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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.

If $F$ is a cumulative distribution function, the quantile function $Q$ or generalized inverse $F^{-1}$ of $F$ is defined as $$Q (t) = F^{-1} (t) := \inf \{ c \in \Bbb R : F(c) \geq t \}$$

One of its properties is that given a uniform on $(0,1)$ distributed random variable $U$, then $$Q(U)$$ has the distribution according to $F$.

55 questions
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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 T distribution function InvT(x=area,d=degrees of…
Тyma Gaidash
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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 quantile function should be the inverse of $F_X…
W. Volante
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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 know about Langevin dynamics and Fokker-Planck, but…
ViktorStein
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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, thus computing values is subject of numerical…
m-stgt
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Clarification on a proof of the Skorokhod representation theorem

(Skorokhod's representation theorem): Let ${X_1,X_2,\dots}$ be a sequence of real random variables, and $X$ a further random variable. Then ${X_n}$ converges in distribution to ${X}$ if and only if, after extending the probability space model if…
shark
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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 counterexample. I think uniform convergence of…
Stan
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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 $p^{\text{th}}$ is given by $\theta = F^{-1}(p)$ where…
chesslad
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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 got the two-term CDF as…
FragileX
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The $L^1$ distance of two CDF is the $L^1$ distance of the quantile function coupling

In a book I found the following exercise: Let $F,G$ be two cumulative distribution functions. Then $$\int_0^1 \vert F^{-1} (t) - G^{-1} (t)\vert \text d t = \int_\Bbb R \vert F(x) - G(x) \vert \text d x$$ where $H^{-1} (t) := \inf \{ x \in \Bbb R :…
Falrach
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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 distribution function corresponding to the density $f(x)$.…
Fei Cao
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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 $Q_{\mu}(p) := \min\{ x: \mu((-\infty, x]) \ge p \}$, $p…
ViktorStein
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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 to $g_{\#} \lambda$, where $\lambda$ is the…
ViktorStein
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Pointwise convergence of generalized inverse function

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: $$F^{\leftarrow}(y):=\inf\{x:F(x)\ge y\}.$$ In Proposition 0.1 he proofs, that for…
Ilja
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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 $\mathbb R$ and $F: \mathbb R \to [0, 1]$ its…
Analyst
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