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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

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9
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5answers
8k views

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 ...
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2answers
648 views

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 $G:\mathbb{R}\to[0,1]$ ...
3
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2answers
440 views

what i am doing wrong in calculating quartiles

can somebody please help me to understand? Why 1st quartile in this data is equal 4.5 and 3rd quartile is equal 6.5. I am getting 4.25 and 6.25 , but not 4.5 and not 6.5. I use formula (n+1)/4 for ...
3
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1answer
50 views

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+\...
3
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1answer
156 views

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 ...
3
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1answer
399 views

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-\...
3
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0answers
47 views

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 ...
3
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0answers
92 views

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 ...
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1answer
2k views

Approximation for lower incomplete gamma function

Does any one know approximations for the lower incomplete gamma function $\gamma(a,bx)$. The problem is this: I want to find the quantile function for the CDF of the gamma distribution. The CDF of the ...
2
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1answer
67 views

Bound on a normal quantile function $\phi(\Phi^{-1}(x)) \ge \sqrt{\frac{2}{\pi}} \min(x,1-x)$

In this post, the author has mentioned the following lower bound on the composition of a normal pdf and invers cdf \begin{align} \phi(\Phi^{-1}(x)) \ge \sqrt{\frac{2}{\pi}} \min(x,1-x), x\in (0,1). \...
2
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1answer
1k views

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 ...
2
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2answers
39 views

If parametric quantile esimation estimates $p$ by computing the MLE, then how to get non-parametric $p$? [closed]

For non-parametric or parametric quantile estimation. If parametric quantile esimation estimates $p$ by computing the MLE, then how to get non-parametric $p$? Related: https://mathoverflow.net/...
2
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1answer
333 views

What is meant by the Fisher information of a particular of a particular quantity for a quartile function?

My provided definition of the Fisher information $\mathcal{I}(\theta)$ is the expected value of the observed information $I(\theta)$, where $I(\theta)$ is the second derivative of the log-likelihood ...
2
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0answers
42 views

How to treat lower quantiles

A function $q: (0,1) \rightarrow R $ is said to be a quantile function of $X$ if $P[X<q(u)]\le u \le P[X\le q(u)]$ for all $u\in(0,1)$ The lower quantile function of $X$ is given by $q_X^-(u).=...
2
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0answers
62 views

For cdf $F(x)$ ad empirical cdf $F_n(x)$, show that $|F^{-1}\big(F_n(\xi_p)\big)-\hat\xi_p|\overset{a.s}\to0$

Suppose $X_1,\cdots,X_n$ are i.i.d. continuous random variables from distribution with cdf $F(x)$.Let $F_n(x)$ be a random variable defined by $$F_n(x)=\frac{1}n\sum_{i=1}^nI\{X_i\le x\}.$$ Define the ...
2
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1answer
48 views

Expected value of position in subset.

I have set of $n$ elements $[0, ... n-1]$. I randomly pick a subset $S$ of $k$ elements (also ordered). Assume I have $t \in \{1, .. k\}$. What is expected value of $t$-th position in ordered subset?...
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0answers
3k views

Basic Quantile Calculation

I have a slight confusion with the current method of calculation of a quantile for a give ungrouped distribution. To give you an example, i shall refer to calculation of a Quartile, but this doubt ...
2
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0answers
43 views

How to prove that sample of size $O(\epsilon^{-2} log \delta^{-1})$ is enough to predict quantiles?

There is a known problem: You are given a stream of numbers and you need to find it's $q$-th quantile ($0 \le q \le 1$). You may get wrong answer but you need to return answer between $q-\epsilon$-th ...
2
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0answers
61 views

Quantiles when the population contains only one unique value

My apologies for the mixing of the terms quartile and quantile below. I am interested in the general case of quantiles, but I'm using a quartile as a specific example. Also feel free to clarify any ...
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1answer
2k views

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 ...
1
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3answers
30 views

Inversed Cross Median of two arrays

The problem I am facing is as follows : Given two arrays $A$ and $B,$ I would like to find a threshold $t,$ satisfying: the number of elements of $A$ that are less than $t$ equals the number of ...
1
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2answers
33 views

Choosing an interval of the CDF to find each quartile

I have a random variable $X$ which has the following CDF: $$F(y) = \left\{\begin{array}{ll} 0 & : y \lt 0\\ \frac{y}{30} & : 0 \le y \lt 20\\ \frac{2}{3} + \frac{y-20}{60} & : 20 \le y \lt ...
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1answer
178 views

Inverse CDF of a function

I am trying to find the Inverse CDF (quantile function) of this function to create an random number generator: $f(p_a) = (\beta + 1) p_a^{\beta} \text{ where } \beta \geq 1 \text{ and } 0 \leq p_a ...
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1answer
2k views

Empirical quantile definition?

Suppose we have an order set of data $\mathcal X=\{x^{(1)},x^{(2)},...,x^{(n)}\}$ such that $x^{(1)}\le x^{(2)}\le ...\le x^{(n)}$. For some reason in my course's definition of empirical quantile, we ...
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1answer
272 views

Uniqueness of solution to quantile minimization problem

I read here: http://librarum.org/book/11685/31 (p. 51, Ex. 3) that quantiles are solutions to certain minimization problem. Here is the proof: http://www.math.ucla.edu/~tom/MathematicalStatistics/...
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1answer
58 views

How to show that $\Phi^{-1}(1-x) =O(\sqrt{\log{x^{-1}}})$

In the middle of some proof, I have faced an expression $\Phi^{-1}(1-x) =O(\sqrt{\log{x^{-1}}})$, where $\Phi(\cdot)^{-1}$ is a quantile function of the standard normal distribution and $x \in (0,1)$. ...
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2answers
74 views

Is this definition of a quantile proper?

I need to find a proper definition of a quantile. It says: a p-th quantile $x_p$ is a number, that satisfies the following conditions: $$ 0<p<1 $$ and $$ P(X \le x_{p}) \ge p $$ and $$ P(X \ge ...
1
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1answer
66 views

What are the advanatages of CDFs for RNG over simple random sampling?

If I understand Cumulative Distribution Functions (CDFs) correctly, they can be used for random number generation from a given dataset as follows: Build a CDF that maps data points to an ordinal ...
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1answer
12 views

Why use residual plots in linear regression for assessing normality?

Let's take the case of condition simple linear regression for example where we are assuming: $$Y|X=x = \beta_0 + \beta_1 x + \epsilon,$$ where $\epsilon$ represent the random noise. In order to ...
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1answer
24 views

How do you solve for the mean in a Normal Distribution?

I am working on a problem and am a little bit stuck on how to solve it. The problem: Find a Normal Distribution with SD 2.5 and 5% Quantile at -15.2. What I have done so far: $$X=\mu+2.5Z$$ $$.05=P(\...
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1answer
31 views

Implications on the cdf of $\epsilon_1-\epsilon_2$ of conditions on the cdf's of $\epsilon_1, \epsilon_2$

Consider three random variables $\epsilon_1, \epsilon_2, X$. Let $F_{\epsilon_i}(\cdot| x)$ denote the cumulative distribution function (cdf) of $\epsilon_i$ conditional on $X=x$ for any $i\in \{1,2\}$...
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1answer
83 views

How is this statistical index called?

I need to calculate an index that is to be derived like this: If we have some data: 850 700 500 480 300 100 50, we first sort it from the large to small: ...
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1answer
51 views

Calculating Quantile for a specific problem

so I understand how to calculate the 0.5 quantile of the given question. I calculate the CDF of x and then I multiply it to 0.5 But what if there's more than one function for multiple intervals? How ...
1
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1answer
293 views

binomial quantile function

Let $Q(\epsilon, n, p)$ be the $\epsilon$-quantile of a binomially distributed random variable with $n$ trials and success probability $p$. I am interested in the following question: Fix $0 < \...
1
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1answer
231 views

Prove $X(\omega) = \sup\{y \in \mathbb{R}: F(y) < \omega\}$ is a random variable.

Let F be a distribution function. On $(\Omega, \mathfrak{F}, P)=((0,1), \mathfrak{B}(0,1),\lambda)$ where $\lambda$ denotes Lebesgue measure. Define X: $\Omega \to \mathbb{R}$ by $X(\omega) = \sup(y \...
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0answers
17 views

Quantile based Shannon entropy

I'm reading two papers "Quantile based entropy function" and "Quantile based entropy of order statistics". I'm a bit confused whether the quantile based entropy function (Eq 7 in the first paper and ...
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1answer
36 views

Loss of uniqueness of quantiles

We know that, if $X$ is a continuous random variable with a strictly increasing distribution function or DF $F(x)$ then, its $p$th quantile is unique. But if the distribution function is non-...
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1answer
28 views

Tail difference of quantiles of (symmetric) distribution functions

Assume, for example, $z_\alpha$ are $\Phi^{-1}(\alpha)$ quantiles from standard normal distribution, $\alpha > 0$. If we are interested in the sum$$z_\alpha + z_{1 - \alpha}$$ for standard normal ...
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0answers
85 views

Prove that $\min_{\mu}\sum_{i=1}^n|y_i-\mu|=\text{median}\{y_1,\cdots,y_n\}$ [duplicate]

How to prove the equation below in a simple way? $$\min_{\mu}\sum_{i=1}^n|y_i-\mu|=\text{median}\{y_1,\cdots,y_n\}$$
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0answers
61 views

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 \...
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0answers
30 views

Asymptotic behaviour of the process $U_n=U_{n-1}+s(U_{n-1})X_n$, where $X_n$ is iid

Let $s, s^*:\mathbb{R}^+ \to \mathbb{R}^+$ ($0\in\mathbb{R}^+$) such that $0 \le s(x), s^*(x) \le x$ for every $x \in \mathbb{R}^+$ and $X, X_1, \ldots$ is iid with $\mathbb{E}X>0$. Let $U_{0,s}=1$ ...
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1answer
73 views

Explicitly representing a random variable such as $ X(\omega):=\frac{1}{\lambda} \ln \frac{1}{1-\omega}$, which is exponential

Previously: (Dumb question: Computing expectation without change of variable formula) I was wondering how to compute $E[X]$ by $\int_{\Omega} X d\mathbb P$ rather than $\int_{\mathbb R} x d \mathcal ...
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0answers
27 views

Evaluation of $p$th Quantile?

Let $X$ be a random variable having PDF $f(x) = \frac{1}{\lambda} e^{-\frac{\lambda}{x}} , x > 0 , \lambda > 0$. And I was trying to find out the $p$th Quantile, for which we have to set $\int_{...
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1answer
61 views

Calculate t-quantiles and $\chi^2$-quantiles

How are t-quantiles and $\chi^2$-quantiles actually calculated? I find it difficult to find a formula. For example, the t-quantile for 0.975 and 50 degrees of freedom is approximately 2. This is ...
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0answers
44 views

Estimating Quantile Function Using Pseudo Sample

Let $\{X_i\}_{i=1}^n$ be a random sample, $Y_i=\delta(X_i)$ for some function $\delta(\cdot)$, and $Q(\tau)$ be the population quantile function of $Y_i$. We can estimate $Q(\cdot)$ by the empirical ...
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2answers
198 views

Bootstrap method & Confidence Interval

I'm trying to figure out how this method works. My data: 1000 samples from unknown distribution. I need to create 40 vectors from those 1000 samples (each vector with 20 samples) For every one of the ...
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0answers
87 views

How to efficiently find quantile (root) of integral function

Let $f(x)$ be a density function on $\mathbb R$; I want to find numerically the $\alpha$ quantile of the associated distribution, i.e. I want to find $c$ such that $$\int_{-\infty}^c f(x)dx = \alpha$...
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0answers
113 views

Quantile function of the r th order statistics of i.n.n.i.d. random variables

I am repeatedly drawing $n$ random samples where the first sample is drawn from $f(x, \theta_1)$ the second from $f(x, \theta_2)$, $\enspace \dots$ , and the $n^{\text{th}}$ from $f(x, \theta_n)$. ...
1
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1answer
134 views

Sample quantile of order p

My book says the following : "Let $X_{(1)}, X_{(2)}, ..., X_{(n)}$ be a set of values ordered in ascending order ($X_{(1)} \leq X_{(2)} \leq ... \leq X_{(n)})$. For a given $p$ ($0 \le p \le 1$), ...
1
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0answers
187 views

In biweight midvariance, why would the median absolute deviation (MAD) be multiplied by the 0.75 standard normal quantile?

The biweight midvariance $\zeta^2$ is a measure of scale that is more robust to non-normal distributions. It is defined as follows by [1] in three steps. For observations $X_i$, $i = 1,2,\cdots,n$, ...