Questions tagged [order-statistics]

The order statistics of a sample are the values placed in ascending order. The i-th order statistic of a statistical sample is equal to its i-th smallest value; so the sample minimum is the first order statistic & the sample maximum is the last. Sometimes 'order statistic' is used to mean the whole set of order statistics, i.e. the data values disregarding the sequence in which they occurred. Use also for related quantities such as spacings.

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Is there a function that yields the "sorted" PDF of a distribution? [closed]

I am implementing a program where I'd like to compute the indices of the most likely values in a vector where the PDF is known, in descending order. It could be done by discretizing and sorting the ...
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Jacobian of function returning $m$ evenly-spaced order statistics of an $n$-dimensional vector

Let $y\in\mathbb{R}^n$, and let $f:\mathbb{R}^n\to\mathbb{R}^m$ be the transformation that outputs $m$ evenly-spaced order statistics (including the extremes) of $y$. What is the Jacobian of this ...
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Probability of two people being born within the last 200,000 years and within 5 years of each other

I've come up with the number that there's a 1/30 Sextillion chance of being born on Earth. That's considering that you will be born and that there are 30 Sextillion potential planets that could be ...
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Probability that a normal RV is greater than multiple other normal RVs

Let $X_1, X_2, ... X_n$ be independently drawn from different normal distributions, such that $ X_i \sim N(\mu_i, \sigma^2_i) $ For any $j$ what is the probability that $X_j$ is the greater than all ...
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Probability random variable is less or equal to k-th out of two samples when ordered

Given the random variable $X$, $\{X_{i}\}_{i=2}^{n}$, $\{Y_{i}\}_{i=2}^{n}$ all iid and lets denote $X_{(k)}$ as the k-th statistic of $\{X\} \cup \{X_{i}\}_{i=2}^{n}$ and $Y_{(k)}$ for $\{X\} \cup \{...
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Estimator for Range (Length of Stay )

I have sample data ad I want to test the claim that the mean length of stay is 7 days. Data is give as pairs ( Arrival Date, Departure Date) , each date given as the nth day of the year, e.g., Jan 3 ...
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QR interview problem Guessing order of draws from iid U(0,1)

This is for QR at two well know trading firms (think jane street, HRT, Citadel, Jump ...)(not BB bank). Question prompt: Given n iid Uniform distributed r.v.s. $x_i$ ~ U(0,1). $x_1$ is drawn first, ...
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Summing ordered samples

Let's pick a random number between 0 and 1 uniformly 1000 times and put the results in an array for example : [0.176,0.765,0.879,0.234,0.152,0.765,0.645,0.897,0.762,0.087...] and sort this sample from ...
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Estimating density function at specific percentile using empirical cdf

Summary I'm running an experiment where I'm using the empirical CDF of a known random variable to approximate the density of the random variable at a specific percentile. In general I'm interested in ...
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Sum of iid Exponential observations then subtracting the minimum of the observations [duplicate]

Consider $n$ iid $X_1,...,X_n \sim Exp(1)$. My goal is to find the density of $\sum (X_i - X_{(1)})$. My attempt If we write out the entire summation in order statistics, we get $X_{(1)}-X_{(1)} + X_{(...
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PDF of the linear combination of two order statistics

I have just started to learn order statistics so please feel free to correct my notation/terminology. In my field is common to provide data as the median of some sample (for example several ...
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Nonparametric Order Statistics - Does this Exist?

I was reading about Order Statistics (https://en.wikipedia.org/wiki/Order_statistic) : Apparently, if we have a sample with "k" number of elements (e.g. x1, x2, ...xk) and assume a ...
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Statistical test for low treatment sample size, high nb of record and large control group

I am looking to analyse the efficiency of a treatment. The problem is that I have only 3 patients in the treatment group. However: I have thousand of measurement of the value of interest for these 3 ...
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Boolean order statistic

Inspired from this LeetCode question. I authored this question myself however. Please review my answer for accuracy. Let's sort an array ($n \ge 1$) of $1$s and $0$s, so that all $0$s come before $1$s....
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Calculate how many defective parts with p=0.95

After a 2 month test, there is a result of 5 defective parts out of the total sample size of 50 tested parts. How many defective parts can be expected for an annual production of 70000 parts (p=0.95)? ...
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Distribution/estimation of maximum change of a stationary time series

Any help on this would be much appreciated. Let $x_{t} = b x_{t-1} + u_{t}$, where $u_{t} \sim N(0,1)$ and $\lvert{b}\rvert < 1$. What can we say about the distribution of $y_{t} = \max(x_{t+2},x_{...
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Estimate normal distribution parameters from smallest N samples

I have a bunch of small datasets (billions of sets of 7 samples). Each dataset represents the smallest 7 samples of a larger set of 15 values which are normally distributed. Given just the smallest 7 ...
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Probability that a given number falls between the minimum and the maximum of a sample

Let $X$ be a real random variable with absolutely continuous cumulative distribution function $F$. Let $x_{(1)}, ..., x_{(n)}$ be a i.i.d. ordered sample of size $n$ of $X$: $$ x_{(1)} \leq x_{(2)} \...
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Conditional distances in order statistics

Assume I have $n$ points sampled independently from the uniform distribution on the unit interval. After ordering the sample I get the points $X_1, X_2, \dots X_n$ such that $X_1 \leq X_2 \leq \dots \...
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The variance of the weighted median and optimal weights

The median $\tilde{\mu}$ of a sample in many ways is analogous to the sample mean $\mu$. Both are an estimate for the population median or mean respectively, and both approach a Gaussian distribution ...
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Suppose I have 100 integers and I sample 10 without repetition. What is the expected rank of the lowest out of 10 samples?

Suppose I have 100 integers and I sample 10 without replacement. What is the expected rank of the lowest out of 10 samples? i.e. my lowest integer in the 10-sample is kth smallest out of 100. what is ...
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Multinomial Logistic Regression as a latent variable model

I was reading the wiki entry for multinomial logistic regression https://en.wikipedia.org/wiki/Multinomial_logistic_regression#As_a_latent-variable_model and it states that we can view the multinomial ...
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What interval does the median fall into when the values of the numbers before and after the median aren't specified and n is even?

What interval does the median fall into when the values of the numbers before and after the median aren't specified, but are the last and first data points in the intervals sorrounding the median, ...
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Rao Cramèr Lower Bound problem

Let $X_1, · · · , X_n$ be a random sample from the uniform distribution on $[0, θ]$. I want to get the variance of the maximum likelihood estimator of $θ$ and check whether the variance decrease at ...
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Show that Sample Mean and Sample Range are independently distributed for a random sample from Normal Distribution

Let $X_{1},\ldots {, X_{n}}$ be iid random variables with $X_{1} ∼ N(µ,\sigma ^{2}).$Let $\bar{X}= \sum_{i=1}^{n} \frac{X_{i}}{n}$, $R=max_{1\le i \le n} \{X_{i}\}$-$min_{1\le i \le n}\{X_{i}\}$.Show ...
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Finding probability of all success for in an order statistic

𝑓(𝑦) = 5𝑦^4; 0 ≤ 𝑦 ≤ 1 A group of 3 friends order small cups of soda, from the soda dispenser. If the 3 small cups are considered a random sample from the dispenser fills, find the probability ...
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Show that $P(X_k \le x) = P\{N(x) \ge k\}$

Suppose $x_1 ... x_n$ are the order statistics of an iid sample from a continuous distribution $F(x)$. Show that $P(X_k \le x) = P\{N(x) \ge k\}$ where $N(x)$, the number of sample values less than x, ...
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Understanding order of operations in formula

Can someone help me understand the order of operations for this formula? Lets say: y_estimated = 10, 14, 11 #three different estimates that will be subbed into the formula y_true= 12 R=3 I think it ...
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What is the distribution of the $k^{th}$ highest value of a multivariate normal distribution

Let X be an N-dimensional multivariate normal, $X \sim N(\mu,\Sigma)$ where $\mu$ is Nx1 and $\Sigma$ is NxN. If we take a draw of $X$ from this distribution and then sort $X$ from largest to smallest,...
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$P(X_{(1)}+X_{(2)}>X_{(3)})$ for order statistic

I am trying to solve this problem for when $X_1, X_2, X_3$ are independent $U(0,1)$-distributed random variables. The joint density function should then be given by $$f(x_1,x_2,x_3)= \begin{cases} 6, ...
1 vote
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Expected value of the largest order statistic for $Uniform(\theta,2\theta)$ [duplicate]

I'm struggling to find when $X_1,\ldots,X_n \sim Uniform(\theta,2\theta)$, how the expected value of the largest order statistic is $E[X_{(n)}]=\dfrac{2n+1}{n+1}\theta$. I can find that the density of ...
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quantiles (monotonic transformation)

I'm trying to show that, $(-X)_p = -X_{(1-p)}$, that is, the $p$ quantile of $-X$ is equal to the $1-p$ quantile of $X$ after multiplying with $-1$. The results holds when considering quantiles of a ...
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8 votes
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Sum of sample given a priori knowledge of its maximum

Given a sample of discrete random variables $X_1, X_2, \ldots, X_n \sim F$, I am looking to calculate the distribution given by the probability mass function: $$P\left(\sum_{i=1}^n X_i = x~\middle|~\...
2 votes
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Probability bound of the difference of order statistics for i.i.d. Gaussian random variables

I have asked a related question before (with more stronger requirement): Probability bound of the difference of order statistics for correlated and identical Gaussian random variables. Now, I'm pretty ...
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Probability bound of the difference of order statistics for correlated and identical Gaussian random variables

Suppose, there are $n$ identical and correlated Gaussian random variables namely, $X_1, X_2, ..., X_n$ with $X_i\sim\mathcal{N}(0,\sigma^2)$ for all $i\in\{1,2, ...n\}$. The correlation coefficient ...
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MAD & Median of weighted GMM

What is the median and median-absolute-deviation of a weighted GMM in terms of component mean and variance? For example, three normal distributions $A$, $B$, $C$ with means $\mu_a,\mu_b,\mu_c$, ...
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Averages of the two closest pairs out of a set of four observations

Four random numbers are drawn at random from a standard normal distribution. They are grouped in two pairs of closest numbers, $\{x_1, x_2\}$ and $\{x_3, x_4\}$ so that $x_1\le x_2 \le x_3 \le x_4$, ...
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$N \sim \text{Po}(\lambda)$ and $X_1,X_2,....,X_N$ are iid and independent of $N$, what is distribution of $Z_N = \max \{X_i\}_{i=1}^{N}$

I think the title covers most of my concerns. The distribution of the $X_i$ does not really matter, I am just experiencing difficulties in finding an expression for $$\text{Pr}(Z_N \leq x) = F(x)^N$$ ...
2 votes
1 answer
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How do Ordered Target Statistics work for CatBoost?

This question follows closely this paper . I'm trying to fully understand how Ordered Target Statistics (TS) (for CatBoost) works. E.g. the CatBoost algorithm uses this method to group categorical ...
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How many samples are needed to estimate quantiles for an unknown distribution?

I'm trying to evaluate performance of a metric learning model. The model that takes labelled image inputs and maps them to vectors on an N-dimensional unit sphere. The goal of the model is to map ...
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Joint distribution of top order statistics of two independent random samples of Pareto distribution

Suppose $X_1,...,X_n$ and $Y_1,...,Y_n$ are all independent copies of a standard Pareto random variable. For each of the 'two' random samples we can denote the order statistics $X_{n:n} \geq X_{n-1:n} ...
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What is the covariance matrix of the normal order statistics?

I would like to test if a sample comes from a standard normal distribution. I want to do that by sorting the sample values, and measuring the Mahalanobis distance to the expected order statistics from ...
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Total Time on Test (TTT) Statistic

Let $X_{(1)}, X_{(2)}, ..., X_{(n)}$ denote an ordered sample of size $n$ from a life distribution. Let $T_n$ be the total time spent on test by the $n$ sample units until the failure of the longest ...
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What exactly are order statistics?

Suppose $X_1,X_2,X_3.....X_n$ are random sample taken from a population. Then Y(1)<Y(2)<Y(3).....<Y(n) are called order statistics written in increasing order by magnitude where: Y(1)=minimum(...
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what is the expectation of minimum order statistics? [closed]

I want the expectation of minimum order statistics and the variance of minimum order statistics
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Order statistics is minimal sufficient statistics for unknow density function

I'm trying to prove the problem, but there is a problem on definition of term. The theorem that I use to prove it is However, what exact meaning of "family of densities ~ all have common support&...
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Conditional distribution of order statistic of p-values

Suppose we have independent Unif(0, 1) random variables $\{p_1,..., p_n\}$. We sort them by $p_{(1)} \leq p_{(2)} \leq ... \leq p_{(n)}$. For $i \geq 2$ and $x \geq t$, I would like to compute $$f_i(x)...
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1 vote
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What is the covariance of a thresholded random vector

I have a random vector $X = [X_1, X_2, \dots, X_n]^T$. I top-$k$ threshold it so that I get a new vector $Y = [Y_1, \dots, Y_n]$ where $Y_i = X_i$ if $X_i$ is in the top-k entries of $X$, and $Y_i=0$ ...
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If you take the maximum value from two random draws from a normal distribution, what is the mean and standard deviation of the resulting distribution? [duplicate]

Let's say I draw two numbers from a normal distribution with mean 50 and sd 10, and take the maximum of those two numbers. If I generate a distribution of that maximum, what are the resulting ...
4 votes
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Modeling a time series of ordered vectors

I have a series of ordered vectors, $\pmb{x}^o(1), \ldots, \pmb{x}^o(n)$. Here, $\pmb{x}^o$ means the ordered vector of $\pmb{x}$. For example, if $\pmb{x} = (2,5,1)^\top$, then $\pmb{x}^o = (1,2,5)^\...
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