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
3 votes
1 answer
33 views

Conditional distances in order statistics

Assume I have $n$ points sampled independently from the uniform distribution on the unit interval. After ordering the sample I get the points $X_1, X_2, \dots X_n$ such that $X_1 \leq X_2 \leq \dots \...
user avatar
  • 337
0 votes
2 answers
129 views

The variance of the weighted median and optimal weights

The median $\tilde{\mu}$ of a sample in many ways is analogous to the sample mean $\mu$. Both are an estimate for the population median or mean respectively, and both approach a Gaussian distribution ...
user avatar
10 votes
1 answer
643 views

Suppose I have 100 integers and I sample 10 without repetition. What is the expected rank of the lowest out of 10 samples?

Suppose I have 100 integers and I sample 10 without replacement. What is the expected rank of the lowest out of 10 samples? i.e. my lowest integer in the 10-sample is kth smallest out of 100. what is ...
user avatar
0 votes
0 answers
16 views

Multinomial Logistic Regression as a latent variable model

I was reading the wiki entry for multinomial logistic regression https://en.wikipedia.org/wiki/Multinomial_logistic_regression#As_a_latent-variable_model and it states that we can view the multinomial ...
user avatar
0 votes
0 answers
22 views

What interval does the median fall into when the values of the numbers before and after the median aren't specified and n is even?

What interval does the median fall into when the values of the numbers before and after the median aren't specified, but are the last and first data points in the intervals sorrounding the median, ...
user avatar
0 votes
0 answers
29 views

Rao Cramèr Lower Bound problem

Let $X_1, · · · , X_n$ be a random sample from the uniform distribution on $[0, θ]$. I want to get the variance of the maximum likelihood estimator of $θ$ and check whether the variance decrease at ...
user avatar
  • 21
0 votes
0 answers
26 views

Show that Sample Mean and Sample Range are independently distributed for a random sample from Normal Distribution

Let $X_{1},\ldots {, X_{n}}$ be iid random variables with $X_{1} ∼ N(µ,\sigma ^{2}).$Let $\bar{X}= \sum_{i=1}^{n} \frac{X_{i}}{n}$, $R=max_{1\le i \le n} \{X_{i}\}$-$min_{1\le i \le n}\{X_{i}\}$.Show ...
user avatar
0 votes
1 answer
39 views

Finding probability of all success for in an order statistic

𝑓(𝑦) = 5𝑦^4; 0 ≤ 𝑦 ≤ 1 A group of 3 friends order small cups of soda, from the soda dispenser. If the 3 small cups are considered a random sample from the dispenser fills, find the probability ...
user avatar
0 votes
1 answer
144 views

Show that $P(X_k \le x) = P\{N(x) \ge k\}$

Suppose $x_1 ... x_n$ are the order statistics of an iid sample from a continuous distribution $F(x)$. Show that $P(X_k \le x) = P\{N(x) \ge k\}$ where $N(x)$, the number of sample values less than x, ...
user avatar
0 votes
0 answers
23 views

Understanding order of operations in formula

Can someone help me understand the order of operations for this formula? Lets say: y_estimated = 10, 14, 11 #three different estimates that will be subbed into the formula y_true= 12 R=3 I think it ...
user avatar
  • 101
0 votes
0 answers
34 views

What is the distribution of the $k^{th}$ highest value of a multivariate normal distribution

Let X be an N-dimensional multivariate normal, $X \sim N(\mu,\Sigma)$ where $\mu$ is Nx1 and $\Sigma$ is NxN. If we take a draw of $X$ from this distribution and then sort $X$ from largest to smallest,...
user avatar
  • 1
0 votes
1 answer
39 views

$P(X_{(1)}+X_{(2)}>X_{(3)})$ for order statistic

I am trying to solve this problem for when $X_1, X_2, X_3$ are independent $U(0,1)$-distributed random variables. The joint density function should then be given by $$f(x_1,x_2,x_3)= \begin{cases} 6, ...
user avatar
1 vote
0 answers
56 views

Expected value of the largest order statistic for $Uniform(\theta,2\theta)$ [duplicate]

I'm struggling to find when $X_1,\ldots,X_n \sim Uniform(\theta,2\theta)$, how the expected value of the largest order statistic is $E[X_{(n)}]=\dfrac{2n+1}{n+1}\theta$. I can find that the density of ...
user avatar
  • 3,355
1 vote
0 answers
25 views

quantiles (monotonic transformation)

I'm trying to show that, $(-X)_p = -X_{(1-p)}$, that is, the $p$ quantile of $-X$ is equal to the $1-p$ quantile of $X$ after multiplying with $-1$. The results holds when considering quantiles of a ...
user avatar
  • 1,035
0 votes
0 answers
16 views

Probability bound of the difference of order statistics for i.i.d. Gaussian random variables

I have asked a related question before (with more stronger requirement): Probability bound of the difference of order statistics for correlated and identical Gaussian random variables. Now, I'm pretty ...
user avatar
1 vote
0 answers
31 views

Probability bound of the difference of order statistics for correlated and identical Gaussian random variables

Suppose, there are $n$ identical and correlated Gaussian random variables namely, $X_1, X_2, ..., X_n$ with $X_i\sim\mathcal{N}(0,\sigma^2)$ for all $i\in\{1,2, ...n\}$. The correlation coefficient ...
user avatar
1 vote
0 answers
57 views

MAD & Median of weighted GMM

What is the median and median-absolute-deviation of a weighted GMM in terms of component mean and variance? For example, three normal distributions $A$, $B$, $C$ with means $\mu_a,\mu_b,\mu_c$, ...
user avatar
3 votes
1 answer
33 views

Averages of the two closest pairs out of a set of four observations

Four random numbers are drawn at random from a standard normal distribution. They are grouped in two pairs of closest numbers, $\{x_1, x_2\}$ and $\{x_3, x_4\}$ so that $x_1\le x_2 \le x_3 \le x_4$, ...
user avatar
5 votes
1 answer
146 views

$N \sim \text{Po}(\lambda)$ and $X_1,X_2,....,X_N$ are iid and independent of $N$, what is distribution of $Z_N = \max \{X_i\}_{i=1}^{N}$

I think the title covers most of my concerns. The distribution of the $X_i$ does not really matter, I am just experiencing difficulties in finding an expression for $$\text{Pr}(Z_N \leq x) = F(x)^N$$ ...
user avatar
2 votes
1 answer
116 views

How do Ordered Target Statistics work for CatBoost?

This question follows closely this paper . I'm trying to fully understand how Ordered Target Statistics (TS) (for CatBoost) works. E.g. the CatBoost algorithm uses this method to group categorical ...
user avatar
  • 218
2 votes
2 answers
99 views

How many samples are needed to estimate quantiles for an unknown distribution?

I'm trying to evaluate performance of a metric learning model. The model that takes labelled image inputs and maps them to vectors on an N-dimensional unit sphere. The goal of the model is to map ...
user avatar
1 vote
0 answers
27 views

Joint distribution of top order statistics of two independent random samples of Pareto distribution

Suppose $X_1,...,X_n$ and $Y_1,...,Y_n$ are all independent copies of a standard Pareto random variable. For each of the 'two' random samples we can denote the order statistics $X_{n:n} \geq X_{n-1:n} ...
user avatar
  • 779
3 votes
1 answer
202 views

What is the covariance matrix of the normal order statistics?

I would like to test if a sample comes from a standard normal distribution. I want to do that by sorting the sample values, and measuring the Mahalanobis distance to the expected order statistics from ...
user avatar
0 votes
0 answers
35 views

Total Time on Test (TTT) Statistic

Let $X_{(1)}, X_{(2)}, ..., X_{(n)}$ denote an ordered sample of size $n$ from a life distribution. Let $T_n$ be the total time spent on test by the $n$ sample units until the failure of the longest ...
user avatar
  • 41
3 votes
1 answer
104 views

What exactly are order statistics?

Suppose $X_1,X_2,X_3.....X_n$ are random sample taken from a population. Then Y(1)<Y(2)<Y(3).....<Y(n) are called order statistics written in increasing order by magnitude where: Y(1)=minimum(...
user avatar
  • 33
1 vote
1 answer
178 views

what is the expectation of minimum order statistics? [closed]

I want the expectation of minimum order statistics and the variance of minimum order statistics
user avatar
0 votes
0 answers
46 views

Order statistics is minimal sufficient statistics for unknow density function

I'm trying to prove the problem, but there is a problem on definition of term. The theorem that I use to prove it is However, what exact meaning of "family of densities ~ all have common support&...
user avatar
0 votes
0 answers
32 views

Conditional distribution of order statistic of p-values

Suppose we have independent Unif(0, 1) random variables $\{p_1,..., p_n\}$. We sort them by $p_{(1)} \leq p_{(2)} \leq ... \leq p_{(n)}$. For $i \geq 2$ and $x \geq t$, I would like to compute $$f_i(x)...
user avatar
1 vote
0 answers
34 views

What is the covariance of a thresholded random vector

I have a random vector $X = [X_1, X_2, \dots, X_n]^T$. I top-$k$ threshold it so that I get a new vector $Y = [Y_1, \dots, Y_n]$ where $Y_i = X_i$ if $X_i$ is in the top-k entries of $X$, and $Y_i=0$ ...
user avatar
  • 11
0 votes
0 answers
31 views

If you take the maximum value from two random draws from a normal distribution, what is the mean and standard deviation of the resulting distribution? [duplicate]

Let's say I draw two numbers from a normal distribution with mean 50 and sd 10, and take the maximum of those two numbers. If I generate a distribution of that maximum, what are the resulting ...
user avatar
4 votes
1 answer
101 views

Modeling a time series of ordered vectors

I have a series of ordered vectors, $\pmb{x}^o(1), \ldots, \pmb{x}^o(n)$. Here, $\pmb{x}^o$ means the ordered vector of $\pmb{x}$. For example, if $\pmb{x} = (2,5,1)^\top$, then $\pmb{x}^o = (1,2,5)^\...
user avatar
  • 743
0 votes
0 answers
71 views

Unbiased estimator of minimum order statistic

Let $X_1,X_2$ and $X_3$ be a random sample taken from a continuous population with distribution function F. Consider the function $E(X_{1:3})$ , where $X_{1:3}$ is the minimum order statistic. Can $$...
user avatar
3 votes
1 answer
71 views

How to calculate $\int_{-\infty}^{\infty} \Phi(y+c)^{k-1}d(\Phi(y))$

How to calculate $\int_{-\infty}^{\infty} \Phi(y+c)^{k-1}d(\Phi(y))$?When I check the list of integrals of Gaussian functions, I only find k-1=2.
user avatar
  • 41
1 vote
1 answer
57 views

Joint PMF of two order statistics with discrete parent distributions

Let $X_1, X_2$ be i.i.d from a discrete distribution with finite support with cumulative distribution $F(x)$ and probability mass function $f(x)$. Let $X_{1:2}$ and $X_{2:2}$ represent the order ...
user avatar
  • 45
0 votes
0 answers
16 views

Largest order statistics of non-identical distributions when extra information is available

Suppose we have two independent draws, one from a distribution $F_1$ and the other from a distribution $F_2$. The two distributions have the same support, say $[0, 1]$. The distribution of the largest ...
user avatar
  • 381
0 votes
0 answers
21 views

Estimator for a particular statistic involving Order Statistic

Let$ X_{1}, X_{2}, \cdots, X_{n} $ be a random sample from a continuous life distribution $ F $ be with survival function $ \bar{F},$ density $ f $ and finite mean $ \mu. $ While doing some ...
user avatar
2 votes
1 answer
48 views

Prove that $\mathbb E [T_n(x) ~ T_n(y)] = nF_X(x) + n(n - 1)F_X(x)F_X(y)$, for $x<y.$

Let $X_1, \cdots,X_n $ be iid random variables with distribution $F. T_n(x)$ denotes the number of elements $\le x; x \in \mathbb R$. Prove that $\mathbb E [T_n(x) ~ T_n(y)] = nF_X(x) + n(n - 1)F_X(x)...
user avatar
  • 213
0 votes
0 answers
37 views

Proving convergence in distribution between order statistics and quantiles

The random variable of continuous type $X$ has CDF $F(x)$ $X_1, X_2, \cdots, X_n$ is a random sample of size $n$ from the distribution of $X$ Function $h(y)$ is defined as $h(y) = F^{-1}{(1-e^{-y})}I_{...
user avatar
2 votes
4 answers
105 views

Approximate distribution of a complicated function of a random variable

If $X$ is a random variable cdf $F(x)$ such that $F$ is invertible then we have the standard method of finding the pdf of any function of $X$, say, $\sin(X) $ or $ X^3+1 $.However,in many situations ...
user avatar
0 votes
1 answer
176 views

Distribution of Maximum of Geometric random variable

Let $X_1, X_2, ... X_n$ be geometric random variables with density $$P(X=x)=pq^{x-1} , x=1,2,3,...$$ What will be the distribution of $Y=\max(X_1, X_2, ..., X_n)?$ Will the distribution of $Y$ be ...
user avatar
3 votes
0 answers
46 views

Estimate population mean from "best of N" samples

If I have a data set for which I know all measurements represent the largest of N observations, is there a good method for estimating the mean of all observations? So for example if N=10 and I have 3 ...
user avatar
  • 131
1 vote
0 answers
48 views

Distribution of a Minimum of iid random variables to a power

Assume that I have $Z_{1},Z_{2},\dots,Z_{N}$ independent and identically distributed random draws following distribution $F(z)$ which are positive-valued. Define random variables $Y = \min_{i \in N} \...
user avatar
0 votes
1 answer
52 views

Quantitative/statistical comparison of two orderings/permutations

There is a couple, say, Theresa and Robert. They assess their preferences on 5 books by ranking them from the most attractive to the least one. Theresa: [0,4,3,2,1]...
user avatar
  • 2,483
2 votes
1 answer
43 views

Distribution of percentile rank of largest value in sample

Let's imagine that I sample 100 values from some probability distribution $Distribution$ over the real numbers. Out of these samples, I pick the maximum value $m$. It seems intuitive (and apparently ...
user avatar
1 vote
1 answer
91 views

Solved: Relative Efficiency of Average versus Maximum Order Statistic on a Uniform Distribution

$\newcommand{\szdp}[1]{\!\left(#1\right)}\newcommand{\eff}{\operatorname{eff}}$ Problem Statement: Let $Y_1, Y_2, \dots, Y_n$ denote a random sample from the uniform distribution on the interval $(\...
user avatar
5 votes
1 answer
286 views

computing $P\left(\max(U_{(1)}, U_{(2)}-U_{(1)}, \cdots,U_{(n)}-U_{(n-1)} ) <a\right)$

Let $U_{1}, \, ... \, ,U_{n}$ be a random sample of uniform random variables $U_i \sim \mathrm{Uniform}(0,1)$. Let $U_{(1)}, \, ... \, , U_{(n)}$ be the order statistics of the sample. My problem is ...
user avatar
0 votes
0 answers
166 views

What are E(max(X1, X2)) and Var(max(X1, X2)) when the Xs are normal random variables? [duplicate]

Let X = (X1, X2) be normally distributed random variables with mean m = (m1, m2) and covariance matrix S. Y = max(X1, X2) = X1 + max(0, X2 - X1) = X1 + D (X2 - X1), where D = 1 if X2 > X1 and 0 ...
user avatar
  • 9
1 vote
0 answers
46 views

exponential parameter estimtion from the smallest k-th order statistics

Assume $X_1, X_2, X_3,\ldots,X_n$ are i.i.d. samples from Exp($\lambda$). Assume that the integer $k<n$, is it possible to find a an unbiased estimator for $\lambda$ from the k-th smallest ordered ...
user avatar
1 vote
0 answers
22 views

What is the expectation of a random variable satisfying some conditions?

How to find the expectation E[X.I(Y<x,X<x)], where X and Y are independent random variables with respective cumulative distribution functions F(.) and G(.) respectively. x is a positive value. ...
user avatar
9 votes
2 answers
568 views

Convergence in distribution to a degenerate distribution

This question came up based on a disagreement I had with a TA. This was the specific example: Let $X_{1},...,X_{n}$ be an iid random sample from a population with pdf $f(x)=3(1-x)^2, 0<x<1$. The ...
user avatar
  • 587

1
2 3 4 5
9