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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32 views

Uniform Distribution Estimator (not MLE)

Does anyone know where the estimator $\hatθ = X _{(1)} + X_ {(n)}$ for a U(0, θ) distribution comes from? Where: $X _{(1)}$ = min$_i (X_i)$ $X_ {(n)}$ = max$_i (X_i)$ I know it is not the MLE, ...
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1answer
22 views

Order Statistics with two Groups of Draws

Let $X_{1},X_{2},\ldots,X_{m},\ldots,X_{n}$ be independently drawn from a distribution $F$ and let $Y_{k}^{(n)}$ be the $k$-th order statistic (Convention: ...
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0answers
17 views

Reshuffling the order statistic of uniform at midpoint

Let $U_1,\dots,U_n$ be i.i.d. drawn from the uniform distribution on $[0,1]$ and let $U_{(1)} \le U_{(2)} \le \dots \le U_{(n)}$ be their order statistics. Assume that $n$ is even and that we take ...
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7 views

Distribution of sample minimum after bivariate selection (double truncation)

Let $X$ and $Y$ be two RVs with joint distribution $$ (X,Y)\sim \text{Normal}(\mu,\Sigma) $$ Suppose that there is selection on $X$ and $Y$, such that we observe a vector of realisations of $X$, ...
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12 views

Analytical expression for expected the n-th and (n-1)th order statistic of minimum of two uniform random variables

I have two independent random variables $X_1 \sim Uniform(b,1)$ and $X_2 \sim Uniform(0,1)$, where $0<=b<1$. Their CDF's are: $$ F_{X_1}(z) = \begin{cases} 0 & z<b \\ \frac{z-b}{1-b} ...
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16 views

Probability and expectation of three ordered random variables

I am really stuck on the following question during my exam preparation. Let $X_1$, $X_2$ and $X_3$ ...
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11 views

Distribution of a single order statistic (descending order)

I am looking for the distribution of single order statistic ($F_{(r)}(x)$, i.e., the CDF of $X_r$) that is in the form $X_1\geq X_2 \geq \cdots X_r \geq \cdots \geq X_n$. All $X$ are independent and ...
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1answer
21 views

Joint distribution of largest and last sample

Suppose 12 i.i.d. integer samples are taken from 1 to 10 with uniform probability. What is the joint distribution of the 3 last numbers ($x_{10},x_{11},x_{12}$) and the 3 largest numbers ...
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1answer
43 views

Density of maximum of order statistics

Let $X_{1},X_{2},\ldots,X_{n}$ be independently drawn from a distribution $F$ and let $Y_{k}^{(n)}$ be the $k$-th order statistic (Convention: $Y_{1}^{(n)}>Y_{2}^{(n)}>\cdots>Y_{n}^{(n)}$). ...
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1answer
16 views

Marginal Order Statistics

I'm trying to find the marginal statistic of 2 order stats from the full joint and I can't figure out the bounds on the following problem. Consider $0\leq Y_1\leq Y_2\leq Y_3\leq Y_4 < 1$ from the ...
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2answers
38 views

An Estimator Based on Exponential RVs

Let $X_1$, $X_2$, $\cdots$, $X_n$ be $n$ random variables independently sampled from the exponential distribution $\text{exp}(1)$. Suppose $k \leq n$, and $X_{(k)}$ is the $k$-th order statistic, ...
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46 views

upper bound of the expectation of the maximum of a bunch of random variables

I met one technical problem in my project, and I wish some reference here.. Suppose $\{X_i\}_{i=1}^n$ are independent random variables. Are there some inequalities that give upper bound of ...
2
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1answer
51 views

On expectation of maximum of gaussians

Let $X_1,\ldots,X_n$ be i.i.d $\mathcal{N}(0,1)$ random variables. I am trying to prove that \begin{align} (a)\ \ \mathbb{E} \left[ \max_{i}X_i\right] & \asymp\mathbb{E} \left[ ...
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4answers
26 views

Density function of ordered statistic of a uniform distribution from $0<t<\theta$

Assume $X_1, X_2, ..., X_n$ is a random sample of a uniform random variable $X$ on the interval $(0,\theta)$. What is the density function for $X$ and the density function for $X_{(n)}$ where ...
2
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0answers
34 views

Ratio of Order Statistics goes to infinity?

Is there a sequence of distributions $(F_n)$ with $F_n(0)=0$ such that $\frac{E[\max(X_{1,n},X_{2,n})]}{E[X_{1,n}]} \to +\infty $ as $n\to +\infty$, where $X_{1,n}$ and $X_{2,n}$ are ...
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1answer
17 views

Correlation between a random variable and its rank

Let $X_1,\ldots,X_n$ be a random sample from $U(0,1)$ and $X_{(1)}<\ldots<X_{(n)}$ be the corresponding order statistics. Define, $$ R(X_1) = r\quad \text{if}\quad X_{(r)} = X_1;\quad r = ...
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0answers
26 views

Find the distribution of $ \sum_{i=1}^{4} Y_i $, the first four order statistics of five exponential RVs.

The Problem: Consider $X_1, ..., X_5$, $5$ iid, exponentially distributed random variables with $\lambda = 1$. Let $Y_1 \lt Y_2 \lt Y_3 \lt Y_4 \lt Y_5$ be their order statistics. Find the ...
5
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2answers
47 views

What proportion of my friend's calzones were the best he had made so far?

(True story:) A friend of mine was making a calzone and said it wasn't the best he had ever made. It got me wondering, what fraction of the calzones he has made were the best he had made at the time? ...
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1answer
25 views

Showing $X_{(n)}$ is an unbiased and consistent estimator for $\theta$.

Let $X_1,X_2,\ldots, X_n$ is distributed iid $\mathrm{Uniform}(0, \theta)$ with $\theta$ in being real positive. Show $\frac{n+1}{n}X_{(n)}$ is an unbiased and consistent estimator for $\theta$. I ...
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2answers
33 views

Modeling “Time Until Event Occurs”

I'm having trouble understanding why "the time until an event occurs" is modeled as: $S(t) = P[T > t]$ where $T$ is time until event and $t$ is some time. I don't see how $P[T<t]$ or $P[T=t]$ ...
6
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3answers
167 views

Are there order statistics for a Gaussian variable raised to a power?

Let $X$ be a random variable with a standard normal distribution. Let $Y = |X|^{2p}$. I am trying to find the distribution for $Y_{(n)}$, i.e., the largest value of $Y$ out of $n$ samples. I have ...
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2answers
34 views

Probability distribution of the difference of two uniform variables

What is the PDF or CDF of $|X_1 - X_2|$ (absolute value of the difference of two variables). When both $X_1$ and $X_2$ has different Uniform distribution. Note: We can look at the as the distribution ...
2
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0answers
21 views

continuity of the sort operator

here's my question. Let $\mathcal{S}\colon\mathbb{R}^n\rightarrow \mathbb{R}^n$ be the function that map a real vector into a vector having its components ordered either way, ascending or descending. ...
0
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1answer
53 views

How to find this integral

I want to fin the following integral: $$\int_o^{x_n}\int_0^{x_{n-1}}\cdots\int_0^{x_2}n!\lambda^ne^{-\lambda(x_1+x_2+\cdots+x_n)}dx_1dx_A\cdots dx_{n-1}$$ I think that the answer is $\lambda ...
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2answers
83 views

Is Schoolyard Team Picking Fair?

Two captains are picking teams for dodgeball. They both will pick players in order of their skill level, best to worst. John gets first pick, Sarah gets the next two picks, and John gets the 4th. If ...
0
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0answers
18 views

Median of the sample median equal to the median of the distribution?

I'm taking a course in mathematical statistics this semester. I don't know how much consensus there is among mathematicians about notations and definitions, so I'll just define the essentials. Let me ...
0
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0answers
23 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 ...
0
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1answer
22 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
49 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
109 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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1answer
53 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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0answers
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
68 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 ...
0
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0answers
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
52 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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48 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 $E(|X_r - X_r'|)$, ...
0
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1answer
56 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 ...
0
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0answers
45 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. ...
0
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1answer
35 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
34 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 $T(x) = (X_1,...,X_n)$ ...
2
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3answers
70 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 ...
0
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0answers
28 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}, ...
1
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0answers
24 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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0answers
10 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 ...
2
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2answers
47 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 $P(\min(X^2_1,\ldots,X^2_n)\leq .002) \geq .8$ ...
0
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1answer
37 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
27 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
votes
1answer
45 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
votes
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 ...
0
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0answers
12 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) ...