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Questions tagged [quantile]

one of several equally-frequent subranges of a data set or random distribution; for example, a percentile or quartile

A set of quantiles is a partition of the range of a random variable or set of data into subranges of equal likelihood or frequency (or approximately equal likelihood or frequency if an equal partition is not possible). An $n$-quantile is one of a set of quantiles that partition the range into $n$ such parts.

Common examples of quantiles are quartiles ($4$-quantiles), quintiles ($5$-quantiles), deciles ($10$-quantiles), percentiles ($100$-quantiles), and the median (a $2$-quantile).

147 questions
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Are there 3 or 4 quartiles? 99 or 100 percentiles?

So I understand that a quartile is a quantile where the data is divided into four groups. 1 2 3 ---|---|---|--- And 1, 2, and 3 are the quartiles. The second quartile is the median, etc. But while studying for the GRE, I read this as part of…
jds
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Construct a random variable with a given distribution

Suppose that ($\Omega$, $\mathcal{F}$,$P$) where $\mathcal{F}$ is the $\sigma$-algebra of Lebesgue measurable subsets of $\Omega\equiv[0,1]$ and $P$ is the Lebesgue measure. Let $G:\mathbb{R}\to[0,1]$ be an arbitrary distribution function. My…
yurnero
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Multivariate Quantiles

I am interested whether a concept for the multivariate equivalent to quantiles exists. In the univariate case, a $p$-quantile can be computed via the inverse of the cumulative density function, however, this cannot easily be translated into the…
Stefan Voigt
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Quantile function for binomial distribution?

A test will succeed with a certain percentage. Now this test is repeated X number of times. I want to be able to get an estimate of the total number of succeeded test. Given that I know both the probability of success and the X number of attempts,…
user3619622
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Quantile(X + constant) = Quantile(X) + constant?

I would like to 'prove' that $$q_{\alpha}(X + c) = q_{\alpha}(X) + c $$ For c $\in \Bbb{R}$, $X$ a random variable, and $q_{\alpha}$ the quantile of order ${\alpha}$. I would actually like to prove this for conditional quantile (but I think it does…
G. Ander
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Probability distribution of resampled quantile

There is a list $X$ of $N$ different numbers, sorted in ascending order. First, we perform resampling with replacement, constructing a list $Y$. This means that we consider a uniform distribution over the elements of $X$, and sample from that…
Aleksejs Fomins
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Upper bound for distance between actual and sample quantiles

Let $\xi_p$ be pth quantile of the distribution $F(x)$ with derivative at $\xi_p$, $f(\xi_p) >0$. Then, $$ |\hat\xi_{p,n} - \xi_p| \leq \frac{2}{f(\xi_p)}\sqrt{\frac{\log n}{n}} $$ almost surely for large enough $n$ and any $p \in (0,1)$, where…
Coleman
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Upper bound of difference of squares of quantile standard normal

Let $\Phi$ denotes the cummulative standard normal distribution and $\Phi^{-1}$ denotes its inverse. Given $u,v\in[0,1)$. I'am going to find an upper bound of $$ \left|\left\{\Phi^{-1}(v)\right\}^2-\left\{\Phi^{-1}(u)\right\}^2\right| $$ in term of…
Jlamprong
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Derivatives of quantile functions for continuous distributions

Suppose that $F$ is a distribution function that is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}$ with density $f$. Let $F^{-1}$ be the associated quantile function and assume that $F^{-1}(t)$ is a singleton for all $t \in…
evencoil
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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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Calculating quantiles of weighted array

First the proviso I'm only an aspiring mathematician. Secondly forgive how I've articulated this question in Python, but it is essentially a computational problem I'm looking to solve. I wonder if there's an efficient way to calculate quantile…
geotheory
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expectation of upper quantile proportion

We have a collection $\boldsymbol{S}$ of $n$ discrete random variables $X_1$, $X_2$, $\dots$, $X_n$ $\overset{\small \text{i.i.d.}}{\small \sim}$ $\mathcal{D}$, where $\mathcal{D}$ is a distribution over $\{1, 2, \ldots, U\} \subset \mathbb{N}$ with…
Amit Portnoy
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Calculate theoretical quantiles with calculator (qq-plot)

Let's say we have the following data: $-1.8, -0.82, 0.3, 1.2, 1.6$ Now I want to make a qq-plot out of it by hand, just with a calculator (Casio fc 991). I start by sorting the values in ranks j and calculate how many observations are less than or…
Marl
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Pushforward-change-of-variable with quantile function

I’ve been dealing with an issue about change-of-variable formula. Let $\mu$ be a probability measure on $\mathbf R_+$. Let $F(x) = μ([0,p])$ and $Q$ its quantile function, ie $Q(p) = \inf \{q \in \mathbf R_+ : F(q) \ge p\}$. As $Q$ is increasing, it…
Valentin Melot
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Inequality involving quantiles

Suppose that $X$ and $Y$ are r.v.s such that $F_X$ (the cdf of $X$) is continuous and $$ \sup_{r\in\mathbb{R}}|F_X(r)-F_Y(r)|\le \epsilon. $$ Is it true that $\mathsf{P}(X\le q_Y(\alpha))\le \alpha+\epsilon$, where $q_Y(\alpha)$ is the…
Robert W.
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