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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. Sometimes 'order statistic' is used to mean the whole ...

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Minimum of i.i.d. Random Variables [duplicate]

What importance does the minimum of an identical and independently distributed random variables play in probability distribution?
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1answer
49 views

sampling distribution of order statistics

Let, $X_{1},X_{2},\ldots,X_{n}$ be an i.i.d. sample from $R(1-\theta,1+\theta)$. Show that, $(X_{(1)},\bar{X},X_{(n)})$ is sufficient for $\theta$. Ans: $f(x_{1},x_{2},\ldots,x_{n})=\frac{1}{(2\theta)...
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Cumulative distribution of X_{1} = min{X_{1},X_{2}}

So I have $X_{1},X_{2} \sim^{iid} Unif(\theta,1)$, hence the density function is $f_{X_{i}}(x) = \frac{1}{1-\theta}\mathbb{1}_{(0,1)}(x)$ What is the cumulative distribution of $X_{1} = min\{{X_{1},...
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1answer
32 views

Limiting distribution of first order statistic x

Good evening everyone, I am attempting a problem on limiting distributions. The problem is as seen in the picture above. Could you please help me with part c) of this problem. The answer as per the ...
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1answer
64 views

$X_i \sim \text{Uniform}(0, \theta)$ iid; $Y = \max{(X_1,..,X_n)}$. Why is $\theta$ necessarily larger than $y$?

I'm going through Statistical Inference by Casella & Berger, and on page 419, in the intro section of interval estimation there is the following example (note: most of the text was left out as it'...
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1answer
37 views

maximum likelihood estimator

Suppose, $X_1,X_2,\ldots,X_n$ be a random sample from $U(\theta-2,\theta+2)$. Define, $X_{(n)}=\rm{max}\{X_1,X_2,\ldots,X_n\}$ and $X_{(1)}=\rm{min}\{X_1,X_2,\ldots,X_n\}$. Then which of the ...
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1answer
16 views

Is there a canonical probability distribution on an ordering/permutation

What is a good way of defining a non-uniform probability distribution on a permutation of k objects? For example, suppose the parameter was an ideal ordering, and the probability of an ordering was ...
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56 views

Distribution of maximum over minimum of Weibull(alpha, 1)

Let the PDF of Weibull $(\alpha, 1)$ be $$ f_X (x) = \alpha x^{\alpha-1} \exp\left( - x^{\alpha}\right) $$ I know from probability integral transfrom that $$ X_{(1)}^{\alpha} \overset{d}{=} Z_{(1)...
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1answer
36 views

CDF of top $k$ value among $n$ RV's when the value of one of them is known

Suppose that $X_i$ is a random variable whose CDF if $F$ and pdf is $f$, $i=1,\cdots, n$. Assuming independency, the $m$-th highest value can be found using order statistics. What would be the CDF ...
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PDF for the ith ORDERED uniformly random sample compared to an evenly spaced sample

Let $r_1 ≤ r_2 ≤ ... ≤ r_N$ denote an ORDERED set of N realizations of real numbers that are uniformly random on the number line from 0 to 1. Let $R_1 < R_2 < ... < R_N$ denote a set of ...
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225 views

a fast uniform order statistic generator [closed]

Can someone provide me with the mathematical expression for this code/function as a fast way to generate $n$ sorted $U[0,1]$ random numbers: ...
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17 views

Distributing elements over multiple buckets of a varying size

If one has b buckets that each have a size of b_i and one has to place n elements in the buckets. How can I mathematically described the amount of different permutations the elements can be placed in ...
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1answer
102 views

Density estimation using (different) order statistics

I need to estimate a univariate distribution $F$ as flexibly as possible. However, I do not observe draws from $F$ directly. Each observation $x_i$ is the minimum of $a_i$ draws from $F$, where $a_i$ ...
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1answer
89 views

Mean of maximum of exponential random variables (independent but not identical)

I am looking for the the mean of the maximum of N independent but not identical exponential random variables. I found the CDF and the pdf but I couldn't compute the integral to find the mean of the ...
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33 views

Order Statistics invovling Continuous Distribution [closed]

I am reading a book Introduction to Probability by Joe Blitzstein, Jessica Hwang. I was going though a section on Order Statistics, which I have mentioned below. Let $X_1, X_2, \cdots, X_n$ be i.i.d ...
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1answer
100 views

Distribution of $\sum_{j=1}^n\ln\left(\frac{X_{(j)}}{X_{(1)}}\right)$ when $(X_1,\cdots,X_n)\stackrel{\text{i.i.d}}{\sim}\text{Pareto}(k,a)$

Let $(X_1,X_2,\cdots,X_n)$ be a random sample drawn from a $\text{Pareto}(k,a)$ population with density $f(x)=\frac{ak^{a}}{x^{a+1}}\mathbf1_{x>k}$ where $a,k>0$. What is the distribution of $\...
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1answer
63 views

Expectation and variance of range (x(n)-x(1)) uniform [closed]

I am working on calculating the expectation and then variance of the range from a Uniform(-theta, theta) distribution, but have gotten stuck. Basically the first page I show how I get the pdf and ...
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1answer
75 views

Asymptotics for the expectation of the standard normal order statistics

What is an asymptotics of the expectation of the order statistics of the standard normal distribution $$e(r:n) \approx \Phi^{-1}\Big(\frac{r-\alpha}{n-2\alpha+1}\Big)$$ as $n\rightarrow\infty$? By ...
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67 views

What is the motivation behind this sampling algorithm

I was recently reading on methods for sampling from linear distributions of directions. This distribution on a direction $X$ is defined by the density $$ p(x)=\frac{1}{2}(1-3\kappa)+3\kappa\frac{1+x}{...
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28 views

What is the probability of a random variable larger than or equal to any other independent random variables? [duplicate]

Suppose random variables $X_1$, $X_2$,..., and $X_n$ are independent and normally distributed (with varied means and variances), what is $P(X_i\ge \max_{j\in [1,n], j\neq i} \{X_j\})$?
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38 views

Order statistic, probability

Let $X_{1},X_{2},..., X_{10}$ be i.i.d. with continued, strictly increasing distribution function $F$. Let $X_{1:10},X_{2:10},...,X_{10:10}$ be the order statistic. Let $m_{X}$ be the median of random ...
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77 views

Non-parametric version of German Tanks problem?

The German Tank Problem assumes that we observe k i.i.d. random variables $X_1,...,X_k$ uniformly distributed over $[0,N]$, where $N$ is unknown. The goal in this problem is to estimate $N$ given the ...
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32 views

Conditional Distribution of Order Statistics [closed]

Let $X_1,...,X_n$ be independent and identically distributed random variables with commom cdf F. Let $X_{(i)}$ be the ith order statistics of $X_i^,s$.Then prove that for $j>i, \{X_{(j)}|X_{(1)}=...
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140 views

Closed-form multivariate Gaussian probability

I am looking for a closed-form version of this formula: \begin{align*}\text{P} &= \int_{-\infty} ^{\infty} \left[1 - \left(\int_{-\infty} ^{t}\right.\right.\ldots\\ &\left.\left.\ldots\int_{-...
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1answer
75 views

Covariance of order statistics convergence?

Suppose I have a sample $(X_1 \dots X_n)$ and $(Y_1 \dots Y_n)$, all of which are $N(0,1)$ random variables. I am interested in the asymptotic behaviour of $$\frac{1}{n} \sum_{i=0}^n X_{(i)}Y_{(i)} $$...
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26 views

Derivation of an estimate for CDF given ordered sample

Given a sample $x_1,\ldots,x_n$ one may order them $x_{(1)},\ldots,x_{(n)}$ and an estimate for the CDF $P(X\leq x_{(i)})$ is $\frac{i}{n}$ which is valid given that the CDF of a sample is uniform. ...
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36 views

What is the pdf of the sampling distribution of the sample maximum of iid standard normal?

I found here that The cdf for the max is the cdf for the normal raised to the power of the sample size. [say, $n$] Since the cdf for the normal is the error function, $$F_X(x)=\frac{1}{\sigma\...
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How to find approximation of variance of $i^{th}$ order statistic [duplicate]

Given PDF and CDF of a distribution, how does one find an approximation of $(\operatorname{Var}(X_i))$ using a normal approximation of $(X_i)$? According to "Mathematica Laboratories for ...
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Can you improve an estimate of the right-tail mean with extreme order statistics?

Suppose I have a sample of values drawn from a much larger but finite population, which in turn is drawn from some heavily skewed distributional family with finite mean. I wish to estimate the mean of ...
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432 views

Using Chebyshev's inequality to obtain lower bounds

Let $X_1$ and $X_2$ be i.i.d. continuous random variables with pdf $f(x) = 6x(1-x), 0<x<1$ and $0$, otherwise. Using Chebyshev's inequality, find the lower bound of $P\left(|X_1 + X_2-1| \le\...
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81 views

Finding the mean of the max order statistic drawn from standard normal

Let $X=\max\{X_1, X_2, \cdots, X_N\}$, where each $X_i \sim N(0,1)$ and are independent. What is the approximate value of $X$ for large $N$. The term "approximate" isn't defined very clearly. I'm ...
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expectation of exponential order statistics

Suppose $x_1, x_2, ......, x_n$ be i.i.d. random variable of exponential distribution $exp(1)$, i.e., $f(x)=e^{-x}, x>0$. Given the order statistics $x_{(1)} \le x_{(2)} \le......\le x_{(n)}$, it ...
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Percentiles of $Z=\max(X,Y)$, where $X$ and $Y$ are correlated bivariate normal random variables

Suppose $X \sim N(\mu_x,\sigma_x^2)$, $Y \sim N(\mu_y,\sigma_y^2)$, and $\operatorname{Corr}(X,Y)=\rho$. I am interested in calculating percentiles of $Z = \max(X,Y)$. We can assume bivariate ...
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Densitiy of order statistic when in a certain interval

$X$ is distributed with $F$, i.i.d. and with densities. I am trying to discern an expected value for a certain order statistic $X_{k}$ under the condition that $X_{k}$ is closest to some value $\...
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Groups of Points in QQ Plots

While using QQ Plots, I always notice these 'groups' of points; many sets of points arranged in a straight line and sometimes in a curve as shown below: exponential: normal: The theoretical ...
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Joint distribution of maximum of several rvs and one particular one

There are two random variables say X and Y which are independent and they follow Uniform(0,1). Then how to find joint distribution of max(X,Y) and X?
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Conditional Expectation of Order Statistic [duplicate]

Given a a random sample $X_1, X_2, X_3, X_4$ and family of densities $\mathcal{P} = \left\{ f_\theta: \theta \in \Theta \right\}$, where $f_\theta(x) = \frac{1}{2}\mathbb{I}_{[\theta-1, \theta + 1]}$, ...
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36 views

Is there a closed form solution to the expected value of the maximum of iid random variables?

Let $f_{Xmax}(x)=nf_X(x)F_X(x)^{n-1}$ be the usual pdf of the maximum of iid random variables with density and cumulative distributions f_x and F_X respectively. Is it possible to obtain a closed ...
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Relationship of skew and order statistics

I am trying to understand an expectation that I was able to write as a linear combination of order statistics of an i.i.d. sample $F$. Specifically, the linear combination of smallest and largest ...
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1answer
59 views

Ratios of order statistics of Pareto distributed random variables

Let $X_1, X_2, \dots$ be i.i.d. with $X_i \sim 1-1/x$. It's well known that $$ \mathbb{P} \left[ \frac{X}{t} \leq a \: \middle| \: X \geq t \right] = \mathbb{P}\left[X \leq a \right]. $$ Let $X_{1,n} \...
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Asymptotic joint distribution of the sample medians of a collection and a sub-collection of i.i.d. random variables

Let $X_1,\dots, X_n$ denote i.i.d. random variables, with a smooth density $f$ having unique median $m$. It is known that the sample median $M_n$ of $X_1,\dots, X_n$, $M_n = \operatorname{med}(X_1, \...
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Testing the randomness of the order of values

I would like to test the following values for randomness (or lack thereof). In several years observed values of say rainfall were following: ...
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185 views

Order Statistics w/ Maximum

Let $X_1,X_2,\dotsc$ be i.i.d. r.v.s with CDF $F$, and let $M_n = \max(X_1,X_2, \dotsc,X_n)$. Find the joint distribution of M$_n$ and M$_{n+1}$, for each n ≥ 1. So, CDF of $M_{n+1}$ is given as, $P(...
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30 views

The diagonal section of copula function

The secondary diagonal section of copula function is given by C(t,1-t). I want to show that C(t,1-t)=0 for all t in [0,1] implies C(u, v)=max(u+v-1,0)
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394 views

Compute moments of maximum of multivariate normal distribution

I want to compute the first and second order moment of the maximum of a random vector from multivariate normal distribution, i.e., compute $ E[\,\,Y\,],\,\,E[\,Y^2\,], $ where $ Y=\max (X),\,\,\\ ...
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36 views

What is the distribution of order statistic of Uniform distribution? [duplicate]

I know from a theory that if $X_1,X_2,...,X_n$ be iid from continuous Uniform(0,1), then k-th order statistic $X_{(k)}$ is Beta$(k,n-k+1)$. Now, $X_1,X_2,...,X_n$ be iid from continuous Uniform$(a,b)$...
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1answer
133 views

CDF of standardized order statistics

I am having some trouble with a textbook problem. I am given that $U_1, ... U_n$ are independent random variables with uniform distribution over $[0,1]$ and that $U_{(n)}$ is the maximum. I have to ...
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2answers
393 views

asymptotic distribution of $\sqrt{n}(\hat\theta_n-\theta)$ to determine the efficiency of $\hat\theta_n$?

I want to know the asymptotic distribution of $\sqrt{n}(\hat\theta_n-\theta)$ to determine the efficiency of $\hat\theta_n$. I know there is a theorem with lots of assumptions that immediately ...
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2answers
83 views

Trying to prove consistency, but getting non-sensical limit probabilities

I have $$X_1 \dots X_n \sim f_\theta(x) = \begin{cases} \exp(\theta-x) & x\geq\theta\\ 0& otherwise \end{cases}$$ And I have the estimator $\hat\theta_n = X_{(1)}=\min\{X_1 \dots X_n\}$ I ...
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92 views

Argmax order statistics for multivariate normal

The Problem Given a gaussian random variable $\mathbf{x} = (x_1, ..., x_N)^T \sim \mathcal{N}(0, \Sigma)$ what is the probability that $i = \underset{i\in\{1, \dots, N\}}{argmax}\{|x_1|, \dots, |x_N|\...