such that $\lim_{n \to \infty} \operatorname{Var}(X_n) = 0$.
Can I claim that for large enough $n$ ...
- 446
3
votes
1
answer
1k
views
Proof of the multivariate Cramer-Rao inequality
I search a detailed proof of the multivariate Cramer-Rao inequality in the general case where the estimator is not necessarily unbiased.
Let $T(X)$ be an estimator of the parameter $\theta\in\mathbb{R}...
- 373
2
votes
0
answers
28
views
How to test that a sequence of variances rank ascendingly?
I am investigating forecast optimality. Diebold (2017, p. 334, list item d) indicates that one of the desirable properties of a good forecast is
Optimal forecasts have $h$-step-ahead errors with ...
- 63.6k
1
vote
1
answer
582
views
Chebyshev's inequality for Pareto distribution (3 sigma rule)
According to the Chebyshev's inequality, if we take any distribution, we get >88.8889% of data in +-3 sigma interval.
For a normal distribution it is 99.97%.
How to calculate the interval for a ...
- 15
3
votes
2
answers
99
views
Boundary of $E\left[\frac{\prod_{i=1}^n x_i}{\prod_{i=1}^n x_i+\prod_{i=n+1}^m x_i}\right]$
Suppose $X_i$ are i.i.d. In addition, $X_i>0$ and $E[X_i]>1$. Suppose $E[X_i]$ is known, could we find upper bound or lower bound for the following expectation:
$$ E\left[\frac{\prod_{i=1}^n x_i}...
- 51
3
votes
1
answer
1k
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 ...
- 622
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votes
1
answer
105
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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 ...
- 101
2
votes
1
answer
50
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 ...
- 524
0
votes
0
answers
86
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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 ...
- 883
1
vote
0
answers
29
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{\...
- 111