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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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In a sum of high-variance lognormals, what fraction comes from the first term?

If $X_i \overset{\textrm{iid}}{\sim} \text{Lognormal}(0, \sigma^2)$ for $i=1,\ldots,n$ and $Y_1 = X_1 / \sum_{j=1}^n X_j$, then I would expect that a particular* limiting distribution of $Y_1$, ...
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Approximation of the expected value of the $i$-th standard normal order statistic in a sample of size n

For random variables $X_1, \cdots, X_n$, we denote the order statistics by \begin{align} X_{(1)} & = \min (X_1,\ldots, X_n) \\[6pt] X_{(2)} & = \text{second-smallest of } X_1,\ldots, X_n \\ &...
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Rough answer for the maximum of absolute value of $n$ standard gaussians (Computer Age Statistical Inference Problem 1.3)

I am working through "Computer Age Statistical Inference" as a self-study and am stuck on the follow exercise (1.3): The details of equation 1.6 are unimportant for the exercise, so far as ...
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Need help in calculating $\mathbb{E}(\frac{1}{x_{(2)}-x_{(1)}}\int_{x_{(1)}}^{x_{(2)}} f(t) \ dt)$, where $x_{(i)}$ are related Beta distribution

Suppose $Y, Z \stackrel{\text{iid}}{\sim}\mathrm{Uniform}(0,1)$. Let $a = g(\min(y,z)),\ b=g(\max(y,z)).$ How can I calculate the expectation $$\mathbb{E}\left[\frac{1}{b-a}\int_a^b f(t) \ dt\right]$$ ...
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constant approximation based on "sorted uniform distribution" and beta distribution [closed]

Let $X_1, X_2 \stackrel{\text{iid}}{\sim}\mathrm{Uniform}(0,1)$ and then sort $X_1,X_2$ to get $X_{(1)} < X_{(2)}$. Based on the pdfs of $X_{(i)}$, we know $X_{(1)} \sim \mathrm{Beta}(1,2)$ and $...
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MLE and UMVUE from an ordered sample of the exponential distribution

I am having a lot of trouble with every part of the problem below. Now, finding MLE's is simple in principle. I just find the distribution for $Y$ and then use calculus to find the value of $\sigma$ ...
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Exercise about Order statistics from uniform distribution

I'm trying to solve an exercise about order statistics. The exercise is the following: Let $U_{(1)}< \ldots <U_{(n)}$ be the order statistics from Uniform distribution U(0,1). Show that $(-\log[...
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Sufficient Statistic for Truncated Normal

I am doing exercise 3.18 of "The Bayesian Choice": Give a sufficient statistic associated with a sample $x_1,...,x_n$ from a truncated normal distribution $$ f (x|\theta) \propto \exp(-(x ...
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Hoeffding’s formula for Locally most powerful rank tests

Suppose we have a testing problem with $$H_0: X_1,X_2, . . . ,X_n \ \text{are i.i.d. random variables with a continuous cdf} \ F(x) \ \text{and pdf} \ f(x)$$ and $$H_1: X_1,X_2, . . . ,X_n \ \text{are ...
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How should I best to use reported stats on the Tippy-top?

Suppose I have a large population, in the millions, drawn from some underlying distribution, which we will take as a member of a known distributional family with unknown parameters. Assume the ...
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Can I estimate the mean of a dataset if I have its standard deviation and a portion of the full data that is higher than some threshold?

I have a partial set of measurement data that is limited due to my tool's sensitivity. I know that the data is approximately normally distributed and I have a standard deviation from another data set ...
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Showing that $X_{(1)}$ is sufficient for shifted exponential distribution

If the pdf of a random sample is $f(x)=e^{-(x-θ)}$ where $x \geq θ$, Show that $T=X_{(1)}$ is a sufficient statistic for $θ$. Can one show that $T$ is a sufficient statistic for $θ$ in the following ...
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Lower bound on distance between ordered statistic

I have found the following inequality in a manuscript: Let $S_{t}^{i}$ and $S_{t}^{i^{\prime }}$ denote the $i/N$-th and $i^{\prime }/N$-th cross sectional order statistics. We require: \begin{align} \...
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Distribution of minimum gap of $n$ points in the unit interval

I am trying to find the expected minimum and maximum distance between consecutive points on the unit positive rea interval .I have tried the following so far :Given $n$ uniform random variables on the ...
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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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1 answer
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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 ...
anatolvitold's user avatar
2 votes
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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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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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2 answers
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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 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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10 votes
4 answers
354 views

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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1 answer
434 views

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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1 answer
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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[...
Enrico's user avatar
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1 vote
1 answer
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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 ...
Galen's user avatar
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2 votes
1 answer
146 views

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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2 answers
66 views

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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5 votes
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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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2 votes
1 answer
136 views

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 ...
The Pointer's user avatar
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1 vote
1 answer
423 views

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 $...
Samuel Nelson's user avatar
9 votes
1 answer
22k views

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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11 votes
2 answers
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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 = \...
Michael Baudin's user avatar
2 votes
0 answers
237 views

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 ...
oyy's user avatar
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3 votes
0 answers
70 views

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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3 votes
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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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