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.

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

Multi-variable Rational Function that is always between the largest and second largest of the arguments.

This is a problem I came up for HMMT a few years ago, but I cannot find a solution. Question: Does there exist a rational function $f(x_1,\cdots, x_n)$ such that for any positive $x_i$, $f(x_1,\cdots,...
user avatar
0 votes
0 answers
10 views

The upper bounds of integrals in the expression of marginal pdf of order statistic

The question comes from an expression of marginal pdf of order statistic in a text that I am reading. Two related pages are in the link below. https://imgur.com/a/yamubX7 (you need to scroll down for ...
user avatar
  • 1
2 votes
1 answer
43 views

Expectation of the largest order statistic from uniform random variables

If $X_1, ..., X_n$ are iid random variables from the Uniform[$0,\theta$] distribution, where $\theta >0$, compute the expectation of the largest order statistic denoted $X_{(n)}$. I am looking to ...
user avatar
  • 991
1 vote
0 answers
17 views

Tight bounds for the expected maximum value of k IID Binomial(n, p) random variables

What is the tightest lower and upper bound for the expected maximum value of k IID Binomial(n, p) random variables I tried to derive it : $$Pr[max \leq C] = (\sum_{i = 0}^C {n \choose i}p^i(1 - p)^i)^...
user avatar
-2 votes
0 answers
9 views

How to have a constant error[b-a] for a range of values [1d array] in decending order that are not linear but are semi-linear? [closed]

I want to distribute n numbers in decending order such that their sum is 100 and more importantly the difference between anytwo numbers consecutively should be same
user avatar
  • 1
0 votes
2 answers
23 views

Finding order statistic of $Y_{\min}$

The question below is asking me to find the $Y_min$ such that it is smaller than $0.2$ and has a probability bigger than $0.9$, if I'm understanding it correctly. Let $Y_1, . . . , Y_n$ be iid random ...
user avatar
-1 votes
0 answers
19 views

Knowing what i is for order statistics?

is this a typo in the book or not? The question below is asking for $P(Y_2<5)$ so I thought $i=2$, but it was solved with $i=3$. Does the inequality mean that there's a plus one to $i$ ? And if it ...
user avatar
0 votes
0 answers
16 views

How can one understand the expression $X_{(i)} - F^{-1}(i)$, where $X_{(i)}$ is order statistics and $F$ is distribution function.

There is a lemma in Jaeckel (19171) paper "Some flexible estimates", which basically states that under some conditions $X_(i) - F^{-1}(i^*)$ is $O(n^{\frac{1}{2}})$ or that $\left[X_{(i)} - ...
user avatar
0 votes
0 answers
15 views

Discrete uniform rejection region

I am currently studying for an exam. There was one problem listed that I currently am puzzled about how to approach. I would like some guidance as to what can be done in part a). I will attach a ...
user avatar
  • 473
1 vote
0 answers
23 views

Maximizing the ratio of expected maximum of n IID and the expected value of the IID for a (WHP) non-negative distribution

I’m looking for a distribution that is non negative , or has good tail bounds (so non negative with high probability) and maximizes the following property: $X_1, X_2, …, X_n$ are n IID samples of the ...
user avatar
0 votes
0 answers
14 views

Order statistics when random variable are not identical

I'm trying to find the pmf and cdf of the maximum order statistics from independent (but not necessarily identical) continuous random variables $X_1,\dots, X_n$. This was not very helpful: Order ...
user avatar
  • 148
1 vote
0 answers
25 views

If $X\sim G(a,b_{1})$ and $Y\sim G(a,b_{2})$, then what will be the density function for U=min(X,X+Y)?

Let $X$ and $Y$ two independent random variables for gamma distributions with common shape parameter $a$ and different rate parameter $b_{1}$ and $b_{2}.$ If $U=\min(X,X+Y),$ then what will be the ...
user avatar
-1 votes
1 answer
38 views

Expectation of the minimum of two binomial random variables? [closed]

Does anyone know an expression or at least bounds for the expectation of the minimum of a pair of i.i.d. binomial random variables? Ideally I would like to have lower bounds that are on the order of ...
user avatar
1 vote
1 answer
36 views

Distribution of infimum of countably many random variables.

Motivated by my undergraduate class in probability, where we are learning about order statistics: say, you have a countable collection of i.i.d. random variables $X = \{X_1,X_2...\}$. Can we find the ...
user avatar
1 vote
1 answer
66 views

Given X_1,...,X_n iid, compute the probability that X iid is less than the k-th order statistic

I have $n+1$ i.i.d. RV $X_1,\dots,X_n,X$. I compute the order statistics of $X_1,\dots,X_n$: $$X_{(1)},\dots,X_{(n)}$$ I'm reading a paper which says that $P(X \leq X_{(k)})=\frac{k}{n+1}$ (see ...
user avatar
  • 223
5 votes
2 answers
294 views

Expectation of maximum from $n$ draws.

We have $n$ guys walking down the street, and each can find $1, 2, \ldots $ or $n$ dollars in the street. ($n$ is the same number of guys and the same number of dollars in the problem). Each of them ...
user avatar
  • 619
1 vote
2 answers
65 views

what is the variance of difference between max and min of n i.i.d uniform variables : U(0,1)

It is an interview question: calculate the variance of difference between max and min $$variance[\max(\{X_i\}) - \min(\{X_i\})].$$ Here $\{X_i\}$ is n i.i.d uniform variables : U(0,1). I know it is ...
user avatar
1 vote
1 answer
43 views

convergence of the first order Statistic of uniform distribution

Let $U_1,U_2,...U_n,$ be iid samples from uniform distribution $U(0,1)$. And the order Statistic: \begin{equation} U_{n,1}\leq U_{n,2}\leq ...\leq U_{n,n}, \end{equation} (1) Proof $U_{n,1}\...
user avatar
  • 409
0 votes
0 answers
23 views

Probability that two samples from a multinomial distribution intersect in a specific way

The following combinatorially difficult problem has obvious applications in information retrieval and bioinformatics but I haven't seen its solution discussed anywhere. I am interested in computing ...
user avatar
  • 1,463
1 vote
1 answer
27 views

MLE for uniform distribution with parameter dependent support

Let $X_1,...,X_n \overset{iid}{\sim} Uniform(\theta ,2\theta)$, for some positive real $\theta$. Find the Maximum Likelihood Estimator, $\hat{\theta}$, for $\theta$. I know the likelihood function is $...
user avatar
2 votes
1 answer
45 views

Does FOSD + log concavity of $f(x)$ and $g(x)$ imply MLRP?

I am looking for a result on the ordering of distribution functions. The probability density functions $f(x)$ and $g(x)$ bear the Monotone Likelihood Ratio Property (MLRP) if $$ \frac{f(x)}{g(x)} $$ ...
user avatar
0 votes
1 answer
29 views

What happens to the rank when finding if a test-statistic is even?

When I take $X_1,...,X_n$ from $F(x)$. Then the ranks are $X_{(1)}<...<X_{(n)}$ and lets take the test-statistic, \begin{align} t_0(X_1,...,X_n) = \sum_{i=1}^n X_{(i)} \end{align} If I look at $...
user avatar
  • 779
-1 votes
1 answer
27 views

Distribution of $Z=\min \left\{U_{1}, \ldots, U_{X}\right\}$

Let $U_{i}, i=1,2, \ldots$ be independent uniform random variables in $(0,1)$. Also $X$ is a discrete random variable whose pdf is given by: $$P(X=x)=\frac{1}{(e-1)x !}, x=1,2,3, \ldots$$ Find the CDF ...
user avatar
0 votes
0 answers
9 views

Computationally efficient way of calculating the linear combination of order statistics

Given a dicrete random variable $X$, and the order statistics $X_{(1)}, X_{(2)}, \cdots, X_{(N)}$. What is the PMF of: $$Y=\sum_{i=1}^N \alpha_i X_{(i)}$$ Where $\alpha_i=0\ \text{or}\ 1\ \forall i$ ...
user avatar
1 vote
1 answer
34 views

Expected value of the Max of IID random variables that follow the Discrete Uniform Distribution

I'm trying to find the expected value of $X_{n}$, where $X_{n}$ is the MAX of {$X_{i}$, ..., $X_{n}$} and X ~ U(0, 2Θ), with Θ > 0. I don't know if what i'm doing is right, but so far what I got is;...
user avatar
  • 13
0 votes
0 answers
28 views

Order Statistics: Expected order statistic of a distribution given an order statistic of a linearly dependent distribution.

say I have $X \sim \texttt{Binomial}(n, p)$ and $Y \sim \texttt{Binomial}(cn, p)$ where $0 \leq c \leq 1$. Also, I know what $E[W_{X_1}], E[W_{X_2}, ...]$ where $W_{X_i}$ is the $i$th largest sample. ...
user avatar
2 votes
1 answer
28 views

where did the sum come from in formula of order statistic $ P(X_{(r)}\leq x) = \sum_{j=r}^n C^n_j F(x)^j (1-F(x))^{n-j} $

the order statistic formula is given as follows $ P(X_{(r)}\leq x) = \sum_{j=r}^n C^n_j F(x)^j (1-F(x))^{n-j} $ I undestand that the combination are from the picking $r$ of the $n$ $X$-s to be less ...
user avatar
  • 519
0 votes
1 answer
41 views

Obtaining Distribution of $(X_{(1)},X_{(n)})$ using density function

I am given $X_1, X_2, ..., X_n \overset{iid}{\sim} U(\alpha, \beta)$, and asked to find joint distribution of order statistics- $X_{(1)}, X_{(n)}$. Though I know there exists a much simpler solution, ...
user avatar
  • 15
1 vote
0 answers
33 views

Proof that Mann-Whitney is just translated sum of ranks

Setup Given the iid random sample $X_1,...,X_m$ with ranks $Q_1,...,Q_m$ and the iid random sample, $Y_1,...,Y_n$ with ranks $R_1,...,R_n$. Then the sum of ranks is, \begin{align} W = \sum_{i=1}^{n}...
user avatar
  • 779
1 vote
0 answers
49 views

Asymptotic lower bound for the maximum of $n$ iid binomial random variables

Let $X_i \stackrel{iid}{\sim} \text{Binom}(n,p)$ for any $1 \leq i \leq n$ (assume that $p=o(\log n/n)$). Prove that $$ \Pr(\max_{1 \leq i \leq n} X_i \geq k^*) \rightarrow 1 \text{ as } n \rightarrow ...
user avatar
  • 315
0 votes
0 answers
42 views

Order Statistics and pdfs questions

Sorry if this looks stupid, I'm currently trying to pick up statistics and i can't understand pdfs in Order Statistics, i got several questions regarding the concepts and definitions: Order ...
user avatar
0 votes
1 answer
26 views

Why does the CDF of a k-th order statistic have an ${n \choose i }$ term?

The CDF of the k-th order statistic is given by- \begin{align*} F_{X_{(k)}}(x) & = \sum_{i=k}^n {n \choose i} [F_X(x)]^i\cdot [1 - F_X(x)]^{n-i} \end{align*} But I do not understand why there is ...
user avatar
  • 15
0 votes
0 answers
58 views

Mathematical expression for $i^{th}$ largest element of a set

Suppose I have a finite set of real numbers $S$, and I want to denote the $i^{th}$ largest element of $S$. Is there a nice mathematical expression to express that value?
user avatar
0 votes
0 answers
51 views

Order of fisher information

The Fisher information of n iid r.v.’s is always of order n. I am unsure of how to convince myself this is true or false - I cannot find any information about how to calculate the order of fisher ...
user avatar
0 votes
0 answers
14 views

Estimator for minimizing mean square error for order statistic

Question Context Let $X_{(1)}\leq ... \leq X_{(n)}$ have CDF $F(x)$ and $\tau_{(i)}=\textbf{E}(X_{(i)}),i=1,...,n$. Let the random variables, $Y_{(1)}\leq ... Y_{(n)}$ have CDF $F[(x-\mu)/\sigma]$ ...
user avatar
  • 779
-1 votes
1 answer
46 views

Evaluating the probability of a Distribution [duplicate]

$X_1 ∼ N(µ = 2, σ = 2), X_2 ∼ N(µ = 1, σ = 4), X_3 ∼ N(µ = −4, σ= 3):$ $X_1, X_2,$ and $X_3$ be independent and $Y = (X_1 + 2X_2 + X_3)^2.$ Determine $P(Y > E(Y)).$ My solution: I got the value of $...
user avatar
  • 9
1 vote
2 answers
55 views

Do continuous functions with small uniform norm preserve order?

Consider a sequence $x_1,\dots,x_n\in [0,1]^d$. Suppose that $\mathcal{F}$ is a subset of the space of continuous functions from $[0,1]^d$ to $\mathbb{R}$. Moreover, let $f\in\mathcal{F}$ such that $$...
user avatar
  • 361
1 vote
0 answers
38 views

Conditional expectation involving Order Statistics [duplicate]

Let $X_1,X_2,...,X_n$ be a random sample of size $n \geq 2$ from $\exp(1/\theta)$, where $\theta \in (0,\infty)$. Let $$Y = \min\{X_1,X_2,...,X_n\} \qquad \textrm{and} \qquad T = \sum_{i=1}^{n} X_i .$$...
user avatar
1 vote
1 answer
43 views

Are the probability distributions of order statistics log concave?

If some distribution is log concave, then would the distribution of order statistics sampled from that distribution also be log concave? Specifically, I'm thinking of a case where I have a random ...
user avatar
  • 99
1 vote
1 answer
59 views

finding the marginal pdf of second order statistics

Suppose that $X_1$, $X_2$, $X_3$ is random sample (iid) from a population with pdf $f(x) = 3x^2$ if $0<x<1$ I would like to compute $g_2(y_2)$ which is the marginal pdf of order statistics $Y_2$ ...
user avatar
  • 6,153
0 votes
0 answers
33 views

Expectations of Max Order Statistic on Exponential distribution

Let $X_1,...,X_n$ be iid exponential with scale parameter $\theta$. That is for each $i \in \{ 1,2,...,n\}$, $X_i \sim f_{X_i}(x) = \frac{1}{\theta}\exp(-x/\theta) $. I am interested in the max order ...
user avatar
-1 votes
1 answer
89 views

What is the closed form solution for this infinite sum? [closed]

We are working on a problem related to order statistics. This requires the computation of the following infinite sum. Let i be a positive integer. Let real numbers $$\alpha >0$$ and $$1>\beta>...
user avatar
1 vote
1 answer
63 views

Max and Min have the same variance [closed]

Let $X_i$ be i.i.d uniform variables between 0 and 1. Define $Min_n = \min(X_1, \ldots, X_n)$ $Max_n = \max(X_1, \ldots, X_n)$ How can we show that the variance of $Min_n$ and $Max_n$ are the same? ...
user avatar
  • 477
1 vote
1 answer
106 views

Computing $A=\sum_{k=0}^{\infty}\frac{\alpha(\alpha+k\beta)^{k-1}e^{-(\alpha+k\beta)}}{k!}f^{k}$

I'm working on a problem in order statistics. I am hoping to obtain a closed form solution for the following infinite sum: $$A=\sum_{k=0}^{\infty}\frac{\alpha(\alpha+k\beta)^{k-1}e^{-(\alpha+k\beta)}}{...
user avatar
-1 votes
1 answer
36 views

Convergence in distribution of order statistic random variables with uniform distribution. [closed]

I am trying to find the asymptotic distribution of an order statistic $X_{(1)}$ for iid RVs $X_1, ..., X_n \sim \mathrm{Unif}(0,1)$, The distribution for $X_{(1)}$ $F_{X_{(1)}}(x) = 1 - \left(1 - {x}\...
user avatar
  • 101
1 vote
1 answer
37 views

Density function of average of max and min of uniform random variables

Suppose $X_1, \ldots, X_n \sim U(\theta - \frac{1}{2}, \theta + \frac{1}{2})$. I want to find the density of $Z = \frac{X_{(1)} + X_{(n)}}{2}$. My strategy is to use the transformation $F(x, y) = (\...
user avatar
0 votes
0 answers
16 views

Subsampling from different distribution data?

I have a simple question. Thanks for helping me Is there any difference between simple random subsampling a set with uniform distribution and a set with normal distribution? How can I subsample in ...
user avatar
4 votes
1 answer
94 views

Maximum likelihood estimator and asymptotic distribution

Let $X_1,\dots,X_n$ be a random sample from X whose density is given by $$f(x,\theta) = c(\theta)(1-\exp(-|x|))I\{|x|\leq\theta\}$$ Find the maximun likelihood estimator of $\theta$ and show that $n(\...
user avatar
2 votes
1 answer
94 views

Why does a beta distribution between two uniform order statistics have the distribution it does?

Per the article on order statistics, the $k$-th order statistic of a uniform distribution ($U_{(k)}$) is Beta distributed with parameters $k$ and $n-k+1$. And the distribution of $U_{(k)}-U_{(j)}$ is ...
user avatar
  • 5,509
1 vote
2 answers
113 views

Why are the order statistics of uniform Beta?

For a beta distribution with parameters $a$ and $b$, we can interpret it as the distribution of the probability of heads for a coin we tossed $a+b$ times and saw $a$ heads and $b$ tails. At the same ...
user avatar
  • 5,509

1
2 3 4 5
20