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

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if $f(x) = a exp(-a (x-b))$ . Find sufficient statistic for b

Intuitively I feel that the answer should be $\min(X_1,X_2,\ldots,X_n)$ where $X_i$'s are iid, but I don't know how to prove it.
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Finding more information after finding MLE with Indicator functions.

Ex: $X_1 , X_2 , ... , X_n$ ~ $U(-\theta, \theta); f(x; \theta) = \frac{1}{2\theta}; -\theta \leq X \leq \theta; \theta > 0$ I believe this is the correct approach to finding the MLE in this ...
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Limiting distribution of $nF[X_{(2)}]$?

Hello. I am trying to work on a problem above. What I know so far is that $$\begin{align} nF_{Y_2}(y)&=n\left([1-F(x)]^n+nF(x)[1-F(x)]^{n-1}\right)\\ &= n[1-F(x)]^{n-1}[1+(n-1)F(x)] \\ \...
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Expected value of a minimum and maximum of a collection of independent random variables

Suppose I have two independent random variables $X$ and $Y$ with probability density functions $f_X(x)$, where $0 \leq x \lt a~$, and $f_Y(y)$ with $0 \leq y \lt b$. Let $T = \min\{X, Y\}$ and $W =...
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Computing $\mathbb{P}\{X_{(1)}+X_{(2)} > X_{(3)}\}$ where $X_1,X_2,X_3$ are i.i.d $U(0,1)$

Suppose that $X_1 , X_2 , X_3$ are independent $U (0, 1)$-distributed random variables and let $(X_{(1)} , X_{(2)} , X _{(3)} )$ be the corresponding order statistic. Compute $\mathbb{P}\{X_{(1)}+X_{(...
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Compute $P(X_{(2)} ≤ 3X_{(1)})$ by using the integration technique

Suppose that $X_1,X_2,X_3.X_4$ are independent $U\in(0,1)$-distributed random variables and let $(X_{(1)}X_{(2)}X_{(3)}X_{(4)})$ be the corresponding order statistic. Compute $P(X_{(2)} ≤ 3X_{(1)})$ ...
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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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In how many positions is one order statistic vector greater than another?

Consider two different samples $x_1$ and $x_2$ of size N from a continuous uniform distribution over $[0,1]$. Now consider the sorted values $z_1$ and $z_2$ of those samples, also known as the order ...
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Conditional distribution of the nth order statistic for a $Unif(0,1)$ population [duplicate]

We have $X_1,...,X_n\sim unif(0,1)$ and we are given the value of the first $n-1$ order statistics: $X_{(i)}=s_i$, $i=1,2,...,n-1$ ($s_1<s_2<...<s_{n-1}$). Using this information, we want to ...
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Spearman's Correlation Coefficient for Bivariate Normal Distribution

Referring to the answer here https://stats.stackexchange.com/a/66617 It is written that $\rho_s(X_1,X_2) = \rho(F_1(X_1),F_2(X_2))$ My Questions are :- Is that forumla correct? Because I am not ...
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Average of maximum among order statistic

We have G groups. Each group has $M$ (i.i.d) variables $X_{g,1}$,...,$X_{g,M}$ following the exponential distribution exp(-$\mu$) ($\mu$ is a constant). Focus on each group g (g=1,...,G), we take the ...
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How to get probable rank of a value based on statistical data?

Lets say I have the following data: Total number of elements Maximum value Minimum value Arithmetic Mean Mean Deviation Now if I am given an element already present in table, what is the best way to ...
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Probability of being in the kth place

Suppose we have $N$ indenpendent normal random variables, $X_1, \ldots, X_N$. Suppose $X_i \sim N(0, \sigma_i^2)$ for all $i$. Suppose $X_{(k)}$ is the $k$th order statistic. Then what is $Pr(X_i = X_{...
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How to derive the distribution of mean estimator based on ranked set sampling from normal distribution

I am studying about ranked set sampling from normal distribution. From wolfe,2004 the joint pdf of RSS is $$f_{1,2,...,n:n}(x_1,x_2,...,x_n)=\prod_{i=1}^nf_{i:n}(x_i)$$ where $$f_{i:n}=\frac{n!}{(i-...
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Show that $X_{(1)}$ and $X_{(2)}−X_{(1)}$ are independent, and determine their distributions.

Let $X_1$ and $X_2$ be independent, $\text{Exp}(a)$-distributed random variables. Show that $X_{(1)}$ and $X_{(2)}−X_{(1)}$ are independent, and determine their distributions. Although it looks like ...
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Compute the correlation coefficient $r(X_{(1)},X_{(3)})$ .

I got this problem where: The random variables $X_1, X_2,$ and $X_3$ are independent and $Exp(1)-$ distributed. Compute the correlation coefficient $r(X_{(1)},X_{(3)})$ . I know through research ...
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Compute $P(X_{(1)} + X_{(3)} \le 1),$

Any Idea why I keep getting $3/4$ ? Let $X_{1}, X_{2}, X_{3},$ be independent, $U(0, 1)$-distributed random variables and $X_{(1)}, X_{(2)}, X_{(3)}$ be the corresponding order variables. Compute (a)...
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probability that a random variable is greater than a limit in given ordering of random variables

I am currently working on a modified version of the classic greedy algorithm for the 0/1 knapsack problem. Suppose that one has $N$ items with given weights and profits that are iid random variables ...
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Finding PDF of Range of Order Statistic

Let $Y_1, Y_2,..., Y_n$ denote a random sample from the uniform distribution $f (y) = 1, 0 ≤ y ≤ 1.$ Find the probability density function for the range $R = Y_{(n)} − Y_{(1)}$. First Attempt: ...
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Express d.f. of $Y = \min \{ X1,X2 \}$ in terms of joint d.f.

Example: Express d.f. of $Y = \min \{X1, X2\}$ in terms of joint d.f. $$H(x_1,x_2) = P(X_1 \le x_1, X_2 \le x_2)$$ In this exercise, they have an answer $P(Y \le x)= 1 - H(x,\infty) - H(\infty, x) + ...
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Partial permutation of time sequence data that keep order of events

Suppose you have sequence S of N elements that are descending ordered by time. How many ways can you take K element subsets from S preserving time descending ordering? example for sequence S={A,B,C,D,...
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Statistically Ranking Data

We are sampling the rates of $n$ elevators rising up floors in order to see which elevators are the fastest, and which are the slowest. Say we have $k$ time samples from each elevator, $S_1, S_2, ... ,...
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27 views

Order statistics for uniform random variables

I am trying to develop some intuition for problem 4.6.3a in Pitman (1993) (self study not HW) and need some help confirming my solution to a simpler analogous problem: Simpler analogous problem: Let $...
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1answer
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Find $P(|Y_{(1)}-0.5| < 0.3)$.

Assume that $Y_1,Y_2$ ~ $Uniform(0,1)$. Assume that $Y_1,Y_2$ are independent. Find $P(|Y_{(1)}-0.5| < 0.3)$, with $Y_{(1)}$ = $min{(Y_1,Y_2)}$. I am currently stuck on this question while ...
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Density Function of Series Connection

The random lifetime of a certain electronic component is given by- $ f_Y(y) = \dfrac{1}{a} e^{-(y/a)}, y>0 $. Two such components are connected in series. The system fails when the first component ...
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finding the pdf and expected value of first order statistic

Let $Y_1,Y_2,...,Y_n$ be independent and identically distributed with the following probability density function $f(y) =4(1−y)^3$ for $y$ between 0 and 1 (a) Find the probability density ...
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How to define closeness measure between a matrix and its permuted smoothed version.

I have a matrix $A$ with $n$ rows and $2$ columns. I want to smooth (let's say moving average) each column resulting in a smoothed matrix $B$ with $p$ rows and $2$ columns with $p \lt n$. I can smooth ...
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Self-study Order Statistics

So I got this exercise from a book and I'm confused by a statement they made. Example: In a 100-meter Olympic race, the running times can be considered to be $U$~$(9.6, 10.0)$-distributed. Suppose ...
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STA 3033 - Tchebysheff’s theorem

I'm learning how to apply the theorem and I saw the example below and I'm not sure if that person got it wrong or it's just like that: The time that takes to complete a certain type of construction ...
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Show that the statistics $T(X_1,\ldots,X_n)=(X_{(1)},X_{(n)})$ is a sufficient statistics for $\theta$

Let $X_1,\ldots,X_n$ be a random sample from $U(\theta-\frac{1}{2},\theta+\frac{1}{2})$. Show that the statistics $T(X_1,\ldots,X_n)=(X_{(1)},X_{(n)})$ is a sufficient statistics for $\theta$. Can ...
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How to find the expected value of the first order statistic

There are $Y_1, Y_2, \dots ,Y_n$ which are identically and independently distributed with pdf $4[(1-y)^3]$ for $0<y<1.$ We were asked to find the pdf of the first order statistic of which I got $...
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Difference of order statistics between independent samples.

Let $F$ be some cumulative distribution function. Suppose we draw independently two i.i.d. $n$-samples with distribution $F$, say $(X_k)_{1\leq k \leq n}$, $(Y_k)_{1\leq k \leq n}$. Denote $X_{(i)}$, $...
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What is an intuitive application of estimators?

So we're currently studying Estimators and we just proved Cramér-Rao's inequality and that when it is an equality, then whatever estimator we have is a unique MVUE. All of this to me just sounds like ...
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Limiting Distribution on order statistics

Here is the question: Find the limiting distribution of the quartile ratio defined as $\frac{X_{(3n/4):n}}{X_{(n/4):n}}$ (the third quartile divided by the first quartile)for the exponential ...
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Likelihood Function Given Maximum of data, but not actually data points

I was just wondering how I would go about creating a likelihood function if I have a $N( \theta,1)$ distribution and know $x(n)$ the maximum of n observations, but not the actual observations ...
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One modification of percentiles: is it known? What is the name of it?

Suppose that we are given a sample of already ordered points $x_1\geq ...\geq x_n$. We add them all and form the value: $X=\sum_{i=1}^N x_i$. Suppose that $0<p<1$. Let us denote $X_p$ the ...
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How to find the critical value of a test given that the test statistic is derived from a normal distribution (and is an order statistic).

Let $X_1,...,X_n$ be i.i.d from a normal distribution with expectation $\theta$ and variance 1. The test for testing $H_0:\theta=\theta_0 $ v.s. $H_1:\theta=\theta_1$ where $\theta_0<\theta_1$ has ...
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Derive the Hajek pojection of $T_n$.

Let $X_1, \dots , X_n$ i.i.d. copies of $X$ with distribution $F$ and density $f$. Let $(X_{1:n}, \dots , X_{i:n}, \dots , X_{n:n})$ be the order statistic. For a given $p \in (0, 1)$ consider the ...
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Question on non-parametric statistics

How to estimate quartile function Q(p)=inf{x∣F(x)>p} and survival function F(x):=1−F(x) by non-parametric estimation method
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Find $P(2Y_{(1)} < Y_{(2)})$ of a Uniformly Distributed Random Variable

Denote $Y_{(1)} = \min(Y_1,Y_2)$ and $Y_{(2)} = \max(Y_1,Y_2)$. Let $Y_1$ and $Y_2$ be independent and uniformly distributed over the interval $(0, 1)$. Find $P(2Y_{(1)} < Y_{(2)})$. Attempted ...
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Order statistics, what am I doing wrong

From SOA sample 138: A machine consists of two components, whose lifetimes have the joint density function $$f(x,y)= \begin{cases} {1\over50}, & \text{for }x>0,y>0,x+y<10 \\ 0, &...
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1answer
41 views

Proving mean of sample minimum of U[0,1] is 1/(n+1) without calculus

Let $U : \mathbb{R} \times \mathbb{R} \nrightarrow \mathrm{dist}[\mathbb{R}]$ denote the parametrized family of uniform distributions where $U(a, b)$ is the uniform distribution with minimum $a$ and ...
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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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Order Statistics and High Dimension Geometry

Suppose that I have iid random variables $\mathrm U_n \sim \mathrm U(0,1)$. Then, for $\mathrm Y_m$ defined as, $$\mathrm Y_m = \min_{n \in [1,m]} \mathrm U_n$$ it is easy to compute $\mathbb{E}[\...
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60 views

Upper bound of expected maximum of weighted sub-gaussian r.v.s

Let $X_1, X_2, \ldots$ be an infinite sequence of sub-Gaussian random variables which are not necessarily independent. My question is how to prove \begin{eqnarray} \mathbb{E}\max_i \frac{|X_i|}{\...
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54 views

Proof of a Probability Problem by Induction

Dear Statisticians and Mathematicians, I am interested in proving the following lemma by induction I have shown that it holds true for $n=2$ which I don't provide its proof here. We assume it is true ...
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Mean of the product of order statistics.

Let $X,Y$ be two real valued random variables with first and second moments. Let $(X_1,\dots,X_n)$ i.i.d $X$ and $(Y_1,\dots,Y_n)$ i.i.d $Y$ be to independent $n$-samples. Denote $\left(X_{(1)},\dots,...
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A Conditional Probability Problem

I am interested in finding the following problem: Let $\tau_1$ and $\tau_2$ are ordered statistics from a set of 2 independent uniform $(0,t)$ R.V. and let $Y_1,Y_2,Y_3$ are nonnegative iid R.V. that ...
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38 views

A simpler proof for a probability problem

I am trying to prove the following lemma 2.3.5 from Stochastic Processes, Sheldon Ross, 2nd ed, page 77 which I have provided it here for convenience. The proof is provided in the book based on a ...
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3answers
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What is the rate of growth of $M_n := \max_{1 \le i \le n} U_i^{(n)}$, where $U_i^{(n)} \sim \operatorname{Uniform}[0,n]$? Is it “constant”?

On pp. 370-374 (see this previous question) of Cramer's 1946 Mathematical Methods of Statistics, the author shows that for any continuous distribution $P$, if $X_1, \dots, X_n \sim P$ are i.i.d., then ...