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Questions tagged [quantile-function]

For questions related to the so called quantile function of a cumulative distribution function or generalized inverse of a cumulative distribution function.

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Adequate Root Finder To Compute The Quantile Function

The cumulative distribution function of the standard normal distribution $\Phi(z)=\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^z e^{-t^2/2}dt$ cannot be expressed in terms of elementary functions, ...
m-stgt's user avatar
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1 answer
19 views

Computing inverse of sums of strictly monotanically function (e.g CDF of random variables)

I'm trying to compute the general case of inverse of CDF of $Y=X^2$, where $Y,X$ are random variables. Given that a CDF $F_X$ is a strictly increasing function, also has to be it's inverse. The CDF $...
Daniel Muñoz's user avatar
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18 views

What is the probability of $\alpha$- quantile

I didn't understand why the $P(X \leq q_{\alpha}^- (X))=\alpha$ for $0<\alpha<1$. Please help me with this. The setup for it: X is a real random variable and the lower $\alpha$-quantile defined ...
Pirsu TURAN's user avatar
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17 views

the value of the finite sum $F(n)=\frac{1}{n}\sum_{z=1}^nq(z/(n+1))$ for the Pareto quantile function $q$

Let $q:(0,1)\mapsto\mathbb{R}_{\ge0}$ be the quantile function associated to the Pareto distribution, $q(x)=a(1-x)^{-1/b}$. I am interested in the value of $F(n)=\frac{1}{n}\sum_{z=1}^n q(z/(n+1))$. ...
amanwithnoname's user avatar
1 vote
1 answer
41 views

Uniform convergence of cdf implies uniform convergence of quantile functions

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 ...
Stan's user avatar
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1 answer
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When is the quantile function of a gamma distribution concave?

I am thinking about the consequences of adding prediction intervals and the consequence it has on the resulting interval. For example, I am considering when to expect the sum of two such intervals to ...
Galen's user avatar
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41 views

Quantile function

I have the following problem: I have the function $f(x,y)=\frac{1}{(2\pi)}(1+x^2+y^2)^{(-3/2)} $. I have found the quantile function $Q_{T}$ of $T$, where $T=|Y|$, to be: $\tan(\frac{\pi \cdot y}{2})$....
Amy A's user avatar
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0 answers
67 views

Using quantile function of eventually monotonic pdf to get something like an expectation

Consider a random variable on $\mathbb{R}$ that may or may not have an expectation. (E.g., if its pdf is a Cauchy distribution, it won't.) Let $p(u)$ be a probability density function on $\mathbb{R}$. ...
HW.'s user avatar
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Prove $E(X)=\int_0^\infty1-F_X(x)\,dx$ using quantile functions [duplicate]

I’ve encountered following problem: Let $X$ be a positive random variable with distribution $P^X$, cumulative distribution function $F_X$ and quantile function $q_X$. Show that $$E(X)=\int_0^\infty1-...
Papillus's user avatar
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Why does the quantile function of bivariate normal variables become non-elementary in one dimension?

I have been studying a bivariate random process where $X \sim N(0, \sigma_x), Y \sim N(0, \sigma_y)$. It turns out that finding an ellipse that covers proportion p of samples on this process is given ...
feetwet's user avatar
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0 answers
43 views

Monotonicity of quantile function divided by its derivative and argument

Let $F$ be a CDF with $f\equiv F'$ having a positive support (edit), i.e. $\text{supp}(f) \subseteq \mathbb{R}_+$. Then $Q\equiv F^{-1}$ is its quantile function and $q\equiv Q'$, where we know $q(p) =...
John Ritz's user avatar
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2 answers
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Is quantile just quantile function evaluated at specific values?

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'...
ashpool's user avatar
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Sampling Normal Distribution; Box-Muller, Inverse Transform, Rejection, Approximations?

I assume $X\sim\mathcal{N}(\mu,\sigma)$ and wish to sample values but I am confused about different approaches and concepts that seem to be relevant for this problem. It appears to me that this ...
Ronnie Marksch's user avatar
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0 answers
31 views

How to prove this inequality regarding the quantiles of F-distribution?

I'm working on a problem comparing the width of two confidence intervals. The problem boils down to proving this inequality: $$p \cdot F_{p, \,n-p} \,(1-\alpha) > F_{1, \,n-p} \,(1-\alpha),$$ where ...
Jamie Carr's user avatar
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1 answer
154 views

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 $\mathcal{O}(n^{-r})$

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 ...
Fei Cao's user avatar
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Does the vector space of differences of quantile functions have a neat characterization?

Consider the convex cone of quantile functions of random variables on the real line (with finite second moment), that is $$ C := \{ Q_{\mu}: \mu \in \mathcal P_{(2)}(\mathbb R) \}, $$ where $Q_{\mu}(p)...
ViktorStein's user avatar
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5 votes
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Solving $\partial_t \gamma_t(x) = - \gamma_t(x) + \frac{\gamma_t''(x)}{\gamma_t'(x)^2}$, a nonlinear PDE on quantile functions

While pondering Wasserstein-2 gradient flows of the Kullback-Leibler divergence functional $\text{KL}(\cdot \mid \nu)$, where $\nu \sim \mathcal N(0, 1)$ is the standard normal distribution (yes, I ...
ViktorStein's user avatar
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2 votes
1 answer
117 views

Radon-Nikodym derivative of pushforwards: $\frac{d f_\# \mu}{d g_{\#} \mu}$

Let $f, g \colon (0, 1) \to \mathbb R$ be two functions (both spaces are equipped with their respective Borel $\sigma$ algebras). What is the Radon-Nikodym derivative of $f_{\#} \lambda$ with respect ...
ViktorStein's user avatar
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1 answer
113 views

Pointwise convergence of generalized inverse function [duplicate]

I am reading Resnick's Extreme Values, Regular Variation, and Point Processes. In chapter 0.2 he writes about the generalized inverse of a non-decrasing function F: $$F^{\leftarrow}(y):=\inf\{x:F(x)\...
Ilja's user avatar
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48 views

First Order stochastic dominance, but only for the upper-quantiles?

Suppose I have two real-valued independent random variables $X$ and $Y$. What are conditions on $X$ under which the $\gamma$-quantile of $X + Y$ (weakly) exceeds the $\gamma$-quantile of $Y$, for $\...
Nick's user avatar
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1 answer
36 views

Prove the equality: $\sup\{x\geq0 : F_X(x)\leq t\}=\int_0^{+\infty} 1_{\{ F_X(x)\leq t\}}\,dx$. [closed]

I was trying (unsuccessfully) to prove the following equality $$\sup\{x\geq0 : F_X(x)\leq t\}=\int_0^{+\infty} 1_{\{ F_X(x)\leq t\}}\,dx.$$ Can anyone give me an hint?
MathLover's user avatar
  • 155
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1 answer
42 views

Function of quantile function

I stumbled across this Theorem in a paper, but I am not able to obtain its reference or solve it. Please help. Let $X_1, ..., X_n$ be independent random variables with CDF $\Phi_1,...,\Phi_n$ (which ...
Cherryblossoms's user avatar
7 votes
2 answers
393 views

List of closed form special cases and transformations of Wolfram language’s inverse beta regularized $\text I^{-1}_x(a,b)$.

The Wolfram Language’s Inverse Beta Regularized $\text I^{-1}_z(a,b)$ is a quantile function. This applicable yet obscure function appears in Excel as BETA.INV and a special case of it as the Inverse ...
Тyma Gaidash's user avatar
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1 answer
80 views

$\mathcal{N}(0,1)$ c.d.f. and quantile

It is often useful to estimate the quantile for $\Phi^{-1}(d)$ for some real number $d$. Here $\Phi\sim \mathcal{N}(0,1)$. We cannot use for some unknown reasons the Taylor expansion as for estimating ...
user122424's user avatar
  • 3,978
1 vote
1 answer
182 views

Quantile function, bivariate joint density.

Consider random variables $X,Y$ with joint density function $$ f(x, y)=\frac{1}{2 \pi}\left(1+x^2+y^2\right)^{-3 / 2} $$ I want to find the quantile function of $|Y|$. I have learned how to find the ...
Logi's user avatar
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1 vote
0 answers
91 views

Question about the range of the quantile function

Let $X$ be a random variable with probability distribution $F$. To define the question. We have to define to notions. First, we define the points of support of $X$ as \begin{align} {\rm supp}(X)= \{x: ...
Boby's user avatar
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0 votes
1 answer
43 views

Question on definition for proving a theorem regarding simulation.

Theorem. Suppose the random variable $U$ has a uniform $(0,1)$ distribution. Let $F$ be a continuous distribution function. Then the random variable $X=F^{-1}(U)$ has distribution function $F$. In the ...
Paul Ash's user avatar
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3 votes
1 answer
75 views

Convergence of $p^{\text{th}}$ quantile estimator for sample from Exponential distribution

Convergence in probability of $p^{\text{th}}$ quantile estimator for iid sample $X_1, \ldots X_n$ from Exponential distribution given by $f(x, \lambda) = \lambda e^{-\lambda x}$ The $p^{\text{th}}$ ...
chesslad's user avatar
  • 2,533
1 vote
0 answers
39 views

If $\mu$ is atomless then $F \circ F^{-1} (t) = t$ for all $t \in (0, 1)$

I'm trying to prove this property. Could you have a check on my attempt? Let $\mu$ be a Borel probability measure on $\mathbb R$ and $F$ its c.d.f. Then $F$ is right-continuous and non-decreasing. ...
Analyst's user avatar
  • 5,817
1 vote
0 answers
93 views

The quantile function $F^{-1}: [0, 1] \to \mathbb R \cup \{\pm \infty\}, t \mapsto \inf \{x \in \mathbb R \mid F(x) \ge t\}$ is Borel measurable

In optimal transport, I have encountered an integral in which the integrand is the quantile function. Could you confirm if my below attempt is fine? Let $\mu$ be a Borel probability measure on $\...
Analyst's user avatar
  • 5,817
1 vote
1 answer
130 views

linear objective function with linear constraints and one quadratic constraint

I have an optimization problem in which the objective function and most constraints are linear, but I have one constraint is quadratic. I know my problem can not be reformulated as a convex , but if ...
A.F.R.S2022's user avatar
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0 answers
53 views

What is the meaning of "=min!"

Specifically, it appears in the following definition $$ \hat\xi_\tau = \inf_\xi \left\{ \xi \in \mathbb{R} \bigg| \sum_{i=1}^n \rho_\tau(Y_i - \xi) = \min! \right\} $$ where $\rho_\tau(.)$ is the ...
Stéphane's user avatar
  • 222
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0 answers
30 views

How to prove an inequality involving second moments and inverse distribution functions

I have recently encountered a problem requiring dealing with two inverse distribution functions simultaneously. For a cumulative distribution function $\Psi$, the function $\Psi^{-1}:(0,1)\rightarrow \...
Cofield's user avatar
  • 123
0 votes
1 answer
146 views

Quantile function: $F(x) \ge t \iff x \ge F^{-1}(t)$ is not true if $F^{-1} (t) := \inf \{x \in \mathbb R \mid F(x) \color{red}{>} t\}$?

I'm reading about quantile function of a probability measure. However, I could NOT prove the property (c) $F(x) \ge t \iff x \ge F^{-1}(t)$ if $F^{-1}$ is defined with the strict inequality, i.e., \...
Analyst's user avatar
  • 5,817
1 vote
0 answers
578 views

Integral of Quantile Function is Expected Value?

Take a distribution function $F$. If $Q_F : [0, 1] \rightarrow \mathbb{R}$ is defined by $Q(p) = \inf\{x | p \leq F(x)\}$, is it true that $$E_F[X] = \int_0^1 Q_F d\mu$$ where $\mu$ is the Lebesgue ...
housed_off_space's user avatar
0 votes
0 answers
59 views

Equality in distribution for quantities involving empirical and regular quantile functions

Let $U_1,...U_n$ be i.i.d. random variables of uniform distribution on $[0,1]$ and $X_1,...X_n$ i.i.d. real random variables with common cumulative distribution function (cdf) $F$ and empirical cdf $...
Skywear's user avatar
  • 192
1 vote
0 answers
49 views

Convergence of a type of Monte Carlo integration which is different from the common one

$a_i \in (0,1)$, $I = 1,\cdots,N$ are $N$ random samples from uniform distribution $U(0,1)$. $a_i$ is in ascending order $a_1 < a_2 < \cdots < a_N$. $Q(p)$, $p\in(0,1)$ is a differentiable ...
Chp's user avatar
  • 123
0 votes
1 answer
165 views

Quantile of the alpha level for an absolute continuous random variable

Absolutely continuous random variable X can take values only in the interval [4,9]. On this segment, the distribution density of the random variable $X$ has the form: $f (x) = C (1 + 7x^{0.5} + 8x^{0....
Ben's user avatar
  • 93
2 votes
0 answers
67 views

Is the quantile function of a differentiable function of absolutely continuous random variables differentiable?

Let assume $n$ absolutely continuous random variables $X_1, \dots X_n$, and a differentiable function $$g: \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}; (x_1,\dots,x_n \, ; \, y) \mapsto g(x_1,\...
Julien V's user avatar
  • 145
7 votes
0 answers
153 views

Sum of two independent random variables: distribution function and quantile function

If $X,Y$ are two independent random variables with CDFs $F_X,F_Y$, their sum has CDF $F_X \star F_Y$ ($\star$ is the convolution product). What can be said about the quantile function of $X+Y$ ? The ...
W. Volante's user avatar
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3 votes
0 answers
82 views

Quantile function of two-term gaussian

I'm trying to find the quantile function of the two-term gaussian. From https://statproofbook.github.io/P/norm-qf.html, I've got that I can take the inverse of the CDF of the two-term gaussian. I've ...
FragileX's user avatar
  • 133
1 vote
1 answer
97 views

How to calculate quantile function for Birnbaum–Saunders distribution? [closed]

According to wikipedia the quantile function of for Birnbaum–Saunders distribution, $ G(p)$, depends on the quantile function of the standard normal distribution. For example, in the paper https://...
Itzhak's user avatar
  • 19
0 votes
0 answers
108 views

Convergence of normal quantile approximations (or convergence of QQplot)

Suppose we have standard normal random variables: $X_1, X_2, ..., X_n \sim N(0,1)$ and we denote by $X_{(i)}$ the corresponding ordered statistics, i.e. $$X_{(1)} \leq X_{(2)} \leq ... \leq X_{(n)}.$$ ...
Julien's user avatar
  • 153
0 votes
1 answer
111 views

Student $t$ distribution table

Why is it impossible to find a table of the Student $t$ distribution without the confidence area ? The normal distribution $\mathcal N(\mu,\sigma^2)$ for example, has these two tables : $\mathbb P(Z\...
Hamdiken's user avatar
  • 1,513
0 votes
0 answers
102 views

quantiles with convex combination

I have define the right quantile and left quantile, and I want to study the quantiles with convex combination. that's $$F^{-1}(q) = \inf\{x: F(x) \geq q\},$$ and $$F^{-1+}(q) = \inf\{x: F(x) > q\}.$...
ZFZou's user avatar
  • 31
0 votes
0 answers
65 views

About continuity of the distribution function of $X+Y$ when they are independent and one has continuous distribution.

I want to prove that if $X, Y$ are independent random variables and the distibution function of $X$ is continuous (it doesn't really matter wether this is for $Y$) then the distribution function of $X+...
Fubini's user avatar
  • 159
0 votes
1 answer
162 views

Help with a covariance inequality

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of real-valued random variables. In a proof of a theorem, the author starts with $$\lvert Cov(X_i,X_j)\rvert\leq \int_0^{\alpha_{\lvert i-j\rvert}}[Q_i(u)]^2+...
Celine Harumi's user avatar
0 votes
1 answer
320 views

Integrals of quantile function

Suppose $X = [0,1]$ and we have two cumulative distribution functions $F$ and $G$ on $X$ of two random variables. I've just read the claim $$\int_0^1 |F^{-1}(x) - G^{-1}(x)|dx = \int_0^1|F(y) - G(y)|...
qp212223's user avatar
  • 1,662
0 votes
1 answer
86 views

Quantile Function of a Normal RV

I am wondering if I can solve this with Quantile function. Suppose LCD screens have lifetimes that are normally distributed with a mean lifetime of 18000 hours with a variance of 1000000 hours. (c) ...
Lena's user avatar
  • 1
0 votes
1 answer
340 views

Quantile given Characteristic function

For a random variable defined by the PDF $f(x)$ and CDF $F(x)$, Characteristic function $CF(x)$ (if it exists) is given by the Fourier transform of $f(x)$. Quantile $Q(x)$ (if it exists) is given by ...
user400479's user avatar