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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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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
558 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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389 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 ...
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1answer
49 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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1answer
142 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 ...
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0answers
82 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
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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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2answers
34 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/...
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1answer
311 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 ...
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1answer
361 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-\...
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0answers
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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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1answer
989 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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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
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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 ...
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0answers
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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
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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 ...
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2answers
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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
141 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
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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
249 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
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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
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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 ...
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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
21 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
78 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 ...
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1answer
273 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 < \...
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1answer
196 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
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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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0answers
66 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
41 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
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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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1answer
62 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
58 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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42 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
185 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
80 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
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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 ...
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0answers
107 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 $f(x, \theta_n)$. ...
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1answer
120 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 $p$ ($0 \le p \le 1$), ...
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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 $X_i$, $i = 1,2,\cdots,n$, ...
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0answers
258 views

Is there a generally unbiased estimator for the quantiles of a distribution?

1) Is there a generally unbiased estimator for the quantiles of a distribution? If not - I would be glad for an explanation (proof?) of why not. 2) Also (if not), is there a specific family of ...
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1answer
334 views

Calculate the VaR at level alpha of the given CDF

I have to compute the mean value, the variance, the value-at-risk $\mathrm{V@R}_{\alpha}$ and the expected shortfall $\mathrm{ES}$ of a random variable with CDF $$ F(x) = \begin{cases} 1- (\frac{3}{...
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0answers
18 views

q-quantile and order statistic

$X_1, \ldots, X_n$ are independent random variables from a continuous distribution function $F$. I would like to calculate the probability that $P(((X_{(1)},X_{(n)})\subset(q_{1/4},q_{3/4}))$. ...
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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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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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0answers
45 views

Question about your function,

I'm Xavier Vigan, a physical oceanographer. I've been using your $f(x)=\dfrac 12 \times \left(X+C-\sqrt{S+(X-C)^2}\right)$ function to calibrate quantile vs quantile plots. Because of the shape of ...
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1answer
100 views

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, ...
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0answers
46 views

Appropriate regression test apart from MLR for crime data?

thanks in advance. I'm looking to run some statistical methods to find the correlation of crime rates to crime factors. I know about MLR, which is pretty simple to run in SPSS, but what are the other ...