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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9 votes
2 answers
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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 ...
Martund's user avatar
  • 535
2 votes
0 answers
29 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 ...
Richard Hardy's user avatar
1 vote
1 answer
800 views

Linear regression question on Idempotent matrix and leverage points

I am considering a linear regression model $Y_i = X_i^T \beta + \epsilon_i, i = 1,2,\dots,n$. where $X_i \in \mathbb{R}^p$. $\epsilon_i$'s are independent copies of random error $\epsilon \in \mathbb{...
user111092's user avatar
0 votes
1 answer
1k 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 ...
chole's user avatar
  • 3
7 votes
1 answer
202 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
5 votes
1 answer
682 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
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
  • 935
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/...
Robin Andrews's user avatar