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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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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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79 views

basic Quantile proves

Let this be my definition of a quantile funktion. X is a real-valued random variable. And let F be it's distribution function. then \begin{align*} F^{-1}(a):=\inf\{x\in \mathbb{R}: F(x) \ge a\}. \end{...
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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 ...
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1answer
119 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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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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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
132 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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1answer
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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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A question about quantile function of normal distribution

Suppose that I know a normal distribution of randome varible $X$ satisfies the property that $P(20<X<30) = 0.9$, is that true that the lower quartile of $X$ must be between $20$ and $25$? I ...
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72 views

Calculation of quantiles of a uniform distribution over a sphere

How do we calculate quantiles of a uniform distribution over a sphere ? Can anyone provide me with a tutorial ?
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51 views

Is there any probability distribution that can fit based on 3 quantiles?

Assuming that we know quantiles 0.25, 0.5 and 0.75, is it possible to fit a distribution from these values ? What distribution ? How ? Thank you Edit : I forgot to mention that I would like the ...
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1answer
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how binary quantile regression divides the dependent variable into quantiles

I am not very clear with binary quantile regression. As if it was ordinary quantile regression, it would divide the dependent variable's value by its ascending value into quantiles. But I cannot ...
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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 ...
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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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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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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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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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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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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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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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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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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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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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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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256 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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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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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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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 ...
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About moments in a quantile processes

Let $q_{n}(t)$ be the $nth$ quantile processes ($t\in (0,1)$) based on the distribution F: $$q_{n}(t) = \{\sqrt{n}[F^{-1}_{n}(t)-F^{-1}(t)]\}.$$ In this case, $F^{-1}$ is the (generalized) inverse of $...
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A question about solving directional derivatives

The question comes from the paper ``Regression Quantiles'' by Roger Koenker and Gilbert Bassett(1978). $0< \theta <1$. Define $\psi(b;\theta,y,X)=\sum^{T}_{t=1}[\theta-1/2+1/2 \; \text{sgn}(y_{...
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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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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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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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29 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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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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Calculate Quantile function Q(p) for the F(x)

Given F(x) = 1/26(2x^2 + x - 10) and I need to find the Quantile function of Q(60). I have tried with Q = F^-1(p) but still, I have not got the correct answer. Note: Correct answer is 3.3364
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Inverse distribution function (quantile function)

From Wiki: If the CDF ''F'' is strictly increasing and continuous then $ F^{-1}( p ), p \in [0,1], $ is the unique real number $ x $ such that $ F(x) = p $. In such a case, this defines the "inverse ...
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find scale and shape parameter of Weibull distribution by having the 50 and 90 percent quantiles

I have the 50 and 90 percent quantile of a Weibull distribution. Is it possible to get the shape and scale parameter just with this information? Mathematically I have this formula for calculating the ...
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Quantiles of a Brownian Motion

I am looking for the p-th percentile of a stochastic process $X_t$ that satisfies $dX_t = \mu(X_t) dt + \sigma(X_t) dW_t$ where $W_t$ is a standard Brownian motion. I believe that the p-th percentile ...
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consistency of inverse cdf

If $\hat{F}(x)$ is a consistent estimator of $F(x)$ where $F(x)$ is a cdf, can we state that $\hat{F}^{-1}(x)$ is also a consistent estimator of $F^{-1}(x)$? Is that straightforward? Why? You can make ...
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Continuity of quantiles as function of measure.

Let $P(R)$ be the probability measures on the real numbers $R$ and fix $\alpha \in (0,1)$. Define $$Q_{\alpha} : P(R) \to R $$ as the function taking a measure $\mu \in P(R)$ to its $\alpha$-th ...