Questions tagged [quantile]

Question 1

For multi OLS regression. I normally use variance attribution to explain how much contribution an independent/predictor variable has to the dependent variable. For example, a standard two factor ...

Question 2

Result. If $g:\mathbb R\rightarrow\mathbb R$ is non-decreasing and left continuous, then $$V_\alpha(g(L))=g(V_\alpha(L))$$ for all $\alpha\in[0,1]$ where, $V_\alpha(L)$ is the value at risk of loss ...

Question 3

Let $V_\alpha(L)$ be the value at risk of a function $L$ at $\alpha\in[0,1]$, $$V_\alpha(L)=\inf\{x:P(L\le x)\ge\alpha\}.$$ Prove that for any $\alpha$, $$V_\alpha(-L) =-\lim_{\epsilon\rightarrow0^+}...

Question 4

I was trying to show that a push forward operation of a vector field $\phi$ in a density $p$ $( \phi \# p )$ preserves the quantiles. For example, I want to show that: \begin{equation} F_{p}(x) = \...

Question 5

Fisher had pointed out that $\sqrt{2\chi^2_n}$ to be normally distributed about $\sqrt{ (2n-1)}$ with unit standard deviation (when n is large). But how did he get to that conclusion? We know $E(2\chi^...

Question 6

I’ve recently encountered specific problem and I am ashamed to admit that I am quite stuck. Suppose that I have random sample of data for which I want to calculate e.g. 95th quantile. Suppose that it ...

Question 7

Let $X$ and $Y$ be independent, centered random variables. Let $\alpha \in (0,1)$ be fixed (e.g., $0.95$), and denote by $Q(X)$ the $\alpha$-quantile of $X$. I would like to determine under what ...

Question 8

Assume we have $F_X$ and $F_Y$ as cumulative distribution functions of the two variables $X$ and $Y$, and $F^{-1}_X$, $F^{-1}_Y$ are the inverse of them. I am trying to show the following: We take an ...

Question 9

I found the following problem in a step of a proof of a certain theorem. Given a random variable $X$ with bounded expectation and $\alpha\in(0,1)$, the problem is $$\min_{z\in \mathbb{R}}\left\{\alpha\...

Question 10

Let $X_1,\ldots,X_K$ be possibly dependent random variables that are marginally $\mathrm{Uniform}[0,1]$ distributed, and define $\bar X := \frac{1}{K}\sum_{i=1}^K X_i$. Is it possible for the $\gamma$...

Question 11

Problem 21.1 of the book 'Asymptotic Statistics' by Aad van der Vaart reads the following Suppose that $F_n \to F$ uniformly. Does this imply that $F_n^{-1} \to F^{-1}$ uniformly or pointwise? Give a ...

Question 12

I'm rebuilding the methods described in the paper Strong statistical parity through fair synthetic data, and on page 3 it describes the following methodology: We align both distributions by learning ...

Question 13

According to Wikipedia, the quantile function is defined by $$Q(p)=\inf \{x\in\mathbb{R}:F(x)\geq p \}.$$ But if I apply this to equally likely data set 10, 11, 12, 13, I get $Q(0.5)=11$. But shouldn'...

Question 14

Let $F$ be the CDF of $X$ and $p \in (0, 1)$, and $F_n$ be the empirical CDF of $X_1, ..., X_n$; $F_n(x) = \frac{1}{n}\sum_{i = 1}^nI(X_i \le x)$. The $p$ th quantile of $F$ and $F_n$ are defined as ...

Question 15

Suppose that $f(x)$ is a smooth probability density function on $\mathbb R$ and denote by $a_i$ the $\frac{2i-1}{2n}$-th quantile of $F$ for $1\leq i \leq n$, where $F(x)$ is the cumulative ...

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Questions tagged [quantile]

Quantile Regression variance attribution

Show that $V_\alpha(g(L))=g(V_\alpha(L))$ for non-decreasing, left continuous $g$ [closed]

Prove for value at risk $V_\alpha(-L)=-V_{1-\alpha}(L)$

Does the push forward operation of a vector field in a probability density preserve quantiles?

fisher’s square root transformation

Using Student’s distribution quantile for Normal distribution

Subadditivity of Quantiles: For Which Distributions Does $Q^2(X+Y) \leq Q^2(X) + Q^2(Y)$ hold?

Quantile to Quantile function when two random variables are linearly related

Does this probability theory problem have a name?

Quantile of the average of dependent Uniform[0, 1] variables [closed]

Uniform convergence of cdf implies uniform convergence of quantile functions

A linear function that transforms the set of a quantile distribution to match another quantile distribution

Is quantile just quantile function evaluated at specific values?

Quantile of Empirical CDF and Tail Bound

Show that $\int_{-\infty}^{a_1} (a_1-x)^r f(x) \mathrm{d} x$ and $\int_{a_n}^\infty (x - a_n)^r f(x) \mathrm{d} x$ are of order $\mathcal{O}(n^{-r})$

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