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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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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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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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Relation between a Gamma prior and posterior in terms of paramters [migrated]

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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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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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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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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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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What is the rate of growth of $M_n := \max_{1 \le i \le n} U_i^{(n)}$, where $U_i^{(n)} \sim \operatorname{Uniform}[0,n]$?

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 ...
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Chi square test or Exact test

This data is about cases with convulsive disorders. Among the cases there were $82$ females and $118$ males. At the $5\%$ significance level, test the hypothesis that a case is equally likely to be of ...
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Conditional expected value of order statistic

Let $\theta_1,\dots,\theta_N$ be a collection of independent RVs, ditributed uniformly on $[0,1]$. Further let $\theta^{(r)}$ be the $r$th order statistic where $\theta^{(1)}\leq\dots \theta^{(r)}\leq\...
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maximum of variables conditioned on geometric distribution

Let $X_1$, ... be independent random variables with the common distribution function $F$, and suppose they are independent of $N$, a geometric random variable with parameter $p$. Let $M = \max(X_1,...,...
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Random sampling with variance minimising estimator

Consider a population $\{1,\ldots, N\}$ and a set of properties represented by numbers $\{ a_1,\ldots, a_N\}$. Now you pick $n$ out of $N$ without laying back, such that you will get a vector like $$\...
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Order statistic of i.i.d exponential($\lambda$) random variables, $X_{(n, k_n)}$ convergence in probability

Suppose that $X_1,X_2$,....are iid from exponential($\lambda$).For n $\geq$ 1, let $X_{(n,1)}\le X_{(n,2)}\le X_{(n,3)}\le.......\le X_{(n,n)}$ be the order statistics of $X_1,X_2....X_n$. Suppose ...
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Expectation of k-th order statistic of Negative Binomial Distribution

Let $ X_1, X_2,...$ be i.i.d $NB(k,q)$. I am interested in calculating the expectation of their k-th order statistic $X_{k:n}$. From my understanding of order statistics, the CDF of $X_{k:n}$ is given ...
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estimating the sum of Cauchy variables by its biggest terms

Let $(X_1,...,X_n,...)$ be Cauchy variables. Let $(Y_{1,n},...Y_{n,n})$ be the same variables as $(X_1,...X_n)$, ordered by decreasing absolute value. The sum $\sum_{i=1}^n Y_{i,n}/n $ converges in ...
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Finding probability of a random variable dependent on other random variables

Let H,C G, D be three independent random variables where H belongs to Gamma distribution and C follows an exponential distribution, G and D belong from similar distributions with different mean. Rest ...
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1answer
24 views

Inequality between Expectation and Quantil

For a sample of independent observations $X_1,X_2,...,X_n$ on a continuous distribution $F$, let the ordered sample values be $X_{(1)},X_{(2)},...,X_{(n)}$. From the theory of order statistics, the ...
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35 views

Most powerful test for discrete uniform Neyman Pearson Lemma

This is with regard to the question whose link is given below- Most powerful test for discrete uniform I obtained the most powerful test function as- $\phi(x)$ = 1 if X < 3 ; ...
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202 views

Why is the maximum of i.i.d. Gaussians asymptotically $\sqrt{2 \log n}$?

Assuming that $\xi$ is bounded (as a function of $x$?), the claim is that given the equation: $$\xi \frac{\sqrt{2\pi}}{n} = \frac{1}{x} e^{-\frac{x^2}{2}} \left( 1 + O\left(\frac{1}{x^2} \right) \...
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Normal order statistic. Expected Values (Fisher-Yates)

We have $\xi_{i} \sim \mathcal{N}(0,1)$ iid, $i= 1,\dots N$. We look for $a_i = \mathbb{E} \xi_{i:N} $, where $\xi_{i:N}$ is order statistic. I already checked in R, that ...
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Expected value of order statistics $X_{(i+1)}$ conditional on $X_{(i)} < t$

Consider $N$ random variables $X_1, X_2, \ldots, X_N$ that are i.i.d. distributed according to some cumulative distribution function $F$. Assume we receive a signal that says that $n$ number of the ...
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1answer
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How do I prove the measurability of the $j$-th smallement element mapping function?

Related : How do we formally define "j-th smallest element"? Let $F_j:\mathbb{R}^n\rightarrow \mathbb{R}$ be a $j$-th smallest element picking function. How do I prove that $F_j$ is Borel ...
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Probability and Statistic

I have a question about how to calculate the confidence interval. The problem is: I have a model which gives the probabilities of 4 genotypes AB, Ab, aB and ab as (1/4)(2 + theta), (1/4)(1 - theta), (...
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expected value of conditional order statistics

Let X1, X2, . . . , Xn be i.i.d. random variables and let X(1),X(2), . . . ,X(n) be the order variables. Show: E(X1 | X(1),X(2), . . . ,X(n)) = $\sum_{k=1}^n \frac{Xk}{n}$ This is a question from a ...
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Expectation of sample mean, given maximum and minimum order statistics [duplicate]

Let $X_1, · · · , X_n$ be i.i.d. $\mathrm{Uniform}[\alpha, \beta]$, where $\alpha$ and $\beta$ are unknown. Show that $$ E(X_\ast|X(1), X(n)) = \frac{X(1) + X(n)}{2} $$ where $X_\ast$ is the ...
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stochastic ordering of counting processes/vectors

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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Concentration of highest order statistic of bounded distribution

I have a question about how the highest order statistic of a bounded distribution concentrates around the upper bound of the distribution. Intuitively, as sample size increases, the statistic should ...
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Absolute moments of order statistics

I had the following question on a test some time ago and was unable to find a solution: Let $X_1,\dots,X_n$ be i.i.d. with distribution function $F$ such that $E|X|^\alpha<\infty$ for some $\alpha&...
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Uniform distribution, chi square test

What test or procedure can I use to determine the best estimate $\alpha\in [0,1]$ whether given $N$ numbers come from the uniform distribution in the interval $[0,\theta]$ for a given $\theta>0$? I'...
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(Self-Study) UMVUE of the mean of a normal distribution [duplicate]

Let $X_1,...,X_n$ be a random sample from normal(θ,1). Is there an UMVUE of $θ^2$ here? $X^2-1$ is an unbiased estimator of $θ^2$. First thing that came to my mind is to use Lehmann-Scheffe Theorem. ...
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57 views

Minimal sufficient statistics for Cauchy distribution

I'm trying to find the minimal sufficient statistics for a Cauchy distributed random sample $X_1,...,X_n$, here \begin{equation} f(x|\theta) = \frac{1}{\pi[1+(x-\theta)^2]} \end{equation} I begin by ...
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Using independence to compute exponential order statistics

I am in a stochastic processes course, and I am trying to apply a result about the minimum of iid exponentials. Here is the result: Let $X_1, \ldots, X_n$ be independent exponential random ...
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Conditions for “$\mathbb{E}\{X|Y = k\}$ is monotone in $k$ if and only if $\mathbb{E}\{Y|X = k'\}$ is monotone in $k'$” to hold?

Are there conditions on the joint distribution f(X,Y) for the following statement to hold: "$\mathbb{E}\{X|Y = k\}$ is monotone in $k$ if and only if $\mathbb{E}\{Y|X = k'\}$ is monotone in $k'$"? I ...
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Find the probability that a rv is equal to its order statistics $P(X_i = Y_j)$

Let $X_i$, $i=1...n$ be independent non identically distributed random variables and let $Y_i$ be its order statistics. I want to find the probability that $X_i$ is equal to $Y_j$ for each i, j $i....
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Expectation of exponential variable

What approach do I have to use in order to solve the following integral: $$\sum_{i=1}^N\int_0^\infty \log_2(1+g_ip_i)dg_i, $$ where $\mathbf{g}$ has an exponential pdf. And $$g_1\geq g_2\geq , \dots,...
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Intrinsic Bayes factor

Here on the page 238 in the formula: the arithmetic intrinsic Bayes factor, $$B^A_{10}=\frac{1}{L}\sum_{x(\ell)}B_{10}^{(\ell)}=B_{10}(x)\frac{1}{L}\sum_{x(\ell)}B_{01}\left(x_{(\ell)}\...
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Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution [duplicate]

Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution having p.d.f. f(y) = e^(-y), 0 < y < ∞, zero elsewhere. Show that Z1 = Y(2) and Z2 = ...
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Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution having p.d.f.

Help me to solve this problem please.. Let $Y_{(1)}, Y_{(2)}, Y_{(3)}, Y_{(4)}, Y_{(5)}$ denote the order statistics of a random sample of size 5 from a distribution having p.d.f. $f(y) = e^{(-y)}, 0 ...
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22 views

Finding a fair weighting for “most popular responses”

apologies if this has been asked before - I am not sure how pose my question in a searchable way! I want to collect votes on people's "favorite album" for a band, and then I want to show the band's ...
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64 views

Fundamental theorem of calculus on double integrals

The problem: Let $ X_1,X_2,\cdots ,X_n $ be a random sample from the uniform distribution with pdf $f(x;\theta _1,\theta _2)=\frac{1}{2\theta _2} , \theta _1-\theta _2<x<\theta _1+\theta , $ ...
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1answer
79 views

Joint Distribution of Uniformly Distributed Independent Random Variables

The problem is as follows: Suppose we have have independent random variable $Y_1, Y_2, ... ,Y_n$, and they are uniformly distributed over the closed interval [0,1]. If $V$ and $W$ are the smallest ...
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2answers
41 views

Transformations of Order Statistics

Consider the ordered sample $X_{(1)} < X_{(2)} < X_{(3)}$ from a distribution with PDF $f_X(x) = 2x, 0<x<1$. Show that $Y_1 = \frac{X_{(1)}}{X_{(2)}}$, $Y_2 = \frac{X_{(2)}}{X_{(3)}}$ and $...