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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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Order statistics uniformly spaced?

Suppose you have $n$ i.i.d. random variables $X_i$ that take values in $[0,1]$, and have an absolutely continuous distribution. Let $X_{(1)}\le X_{(2)}\le \dots \le X_{(n)}$ be the random variables ...
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Prove that $X_{(n)} - X_{(1)}$ is an ancillary statistics

Let $X_{1},X_{2},\ldots,X_{n}$ be an independent and equally distributed random sample whose distribution is uniform on the interval $(\theta,\theta+1)$, $-\infty<\theta<+\infty$. Then consider ...
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Concentration inequality for max component of a multivariate Gaussian in the general case

I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
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Probability of having an increasing trend in normal variates

Let $x\sim N(\mu,\sigma)$ and $x_i$ is ordered instances of random variate of $x$ for $i=1...n$. What is the probability that the series is in increasing (or decreasing) order? The problem is finding ...
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The joint pdf of sample maximum and sample mean for uniform distribution?

Assume $$\{X_i\}\stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{Uniform}(0,1)$$ Find the joint p.d.f. of $$X_{(n)} \hat= \max \{X_1,X_2,\ldots,X_5\}\quad\text{ and }\quad \bar X\hat=\sum^n_{i=1}{X_i}$$ ...
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Understanding the solution to a problem about a homogeneous Poisson process

This is probably easy, but right now I can't figure it out, so bear with me. The question is: Let $\{N(t),t\ge 0\}$ be a homogeneous Poisson process on $(0,\infty)$ with rate $\lambda$. Let $\{S_i, i=...
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Meaning of Extreme Value distribution vs. lowest/highest Order Statistic

How exactly does the meaning of the Extreme Value Distribution differ from the distribution of the lowest/highest (extreme) order statistics? I understand that the extreme value distribution (EVD) ...
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Proving the MVUE is the following

I am stuck on the following question and I was wondering if can get some help. Let $f(x;\theta) = g(\theta)h(x),\ a(\theta) \leqslant x \leqslant b(\theta)$ with $a(\theta)$ decreases and $b(\theta)$...
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Order Statistics of Poisson Distribution

I have been given the following question, Let $n ≥ 2$, and $X_1, X_2, . . . ,X_n$ be independent and identically distributed $Poisson (λ)$ random variables for some $λ > 0$. Let $X_{(1)} ≤ ...
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All Order Statistics are statistics but not estimators [closed]

Any order statistic is a statistic but not an estimator. Discuss along with an example. Can anyone help me out with this question, please?
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Is there a random variable $X$ with positive support such that the ratio of the two smallest realizations of an iid sample goes to one?

Imagine I have given a random variable $X$ with supp$(X)=(0,\infty)$ and $\mathbb P(X \in (0,a))>0$ for any fixed $a>0$ Now given an iid sample $X_1,...,X_n$ - is it possible that $$X^{(2)}/...
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Prove that the order statistics are minimal sufficient for a random sample from an unknown density $f$ [duplicate]

This is Exercise 6.29 from Casella and Berger's Statistical Inference, so I'll just post the question in full, and I'll also post the answer included in the solutions manual. I'll make the part that I ...
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Prove the maximum order statistic $X_{(n)}$ is a minimal sufficient statistic for the uniform$(0,\theta)$ family using a particular theorem

I'm doing Exercise 6.26 in Casella and Berger's Statistical Inference, and I'm trying to prove the following: "Use Theorem 6.6.5 to establish that, given a sample $X_1,...,X_n$, the maximum order ...
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Distribution of the midrange in the general case

Given an iid sample $X_1, \cdots, X_n$ drawn from a sufficiently nice (finite expectation, maybe L2 integrable, etc.) distribution, I want I want to know the population CDF of the mid-range or ...
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Order Statistics; Finding the probability that the first sample is < 0.6, and the last sample is > 0.6

Here is the problem statement below: A random sample of size 5 is drawn from the pdf $f_Y(y)=2y, 0\le y \le1$. Calculate $P(Y_1^{'} < 0.6 < Y_5^{'})$. Here, using formulas for order ...
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Expectation of kth order statistic of Pareto distribution

I am trying to find the expected value of $X_{(k)}$ Given cdf $$ F(x) = \begin{cases} 1-\left(\frac{\sigma}{x}\right)^\alpha, & x > \sigma\\ 0, & \text{else.} \end{cases}$$ My attempt: $$...
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Conditional expectation of uniform random variable given order statistics

Assume X = $(X_1, ..., X_n)$ ~ $U(\theta, 2\theta)$, where $\theta \in \Bbb{R}^+$. How does one calculate the conditional expectation of $E[X_1|X_{(1)},X_{(n)}]$, where $X_{(1)}$ and $X_{(n)}$ are ...
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Association between an ordinal Likert variable and a nominal variable

I have responses from a 7 point Likert semantic differential scale -let's say from 'Completely Agree' at one end to 'Completely Disagree' to the other- and a nominal/semi-ordinal age groups variable ...
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1answer
108 views

Variance of Normal Order Statistics

Suppose we have $X_1, \cdots, X_n \overset{\textrm{i.i.d.}}{\sim} \mathcal{N}(0, 1)$ with $n > 50$, and let $X_{(1)}, \cdots, X_{(n)}$ be the associated order statistics. Are there any references ...
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What's the probability that all three parts would fail within 2 years of each other? (joint PDF)

Suppose an instrument has three independent parts, all of whose lifetimes (in years) are modeled by an exponential pdf which is $f_Y(y)=e^{-y}, y>0. $ What's the probability that all three parts ...
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Estimation of an exponential parameter

I´m trying to figure out the pdf $f_\min(X_i)$ of $\min(X_i)$, where the distribution of the sample $X_1,...,X_n$ is $\mathcal{E}xp(\lambda)$, where $\lambda$ is the unknown parameter. I tried with ...
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Order statistics: What's the probability that all three components will fail within 2 years of each other?

Suppose an instrument has three independent parts, all of whose lifetimes (in years) are modeled by an exponential pdf which is $f_Y(y)=e^{-y}, y>0. $ What's the probability that all three parts ...
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37 views

Order statistics of a single r.v

I have trouble understanding the following question: We have N i.i.d random variables from the uniform distribution between 0 and 1. If N=1, what is the probability that the n^th order statistic ...
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$k$-th order statistics when the value of $j$-th one is known

Suppose there are $n$ random variables $X_i,~i\in\{1,\cdots,n\}$ which are independently drawn according to a CDF $F$ and pdf $f$. Suppose also that we know one of the realization, say $X_{(j)}=x_{(...
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How to see this order statistic result and find my error

Let W be a random variable with pdf $f(w)=\theta B^{-\theta}w^{\theta-1}$ for $0 \lt w \lt B$ and 0 otherwise. Assuming Independence, Show that , $W_{n:n} \to B$ as $n \to \infty$ where $W_{n:n}$...
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Comparing ordering of sample and conditional distribution based on that sample

I don't have a specific question, but just a request for resources and/or guidance. Suppose we take $n$ draws from a distribution $F$. Suppose a particular sample is given by $\vec{\theta} = (\...
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26 views

Is there a closed-form solution for ratios of order statistics?

Is there a closed-form solution for the expected value and variance of the ratios between specified order statistics drawn from a large sample from a known parametric distributional family? Actually, ...
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Ordered statistic calculation with a different kind of cdf [closed]

Let $X_1, X_2,...,X_n$ be I.I.D random variables, and let $Y_{(1)}\leq Y_{(2)} \leq ... \leq Y_{(n)}$ be the ordering of the random variables.The distribution function is given as- \begin{align*} F(x) ...
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Proving completeness of highest-order statistic using Leibnitz' Rule

Suppose that $X_1,...,X_n$ are iid with common pdf given by $$f(x;\theta)=2e^{2x}\theta^{-2}I( x<log(\theta)).$$ I am tasked with finding a complete-sufficient statistic for $\theta$, and I have ...
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71 views

Basic calculations with Order Statistics

I've come across the following problem, and I am tempted to delve into order statistics to solve this. I would greatly appreciate any help! Suppose you draw 6 independent samples from a continuous ...
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Reference line for QQPlots: robust regression vs quartiles method

Using the car package, I drew two qqPlots of 144 p values from correlated tests on one dataset. The top plot uses a reference line drawn by "quartiles" and the bottom plot uses a line drawn by "robust"...
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Relation between a Gamma prior and posterior in terms of paramters

I am doing a maths exercise and I have found out that the prior of my parameter is is a inv.gamma (alpha, beta), the likelihood is an exponential distribution. Finally I have discovered that my ...
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Tail probability of minimum to maximum ratio among $n$ i.i.d. half-normal random variables

I have $n$ i.i.d.$\sim\mathcal{N}(0,1)$ random variables $X_1,\cdots,\ X_n$. For my research, I am interested in finding bounds (upper and lower) of the tail probabilities of the ratio $\frac{|X|_{(1)}...
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Does having the same order statistics imply the same distribution?

I'm stuck at the following problem: Say $X_i \overset{\text{iid}}{\sim} \operatorname{Exp}(\lambda)$ for $i = 1, \ldots, n$. Denote $X_{(1)}, \ldots, X_{(n)}$ the order statistic from the $n$ ...
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Correlation between an observation and its rank in a random sample

Suppose $X_1,X_2,\ldots,X_n$ are i.i.d random variables with an absolutely continuous distribution. We say the observation $X_i$ has rank $R_i$ if $$X_i=X_{(R_i)}\quad,\,i=1,2,\ldots,n,$$ where $X_{(...
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150 views

Finding the distribution of sample range for a Beta population

Let $X_1,X_2,\ldots,X_n$ be i.i.d random variables having density $$f(x)=2(1-x)\mathbf1_{0<x<1}$$ I am trying to derive the distribution of the sample range $R=X_{(n)}-X_{(1)}$. The ...
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stochastic ordering of counting processes

Let $N_1(t)$ be a delayed renewal process and $N_2(t)$ be an ordinary renewal process such that $N_1(t)\geq_{st}N_2(t)$. Consider a renewal process $Z(t)$ with the same inter-arrivall distribution as $...
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Order statistics for log series distribution

I am trying to obtain the probability mass function for various order statistics of a log series distribution for a given n. To do so, I tried modifying the code given in this question: Simulating ...
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Order statistics are minimal sufficient for the set of all continuous distributions

Previous question exists here, but no answer has been posted. I don't believe this is a duplicate of existing questions concerning minimal sufficiency of order statistics that have an answer. Problem:...
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Estimating Expected Order Statistics

I have a fairly basic question that I'm looking for a reference for. First, a couple definitions. Let's say $X_1,\ldots,X_n$ are IID samples from a distribution $F$ over $[0,1]$. For any $k\in\{1,\...
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Minimum of i.i.d. Random Variables [duplicate]

What importance does the minimum of an identical and independently distributed random variables play in probability distribution?
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sampling distribution of order statistics

Let, $X_{1},X_{2},\ldots,X_{n}$ be an i.i.d. sample from $R(1-\theta,1+\theta)$. Show that, $(X_{(1)},\bar{X},X_{(n)})$ is sufficient for $\theta$. Ans: $f(x_{1},x_{2},\ldots,x_{n})=\frac{1}{(2\theta)...
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Cumulative distribution of X_{1} = min{X_{1},X_{2}}

So I have $X_{1},X_{2} \sim^{iid} Unif(\theta,1)$, hence the density function is $f_{X_{i}}(x) = \frac{1}{1-\theta}\mathbb{1}_{(0,1)}(x)$ What is the cumulative distribution of $X_{1} = min\{{X_{1},...
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Limiting distribution of first order statistic x

Good evening everyone, I am attempting a problem on limiting distributions. The problem is as seen in the picture above. Could you please help me with part c) of this problem. The answer as per the ...
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1answer
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$X_i \sim \text{Uniform}(0, \theta)$ iid; $Y = \max{(X_1,..,X_n)}$. Why is $\theta$ necessarily larger than $y$?

I'm going through Statistical Inference by Casella & Berger, and on page 419, in the intro section of interval estimation there is the following example (note: most of the text was left out as it'...
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1answer
54 views

maximum likelihood estimator

Suppose, $X_1,X_2,\ldots,X_n$ be a random sample from $U(\theta-2,\theta+2)$. Define, $X_{(n)}=\rm{max}\{X_1,X_2,\ldots,X_n\}$ and $X_{(1)}=\rm{min}\{X_1,X_2,\ldots,X_n\}$. Then which of the ...
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Is there a canonical probability distribution on an ordering/permutation

What is a good way of defining a non-uniform probability distribution on a permutation of k objects? For example, suppose the parameter was an ideal ordering, and the probability of an ordering was ...
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1answer
69 views

Distribution of maximum over minimum of Weibull(alpha, 1)

Let the PDF of Weibull $(\alpha, 1)$ be $$ f_X (x) = \alpha x^{\alpha-1} \exp\left( - x^{\alpha}\right) $$ I know from probability integral transfrom that $$ X_{(1)}^{\alpha} \overset{d}{=} Z_{(1)...
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CDF of top $k$ value among $n$ RV's when the value of one of them is known

Suppose that $X_i$ is a random variable whose CDF if $F$ and pdf is $f$, $i=1,\cdots, n$. Assuming independency, the $m$-th highest value can be found using order statistics. What would be the CDF ...
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PDF for the ith ORDERED uniformly random sample compared to an evenly spaced sample

Let $r_1 ≤ r_2 ≤ ... ≤ r_N$ denote an ORDERED set of N realizations of real numbers that are uniformly random on the number line from 0 to 1. Let $R_1 < R_2 < ... < R_N$ denote a set of ...