Questions tagged [quantile]

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

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

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 ...
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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'...
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Quantile of Empirical CDF and Tail Bound

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 ...
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Sequence of sets of p-quantiles converging to a given set of p-quantiles

One of my homework problems asks us to show that a sequence of sets of p-quantiles converges to a given set of p-quantiles. I can start the question but don't know how to continue. Here's the problem ...
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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 ...
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Limit of the quantile function

I hope you can help me with the following, please Let $X$ be a random variable with c.d.f. $F$. For each $p\in (0,1)$. Define $F^{-1}(p)$ to be the smallest value $x$ such that $F(x)\geq p$. I want to ...
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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 ...
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Linear quantile regression v.s. two-step OLS regression

Assume we have a linear model $y = 10 + 0.5x + \epsilon$ where the $\epsilon$ is a random noise. We have $n$ samples $(y_1, x_1),\cdots,(y_n, x_n)$ and want to calculate the 90th percentile y ...
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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 ...
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The Method of Quantiles

I am reading a book about quantile regression and at some point, they come to the method of quantiles, which is pretty much like the beloved method of moments. The case is that I do not understand the ...
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How to compute $VaR_\lambda(X_n)$?

I have the following problem. Let $X\in L^\infty$ and $\lambda\in(0,1)$, we define $$VaR_\lambda(X)=\inf\{m\in\mathbb{R} : \mathbb{P}(X+m<0)\leq\lambda\}.$$ I have to prove that $VaR_\lambda(X_n)=0$...
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Is it possible to consider the outlier detection criterion as the lowest price?

I am having difficulty solving the following problem I need to get the expected lowest price of items. I have date-by-date price data for items. Regression analysis or machine learning cannot be ...
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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 ...
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Finding the median value from a PDF

Suppose that the time in days until hospital discharge for a certain patient population follows a density $f(x) = (0.5)\exp(-x/2)$ for $x > 0$. What is the median discharge time in days? I reason ...
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Solving for the quantile of a probability density function.

What is the quantile, p, from the density $e^{-x}(1+e^{-x})^{-2}$? I believe I am on the right path to the solution, but I am stuck part way through. I figure that the CDF is almost certainly $(1+e^{-...
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M-estimator as a quantile estimator

According to the answer https://stats.stackexchange.com/a/497785/310702, $\alpha$-quantile sample estimator can be considered as M-estimator with $\rho(y_i,\theta)=\alpha(y_i-\theta)_+ + (1-\alpha)(\...
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Find the value of the $5$th decile, $D_5$.

For the data set: $$18,15,12,6,8,2,3,5,20,10$$ Find the value of the $5$th decile, $D_5$. I computed this 2 ways and each time I got a different answer. If the sample size is $n$, then the rank of ...
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Frequentists tests to check for normality

Let $X_1,...,X_n\sim X$ be $n$ i.i.d. random variables. I want to to test if they follow a normal distribution, in other words, check if their distribution belongs to the Gaussian family. These are ...
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Quantile estimation with apriori known expectation

I have the following problem: If we have a typical random sample $\{X_1, X_2, ...\}$ from some unknown distribution and we want to estimate a quantile we just need to sort our observations and take a ...
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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 ...
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Gaussian tail probability

Let $z(q)$ be the quantile of a standard normal random variable $Z$, i.e., $z(q) = k$ when $\Pr(Z\geq k) = q$. Then I would like to know why the following two results hold. (a) If we hold $\alpha$ ...
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What is the notation for the mean of $y$ in the $i$-th quantile of $x$?

I would like to add the appropriate mathematical notation to use in a figure. I have two original quantitative variables, $y$ and $x$, named “long variable name $y$” and “long variable name $x$”. I ...
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Showing $\Phi^{-1}(x) = - \Phi^{-1}(1-x)$ for a normal distribution

Let $\Phi$ be the CDF of a normal random variable. Let $x \in (0,1)$. Why do we have $$\Phi^{-1}(x) = - \Phi^{-1}(1-x)?$$
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Quantile of Centered Binomial

Let $X$ be a Binomial distribution with $n$ trials and success probability $p$ in $(0,1)$. It is clear that the quantiles of $X$ are an increasing function of $p$. Let $Y$ be a centered Binomial, i.e. ...
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How do you differentiate the unconditional quantile function for y wrt to P(x=1) where x is a Bernoulli random variable?

Suppose $X$ is a Bernoulli random variable (0,1). The marginal cdf of $Y$ can be written. $F_Y(y)$ = $Pr(X=1)*F_{Y|X=1}(y)$ + $Pr(X=0)*F_{Y|X=0}(y)$ (Eq. 1) Replace $y$ with $q_Y(\tau;p)$, the ...
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Lower Quartile, Median and Upper Quartile

Hello I want to calculate the $Q_1$, $Q_2$ and $Q_3$ for my Cumulative Probability Function $F(x)=\dfrac{e^x}{1+e^x}$. So if $P=\dfrac{e^x}{1+e^x}$ how do I convert this expression to the form $x=...$ ...
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How to resolve ties in quantile distribution

What rules are available for deciding which quantile a particular value would fall into when the boundaries of more than one quantile are identical? My data set includes score ranging between 1 and 20....
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Calculate the first and third quartiles for the lognormal distribution

I intend to calculate the first and third quartiles of a lognormal distribution with mu and sigma (two lognormal parameters) equal to -0.33217492 and 0.6065058. The expected value and the standard ...
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p-value based on quantile of standard normal distribution for mean inequality hypothesis

Let $z$ be a quantile of standard normal distribution. Find p-value for $z=1.89$ and $H_a : \mu>\mu_0$. The way I tried solving this was by getting the area right of $x=1.89$, which is $0.02938$ ...
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99% of all readings are within WHAT temperature range?

I am a bit stumped. "The temperature reading from a thermocouple placed in a constant-temperature medium is normally distributed with mean of 82.5C and standard deviation .1C. 99% of all readings ...
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Give an upper bound on the probability that you wait more than 30min for the bus.

Consider the time to wait for your bus is modeled by $X\text{~}Exp(\lambda)$. What is the $90/100th$ quantile for $\lambda=1$? Given $90/100th$ quantile is 20 min ($\lambda$ is unknown). Give an upper ...
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Neat proof to show that $\sup\{x\; : P(X<x)< \lambda\}=\inf\{x\; : P(X\leq x)\geq \lambda\}$

Consider a probability space $(\Omega,\mathcal{F},P)$ and some real-valued random variable $X$. I want to show that $\sup\{x\; : P(X<x)< \lambda\}=\inf\{x\; : P(X\leq x)\geq \lambda\}$ for any $...
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relation of medians of two way partition

Given a set of finite real numbers $X \subset \mathbb{R}$, one can obtain the median using $M: X \to \mathbb{R}$. And $X$ can be partitioned into two subsets $A_i, B_i \subset X, A_i \cup B_i = X , ...
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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\}.$...
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Ranking of value based on given quartiles of a range

May be a noob question here, but my query is: I have a range of values of which I have quantile values: example: q = [-10, 0.25, 0.5, 0.75, 10] I have a score value say .85 which i wish to rank but ...
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Non-uniqueness of optimization problem to estimate sample statistics

Sample statistics can be estimated by solving an optimization problem. Is the optimization problem unique? As I know, the optimization problem for expectile is not unique. If not, is there a best one?...
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How do I estimate a percentile for a value given several other percentiles?

Given the length percentiles data the WHO has published for girls. That's length in cm at for certain months. e.g. at birth the 50% percentile is 49.1 cm. ...
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quartiles of box plot

I'm struggling with the meaning of quartiles , in this box plot how to compare these series like should I compare the upper $50$% of each series or the lower or the $50$% between $1$st quartile and $...
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Recovering a pdf from quantile function

Suppose I have $q(x)$ which represents the quantile function of a distribution. I can query the function for its value and derivative at any point (e.g. I can ask for $q(0.6)$ and $q'(0.221)$). Is ...
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Notation for a set of quantiles

Is there a formal notation for a set of quantiles of for instance x, a numerical array? This answer suggests the notation for percentile is Pi (though what to do if you already have a variable P is ...
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On describing percentiles within quartiles

In describing the relationship between percentiles and quartiles of a data set, is it accurate to say that the percentiles belonging to the first quartile range from the zeroth to the 24th percentile? ...
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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 ...
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quantiles of a monotonic function of two independent R.V.s

Consider two independent random Variables $X, Y$, and $X>=0, Y>=0$. $f(X, Y)$ is monotonic to $X, Y$ respectively. Suppose that we know the 0%, 25%, 50%, 75%, 100% quantiles for both $X, Y$ and ...
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cumulative distribution function, inverse and quantile

I have some issues with the cumulative distribution function ( denoted by $F$ of a random variable $X$, his quantile $Q_X(\alpha)$ and the inverse of the cumulative distribution function $F^{-1}$. ...
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Finding quartile of normal distribution

Find the quartiles :$Q_{1},Q_{2},Q_{3}$ of $X \in N(2,9)$ What I have tried : So for $Q_{1}$ for example , $Q_{1} =\frac{1}{4}$ $ X ~ (\mu,k^2) = P(X \le x)= P(kx+\mu \le x) = P(Z \le \frac{x-\mu}{...
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Finding Quantile Function from CDF

Let $f\left( x \right) = {{\left( {1 + \alpha x} \right)} \over 2}$ for $ - 1 \le x \le 1$, and $0$ otherwise. We further assume that $\left| \alpha \right| < 1$. Show that (i) $f$ is a ...
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If $F$ is a distribution function and $x_0\in F^{-1}([0,1])$, can we characterize $\left\{t\in[0,1]:F^{-1}(t)=x_0\right\}$?

Let $F:\mathbb R\to[0,1]$ be a distribution function$^1$ and $$M_t:=\left\{x\in\mathbb R:F(x)\ge t\right\}$$ and $$F^{-1}(t):=\inf M_t$$ for $t\in[0,1]$. Let $x_0\in F^{-1}([0,1])$. Can we ...
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If $F$ is a distribution function and $t>0$, can we show that $F(F^{-1}(t))\ge t$?

Let $F:\mathbb R\to[0,1]$ be a distribution function$^1$ and $$M_t:=\left\{x\in\mathbb R:F(x)\ge t\right\}$$ and $$F^{-1}(t):=\inf M_t$$ for $t\in[0,1]$. If $t>0$, can we show that $x_0:=F^{-1}(...
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Show that if the inverse cumulative distribution function is continuous, it is invertible

Let $F:\mathbb R\to[0,1]$ be a distribution function$^1$ and note that $$F^{-1}(t):=\inf\left\{x\in\mathbb R:F(x)\ge t\right\}\;\;\;\text{for }t\in(0,1)$$ is left-continuous and nondecreasing with $$F^...
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Quantile function of log-normal distribution

The quantile function for log-normal distribution is given by $$F^{-1}(p)=\exp(\mu+\sigma\Phi^{-1}(p)),$$ where $0<p<1$ and $\Phi(p)$ is the CDF of a normal distribution. I am trying to derive ...
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