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. Order statistics are widely used in non-parametric inference.

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8 views

Conditional Expectation of RV given the minimum of a Order Statistic

Given an exponential distribution with $X = (X_1,X_2,..,X_n)$ is i.i.d and order statistics $X_{(1)}\le X_{(2)}\le...\le X_{(n)}$, how does one compute $E[X_1|X_{(1)}]$? Intuitively, I know the ...
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Finding the probability density function of the $n$th largest random variable.

Let $X_1,...,X_{25}$ be independent Unif $[0,1]$ random variables. Let $Y$ be the $13$th largest of the $25$ random variables. Find the probability density function of $Y$. I already know the answer ...
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Why the median p.d.f. of the uniform distribution is not a p.d.f?

Let $X$ be uniformly distributed on interval $[\theta-2, \theta+2]$, $\theta\in\mathbb{R}$. Let the sample size of $3$, find the p.d.f. of median! I have tried as follows. The p.d.f. of $X$ is \begin{...
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Prove the CDF of kth Order Statistic

I know the general form of the CDF of kth Order Statistic for n iid random variables is given by $$F_{k}(x) = \sum_{j = k}^{n} {n\choose j}F(x)^j[(1-F(x)]^{n-j}$$ And I am tring to get PDF of the ...
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Find $E(X_{(1)}\mid T)$ where $T=\sum_{i=1}^n X_i$

Let $X_1,X_2,\ldots,X_n$ be a random sample with $n\geq 2$ from an exponential distribution. $X_{(1)}=\min(X_1,X_2,\ldots,X_n)$. Find $E(X_{(1)}\mid T)$ where $T=\sum_{i=1}^n X_i$. I was able to find ...
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Correlation coefficients ("Eta, Biserial, Point-Biserial, Gini) [closed]

I want information about the following correlation coefficients. Correlation coefficients ("Eta, Biserial, Point-Biserial, R^2, Gini)
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Simplifying or approximating $\sum_{k=1}^{\infty}\left(1 - \left(1 - 2^{-k}\right)^n\right)$?

Consider a game in which you flip a coin until you flip tails. Your score is then the number of heads you flipped. So, for example, the sequence $H$, $H$, $H$, $T$ has a score of three, while the ...
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1answer
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$F(X_{(n-k_n)})\overset{n\to\infty}{\to} 1$?

Let $X_1, X_2, \dots$ be iid (possibly heavy tailed) with their df $F$. Notation $X_{(k)}$ represents the $k-$th order statistic, i.e. $X_{(1)}=\min_{i\leq n} X_i$. Let $k_n\in\mathbb{N}$ fulfill $$...
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$F(X_{(n-k_n)})\overset{n\to\infty}{\to} 1$ for time series?

Let us have a stationary time series $X=(X_t, t\in\mathbb{Z})$ following e.g. AR(p) model (i.e. there are $a_1, \dots, a_p$ such that $X_t=a_1X_{t-1}+\dots + a_pX_{t-p} + N_t$ where $N_t$ are some iid ...
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42 views

Order statistics probability

Suppose that $X$, with a random sample $X_1, \ldots, X_n$, is a random variable with the following pdf: $f(x; t) = xte^{\frac{-x^2t}{2}}$ with the following support: $x \in [0, \infty)$, where $0 < ...
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Sufficient statistic as an estimator

Suppose that $X$, with a random sample $X_1, ..., X_n$, is a random variable with the pdf $f(x;t) = 2xt^{-2}$ and support $x\in [0, t]$, where $0 < t < \infty$ is unknown. (i) Show that $W = max\...
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Likelihood estimators and order stats

Suppose that $X$, with a random sample $X_1, ..., X_n$, is a random variable with the following pdf: $f(x; t) = xte^{\frac{-x^2t}{2}}$ with the following support: $x \in [0, \infty)$. Note that $0 <...
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33 views

Probability distribution of total life time of a machine with two parts in parallel system

Two identical components having lifetimes $A$ and $B$ are connected in parallel in a system .Suppose the distributions of $A$ and $B$ independently follow exponential with mean $\frac 1a, a>0$. ...
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1answer
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What is the simplest way to compute $f_Z(z)$ when $Z = \min(X, Y), X \sim U(0, 5), Y \sim U(0, 10)$?

Note that $Z \in (0, 5)$, so we have $$ F_Z(Z \leq z) = \begin{cases} ? & z \leq 5 \\ 1 & z > 5 \end{cases} \\ $$ Now need to compute $F(Z \leq z | z \leq 5)$. $$ F(Z \leq z ...
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58 views

Mean squared error of order statistic [duplicate]

Let $\theta \in \mathbb{R}$ and $X_1, X_2,..., X_n$ be independent and identically distributed with density \begin{equation} f(x; \theta) = I\left(|x-\theta| \leq \frac{1}{2}\right) \end{...
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Joint distribution of the $n-1$ nontrivial terms $X_{i}/(\min_{i}X_{i})$ conditional on $\min_{i}X_{i}$

If $X_{i}\sim$Pareto, how can I obtain the joint distribution of the $n-1$ nontrivial terms $X_{i}/(\min_{i}X_{i})$ conditional on $\min_{i}X_{i}$?
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Correlation Coefficient of two Order Statistics [duplicate]

My problem is exactly the same as asked in here with a change in the notation of the two order statistics. Reframing the question: If $\left(X_1,X_2,…,X_n\right)$ are a random sample from Uniform(0,1)...
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1answer
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Picking 5 random numbers on average how big is the largest gap between them?

We pick 5 random numbers from 1 to 100 with repetition. We order them. On average what is the largest difference between two consecutive numbers? What is the smallest difference? Example: 4, 22, 47,55,...
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Question about order statistics of Uniform distribution

Let $X_i$ be i.i.d $\operatorname{Uniform}(-\theta,\theta)$ random variables, and let $X_{(1)},\dotsb, X_{(n)}$ the order statistics of the random variables. Then find distribution of $Y = X_{(n)} - ...
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Computing the UMVUE for $U(0,2\theta+1)$

This question is from one of the introductory books in Mathematical Statistics. Let $X_1,...,X_n$ be a random sample from a pdf $f(x;\theta)=\frac{1}{2\theta+1},0<x<2\theta+1,$ zero elsewhere. (...
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Joint Distribution of two Order Statistics

Let $\left(X_1,X_2,...,X_n\right)$ be a random sample from a population having absolutely continuous distribution function $F\left(x\right)$. Suppose, I want to find the joint distribution of two ...
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Is the convex mixture of two exponential density functions log-concave?

Consider an exponential distribution with density $f(x)=\lambda\exp(-\lambda x)$ and distribution function $F(x)=1-\exp(-\lambda x)$. Denote $f^{(1)}(x)=n F^{n-1}(x)f(x)$ the density of the highest ...
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Find the joint distribution of $Y_1$ and $Y_n$

Let $X_1, \cdots , X_n$ be identically distributed random variables, with probability function $~ Exp (θ)$ with $θ = 1$. Let $Y_1, \cdots, Y_n$ be the corresponding order statistics: Obtain the joint ...
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1answer
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Order statistics: probability that one statistic is larger than the other; two sets of data drawn from a uniform distribution

I'll preface the question quickly by saying that I think I already have an answer, however, I'm looking to solve the question in a specific way, and I'm not sure how to. (Unfortunately, I'm finding my ...
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157 views

We choose $5$ numbers from $1$ to $100$. We order them by value. What is the expected difference between the second and the third?

I came across this peculiar problem. We choose $5$ numbers from $1$ to $100$ (with repetition). We order them in decreasing order by value. What is the expected difference between the second and the ...
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29 views

What David's Score does one use for new observations?

I'm examining dominance rankings in my data using a variety of measures and I'm exploring (un-normalized) David's Score (Ranking from unbalanced paired-comparison data (David (1987)) as an alternative....
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1answer
30 views

Reference algorithm/formula for the distribution of the median of random variables?

The distribution of the mean of two random variables can be calculated using a convolution. I have a collection of $n$ independent random variables each with PDFs that are simple functions on $[0,1]$. ...
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19 views

Non-asymptotic tail bounds for the gap between the largest and the second largest value in iid sample from $N(0,1)$

Let $n$ be a positive integer and consider the probability density $f_n$ on $\mathbb R_+$ given by $f_n(z):=\int J_n(u,u+z)du$, where $J$ is a probability density on $\mathbb R^2$ given by $J_n(u,v):=\...
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1answer
35 views

Expected value and variance of kth order statistic given maximum value

Let X(1) < X(2) < X(3) < X(4) < X(5) be the order statistics corresponding to a random sample of size 5 from a uniform distribution on [0, θ], where θ ∈ (0, ∞). Prove that the variance of ...
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Can the moment estimator be not sufficient?

As the title, can the moment estimator sometimes be NOT sufficient? Consider the following example where Y ~ Uniform(1-b, 5+b): From method of moment estimator: E[Y] = 1/2((1-b) + (5+b)) = 3 => ...
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Order statistics: Distribution of order statistics for dependent variates

Suppose that we have a sequence of $n$ i.i.d. random variates $X_1, X_2, ..., X_n$ with cdf $F_X(x)$ and pdf $f_X(x)$. Now define $Y$ as the sum of $k$ consecutive realizations of $X$ $Y_i = \sum_{j=i}...
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Order statistics form exponential distribution

Let $X_1,...,X_n$ be a random iid sample from $Exp(\lambda)$. By $X_{(n)}$ we denote $n-th$ order statistic. Prove that $X_{(k)}, X_{(m)} - X_{(k)}$ are independent ($1\leq k\leq m \leq n$). I dont ...
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The expectation of the number of occurrences of the most frequent element in a random sequence

We generate the sequence $\{a\}$ in the following way. $n$ is a given fixed integer, and for each position $i, a_i$ will be chosen from $\{1,2,...n\}$ with equal probability and the sequence ends ...
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1answer
52 views

Gini index different expressions

Let $X$ be a non-negative random variable with positive finite expectation. The Lorenzcurve $L_X$ is defined by $$ L_X(u) = \frac{\int_0^u F_X^{-1}(y) dy}{E(X)}, \quad 0 \leq u \leq 1,$$ where $$ F_X^{...
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1answer
38 views

Sufficient statistic for $N(\mu,1).$

So I am answering a problem where $\underline{X}=(X_1,X_2)'$ be a random variable from $N(\mu,1).$ Here is my approach, the joint density is: $f(x_1,x_2|\mu) = (2\pi)^{-1} \textrm{exp}(\frac{-1}{2\...
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Questions about the parts I don't understand during the proof process in the Poisson process.

The Poisson Process Chapter in "Stochastic processes 2nd edition" written by Sheldon Ross. There is a theorem (Thorem 2.3.1) in the book, which is stated as follows: Given that $N(t)=n$, ...
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Find the median of the pdf of the median of the random sample when $n$ is even.

I am trying to find the pdf the $$M = \frac{X_{\left(\frac{n}{2}\right)} +X_{\left(\frac{n}{2}+1\right)}}{2}$$ here we let $X_1, X_2, ..., X_n$ be a random sample from a Uniform$(\theta; \theta + 1)$ ...
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1answer
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If $X_i$ follows $ U( \theta, \theta+1)$ and n is even. How do I find the probability distribution of median?

I have used the following transformation to find the joint pdf: $u= X_\frac{n}{2}$ and $ v = \frac{X_\frac{n}{2}+X_{\frac{n}{2}+1}}{2}$ The joint pdf I have found is like below: $f_{(U,V)}(u,v)= \...
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Expected value of top $k$ values from $n$ samples of a zeta distribution

Suppose we have $n$ values $X_1,...,X_n$ sampled i.i.d. from a zeta distribution with scale parameter $s$. Let $k\le n$. I want to know the expected value of the sum of largest $k$ values in $X_1,...,...
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1answer
62 views

Find the joint pdf of $X_{(n/2)}$ and $M$.

Let $X_1, X_2, \ldots , X_n$ be a random sample from a Uniform$(0, θ)$ population with $θ > 0$, where $n$ is an even integer. Let $M=\frac{X_{\left(\frac{n}{2}\right)}+X_{\left(\frac{n}{2} + 1\...
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60 views

Finding conditional distribution in matching ordering situation

Suppose we draw two values $x_1,x_2$ according to a CDF $F$. Independently, we draw another two values $y_1,y_2$ according to another CDF $G$. Both $F$ and $G$ has support $[0,1]$. Among those four ...
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Question about asymptotic distribution of order statistics.

Let $U_i$'s be i.i.d. Uniform$(0,1)$ random variables, and let $X_n = min\{U_1, \dots , U_n \}, Y_n = max\{ U_1 , \dots , U_n \}, Z_n = Y_n - X_n $. Then find asymptotic distribution of $n(1-Z_n)$. I ...
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12 views

Max of Dependent RVs vs CDF

Suppose I have $N$ random variables $X_{1}, X_{2}, \ldots, X_{N}$ and they are dependent in some way. Is there anything we can say about the values of their CDF $P(X_{1} \leq \gamma, X_{2} \leq \gamma,...
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2answers
77 views

Show that $T(X)=(R,V)=\left( X_{(n)}-X_{(1)},\frac{X_{(n)}+X_{(1)}}{2} \right)$ is a minimal sufficient statistic for $\theta$.

Let $X_{1}, X_{2}, ..., X_{n}$ be a random sample from $\text{Uniform}(\theta,\theta+1)$ population with $-\infty<\theta<\theta+1< \infty$ show that $T(X)=(X_{(1)},X_{(n)})$ is a minimal ...
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1answer
38 views

Probability of Maximum of Subset of Multinomial Random Variable

Suppose that I have a random variable $X \sim \text{Multinomial}(N, M, \mathbf{p})$, where $\mathbf{p} = \frac{1}{M} \mathbf{1}$, $N$ is the number of trials, and $M$ is the number of bins. I want to ...
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0answers
32 views

Obtaining the likelihood function from cdf in paper

I'm studying the Hill's estimator in Here and I'm trying to find out how he got his result. So, given $Z_1,\dots, Z_k$ a sample from a cdf $F(x)$ and $Z_{(1)}\geq\dots \geq Z_{(k)}$ the order ...
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1answer
95 views

Convergence in distribution for joint vector and maximum

I know that $X_n\xrightarrow{d} X$ and $Y_n\xrightarrow{d} Y$ does not necessarily imply $\max(X_n,Y_n)\xrightarrow{d} \max(X,Y)$, where $\xrightarrow{d}$ is convergence in distribution. However, if ...
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0answers
21 views

Probability of the maximal marginal of a multinomial distribution

So let us suppose I have a set of random variables described by the multinomial distribution: $$ f(x_{1}, x_{2}, \ldots, x_{M}) = \frac{N!}{x_{1}! x_{2}! x_{M}!} \prod_{k=1}^{M} p_{k}^{x_{k}} $$ I ...
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0answers
44 views

Question about order statistics. [duplicate]

Let $U_i$ be i.i.d Uniform(0,1) random variables, and let $X_{(1)},\dotsb, X_{(n)}$ the order statistics of the random variables. Then find distribution of $Y = U_{(n)} - U_{(1)}$. I used two methods ...
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45 views

Independence of Order Statistics

Let $X$ be the minimum and $Y$ the maximum of two independent, nonnegative random variables $S$ and $T$ with common continuous density $f$. Let $Z$ denote the indicator function of the event $(S < ...

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