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 ...

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22 views

Probabilities of each waitlist person

Coming from this, $10$ Applicants for a exclusive club membership. I found that you can use total probability to consider existing and old members leaving/returning as members in the club, ended up ...
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1answer
15 views

Find the probability that the range of a random sample of size $3$ from the uniform distribution is less than $0.8$.

What do i have to use to calculate the probablity that the range of a random sample of size $3$ from the uniform distribution is less than $0.8$? I have the pdf of the range : $$f_W(w)= n(n-1) ...
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3answers
37 views

Find the joint distribution of the continuous order statistics?

Take $n$ independent variables, ${X_1, X_2,\dots, X_n}$, which are uniformly distributed over the interval $(0,1)$. Then, introduce the variable $M=\min(X_1, X_2,\dots, X_n)$ and the variable ...
3
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1answer
93 views

The expected value of the $k$th order statistic of iid geometrically distributed rvs, and its asympotic expansion.

I have read the paper Combinatorics of geometrically distributed random variables: Left-to-right maxima. In the paper, the largest order statistic $X_{n:n}$ (i.e., $\max\{X_1,X_2,\ldots,X_n\}$) is ...
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27 views

expected value of order statistic

I am having troubles with solving this task 7.4.5. Show that the first order statistic $Y_1$ of a random sample of size $n$ from the distribution having pdf $$f(x;\theta) = e^{-(x-\theta)}, ...
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24 views

What is the asymptotic behaviour of $\max_{n \in \{1, 2,\, \dots N\}} \tan n$?

What is the asymptotic behaviour of $f(N) = \max_{n \in \{1, 2,\, \dots N\}} \tan n$? Any non-trivial bounds above or below would be of interest. Some quick numerical experimentation shows: ...
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3answers
63 views

For $n \geq 2$, let $X_1,X_2,\ldots,X_n$ be independent samples from $P_{\theta}$, the uniform distribution $U(\theta,\theta +1),\theta \in \mathbb R$

For $n \geq 2$, let $X_1,X_2,\ldots,X_n$ be independent samples from $P_{\theta}$, the uniform distribution $U(\theta,\theta +1),\theta \in \mathbb R$. Let $X_{(1)},X_{(2)},\ldots,X_{(n)}$ be order ...
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16 views

Given $X_i \sim Geom(p)$, what is the PDF of $R := \min \{ r:\sum_{j=1}^n \mathbb{1}_{\{ X_j \leq r\}} \geq m \}$?

I am considering a file transfer problem that given a file divided into $n$ blocks, the number of transmission rounds of each block $X_i$ satisfies $X_i \sim Geom(p)$. Thus the number of transmission ...
2
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1answer
48 views

Show that $(\frac{S_1}{S_n+1},\frac{S_2}{S_n+1},…\frac{S_n}{S_n+1})=_d (U_{(1)},U_{(2)},…,U_{(n)})$.

Let $(X_1, X_2,...,X_n) \in \mathbb R^n$ have density function $p(x)$. (1) Find the density of $(U_{(1)},U_{(2)},...,U_{(n)})$, the order statistics from a sample of iid $\mathbb U[0,1]$ (uniform ...
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18 views

subsample order statistics

I am interested in characteristics depending of the $r$-th order statistics $X_r$ of a distribution with unknown pdf. For example, I would like to estimate the Gini mean difference $E(|X_r - X_r'|)$, ...
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1answer
47 views

compare sample mean and sample median as estimators of µ

Under symmetry, $F^{−1} (0.5) = E(X)$. Compare the sample mean as an estimator of µ to that of the sample median ($F^{−1} (0.5)$) for n sufficiently large, assuming that $X_i$ ∼ Z = N(0, 1), i ...
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44 views

Distribution of Interquartile Range on $X_i$ ∼ U(0, 1), i = 1, . . . , 20, iid.

Let Xi ∼ U(0, 1), i = 1, . . . , 20, iid. IQR = $F^{−1}(.75)−F^{−1}(.25)$ = $X_{(15)}−X_{(5)}$ in this example as n = 20. a. Find the distribution of the random variable W = IQR. b. ...
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1answer
32 views

Order statistics for exponential random variables

Let $\tau_1, \tau_2, ..., \tau_K$ be i.i.d. exponential random variables with distribution $P(\tau_k<t) = 1 - e^{-\lambda t}$. Let $\tau^*_i$ be the $i^{th}$ order statistic. The p.d.f. of ...
3
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1answer
23 views

Compute the Maximum Likelihood Estimator of a pareto Distribution for statistic…

Can anyone please help compute the Maximum Likelihood Estimator for: $X_1,...X_n$ iid with $f(x; \theta) = \frac1 \theta(1+x)^{\frac{-\theta +1}\theta}$ For the statistic $T(x) = (X_1,...,X_n)$ ...
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3answers
65 views

Prove uniform distribution

For any random variable $X$, there exists a $U(0,1)$ random variable $U_X$ such that $X=F_X^{-1}(U_X)$ almost surely. Proof: In the case that $F_X$ is continuous, using $U_X=F_X(X)$ would suffice. In ...
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0answers
14 views

how to find asymptotic joint distribution of two linear combination of order statistics?

Suppose I have n order statistics from some unknown continuous distribution funciton F(x), $X_{1}\leqslant X_{2}\leqslant...\leqslant X_{n}$. And I have two linear combination of these order ...
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25 views

Expectations of order statistics of uniform RVs, via exponential formulation

If $U_1,\ldots, U_n$ are i.i.d. uniform random variables, then I know that the order statistics satisfy $$(U_{(1)},\ldots, U_{(n)}) \overset{d}{=} \left(\frac{X_1}{\sum_{i=1}^{n+1} X_i}, ...
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18 views

Order statistics and transformations

Assume random variables X$_1$, ... , X$_n$ and Y$_1$, ..., Y$_n$ are U(0,a)-distributed. Show that Z$_n$ = n*$log\frac{max(Y_{(n)},X_{(n)})}{min(Y_{(n)},X_{(n)})}$ has an Exp(1) Distribution. I've ...
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9 views

Measurability of some statistics

Let's define some framework: Let $X_{i}$ be a real absolutely continuous random variable for all $i=1,\dots,N$. We define the set of observations $X=(X_{1},\dots,X_{N})$ such that it admits ...
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2answers
46 views

Minimum of an Order Statistic with probability

Let $X_1,\ldots,X_n$ constitute a random sample of size $n$ from a normal distribution with $\mu = 0$ and var= 2. Find the smallest value of n such that $P(\min(X^2_1,\ldots,X^2_n)\leq .002) \geq .8$ ...
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1answer
34 views

Density functions for ordered statistics, how to find $\mu$ and the density function of a minimum

Problem: Let Y denote the amount of milk (in gallons) remaining in a 1-gallon container on its expiration date. Suppose that the density function for Y is $f(y)=2y$ for $0 \le y \le 1$. Suppose that a ...
2
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1answer
20 views

Showing a statistic is not complete

Show that the order statistics from an i.i.d sample from the family of all symmetric distributions on the line with known center of symmetry are not complete. So without loss of generality, I ...
2
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1answer
39 views

Sample statistics probability bernoulli trials

Problem There are two restaurants on the campus of a university. Each can feed 120 students. We know that there are 200 students attending the university who will want to eat lunch in one of the ...
2
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1answer
25 views

“Set of observations” is an identity map - why such a framework?

First of all, I do not ask this on CrossValidated since I'm speaking here of mathematical statistics, which explicitly belongs to mathematics. I am currently working on rank tests in order to write ...
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0answers
7 views

Intuition: Distribution Transformations, finding bounds?

Can anyone please give me a normal, intuitive definition of how we find for certain variables when performing transformation of Random Variables. For example: Say $X_1...X_n$ are iid uniform(0,a) ...
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0answers
17 views

Exponential Distribution, Order Statistic

There is a system that requires 4 out 6 comp to work in order. The system breaks as soon as there are only 3 working. X1.....X6 iid with λ= 2 are the failure times of the components 1....6 Denote ...
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2answers
2k views

Inequality concerning sums of ordered variables

With $Y=(Y_1,\ldots,Y_{n_y})$, $n_y \geq 2$, and $Z=(Z_1,\ldots,Z_{n_z})$, $n_z \geq 2$, writing $Y_{(1)},Y_{(2)},..., Y_{(n_y)}$ the ordered values of $Y$ such that: $Y_{(1)} < Y_{(2)} < ... ...
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22 views

Condition Expectation with Order Statistics of Uniform Distribution

Consider the family of uniform distribution $U(a,b)$. 1) Prove that $T = (X_{(1)},X_{(n)})$ is sufficient 2) Calculate the conditional expectation $E(\bar{X}|T)$ using the fact that ...
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1answer
21 views

How to find the median element when each element has three scores?

I know how to find the median with a list of one element but what to do with a list like this one: \begin{array}{|l|cr} & math & info & gestion\\ \hline Nelim & 0 & 4 & 0\\ ...
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4answers
795 views

How to solve this confusing permutation problem related to arrangement of books?

Ms. Jones has 10 books that she is going to put on her bookshelf. Of these, 4 are mathematics books, 3 are chemistry books, 2 are history books, and 1 is a language book. Ms. Jones wants to ...
2
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2answers
13 views

Deriving a statistic of the t distribution with 2 degrees of freedom

Assume I have an independent $X_{1}, X_{2}, \ldots, X_{n}$ with $X_{i} \sim N(i,i^{2})$ and I want to find a statistic that has a t distribution with 2 degrees of freedom. How would I go about ...
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0answers
12 views

Asymptotic Distribution of Sample Quantile

My question is related to the proof below. Why can we say that $F^{-1}(Y_n(x))=X_{[np]}$? I would like to understand better the reason for that. Any help would be appreciated. So
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1answer
49 views

Is $F_{\max(X_1, …, X_n)} = F_{U[0,1]}^n$? Is $F^{-1}=F^{1/n}$?

Suppose that I have that $X_1, ..., X_n$ are iid random variables with distribution function $F$ and $U$ is distributed uniformly on $[0,1]$. If I have that $F(\max(X_1, ..., X_n)) = F_U^n$, that is, ...
1
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1answer
29 views

Proving mean and variance relationships from order statistics

Say we have $ X_1,\ldots,X_n$ be a random sample from a population with mean $\mu$ and variance $\sigma ^2$, How can I got about proving that $$ E\left(\sqrt n \, ...
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1answer
14 views

Finding the limiting distribution of $X^n_{(0)}$

I'm quite confused about how to find a limiting distribution if you're given a minimum order statistic from a random sample. If you can perhaps explain the general steps and theory with this example ...
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0answers
12 views

Sum of the $L$ maximum elements from a collection of RVs

Given a collection of $N$ $\chi^2$-distributed RVs (iid, 2 degrees of freedom), $\{X_1,\dots,X_N\}$ what is the distribution of the sum of the $L$ largest of them? That is, what is the distribution ...
2
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1answer
50 views

The expected value of an order statistic for normal random variables

Let $X_1$ and $X_2$ be a random sample from normal distribution with mean equal to zero and variance $\sigma^2$. Prove $E[X_{(1)}]= \frac{-\sigma}{\sqrt{\pi}}$. May I have to standarize the sample? ...
3
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1answer
42 views

Upper bound on expectation of n non independent random variables

Given $X_i$ are (not necessarily independent) and $\max_{j \leq n} (E|X_j|^p)^{1/p} = \sigma_p < \infty$, $p>1$ Prove that : $E\ \max_{j \leq n} |X_j| \leq n^{1/p} \sigma_p$ Approach: $$ E\ ...
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1answer
38 views

Independence of ratios of order statistics for a sample from uniform distribution [duplicate]

Let $X_1,...,X_n$ be the independent and identically distributed samples taken from an uniform distribution on $(a,b)$. $-\infty<a<b<\infty$. $X_{(1)},...,X_{(n)}$ are the order statistics of ...
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2answers
112 views

NBA Draft Pick for Worst Team Probability

This is actually a question from "A First Course in Probability" by Sheldon Ross, and I have the solution, but am unclear as to why the solution is the case. Would anyone please clarify? The ...
1
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1answer
22 views

Distribution of ordered statistic implies the original distribution?

Suppose we have random variables $Y_1,\dots,Y_n$ valued in $\mathbb{R}$. Denote the order statistics of $Y_1,\dots,Y_n$ as $Y_{(1)}\leq \dots \leq Y_{(n)}$. The order statistics has the distribution ...
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2answers
62 views

Let $U_1, …, U_n$ be independent uniform random variables, and let $V$ be uniform and independent of the $U_i$.

Let $U_1, ..., U_n$ be independent uniform random variables, and let $V$ be uniform and independent of the $U_i$. a) Find $P(V<U_{(n)})$, b) Find $P(U_{(1)}<V<U_{(n)})$ I found that the ...
3
votes
0answers
72 views

Formulating an order statistics problem

I've done some googling, but there are a few different notations employed, and I don't feel I've found an answer I'm comfortable with. Suppose you have a sample of 'i.i.d' observations ...
3
votes
1answer
71 views

Expected maximum absolute value of $n$ iid standard Gaussians?

I have a problem where my errors are normally distributed and I want to know what the expected maximum error is if I repeat the process $n$ times. What is the smallest constant $C$ such that the ...
1
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1answer
65 views

Joint density of first k order statistics

I need to compute the joint density of the first $k$($<n$) order statistics I was trying to do it by getting the marginal density function: $$\int \cdots \int f(x_1)\cdots f(x_n)n!dx_{k+1}\cdots ...
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0answers
41 views

Mean squared error consistency of estimator

Given is the following distribution: $f_\theta(x)=\frac{1}{\theta}$ if $0<x\leq\theta$, and $0$ otherwise; $\theta<0$. I need to show that the maximum likelihood estimator of $\theta$, ...
3
votes
1answer
56 views

Generating a sample of Epanechnikov Kernel

So I am really struggling with this problem and could use some help. Consider the Epanechnikov kernel given by $$f_e(x)=\frac{3}{4}\left( 1-x^2 \right)$$ According to Devorye and Gyofri to generate ...
2
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1answer
68 views

Expectation of largest and smallest order statistic from uniform distribution

Given is a random sample of size n from a uniform distribution with parameters $-\theta$ and $\theta$, $\theta>0$. I'm asked to find a constant $c$ such that $c(X_{n:n}-X_{1:n})$ is an unbiased ...
1
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1answer
18 views

Bound of norm of ordered numbers

Suppose we have two sets of real numbers $x$ and $y$ each with $n$ elements. Suppose I order the sets such that $x_{(1)} \geq x_{(2)} \geq x_{(3)} \geq ... \geq x_{(n)}$ and $y_{(1)} \geq y_{(2)} ...
0
votes
2answers
48 views

Expected values of a sorted sequence

If I have $n$ random variables uniformly distributed on [0,1]. What is the expectance value kth highest value? In other words, what is the expected value of sorting the random variables? For $n=2$ ...