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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Which statistical test is suitable to compare order statistics of two independent samples?
Say I want to compare two order statistics (say the 2nd largest value or min value) of two samples. Let's not make any distributional assumptions except that the variance is finite?
Is something like ...
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How to find the MGF of the max of a set of i.i.d. exponential random variables
As the title suggests, I would like to find the MGF of the max of iid exponential random variables. Assume $Z=\max(x_{1},...,x_{n})$, where $x_{i}$ is distributed as exponential($\beta$) and has pdf $\...
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Max of the running average of the kth through nth elements for a given probability distribution
This question is based slightly on https://www.reddit.com/r/AskStatistics/comments/16bqit0/calculating_probability_when_phacking_is_allowed/
Given a variable $X$, let $A_j$ be the average of $X_1$ ...
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Finding Sample Range of Fisher's z-distribution via Approximating Hypergeometric $\,_2F_1\left(\frac{1}{2},\frac{x+1}{2};\frac{3}{2};-z^2\right)$
Recently, I have encountered Hypergeometric function $\,_2 F_1\left(\frac{1}{2},\frac{x+1}{2};\frac{3}{2};-z^2\right)$ in the context of order statistics.
In particular, I am trying to evaluate an ...
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Covariance between two binomial random variables or expectation of product of binomial random variables
I have an empirical distribution $S_n(x)$ (= proportion of samples less than equal to x) from a random sample $X_1, X_2, ..., X_n$ for a random variable $X \sim F_X$. Consider the random variable $T_n(...
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The difference of $\sum_{i=1}^{n}X_{i}$ and $\sum_{i=1}^{n}X_{(i)}$
Here is a exercise from *Mathematical Statistics. Jun Shao. Second edition. EX2.20
Let $X_1,..., X_n$ be $i.i.d.$ random variables having the exponential distribution $E(a,\theta)$, $a\in R$, and $\...
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Are these conditions sufficient to find the lowest score that could have been scored by more than at least one student, if not, what else?
What other deductions are possible?
In a class of n students, a test was administered and only scored on whole numbers from 0-10 if every possible score was scored at least once, the average score ...
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Order statistics for non independent variables
I have three random variables $X_1, X_2, X_3$ where $X_1$ is from distribution $F_1$.
$X_2=X_3$ are from distribution $F_2$ and they happen to be identical.
My question is what will be the order ...
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Properties of the inverse normal cdf and permutation probabilities as models for horse racing
Let $T_i$ be the running time of horse $i$ and $T_i \sim N(\theta_i,1)$ and the $T_i$'s are independent. Then Henery (1981) showed that the probability $P(T_1<T_2<\cdots <T_n)$ can be ...
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Most probable value for the successor (in ascending order) of a known statistical unit
Let $n$ be an integer $>1$.
Suppose a sample $(x_{k})_{k\in[[ 1;n]]}$ is taken from a known distribution on $\mathbb{R}$.
Given $x_1$ and supposing $\exists k\in[[2,n]], x_k>x_1$, what is the ...
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Proof of convergence of distribution of order statistics
Let $X_i$ be i.i.d with distribution $F$ such that for some $a_n>0$, $b_n \in \mathbb{R}$. $F^n(a_n x +b_n) \to \exp(-(1+\gamma x)^{\frac{-1}{\gamma}}$
Define $X_{1,n} \leq \ldots \leq X_{n,n}$ to ...
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Power of Uniform Order Statistics
I know that if $U$ is a uniform r.v. in $(0,1)$, then $U^a\sim Beta(1/a,1)$ with $a>0$.
On the other hand, if $U_{(1)}\leq \cdots\leq U_{(n)}$ are the uniform order statistics, then, with $U_{(0)}=...
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Proving that if an equally smallest order statistic is exponentially distributed then the random variables are exponentially distributed
The question I'm stuck on states to prove that if $X_{(1)}$ is exponentially distributed, then so is $X_1$, where $X_1,\dots,X_n$ is a sample of i.i.d. random variables and $X_{(1)}\leq\dots\leq X_{(n)...
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Asymptotic efficiency of IQR
I was wondering about the asymptotic efficiency of the Interquartile Range (IQR) in the Gaussian case. I have calculated it empirically using a Monte Carlo estimator, and it appears to be equal to ...
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Asymptotic distribution of $n^r \frac{U_{(1)}}{U_{(n)}}$: figuring possible $r>0$
Consider the i.i.d. sample $U_1, U_2 \cdots, U_n$ from the uniform distribution $U(0, 1)$. I should find a possible values of $r>0$ to have an asymptotic distribution of
$$ n^r \frac{U_{(1)}}{U_{(n)...
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Is there any advanced version of Total Time on Test (TTT) Transform?
One parameter survival distribution with increasing hazard rate is (for example) Rayleigh and Lindley, where Rayleigh's hazard rate increases linearly and Lindley's hazard rate increases with a ...
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Is modeling the extreme value of a distribution a basic probability result?
I was reading briefly about the field of EVT - extreme value theory, and the associated distributions that arise from modeling the maximum of a finite sample. It's not quite clear to me the nature of ...
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Is my logic about comparing the 70th/90th percentile from two respective datasets correct or is there a proof to do this?
Larry and Tony work for different companies. Larry's salary is the $90th$ percentile of the salaries in his company, and Tony`s salary is the $70th$ percentile of the salaries in his company.
Which ...
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Why is median not a sufficient statistic? [duplicate]
Suppose a random sample of $n$ variables from $N(\mu,1)$, $n$ odd. The sample median is $M=X_{(n+1)/2}$, the order statistic of the middle of the distribution.
How to prove that sample median is not a ...
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Order Statistics - Percentile Range of Normal Mixture of Normals
Say I have draw N values from a normal distribution [$\mu_1$, $\sigma_1$]. Below are 10 sampled points compared to the normal distribution they're sampled from
I then create a normal mixture of ...
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How to compute the correlation between the min and max of two random variables?
A joint bivariate function (PDF) with variables $A,B$ is given in the function:
$f_{A,B}(a,b) = Cab(a+b), \quad 0<a<1, \quad 0<b<1.$
where $C$ is just a constant. Assume that $Q=\min(A,B)$...
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Correlation Between Min and Max of Two Different Uniform Distribution
$\textbf{This is a self-study problem that I am interested in knowing the correct answer.}$ $\textbf{However I do not trust my computations and I need help.}$
$Y$ is Uniform(0, 2); $Z$ is Uniform(1, 3)...
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Non-IID Uniform Distribution
$A$ is uniform (0, 2) and $B$ is uniform(1, 3). Find the Cov$(W, Z)$, where $W=\min(A,B)$ and $Z=\max(A,B).$
Since $WX = AB,$ then by independence of $A$ and $B$, $E(WZ) = E(A)E(B),$ so that
$$Cov(WZ)...
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Permutation and order statistic
Let $X_1,…,X_n$ be i.i.d. random variables. Are these two equalities correct?
$P[X_{(2)}<x_2,…,X_{(n)}<x_n| X_{(1)}=x_1]=\\
=n!P[X_{2}<x_2,…,X_{n}<x_n| X_{1}=x_1]=\\
=n! P[X_{2}<x_2]…P[...
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Equality between order statistics and certain generalized means?
Question
Suppose I have a random variable $X$ with CDF, $F(x)$, and I want to model an IID sample $\{X_1, \cdots, X_n \}$ of size $n$.
For any order statistic $X_{(k)}$ for this sample where $1 \leq k ...
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CDF of max of $n$ cauchy variates
Suppose $X_1,X_2,\cdots,X_n$ are iid copies of a standard cauchy variate with pdf
$$ f(x)=\frac{1}{\pi(1+x^2)},0<x< \infty. $$
Define:
$$ Y=1+ \underset{1 \leq i \leq n}\max X_i.$$ I want to ...
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What is the reasoning behind the string of equality $P(x_{(n)} \le t) = P(X_i \le t, \ \text{for each $i$}) = \{\Phi(t - \theta)\}^n$? [duplicate]
I am currently studying the textbook In All Likelihood by Yudi Pawitan. Example 2.4 of chapter 2.2 Examples says the following:
Example 2.4: Suppose $x$ is a sample from $N(\theta, 1)$; the ...
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Statistical notation question: How do I represent sorting variables, individually and by each other, symbolically?
I'd like to write a formula for a correlation coefficient that involves sorting continuous observations, both within a variable and by another variable. For example, I'd like to say that $r$ is ...
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How should I estimate the variance in a sample or population from the sample range?
Suppose I wish to know the variance within a sample or of the population from which it is drawn. However, I do not have true measurements for most of my "observations". Think of them as like ...
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A question about order statistics
Let $X_1,\dots,X_n\overset{iid}{\sim}F_X$ be random variables. Let $b\in(0,\infty)$, and consider the transformation:
$$
Y_i=bF_X(X_i)\\
i=1,\dots,n
$$
How do I calculate the order statistic $Y_{(n)}?$...
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Making sense of this section on likelihood of order statistics
I am currently studying the textbook In All Likelihood by Yudi Pawitan. In chapter 2 Elements of likelihood inference, the author presents the following example:
Example 2.4: Suppose $x$ is a sample ...
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Roll 4 Dice, What's the Expected Value of the Sum of the highest 3?
After writing a simulation in python (code at bottom) I realized my calculations are incorrect but can't figure out where I went wrong.
Let {$D_1$, $D_2$, $D_3$, $D_4$} be the ordered dice rolls. Let $...
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Why does R say 'cannot compute exact p-values with ties' when I can do it with pen and paper?
Suppose I have two sets of three numbers: $x_1, x_2, x_3$ and $y_1, y_2, y_3$ and I want to test the Null hypothesis that they are drawn from the same distribution using the Wilcoxon-Mann-Whitney test....
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Link between the Beta and Exponential distribution
Let $n \geq 1$ be an integer. Let $X \sim \operatorname{Beta}(i, n - i + 1)$ where $i \in \{1, ..., n\}$. Therefore:
$$
X = \frac{A_n}{A_n + B_n}
$$
where
$$
A_n = \sum_{r = 1}^i Z_r, \qquad B_n = \...
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Lower Bound on Expected Maximum and Upper Bound on Expected Minimum of Order Statistics
This question relates to bounds on expectations of order statistics, elaborated upon in the Book by Arnold and Balakrishnan (1989). Let $X_1,\ldots,X_n$ be i.i.d. continuous random variables ...
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Proof that $\sqrt{n}\left( \hat{F}_n(x_1)-F(x_1),\dots,\hat{F}_n(x_k)-F(x_k)\right) \rightarrow \mathcal{N}_k(0,\Sigma)$
By definition \begin{align*} \Sigma = F(\min(x_i,x_j))-F(x_i)F(x_j)\end{align*}
Note: $\hat{F}_n(x) = \frac{1}{n} \sum_{j=1}^n 1_{\{X_j \leq x\}}$
I think that $\mathbb{E}\left[\hat{F}_n(x_i)\right] =...
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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 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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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 on Wikipedia [retrieved 29 June 2022]:
Apparently, if we have a sample with $k$ elements (e.g., $x_1, x_2, ..., x_k$) and assume a probability distribution for ...
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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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