Questions tagged [quantile-function]

Question 1

If I have a measure $\nu$ on $\mathbb{R}$ and $G_\nu$ is its quantile function, i.e. the generalised inverse of the cumulative distribution function $F_\nu$. I'm trying to show that for any $s\in\...

Question 2

This theorem is from Capinsky and Kopp's book on measure theory. Theorem: If a function $F : \mathbb{R} \to [0, 1]$ satisfies the following conditions (i) $F$ is non-decreasing i.e. $y_1 \leq y_2$ ...

Question 3

(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, ...

Question 4

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, ...

Question 5

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 $...

Question 6

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 7

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 ...

Question 8

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 $\mathbb{R}$. ...

Question 9

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-...

Question 10

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) =...

Question 11

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 12

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 ...

Question 13

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 ...

Question 14

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)...

Question 15

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 ...

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Questions tagged [quantile-function]

The measure of the interval between the generalised inverse of two converging points

Proof of the fact that every monotonic right continuous function is a CDF of some random variable.

Clarification on a proof of the Skorokhod representation theorem

Adequate Root Finder To Compute The Quantile Function

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

Uniform convergence of cdf implies uniform convergence of quantile functions

When is the quantile function of a gamma distribution concave?

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

Prove $E(X)=\int_0^\infty1-F_X(x)\,dx$ using quantile functions [duplicate]

Monotonicity of quantile function divided by its derivative and argument

Is quantile just quantile function evaluated at specific values?

Sampling Normal Distribution; Box-Muller, Inverse Transform, Rejection, Approximations?

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

Does the vector space of differences of quantile functions have a neat characterization?

Solving $\partial_t \gamma_t(x) = - \gamma_t(x) + \frac{\gamma_t''(x)}{\gamma_t'(x)^2}$, a nonlinear PDE on quantile functions

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