Skip to main content

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
5 votes
1 answer
131 views

Convergence of Step Functions Generated by Uniformly Distributed Random Points

I have encountered a surprisingly complicated problem to solve and I'm looking for some help. It could be difficult because I don't have a background in probability and so don't know the appropriate ...
Jason Bramburger's user avatar
2 votes
0 answers
12 views

Expectation of a piecewise const approximation based on Beta distribution

Let $X_1, X_2 \stackrel{\text{iid}}{\sim}\mathrm{Uniform}(0,1)$ and then sort $X_1,X_2$ to get $X_{(1)} < X_{(2)}$. Based on the pdfs of $X_{(i)}$, we know $X_{(1)} \sim \mathrm{Beta}(1,2)$ and $...
learner's user avatar
  • 482
0 votes
1 answer
46 views

P(max(x,y) < a) = P(x < a) P(y <a) yielding incorrect solution?

I was working on the below problem which, along with variants of it, has been asked multiple times on this site (here is one such example), but I'm unable to figure out why my approach is incorrect ...
mk0219's user avatar
  • 178
6 votes
0 answers
32 views

Expectation of maximum of normal variables with different means [closed]

Given that ${X_1,...,X_n}$ are $n$ independent normal variables and $X_i \sim N(a_i,\sigma^2)$, can we show that $E[max\{X_1,...,X_n\}]$ is increasing in $\sigma$? Since $a_i$'s are not identical, the ...
Ssssssponge's user avatar
1 vote
1 answer
98 views

Confidence interval with shortest length for location parameter of uniform distribution $U (\theta, \theta+1)$ based on pivot $X_{(1)} - \theta$

$X_i$ is iid from a uniform distribution on $(\theta, \theta + 1)$, $\theta \in R$. Show that $X_{(1)} − \theta$ is a pivot. Showing that the pivot pdf does not depend on theta, i solved by first ...
Maale Faustus's user avatar
1 vote
0 answers
15 views

convolution of an order statistics and an exponential distribution

Is there a simplification to the convolution of the k-th order statistics from an erlang distribution with shape 2: $f_{1_{(k)}}(z) = \frac{n!}{(k-1)!(n-k)!} \cdot \left(1-e^{-\lambda z} (\lambda z+1)...
user9467051's user avatar
0 votes
1 answer
23 views

Order Statistics from a sum of exponential distributions

Let $X_i$ $(X_1, \dots, X_n)$ and $Y_i (Y_1, \dots,Y_n)$ be i.i.d. exponential r.vs with rate $\lambda$. Let $Z_i= X_i+Y_i$. How to write the pdf of the k-th order statistics of the $Z_i$ random ...
user9467051's user avatar
0 votes
1 answer
67 views

Conditional Expectation when variables are drawn from uniform distribution of different domain

I have the following problem - $a$ and $b$ are independently drawn from uniform distribution. $a$ is drawn from uniform distribution $\{m, 1+m\}$ and $b$ is drawn from uniform distribution $\{0,1\}$. ...
Elina Gilbert's user avatar
0 votes
0 answers
9 views

I want to choose the top 60% of animals in my study. 72% of the selection weight is due to random factors, while 28% is due to height.

I want to choose the top 60% of animals in my study. 72% of the selection weight is due to random factors, while 28% is due to their height (both random factors and height are in z scores and normal ...
John's user avatar
  • 5
0 votes
0 answers
21 views

The probability of each specific ordering of a group of independent uniform random variables

Given: A sequence of $n$ positive real numbers (the scales of the uniform random variables) $t_1, t_2, \ldots, t_n \in \mathbb R_{+}$ Let: $x_1, x_2, \ldots, x_n$ be mutually independent uniform ...
Vezen BU's user avatar
  • 2,098
0 votes
0 answers
26 views

On the convergence in probability of the maximum statistic of a random variable according to triangular and uniform

Set up Consider the example in section 2 of Ferguson (1982). Let $X_1, \ldots, X_n$ be i.i.d. with a distribution which with probability $\theta$ is the $U(-1, 1)$, and with probability $(1-\theta)$ ...
ytnb's user avatar
  • 564
1 vote
1 answer
18 views

Revisiting the distribution of mth order statistic using symmetry

Not asking for the distribution of $m$th order statistic, for it has been well documented and covered and in fact well-known. What I am rather confused, perhaps basic but failing to see, is the ...
User1865345's user avatar
0 votes
0 answers
27 views

Non-constant IID distribution with identical order statistics.

Is there a non-constant distribution $\mathbb{D}$ such that if $X_1, ..., X_n \sim \mathbb{D}$ are iid random variables with order statistics $Y_{(1)}, ..., Y_{(n)}$, then the distribution of $Y_{(i)} ...
AspiringMat's user avatar
  • 2,447
0 votes
1 answer
89 views

Average is higher than 75th percentile [closed]

I have a statistical question with mean, median and percentile. I have run my data with over thousand samples. From this run, I can see that the Average result value is higher than the median and ...
Peter Serey's user avatar
2 votes
0 answers
31 views

Almost Sure Convergence of Order Statistics

Let $F$ be a strictly increasing distribution function. For a given $\tau \in (0,1)$, suppose there exists $\epsilon_{\tau}$ such that $F(\epsilon_{\tau}) = \tau$. Considering a set of independent and ...
Florian Huo's user avatar
0 votes
0 answers
23 views

Proving Independence of Complete and Sufficient Statistic

Let $X_1, \ldots, X_n$ be i.i.d r.v. with the p.d.f. $$ f(x; \theta_1, \theta_2) = \begin{cases} \frac{1}{\theta_2} \exp\left(-\frac{x - \theta_1}{\theta_2}\right), & \text{if } x > \theta_1, \...
Brandon's user avatar
0 votes
0 answers
16 views

Order statistics of the gaps

Consider a probability distribution functions $f(x)$ with cumulative density function $F(x)$. I would like to compute the PDF $g_{ij}$ of the gap $x_j-x_i$ where $x_i$ is the ith smallest sample. The ...
Nichola's user avatar
  • 203
0 votes
0 answers
29 views

Multivariate order statistics and rankings

Let $(X_1,Y_1),\dots (X_n, Y_n)$ be an i.i.d. sequence of vectors. Note that $X_i$ need not be independent of $Y_i$ in general. We consider $X_i$ and $Y_i$ continuous so their rankings are almost-...
Julius's user avatar
  • 1,398
0 votes
1 answer
60 views

Understanding $P(\max\{X_{1},\ldots,X_{n}\} \le y) = P(X_{1} \le y,\ldots,X_{n} \le y)$.

I understand the logical statement that: $$\max\{{X_{1},\ldots,X_{n}}\}(\omega) \le y \iff X_{1}(\omega) \le y,\ldots,X_{n}(\omega) \le y$$ But my issue falls with $\max\{\}$ itself. Shouldn't the ...
cDralda's user avatar
-1 votes
2 answers
45 views

Covariance of max and min of independent uniform random variables [closed]

For $X$, $Y$ independent uniform random variables distributed $U[0,1]$, let $H = \max(X,Y)$, $L = \min(X,Y)$. We know that: $\operatorname{Cov}\left\{\min(X,Y),\max(X,Y)\right\} \neq \operatorname{...
herpderp123's user avatar
0 votes
0 answers
37 views

Probability of largest order statistic greater than or equal to the second largest order statistic

Given random variables $X_1, X_2..., X_n$ that are iid from uniform $[0,1]$, the order statistics $X_{(1)}, X_{(2)}, ..., X_{(n)}$ are also random variables, defined by sorting the values (...
Francis's user avatar
  • 121
4 votes
1 answer
168 views

Probability of no friends meeting at a coffee house

I came across what seems to be in my opinion a very challenging problem: A number n of friends each visit Old Slaughter’s coffee house independently and uniformly at random during their lunch break ...
DeadKarlMarx's user avatar
1 vote
1 answer
133 views

Differences of order statistics for symmetric random variables

Take a sequence of $n$ i.i.d. random variables symmetric around zero and with zero expectation: $$ \eta_1,\eta_2,\dots, \eta_n. $$ Use standard order statistic notation and consider $$ \eta_{(1)}\leq \...
Star's user avatar
  • 206
0 votes
0 answers
15 views

Sample variation maximization during sampling

Let's say that I am sampling $\{x_i\}$ from a distribution with CDF $F(x)$, and the samples are always non-negative. For each new sample, let's say the $n$-th sample, provided that we have sampled $n-...
Rebecca Zorichyevna's user avatar
0 votes
0 answers
24 views

6.36 of Theory of Point Estimation, second edition

the question is: $$ X_1, X_2, ..., X_n $$ i.i.d random variables of uniform distribution U(a,b), where a<b. Show that $$ Z_i = \frac{X_{(i)}-X_{(1)}}{X_{(n)} - X_{(1)}}$$ ,i = 2,...n-1, are ...
Wei Li's user avatar
  • 1
0 votes
0 answers
20 views

Solving systems of simultaneous equations with permutations as variables: algebra meets combinatorics and order statistics

We may treat a given permutation as a vector variable, so that permutation s$_n$ is a variable that takes values among all possible number sequences of length $n$, which have elements in the rank ...
virtuolie's user avatar
  • 171
1 vote
1 answer
39 views

Fraction of the largest element of a sum of N i.i.d. random variates sampled from some distribution

I want to figure out how the fraction of the biggest sample of the sum of some iid samples from an arbitrary distribution varies. For example, I have a random variable $X \sim f(x)$, and I sampled $n$ ...
Rebecca Zorichyevna's user avatar
1 vote
2 answers
60 views

Does the sample minimum of $Y_i=h(X_i)$, $X_i\sim$ uniform, converge to $\min_x h(x)$?

Let $h:[a,b]\to \mathbb{R}$ be Borel and continuous. So there exists $x^*$ such that $\min_{x\in[a,b]} h(x) = h(x^*)$. For simplicity assume $x^*$ is unique. Let $X_1,\dotsc, X_n$ be IID $\mathcal{U}(...
Nap D. Lover's user avatar
  • 1,229
9 votes
1 answer
267 views

Expected difference between the largest and second largest observations in a sample of i.i.d. normal variables

Let $X_1,\dots,X_n$ be an i.i.d. sample from the standard normal distribution. Let \begin{align} \mu_n = \mathbb{E}[X_{(n)} - X_{(n-1)}], \end{align} be the expected difference between the largest ...
svonimir's user avatar
  • 359
1 vote
1 answer
44 views

Maximizing Pearson Correlation Coefficient in Ordinal Data: A Constrained Optimization Problem

I am puzzled by a problem in which the mathematical formula for the Pearson Correlation Coefficient ($r$) is an objective function to be maximized, but in a contingency table with specific marginal ...
smoser's user avatar
  • 11
1 vote
1 answer
34 views

Expected Tries to Find the Maximum from $n$ normally distributed numbers

Consider a scenario where you write on $n$ pieces of paper a value obtained from a normal distribution $N(0,1)$. They're flipped over and arranged randomly. One-by-one you begin flipping over the ...
rough-parsley's user avatar
0 votes
1 answer
74 views

exception of order statistics

Assume $X_1,X_2,\dots ,X_n \overset{i.i.d}{\sim} F(.) $ I define $X_{(1)},X_{(2)},\dots,X_{(n)} $ Like that: $X_{(1)}\leq X_{(2)}\leq \dots\leq X_{(n)}$ For $r\in \{1,2,\dots,n\}$ : What is the ...
Mohammadreza Shahriyarkeshe's user avatar
3 votes
1 answer
54 views

Statistics Question of the Day

Motivation: I am a graduate student in the Department of Statistics at Kansas State University. Everyday I create a "question of the day" for myself, and it has been going well for the past ...
aidan kerns's user avatar
0 votes
1 answer
70 views

Can anyone help to solve this task ?In a multiple-choice test with m options, a student knows the correct answer with probability p,...?

"In a multiple-choice test with m options, a student knows the correct answer with a probability p, and in the absence of knowledge, chooses randomly one of the available options. What is the ...
Viktoria 's user avatar
10 votes
2 answers
722 views

Expected difference between largest and second largest of i.i.d. random variables

Let $(X_i)_{i\geq 0}$ be i.i.d. nonnegative random variables with continuous density function $f$. Let \begin{align} \mu_n = \mathbb{E}[X_{(n)}-X_{(n-1)}] \end{align} be the expected difference ...
svonimir's user avatar
  • 359
0 votes
1 answer
46 views

Finding the posterior and Bayes estimator with a beta prior

Let $Y_i, \ i =1,2,\ldots n$ be a random sample from the probability function $$f(y\mid p) = \frac{2y}{p^2}, \quad 0 < y \le p$$ where $p\sim Beta(2n+1, 1)$ is the prior, find the posterior ...
holala's user avatar
  • 731
0 votes
1 answer
34 views

Expected value for amount of nondominated points in a set of n 2d points,

We have a set of $n$ 2-d points ${(x_1​,y_1),(x_2​,y_2​),…,(x_n​,y_n​)}$ where $x$ and $y$ are sampled from independent normal distributions. A point $(x_a, y_a)$ is said to dominate another $(x_b, ...
Michael Wang's user avatar
0 votes
0 answers
28 views

Sum of Cumulative Max of Exponentially Distributed Variables

Let $X_1, X_2, ..., X_n$ be independent, identically and exponentially distributed random variables, $P(x) = k \exp(-k x)$. Define $Y$ as the sum of the sequence of cumulative maxima: $Y = X_1 + \max(...
GingerBreadMan's user avatar
2 votes
1 answer
58 views

Trouble understanding order statistics

Order statistics were introduced in my text as follows: I am trying to understand what this means. $X_1 , \dots , X_n$ is a random sample, i.e. an independent and identically distributed sequence of ...
Bastiza's user avatar
  • 293
2 votes
1 answer
39 views

$X_i\sim \mathrm{UNIF}(0,\theta)$. Show that $S=X_{n:n}$ is sufficient for $\theta$ by the factorization criterion.

Consider a random sample from a uniform distribution $X_i\sim \mathrm{UNIF}(0,\theta)$, where $\theta$ is unknown. Show that $S=X_{n:n}$ is sufficient for $\theta$ by the factorization criterion. ...
Ocean's user avatar
  • 85
0 votes
0 answers
27 views

CDF of sum and subtraction of independent and identically distributed random variables having Gamma distribution with same shape and scale parameter

What is the CDF of the following: P$($$a_m$ $Z_m$ - $K$ $\sum^{M}_{i=m+1}$ $a_i$ $X_i$$ $<$ C$$)$ where $Z_m$ and $X_i$ are independent and identically distributed random variables, continuous ...
Math Explorer's user avatar
0 votes
1 answer
29 views

Find the conditional expectation $E(2Y_2|Y_5)$

My Question: $X_1,X_2,...X_n$ are iid random varables and follow Uniform$(0,\theta)$ We pick up five samples from population.And Let $Y_1<Y_2<Y_3<Y_4<Y_5$. Want to know $E(2Y_2|Y_5)$ My ...
Panda Chou's user avatar
0 votes
0 answers
40 views

$\frac{X_{1:n}+X_{n:n}}{2}$ is unbiased for mean

For $X_{i}\sim UNIF(\theta_{1},\theta_{2})$, we know that $X_{1:n}$ and $X_{n:n}$ are jointly sufficient for $\theta_{1}$ and $\theta_{2}$. Suppose that it is desired to estimate the mean $\mu = \frac{...
JuanFerRp's user avatar
3 votes
1 answer
256 views

Probability that mean is larger than median

Let $X_1,\ldots,X_n$ be i.i.d random variables taking values in $\mathbb{R}$. Suppose that $n$ is odd and the $X_i$ follow a continuous distribution. I am interested in the probability that the mean ...
Idontgetit's user avatar
  • 1,371
1 vote
0 answers
22 views

Disproving the regularity condition of Cramer-Rao Lower bound

Let $X = (X_1,\cdots, X_n)$ where $X_1,\cdots,X_n$ be i.i.d from the uniform distribution $U(0,\theta)$ with $\theta>0$. I was asked to show the regularity condition of the Cramer-Rao lower bound: $...
Nothing's user avatar
  • 1,718
0 votes
0 answers
10 views

given SD and mean find bserved value of the test statistic?

I had this word problem where I am trying to see whether new version of something is more precise so I know that the formula for test stat is $$\sum_{i=1}^9 \frac{(yi-\mu)}{x^2}$$$ where $$\sum_{i=...
fashionable's user avatar
0 votes
1 answer
48 views

Probability of a Minimum out of a Subset of a Set of Integers.

Let us assume a set $\mathcal{A}$ of $K$ unique integers: $\mathcal{A} = (X_{1}, \ldots, X_{K})$ and randomly pick an integer $X_{j}$ from the set $\mathcal{A}$. Now, let us draw a random subset $\...
Christos's user avatar
  • 189
1 vote
0 answers
115 views

Bias of a Maximum Likelihood Estimator

Question: For $\lambda \subset \mathbb{R}$, define the function $$ g_\lambda(y)=e^{-(y-\lambda)} 𝟙_{(y\geq\lambda)} $$ Let $Y=(Y_1, Y_2, \ldots, Y_n)$ with $Y_1, Y_2, \ldots, Y_n$ iid random ...
Lexi's user avatar
  • 11
5 votes
3 answers
321 views

Expected Maximum Value of 10 Randomly Selected Balls from an Urn

There are $20$ balls in an urn labeled from $1$ to $20$. You randomly pick $10$ balls out of this urn. What is the expected maximum value of the $10$ balls you picked out? I was able to solve the ...
Devansh Agarwal's user avatar
0 votes
0 answers
49 views

Joint distribution of the $X_{(1)}, X_{(2)}, X_{(n)}$

Suppose that we have i.i.d. random variables $X_1,...,X_n$ with the CDF $F$ such that $\forall a \in \mathbb{R}~ \mathbb{P}(X_i=a) = 0$. I wanted to find the joint CDF of the $X_{(1)}, X_{(2)}, X_{(n)}...
perepelart's user avatar

1
2 3 4 5
22