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.
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Distribution of the minimum among $N$ random variables
I read this post https://stats.stackexchange.com/questions/220/how-is-the-minimum-of-a-set-of-iid-random-variables-distributed where I can find how to compute the density distribution of the minimum ...
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Probability of the minimun value of n geometric variables be y
I already have the answer to this problem but i would like to understando if and why my resolution is correct or no. Could anyone help me, please?
Question: Let $X_1, X_2, X_3, \dots X_n$ be ...
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Help with a proof regarding first order stochastic dominance
I am struggling with (/failing at) the following proof in one of my theorems (Economics):
$$\int_a^b f(x)^2F(x)^{n-2}[1+(n-1)ln(F(x)] \leq \int_a^b g(x)^2G(x)^{n-2}[1+(n-1)ln(G(x)] $$
Where n>1, $0\...
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Minimum of n iid chi-squared with k degrees of freedom
I am currently trying to compute the expectation (a nice closed form distribution would be even better) of the minimum of n iid chi-squared with k degrees of freedom.
After struggling with the pdf ...
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How to prove $E(|T_{(1)}-T_{(2)}|)=O(n^{-1})$.
Suppose $T_i\sim F$ i.i.d. for $i=1,\,\cdots,\,n$ where $F$ has continuous density function $f$ on $[0,\,1]$ such that $\inf_{t\in[0,1]}f(t)>0$.
Prove that $E(|T_{(1)}-T_{(2)}|)=O(n^{-1})$.
Here ...
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Expectation of runtime
I am running a program and its runtime follows a normal distribution $T_i = N(t,\sigma).$ Now, I am running $n$ programs in parallel ($T_1, T_2, \cdots, T_n$ are i.i.d.), and the finish time for $n$ ...
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Variance of Order Statistics
I have a question about bounding the variance of order statistics.
Given that for $i \in \{1,\cdots,\lambda\}$, denote $Bin(s,\frac{1}{n})$ to be a binomial random variable with success probability $\...
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Empirical CDF and Order statistics
Screenshot of a result with sketch of its proof
I have understood the parts above the red arrow. I'm stuck after that. While I can intuitively understand that the difference between the order ...
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The expectation of the minimum of two IID binomial distribution Bi(m,1/M) only depends on m/M
I want to prove following proposition:
$X,Y \sim Bi(m,1/M)$, $X$ and $Y$ are independent. Prove that $f(m,M)= \mathbb{E} \min(X,Y) $ only depends on $\dfrac{m}{M}$.
I have done the following ...
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Sharp Concentration of General Order Statistics of Gaussian
Suppose $X_{1}, \ldots, X_{n}\overset{i.i.d.}{\sim} \mathcal{N}(0,1)$. Consider the order statistics $X_{(1)} \geq X_{(2)} \geq \ldots \geq X_{(n)}$. Then the Gaussian maxima $X_{(1)}$ has sub-...
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Distribution of a sum of first and last order statistics
I can't find anywhere on the internet a solution to the following exercise:
Let $X = (X_1, \dots, X_n)$ be a sequence of i.i.d random variables from exponential distribution. Find the distribution of ...
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Unjustified trick for computing difference in order statistics of exponential distribution
It is well known that if $X_1,...,X_n \overset{iid}{\sim} Exp(\lambda)$, then $Y_i:=X_{(i)}-X_{(i-1)} \overset{ind}{\sim} Exp((n+1-i)\lambda)$ for $i=1,...,n$ ($X_{(0)}:=0$). The standard way is to ...
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Order of appearance of distribution support in an i.i.d sequence
Consider a discrete distribution $\mathcal{D}$ with probabilities $(p_1,\ldots,p_n)$ over numbers $1, \ldots, n$. In a sequence of i.i.d. draws from $\mathcal{D}$, $X_1, X_2, \ldots$, let $X^{(i)}$ ...
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What is the probability that $\max(X_1, X_2, \ldots, X_n) < \min(Y_1, Y_2, \ldots, Y_n)$ where the $X_i$'s and the $Y_i$'s $\sim U[0, 1]$? [closed]
What is the probability that $\max(X_1, X_2, \ldots, X_n) < \min(Y_1, Y_2, \ldots, Y_n)$ where:
$X_i \sim U[0, 1]$ and $Y_i \sim U[0, 1]$, and
$X$ and $Y$ are independent continuous random ...
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Independence among order statistics
Given i.i.d uniform variables $X_1, ..., X_n \sim U[0,1]$, is it true that the event $X_i \leq \min_{k \in [i-1]} X_k$ is independent of $X_j \leq \min_{k \in [j-1]} X_k$ for any $i > j$?
The ...
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Median:Mean::Percentile:?
Here is a simple question that I am hoping has a simple answer.
The median and mean are both estimates of the central tendency of a random variable. They both have different advantages in different ...
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Conditional probability of order statistics
Let $X_1, \ldots, X_n$ be $n$ independent random variables distributed according to some cumulative function $F$. Let $X_{(1)}, \ldots, X_{(n)}$ denote the order statistics. It is a well-known result ...
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Calculating Probability of a horse winning based on expected position [closed]
My question is this:
If you have 4 horses (A , B , C and D ) in a race:
Horse A is expected to get position 2
Horse B is expected to get position 2
Horse C is expected to get position 3
Horse D is ...
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Show that $K(u)=\sum_{k=0}^r P_k(0) P_k(u) \mathbf{1}_{\{|u| \leq 1\}}$ is a kernel of order $r$
I have a question concerning the construction of kernels wit orthogonal polynomials. The instructor defined the orthogonal polynomials as
$$P_0(x)=\frac{1}{\sqrt{2}}, P_m(x)=\sqrt{\frac{2 m+1}{2}} \...
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Finding the probabity of a minimum from a joint marginal probabity
Suppose I have the following marginal
$P(Y) = \sum\limits_{x} P(Y|X)P(X)$
Assuming $Y_1 \cdots Y_n$ are iid and same for $X$
Thus, to obtain $Y=i$ we compute :
$P(Y=i) = \sum\limits_{x} P(Y=i|X)P(X)$
...
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Reference on the sum of absolute differences between $n$ samples from a random variable
Let $X_1, X_2,\ldots X_n$ be $n$ samples taken of a random variable with a given distribution (so in particular it is i.i.d.). Is there literature on or a name for the random variable defined by $$Y = ...
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Confidence Interval - Meaning and Interpretations
Could someone please help me understand what exactly "Confidence" in confidence interval actual means?
Does it mean, that, (on average or exactly?), we can say, that (for a 90% confidence ...
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Asymptotic decomposition of CGF
Consider a sample of $n$ positive, unbounded and i.i.d. random variates $\omega_1, \dots, \omega_n$ with cumulant-generating function $C^\omega(\kappa) = \log \mathbb{E}[\exp(\kappa\omega)]$. Suppose $...
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can I use central limit theorm with order statiscs of unknown disribution
Given a set of iid variables with unknown underlying distribution Suppose that $\mu = 75, \sigma = 10 $ and the sample size $n=50$ and the minimum of that sample is 58. To compute the probability of ...
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How to obtain confidence Intervals for minimum order statiscs
After computing the minimum order statics for iid normally distributed data as follow :
P(X1< x ) = 1-P(X1 > x )
= 1-P(X1 > x, X2 > x, ... ,Xn > x )
= 1-P(X1 > x) P( X2 > x) ... ...
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Moments of Order Statistics
Problem: Taken from A Probabilistic Theory of Pattern Recognition by Devroye et al.
Let $U_{(1)},\dots,U_{(n)}$ be order statistics of $n$ i.i.d. $\text{Unif}(0,1)$ variables. Show that
$$\mathbb{E}[...
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Looking for distributions for which distribution of maximum has a known distribution where the base distributions has continuous positive supports.
Suppose $X$ and $Y$ are two positive continuous distributions and they are independent. Let
$Z = \max(X,Y)$. The distribution of $Z$ can be calculated using the probability $$P(X \le z, Y \le z)=P(X \...
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Simple treatment of order statistics in the conditional part
Let $X_1,...,X_n$ be iid Uniform(0,1) and $X_{(k)}$ be the kth order statistic in increasing order. It seems intuitive that for any $x\in (0,1)$, $P((X_{(1)},...,X_{(n-1)})\in A|X_{(n)}=x)=P((Y_{(1)},....
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Conditional density of the second highest order statistics
Notation:
$Y_1$: Highest order statistics of $(N-1)$ players' valuation.
$F_n^M:$ The distribution function of the highest $n$th order statistics of $M$ players.
$f_n^M:$ The density of the highest $n$...
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Expectation of the maximum order statistic of Pareto distribution
Suppose X follows the Pareto Type I given,
$$P(X>x) = \Bigl(\frac{\gamma}{x}\Bigl)^\alpha, \quad x\geq \gamma,\; \alpha>0.$$
Then,
\begin{align*}
P(X=x) &= \frac{d}{dx}P(X\leq x)\\
&= \...
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How to find the minimum-maximum of $ N( 0, 1) $
Given that two random variables $J,P∼N(0,1)$ are independent. Show how to compute the following and provide the answer:
(i) $E[\min(J,P)]$ and $E[\min(J^2,P^2)]$ and
(ii) $E[\max(J,P)]$ and $[\max(P^2,...
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Min-Max of Standard Normal Distribution
Given that two random variables $K,L∼N(0,1)$ are independent. Show how to compute the following and provide the answer:
(i) $E[\min(K,L)]$ and $E[\min(K^2,L^2)]$ and
(ii) $E[\max(K,L)]$ and $[\max(L^2,...
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Convergence of the ranking list of $N$ binomial variables.
I have $M$ random variables $v1, v2, ..., v_M$, and $\forall i \in [M]$, $v_i$ follows a binomial distribution with the probability of $p_i$. W.l.o.g., I suppose that $1 > p_1 > p_2 > ... >...
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Multivariate Order Statistics (Binomial Distribution)
If $X_i\sim B(1, p_i), i=1, ..., n,$ all independent. Let $Q=\min(X_1,\ldots, X_n)$ and $W = \max(X_1, \ldots, X_n).$ Find the correlation between $Q$ and $W.$
Computing correlation by the usual ...
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Bapat–Beg theorem for two order statistics
Bapat–Beg theorem gives the joint probability distribution of order statistics of independent but not necessarily identically distributed random variables in terms of the cumulative distribution ...
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Permutation and ordered statistic
Let $X_1,…,X_n$ be i.i.d. random variables. Are these two equalities correct?
$P[X_{(2)}<x_2,…,X_{(n)}<x_n| X_{(1)}=x_1]=\\
=n!P[X_{2}<x_2,…,X_{n}<x_n| X_{1}=x_1]=\\
=n! P[X_{2}<x_2]…P[...
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Why is $\bar X$ mirrored from zero (additive inversion) after the described operation, and what is its maximum value?
Preface:
I am currently preparing for a statistics olympiad qualifiers of the one of my country’s universities. Yesterday I came across one task, in which I do not quite understand the intuition ...
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Upper bound the even momentum of the squared vector length
I am reading a proof the follow lemma
Let $w = \frac{1}{\sqrt{d}} (1, \dots, 1)$. Then, for every unit
vector $a (a_1, \dots, a_d)$ and for all $k=0,1,\dots$ we have
$E[Q(a)^{2k}] \leq E[Q(w)^{2k}]$ ...
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Joint density of $k$th upper order statistics [duplicate]
I'm trying to show that for $X_1,...,X_n$ iid with distribution function $F$ which has density $f$, the corresponding order statistics $X_{1,n}>\cdots>X_{n,n}$, the joint density of the $k$th ...
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conditional distribution of sample given maximum
Given an i.i.d. sample $X_1,\dots, X_n$ from the uniform distribution on $[0,\theta]$, and denoting their order statistics by $X_{(1)} < X_{(2)} < \cdots < X_{(n)}$, it is easy to show that $...
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How to Find the Median of a Probability Distribution Function?
I am interested in learning about how to find out the Median (https://en.wikipedia.org/wiki/Median) for some "generic" Probability Distribution Function (https://en.wikipedia.org/wiki/...
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Is there a formula for the $k$th smallest number of $n$ real numbers?
Let $a_{(k)}$ be the $k$th smallest number of $n$ real numbers $a_1,a_2,\ldots,a_n$. Is there a formula for $a_{(k)}$?
I know $a_{(1)}$ can be found recursively using
$$a_{(1)}=\min\{a_1,a_2,\ldots,...
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Expected value of difference of two ordered statistics
I have some questions about computing the difference between two order statistics.
Given that for $i,j \in \{1,\cdots,\lambda\}$, denote Bin(s,p) to be a binomial random variable with success ...
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Probability of minimum order stats in case of discrete distribution (Poisson)
$X_1$,$X_2$ ~ $P(1)$, $Y$ = Min{$X_1$,$X_2$} $\mathbb\quad P(Y=1)$ =$?$
I tried this in two different ways and I am getting a different answer in both:-
Firstly I made PDF of $X_{(1)}$ = $n(1-F(x))^{...
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Expectation of the maximum of independent poisson random variables
I am trying to prove the following estimation of the expectation of the maximum of independent poisson random variables. I've become interested in this problem while reading Joel A. Tropp's "An ...
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A problem from Mathematical Statistics and Data Analysis [closed]
This problem is from Ch. 5, Mathematical Statistics and Data Analysis, 3rd edition, stated as follows,
In addition to limit theorems that deal with sums, there are limit theorems that deal with ...
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Expected value of $i$'th smallest random variable
Suppose we sample $k+1$ i.i.d. random variables $X_i$ uniformly at random in $[0,1]$.
What is
$$
\mathbb{E}[X_i | X_1 > X_2 > \cdots > X_k \cap X_{k+1} > X_k]?
$$
To start, suppose we ...
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For independent $X_i \sim Exp(\lambda_i)$, why are $\min\{X_1, ..., X_n\}$ and $[\max\{X_1, ..., X_n\} - \min\{X_1, ..., X_n\}]$ independent?
As someone who is trying to pick probability and statistics back up after not using it for the last 4 years, I'd like to ask for some help with a question I encountered on a MIT OCW midterm test (see ...
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Distribution of a difference of an order statistic and the sum
I have rather a pecular problem (that I am afraid is unsolvalble).
Let $X_1,...,X_n$ and $Y_1,...Y_m$ be distributed independently according the same distribution $f(.)$ (e.g. each $X_i$ and $Y_i$ is ...
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Max of two random values distribution
Consider code like this. Briefly. We throw two dice NUMBER_OF_RUNS times. Each dice has BIN_COUNTS faces. We pick the larger value. Then we plot M vs the number of times we see M as the largest value ...
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