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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2
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2answers
44 views

Is $P(n) = \frac{a n }{b}$ or $\frac{(a+1) n}{b + 1}$?

I investigated Some random data and I was a bit confused. Could be Mathematical coincidence but i'm not sure. Consider the integers $1,2,3,...,a$ Randomly Pick $b$ dinstinct element out of them. ...
-1
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0answers
6 views

Does cross sort relate with cross table?

Does cross sort mean sorting the values in columns as well as values in rows in a cross table? Thank you beforehand.
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0answers
12 views

I think cross sort is a statistical term

What does cross sort mean? Is it sorting the values in columns as well as values in rows.
0
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0answers
25 views

Calculating high order moments

All the information I can find about "higher order" moments seems to stop at Kurtosis which is only the 4th moment. What about the 5th, 6th, 7th, etc? I get that the formula is basically the same, but ...
0
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0answers
21 views

inequality involving expectation of the maximum

For $X_i \sim$ i.i.d with cdf $F_x$, and $\forall c \in \mathbb R$, then, letting $M_n$ denote the maximum observation $$ M_n \le c+ \sum_i^n (X_i - c) \mathbb I(X_i > c) $$ I proved this by ...
0
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1answer
24 views

Maximum of Uniforms

I am trying to show that for iid standard uniforms, $\displaystyle\lim _{ n\to \infty} P(X_{(1)} \le 1 + x/n) = e^{x} \mathbb I_{x \le 0}$, where $X_{(1)}$ is the maximum observation. Using the fact ...
3
votes
2answers
48 views

Inequality involving $X_{(n)}$

I am trying to prove that for random variables (not necessarily iid) $X_1,...,X_n$ that $X_{(n)} \le x + \sum_{i=1}^n X_i \mathbb{I}_{X_i > x}$ for all $x \in \mathbb R$. I tried to show this by ...
1
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3answers
49 views

Finding maximum of two variables

Given $X$ is uniform on $[0, 10]$. Let $$Y = \max(5, X).$$ Determine Var(Y). I'm familiar with how to find the variance of a uniform random variable, as well as the max of two random variables. ...
4
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0answers
213 views

Finding a hidden “heavy” subset of random variables.

Let $X_1,\dots, X_n$ be independent non-negative random variables (with finite expectation and variance), and $0 < m < n$ be a fixed integer such that there exists a subset $S\subseteq [n]$ of ...
0
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0answers
17 views

Writing event involving order statistics in terms of actual observations

For $X_1,X_2,X_3 \sim$i.i.d exponential ($\lambda$), how do I write the event: $$ \{X_{(2)}-X_{(3)} > x ~,~ X_{(3)} > y \} $$ In terms of $X_1,X_2,X_3$ Where $X_{(3)}$ is the third largest ...
2
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1answer
41 views

A more elegant approach to proving independence between $X_{(3)}$ and $X_{(2)}-X_{(3)}$

For $X_1,X_2,X_3 \sim$i.i.d exponential ($\lambda$), I am trying to show independence between $X_{(3)}$ and $X_{(2)}-X_{(3)}$ where $X_{(3)}$ is the third largest observation, i.e. the minimum in this ...
2
votes
1answer
22 views

proving converse of equality involving distribution of minimum observation

Suppose constants $v_n$ are such that: $\lim_{n \to \infty} nF(v_n) =d \in [0,\infty]$ where F is the Cumulative distribution function of $X_i \sim $ i.i.d. random variables. Then the question is to ...
0
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0answers
14 views

Meaningful Extreme value distribution

Extreme value theory (EVT) dictates that the limit distribution of the minimum of the set of i.i.d. Chi-square random varibales $\{C_1,C_2,\cdots,C_n\}$ is Weibull. The Weibull distribution has ...
-2
votes
1answer
44 views

How to find PDF of ordered random variables? [closed]

Assumpion: Let $X_1, X_2, \ldots, X_L$ be $L$ independent and identical random variables (RVs). Let $F_{X_i}(x_i)$ and $f_{X_i}(x_i)$ be CDF and PDF of $X_i$. Suppose that $F_{X_i}(x_i) = F_X(x_i)$ ...
0
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1answer
21 views

Extreme value distributions of uncountably infinite set of random variables

Let us suppose that we have an uncountably infinite set $A=\{x_1,x_2, \cdots\}$ of i.i.d. random variables $x_i$, say with gamma distribution. Are minimum and maximum extreme value distributions ...
0
votes
1answer
23 views

Dividing by epsilon and taking the limit of epsilon to zero

In deriving the joint distribution of two order statistics, there is the following step (F(x) is the Cumulative Dist Function at x, f(x) is the PDF at x): $$ ...
1
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0answers
22 views

A Question About Probability of ratio of $\max(\cdot)$?

In My field , I reached to this problem. Assumptions: Consider $x_i,\hat{x}_i$ are iid (identical and independent) samples of a joint distribution (e.g., exponential). And also, assume we have $N$ ...
1
vote
1answer
31 views

Hypothesis testing with order statistics

I know there have been a lot of questions asked on this forum relating to order statistics, so, hopefully, this is not going to be a duplicate. I am trying to understand how I should go about ...
1
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0answers
40 views

Comparing second order statistic of n draws and first order statistic of n-1 draws

Let $x^{(i),n}$ be the $i$-th order statistic of $n$ draws from some distribution function $F$ on $[0,1]$. I have a claim that $$\mathbb{E}[\max \{ x^{(2),n}, m\}] < \mathbb{E}[\max \{ ...
1
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1answer
42 views

Difference of Ordered Uniform Random Variables

Let $X_1, X_2,..., X_n$ be $n$ random variables distributed uniform(0,1) and $X_{(1)},X_{(2)},..., X_{(n)}$ be the ordered statistics of $X_1,...,X_n$ such that: $X_{(1)} < X_{(2)} < ... < ...
1
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0answers
29 views

Median of the mean/max of an iid sample of exponential variables

I would like to know if one can obtain a simple analytic expression of the median of $T_n$ defined by $$ T_n = \frac{\overline X_n}{X_{(n)}}, $$ where $\overline X_n$ is the empirical mean and ...
2
votes
1answer
103 views

what does `ensemble average` mean?

I'm studying this paper and somewhere in the conclusion part is written: "Since this rotation of the coherency matrix is carried out based on the ensemble average of polarimetric scattering ...
3
votes
2answers
42 views

Probability of ordered sequence of Gaussian Distributions

Suppose there are 5 cars racing on a match. Suppose we know that the distributions of their finishing time are roughly in some Gaussian distributions $N(\mu_i,\sigma_i^2), i=1,2,3,4,5$. How can I ...
10
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0answers
101 views

Looking for references related to an inequality in order statistics

I was reading the paper "on the minimum of several random variables". In example 10 item (ii) it states: Let $1\leq k\leq n$. Let $g_i,i\leq n$, be independent $N(0,1)$ Gaussian random variables. ...
2
votes
3answers
45 views

Difference between the largest and second largest observations from a sample of iid normal variables

tl;dr: what can we say about the distribution of s_1 - s_2, where s_1 and s_2 are the largest and second largest draws from a sample of N iid normal variables? "Who's the world's greatest ...
3
votes
1answer
48 views

Sampling distribution of sample trimmed (truncated) mean

It is elementary probability theory that the sample mean of an i.i.d. sample follows normal distribution, if the background distribution is normal. But what about the trimmed mean? Is there any result ...
2
votes
1answer
58 views

Probability calculation involving order statistics

There are $n+1$ independent and identically distributed random variables with the same distribution as $D \sim \text{Exp}(\mu)$, denoted by $D, D_1, D_2, \ldots, D_n$. Define event $E_1$ as "$D$ is ...
0
votes
2answers
50 views

Probability of being highest OR second highest draw from a distribution

In my situation, I have a distribution F(x) over some compact interval. Say I take $n$ iid draws from the distribution. I want to find the probability that one draw, $x_i$, is the highest of the $n$ ...
1
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1answer
58 views

Probability ( kth order statistics <x , (k+1)th order stat >x)

consider $n$ iid draws according to some cdf $F$. What is the probability of the following event: the $k^{\text{th}}$ highest value is smaller than $x$ AND the $k+1$ highest value is larger than $x$. ...
1
vote
1answer
33 views

Conditional distribution of the sum of k, independent ordered draws

Suppose I make k independent draws from the same distribution (say uniform). The distribution of the sum of these order statistics should equal the distribution of the sum of k independent random ...
1
vote
1answer
60 views

Sum of order statistics

Is there a general expression for the pdf of the sum of the order statistics? Suppose they are all drawn independently from the same distribution.
1
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1answer
31 views

Distribution of non-identical exponential order statistics

This is a problem given as practice for an upcoming exam (without solution provided). Let $X$ and $Y$ be independent exponential random variables, with $X = \theta e^{-\theta x}$ and $Y = \lambda ...
0
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0answers
19 views

Expected value of k'th largest Gaussian

Suppose $\{g_i\}_{i=1}^n$ are i.i.d. standard normal. Let us sort them in terms of absolute values to obtain $\tilde{g}_i$ where $\tilde{g}_1$ is the largest. Is there a nice (simple,closed form) ...
0
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0answers
12 views

Empirical likelihood method to Find the order of a parameter given a set of constraints

Firstly we assume that $X_1,...X_n$ are order statistics($X_i\leq X_{i+1}$) from an i.i.d sample of random variables and let $r$ be integer and $r\geq1$. Start with the Equation (1) as following ...
0
votes
0answers
15 views

(Empirical likelihood method) Find the order of a parameter given a set of constraints

Firstly we assume that $X_1,...X_n$ are order statistics($X_i\leq X_{i+1}$) from an i.i.d sample of random variables and let $r$ be integer and $r\geq1$. Start with the equation (1) \begin{equation} ...
0
votes
1answer
52 views

Order statistics when variables have different distributions

Let a,b and c be random variables, where a~U[0,8] and b~U[3,8]. Let c=max{a,b}, What is the mean of c? In general, let $(x_1, ..., x_n)$ be independently distributed in different supports. What is ...
0
votes
0answers
90 views

Convert min to max probability

Assuming $Y=min(w_1,\ldots,w_n)$ , $w_i\sim N(\mu,\sigma^2) i.i.d$ I want to express $Y$ in terms of the $Q$ function. Knowing that $Y=min(w_1,\ldots,w_n)=-max(-w_1,\ldots,-w_n)$ $P(Y\leq y)= ...
0
votes
1answer
49 views

+conditional density of sum of random variables

Let $X$ and $Y$ be two independent random variables distributed uniformly on $[0,1].$ How can I compute the following density: $$f(X+Y|\max\{X,Y\}<a)\ \text{for}\ a\in [0,1]?$$ More generally, ...
2
votes
2answers
108 views

The maximum and minimum of five independent uniform random variables

Let $U_1,\dots,U_5$ be independent, each with uniform distribution on $(0, 1).$ Let $R$ be the distance between the minimum and maximum of the $U_i^{'}$s. Find the joint density of the max and the ...
0
votes
0answers
35 views

Probability Solution for Ordered Statistics

I had a question on a solution I saw and the answer is fairly unclear. The question is below and the answer follows it. Let $X_{(1)}\le X_{(2)}...\le X_{(n)}$be the ordered values of n independent ...
0
votes
1answer
15 views

fair distribution of table tennis players in roster

Say I have 18 players who have been ranked in order of ability 1-18. I need to make up 6 teams each with 3 players. How do I choose the members of each team such that based on the ranking of the ...
1
vote
1answer
28 views

Use of binomial coefficient in order statistics

Appreciate all your help with the following: Let $U_{1}, ... U_{n}$ be n independent and uniformly distributed on (0,1) random variables. Let $U_{(1)} \geq U_{(2)} \geq ... \geq U_{(n)}$ be an ...
1
vote
0answers
20 views

showing $E(X_{(i)}| \sum_{i=1}^5 X_i)=1.8E(Z_{(i)})$

suppose 5.5,3.5, 2.5,4.5,2 be a random sample from of gamma distribution with parameters of $ \beta,\alpha=2$. if $Z_{(i)}$ be i-th order statistic a random sample of size 5 from $\Gamma(2,1)$, how ...
0
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0answers
42 views

two-parameter exponential distribution

Now, I have some problems about the distribution of random variable. In my work, let $X_{i}$ for i= 1 to n be iid random variable from two-parameter exponential distribution. We known the Mgf of X is ...
1
vote
1answer
55 views

How can I can calculate the CDF of the random variable?

Assuming that $H$, $N$ are random variables in which $H$ is distributed following exponential distribution with mean value $\Omega$, and $N$ is a Gaussian random variable with probability density ...
2
votes
1answer
50 views

P-value - test at $5 \%$ if there is significant difference in fuel consumption between the two petrol grades?

A car owners want to investigate if gasoline consumption of his car depends on the fuel octane number. He therefore intend to "premium" and "regular" at random and computes each time the ...
1
vote
1answer
27 views

(bookexercise) sign test of $H_0: \bar m = \bar 25$ against $H_1: \bar m < \bar 25$ there we have 15 known random observation

Follwing data desribes the measured fracture strength of 15 randomly selected units made by a new ceramic material $$20, 42, 18, 21, 22, 35, 19, 18, 16, 20, 21, 32, 22, 20, 24$$ At a previously ...
2
votes
2answers
72 views

Compute the order statistics probability of $P(X_{(3)} < aX_{(2)})$

Suppose $a>1$, $X_1,X_2, X_3,$ and $X_4$ are four iid continuous random variables with $U(0,a)$ random variables so their common density function is $f(x)=\frac{1}{a}, 0<x<a.$ $a.$ joint ...
1
vote
1answer
38 views

determine the confidence level for the confidence interval $I_{\bar m}=(x_{(2)},x_{(9)})$

Let $x_1,...,x_{10}$ be a sample from a continuous random variable $X$ with the median $\bar m$. Determine the confidence level for the interval $I_{\bar m}=(x_{(2)},x_{(9)})$. $x_{i}$ is the ...
0
votes
2answers
46 views

True or False — mutual independence and identical distribution? [closed]

Suppose $X_1, X_2,\ldots, X_N$ are mutually independent and identically distributed. Let $F_X$ denote the cdf of $X_1$. Would this statement be true or false? I'm totally lost as to how to approach ...