# 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. Sometimes 'order statistic' is used to mean the whole set of order statistics, i.e. the data values disregarding the sequence in which they occurred. Use also for related quantities such as spacings.

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### Joint PMF of two order statistics with discrete parent distributions

Let $X_1, X_2$ be i.i.d from a discrete distribution with finite support with cumulative distribution $F(x)$ and probability mass function $f(x)$. Let $X_{1:2}$ and $X_{2:2}$ represent the order ...
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### Largest order statistics of non-identical distributions when extra information is available

Suppose we have two independent draws, one from a distribution $F_1$ and the other from a distribution $F_2$. The two distributions have the same support, say $[0, 1]$. The distribution of the largest ...
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### Estimator for a particular statistic involving Order Statistic

Let$X_{1}, X_{2}, \cdots, X_{n}$ be a random sample from a continuous life distribution $F$ be with survival function $\bar{F},$ density $f$ and finite mean $\mu.$ While doing some ...
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### Approximate distribution of a complicated function of a random variable

If $X$ is a random variable cdf $F(x)$ such that $F$ is invertible then we have the standard method of finding the pdf of any function of $X$, say, $\sin(X)$ or $X^3+1$.However,in many situations ...
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### Distribution of Maximum of Geometric random variable

Let $X_1, X_2, ... X_n$ be geometric random variables with density $$P(X=x)=pq^{x-1} , x=1,2,3,...$$ What will be the distribution of $Y=\max(X_1, X_2, ..., X_n)?$ Will the distribution of $Y$ be ...
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### Estimate population mean from "best of N" samples

If I have a data set for which I know all measurements represent the largest of N observations, is there a good method for estimating the mean of all observations? So for example if N=10 and I have 3 ...
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### computing $P\left(\max(U_{(1)}, U_{(2)}-U_{(1)}, \cdots,U_{(n)}-U_{(n-1)} ) <a\right)$

Let $U_{1}, \, ... \, ,U_{n}$ be a random sample of uniform random variables $U_i \sim \mathrm{Uniform}(0,1)$. Let $U_{(1)}, \, ... \, , U_{(n)}$ be the order statistics of the sample. My problem is ...
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### What are E(max(X1, X2)) and Var(max(X1, X2)) when the Xs are normal random variables? [duplicate]

Let X = (X1, X2) be normally distributed random variables with mean m = (m1, m2) and covariance matrix S. Y = max(X1, X2) = X1 + max(0, X2 - X1) = X1 + D (X2 - X1), where D = 1 if X2 > X1 and 0 ...
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### exponential parameter estimtion from the smallest k-th order statistics

Assume $X_1, X_2, X_3,\ldots,X_n$ are i.i.d. samples from Exp($\lambda$). Assume that the integer $k<n$, is it possible to find a an unbiased estimator for $\lambda$ from the k-th smallest ordered ...
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### What is the expectation of a random variable satisfying some conditions?

How to find the expectation E[X.I(Y<x,X<x)], where X and Y are independent random variables with respective cumulative distribution functions F(.) and G(.) respectively. x is a positive value. ...
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### Convergence in distribution to a degenerate distribution

This question came up based on a disagreement I had with a TA. This was the specific example: Let $X_{1},...,X_{n}$ be an iid random sample from a population with pdf $f(x)=3(1-x)^2, 0<x<1$. The ...
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### Order Statistics: How to calculate expected value of a function involving first and second order statistics

I am currently stuck with a challenging problem. I have n values drawn i.i.d. from a distribution F(x). Let $v_1$ be the nth order statistic (highest value) and let $v_2$ be the n-1 order statistic (...
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### Order statistics and sample size

How do I estimate the odds of the highest value elements of samples from two populations, A and B, exceeding a threshold value, where the same size of A is larger than that of B? Even if A and B have ...
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### Probability that a random variable is the k-th smallest

Suppose we have random variables $X_1, \dots, X_n$ drawn independently from different distributions. How do we calculate the probability of some $X_i$ is the $k$th smallest of those random variables? ...
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### Entire set of order statistics

Let's say that we have $X_{1}$,...,$X_{n}$ iid discrete random variables (let's say that there are 15 possible values in the support of the discrete distribution) and n is a large number (let's say 50)...
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### What is the intuition behind the Independent Spacings Theorem? [duplicate]

The IST states: Let X1, X2, ,,, Xn be iid Exponential(μ). Then the random variables (where Xsubscript(0) = 0) Ysubscript(j) = (n - (j-1)) (Xsubscript(j) - Xsubscript(j-1)) for j = 1, 2, 3, 4, ,,, n - ...
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### Order Statistics Problem: Wackerly/Mendenhall/Scheaffer, 5th Ed., Problem 6.58

Problem Statement: Suppose that the number of occurrences of a certain event in time interval $(0,t)$ has a Poisson distribution. If we know that $n$ such events have occurred in $(0,t),$ then the ...
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### Expected value of sum of normal random variables given one of them is the largest

Suppose $X_1\sim N(\mu,\sigma^2)$ and $X_2,X_3,...,X_n\sim N(0,\sigma^2)$, is the following identity correct? $$E(X_1+X_2+...+X_n|X_1>\max(X_2,X_3,...,X_n))=E(X_1+X_2+...+X_n)$$ It seems that it is ...
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### Expected value of max of two discrete random variables

I'm reading this paper An Efficient PTAS for Stochastic Load Balancing with Poisson Jobs. Which is solving a makespan minimizing job-shop problem for Poisson job sizes. Basically, schedule the minimum ...
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### MLE based on bivariate data

Let $x \sim Exp({\lambda}_{1}) , Y \sim Exp({\lambda}_{2})$ and are independent . We observe Z and W with Z = min(X, Y) and $W = \begin{cases} 1 &, if Z=X \\ 0 &, if Z = Y. \end{cases}$ Now ...
The Kth order statistic for a univariate is equal to its kth-smallest value. For instance, given $\{6,9,3,8\}$, the 2nd-smallest value would be the 2nd order statistic. How does this concept ...