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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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25 views

Define a region $R \subset \mathbb{R}^2$ such that $P(x \in R) = 0.95$

Consider a random vector $X \sim N_2(\mu,\Sigma)$. Define a region $R \subset \mathbb{R}^2$ such that $P(x \in R) = 0.95$ Hint: The $0.95-$quantile of $\chi^2$ is about $5.99$ . I think we are ...
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21 views

Set of solutions to probability quantile equation

Let $X$ be a random variable taking its values over some set $S$, with cumulative distribution function (cdf) $F$. Let $\epsilon\geq0$. We define the set $S^*(\epsilon)$ as follows: $$S^*(\epsilon)\...
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1answer
1k views

Sample median of Cauchy distribution is consistent. How?

When we use chebyshev's inequality to show whether an estimator is consistent or not, we require the mean square error of the estimator and I do not know sample median's probability distribution. So ...
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1answer
15 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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14 views

how to calculate the error for nested numerical methods

I want to calculate some integral $\int_a^b q(p)dp$ where q(p) is a quantile of a probability distribution. These quantiles are approximations with a certain error $\epsilon_q$ and the integral is ...
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21 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
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 ...
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15 views

A maximization problem involving random variables

Consider random variables $X$ and $Y$ that are jointly normally distributed, $$ \begin{pmatrix} X \\ Y \end{pmatrix} \sim \mathcal{N} \left[ \begin{pmatrix} \mu_X \\ \mu_Y \end{pmatrix} , \begin{...
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79 views

Solving the following maximization problem analytically

Given continuous random variables $x$ and $y$ and a constant $\beta$, define a random variable $z$ by $z:=y+\beta x$. Further, define a random variable $t$ as a function of $z$: $$ t:=z-\frac{A}{2}(z-\...
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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 ...
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1answer
2k views

Bootstrap estimation of the 95% confidence intervals for the 95% quantile for gamma distribution

I cant find any where information or algorithm how to apply in steps the bootstrap procedure to estimate the 95% confidence intervals for the 95% quantile from a random sample. Does anyone knows how ...
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39 views

Is it possible to find k-th quantile only from frequencies of a value list?

Suppose we have a discrete continuous interval [0, N-1]. And, there exists $M$ samples where each value belongs to the interval. Now, we want to find a $k$-th quantile (say, 4th quantile) from the ...
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1answer
83 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). \...
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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).=...
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How to improve AIC-based model selection to reduce estimation errors?

I have statistical MC data, and put this in a histogram. In most cases the data is near-normal or slightly skewed. So a model set including a normal 2-parameter fit and a shifted lognormal 3-parameter ...
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1answer
2k 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 ...
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38 views

Quantile function for joint distribution

Please help me formulate a multivariate quantile function to facilitate the random sampling of a distribution. I have an idea, and I want to get feedback if it makes mathematical sense. Let there be ...
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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 ...
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1answer
38 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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79 views

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 ...
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133 views

If $(Q_3-Q_2)=\frac34(Q_2-Q_1)$, then

If $(Q_3-Q_2)=\frac34(Q_2-Q_1)$, then There are more data which are less than the median value There are more data which are less than the modal value There are less data greater than the mean value ...
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28 views

Convexity of a function of generalized inverse CDF.

How can I prove (or disprove) that the following function is convex on $X$. $$\rho(X,Z) = Max \{ F^{-1}_Z(t)-F^{-1}_X(t),0 \},$$ where $F^{-1}_X(t)= inf \{ x : F_X(x) \geq t \}$ with $0 \le t \le 1$....
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Primitive of a function

I have a function $h_{j}(u_j) = 1-z^2 _j $ with $z_j= \Phi^{-1}(u_j)$, $\Phi^{-1}$ is the standard normal quantile function and $u_j \in (0,1)$ I want to show that $ \int_0 ^{u_j} h_{j} (\lambda) \, ...
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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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24 views

What is the lower quartile of the set of data?

I came across this problem asking the lower quartile of the ungrouped data. My answer is 3, but other references say it should be 2.5. Here's the data: 1, 1, 2, 2, 3, 3, 4, 4, 6, 7, 8, 10, 11, 14, 15,...
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Quantile function question

is there a way how to get quantile function from random variable $Y$ defined as: $$ Y = \begin{cases} 0 \text{ ... with probability 0.25}\\ f(x) \text{ ... with ...
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Why is quantile function of uniformly distributed random variable a random variable?

I have the quantilfe function $F^{-1}$ of a random variable which is defined as: $F^{-1}: ]0, 1[ \ni u \rightarrow F^{-1}(u) = inf\{x: F(x) \geq u\} \in \mathbb{R}$. Now I can define a new random ...
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48 views

Taylor series expansion of quantile function

Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $. We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$. Do you have any ...
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34 views

How to calculate the area difference of an integral cut at a line with slope?

Using trapezoidal method (because I have a vector and no function is available) I know how to calculate the area difference one gets when the integral (Fig. 1) is cut by a horizontal line such as ...
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53 views

Quantile-Quantile (QQ) Plots

I understand how to assess whether an exponential/normal distribution is suitable to model a piece of data when the parameters are given, i.e. finding the theoretical quantiles and plotting against ...
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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 ...
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24 views

How to calculate quantiles of sums?

Let $X_1, X_2$ be independent normal distributions. Consider the quantile $x$ such that $P(X_1 + X_2 \le x) = \alpha$ for some $\alpha \in (0,1)$. My question is, how does this quantity relate to the ...
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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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1answer
73 views

Quantiles - supremum and infimum

How to prove that $$\inf\{x \in \mathbb{R}: \mathbb{P}(X \le x) > \alpha \}=\sup \{x \in \mathbb{R}: \mathbb{P}(X<x) \le \alpha \}$$ for any random variable $X$ and $\alpha \in (0,1)$?
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How do you solve for lambda in an exponential distribution?

I am working on a problem and am unsure how to solve it. The problem: Find an exponential distribution such that P(Z $\ge$ 3) = .04 What I have done so far: P(Z$\ge$3) = 1 - P(Z$\lt$ 3) We are ...
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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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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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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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0answers
64 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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1answer
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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
76 views

Uniform transformation of a quantile

Let $x_{\alpha} = \inf \{x \in\mathbb{R}: F_X(x) \geq \alpha\}$, $U \sim Uniform(0,1)$ and $Z=x_{U}$. I need to prove that Z has the same distribution as X. Obviously this is true as can easily be ...
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1answer
51 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+\...
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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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673 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 $G:\mathbb{R}\to[0,1]$ ...
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1answer
149 views

Where can I find examples of Skorokhod representations?

So, I recently (re-)discovered that random variables learned in elementary probability such as the exponentially distributed random variable $X$ with cdf $F_X(x) = 1-e^{- \lambda x}$ can be explicitly ...
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1answer
249 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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1answer
67 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
75 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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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 ...