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.

Filter by
Sorted by
Tagged with
1
vote
0answers
5 views

Ranks of a set of maximal invariant statistics

Let $\{\mathbf{x}_l\}_{l=1}^L$ be a set of i.i.d. (continuous) random vectors sharing the same density $p_X$. Let $\{Q_l\}_{l=1}^L$ be a set of (positive and continuous) random variables representing ...
0
votes
0answers
45 views

How to compute the joint density of three ordered random variables? [duplicate]

I am trying to understand ordered statistics and I don't understand how I can compute the joint density of three ordered random variables. Assume ${X_1,...,X_n}$ are i.i.d with a distribution $F(x)$ ...
1
vote
1answer
39 views

How to graph distribution of Order statistics?

Is there a software that can graph the pdfs and Cds of an arbitrary number of order statistics or is there some code such software? How to do it? I'm trying to understand the distribution of order ...
0
votes
0answers
8 views

Estimating median from quantized data

I have quantized measurements and I would like to estimate the median of the underlying distribution. Can I do better than taking the median of the quantized measurement?
0
votes
0answers
22 views

Hypothesis test for $\theta$ in a $uniform(0,\theta)$ distribution

Suppose $X_1, \ldots, X_n$ is a random sample drawn from a $uniform(0,\theta)$ distribution. We will test $H_o: \theta = 3$ and $H_a: \theta = 2$. Use the test statistic $X_{(n)}$ and reject $H_o$ if ...
0
votes
0answers
31 views

Expectation of the k-th order statistic of a standard Gaussian sample

Let $(X_1,\dots,X_n)$ be independent random variables with common distribution $\mathcal{N}(0,1)$. The order statistics satisfy $X_{(1)} \leq X_{(2)} \leq \dots \leq X_{(n)}$. I am interested in the ...
2
votes
1answer
23 views

Is Dixon's Q statistic ancillary for normal data?

Dixon's Q statistic is the ratio of the "gap" between an outlier and the nearest value, over the range of the data. I would like to know is if this is ancillary to the parameters of the normal ...
3
votes
0answers
37 views

Asymptotic behaviour of order statistic $x_{(n-k+1)}$ when k is $n^{\alpha}$

I am interested in the asymptotic behavior of the top k-th order statistic $x_{(n-k+1)}$ from n i.i.d. standard normal samples, when k is $n^\alpha$ where $\alpha\in (0,1)$. I just wonder if we can ...
1
vote
2answers
34 views

What is the probability that the student who finishes last takes over twice as long as the student who finishes first?

Three students independently attempt to solve a statistics problem. Assume the times taken (in minutes) by each student to solve the problem are identically distributed on $U(0,30)$. What is the ...
2
votes
0answers
15 views

Double Integral involving Beta Functions (about Pareto Distribution)?

I have tried evaluate $(m_i,m_j)$th product moment of $X_{(i)}$ and $X_{(j)}$ order statistics of Pareto Distribution, that is $E[X_{(i)},X_{(j)}]$, where $i\le j$ , $X_1,X_2,...,X_n$ i.i.d. from ...
0
votes
0answers
48 views

PERMANOVA test and its assumptions

I have collected data based on a 5 point Likert scale (very low, low, neutral, high, very high) on factors considered by individuals before making investment decisions. There are five factors (D.Vs) ...
0
votes
0answers
23 views

Identifying interaction terms in nonlinear data whose underlying function may be unknown

This is the data that I am using to frame and ask this question (code written in R): ...
3
votes
0answers
29 views

Distribution of the minimum and maximum order statistics under a partial ordering

Let ${\bf{x}} = (x_{11},x_{12}, x_{13},\ldots,x_{nm})$ and $f({\bf{x}})\propto 1_A({\bf{x}}) \prod_{i,j} {x_{ij}}^{\alpha-1} (1-x_{ij})^{\beta-1}$ for $i = 1,\ldots, n$ and $j = 1, \ldots, m$. That ...
2
votes
0answers
52 views

How to fully estimate a probability density from only a sample of minimum values?

We are given a sample $\{ z_i \}$, $i=1,2,\ldots,N$, such that each value $z_i$ corresponds to the minimum of $n$ random variables $x$, i.e., $z = \min \{ x_1, x_2,\ldots,x_n \}$. By means of ...
1
vote
1answer
52 views

Coin flipping with revision of probability

Suppose two coins $x$ and $y$ have "H" heads probability $p_x$ and $p_y$. $p_x$ and $p_y$ are independently drawn according to a uniform distribution over $[0,1]$. Say that we know $p_x\geq p_y$. So, ...
2
votes
0answers
21 views

Testing against non-dominance of discrete distributions

I have two discrete distributions $A$ and $B$ with independent draws. What tests can I use against $H_0$: $A$ does not first-order stochastically dominate $B$?
3
votes
1answer
72 views

Distribution of ranges of normally distributed variables

I have four independent variables $x_i$, each of them normally distributed with $\mu = 0$ and $\sigma = 1$. What is the distribution of the range of the four variables, i.e., $\max(x_i) - \min(x_i)$? ...
5
votes
0answers
141 views

Probability of selecting maximum in bivariate correlated order statistics?

In a testing, ranking, or selection scenario, we have samples of size n where a measurement is correlated 0<r<1 with some second variable of interest; they are bivariate normally distributed. We'...
4
votes
0answers
32 views

How can a probability densitiy be estimated based on the maximum entropy principle, with constraints in the order statistics?

Let's say we are given a sample $\{ z_i \}$, $i=1,2,\ldots,N$, such that each value $z_i$ corresponds to the minimum of $n$ random variables $x$, i.e., $z = \min \{ x_1, x_2,\ldots,x_n \}$. The ...
1
vote
0answers
17 views

How do I find the expected values and covariance matrix of the order statistics of iid random variables sampled from the standard normal distribution?

Recently I was trying to learn more about Normality tests and came to know about Shapiro-Wilk test for Normality. I understood most part of it but one thing I didn't understand is that how do I find ...
2
votes
2answers
51 views

What is the variance of the mean, conditional on being between two order statistics or quantiles?

Suppose I have a simple random sample drawn from a population with a known distribution on some population characteristic like height or income, with probability density function (pdf) $f(x)$. Order ...
0
votes
1answer
26 views

Is my interpretation of the results of my ordered logistic regression right?

I am currently writing my master's thesis. To analyse the results of my survey, I conducted an ordered logistic regression using Stata. My outcome variable is whether someone wants to start a business ...
0
votes
0answers
13 views

what are “adversarial distributions”

I ran into a paper that talked about some k selection algorithms working better or worse with "adversarial distributions", full excerpt here: 1.3. Organization In this paper we present three ...
5
votes
2answers
80 views

What is the distribution of a bivariate normal component conditional on the max of the other component?

Let $n$ be a large integer, and consider two independent multivariate Gaussian $n$-vectors $x, z$ with $x\sim\mathcal{N}\left(0,I\right),$ and $z\sim\mathcal{N}\left(0,\sigma^2 I\right)$. Let $y=x+z$. ...
2
votes
1answer
46 views

Spacings between continuous uniform random variables

Let $U_1, \cdots, U_n$ be $n$ i.i.d continuous uniform random variables on $(0,1)$ and their order statistics be $U_{(1)}, \cdots, U_{(n)}$. Define $D_i=U_{(i)}−U_{(i−1)}$ for $i=1, \cdots, n$ with $...
1
vote
1answer
29 views

Expected value of CDF evaluated at sample min/max, with respect to sample size

I've been using simulation to investigate the behavior of cumulative density function evaluated at extreme values of a sample. Functionally, my study has $n$ instrumented fish migrating downstream ...
3
votes
1answer
47 views

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 ...
1
vote
1answer
52 views

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 ...
1
vote
0answers
35 views

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 ...
3
votes
1answer
33 views

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 ...
2
votes
1answer
62 views

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 $$X_{(n)} \hat= \max \{X_1,X_2,\ldots,X_5\}\quad\text{ and }\quad \bar X\hat=\sum^n_{i=1}{X_i}$$ ...
2
votes
0answers
25 views

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=...
1
vote
1answer
36 views

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) ...
2
votes
1answer
126 views

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 $b(\theta)$...
5
votes
2answers
171 views

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)} ≤ ...
1
vote
1answer
39 views

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?
7
votes
1answer
171 views

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)}/...
0
votes
0answers
22 views

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 ...
0
votes
1answer
153 views

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 ...
1
vote
1answer
74 views

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 ...
0
votes
1answer
39 views

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 ...
2
votes
0answers
94 views

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: $$...
8
votes
2answers
251 views

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 ...
0
votes
0answers
55 views

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 ...
5
votes
1answer
309 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 ...
0
votes
0answers
19 views

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 ...
-1
votes
1answer
75 views

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 ...
1
vote
1answer
40 views

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 ...
0
votes
0answers
45 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 ...
2
votes
1answer
37 views

$k$-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_{(...