# 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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Consider a sequence $x_1,\dots,x_n\in [0,1]^d$. Suppose that $\mathcal{F}$ is a subset of the space of continuous functions from $[0,1]^d$ to $\mathbb{R}$. Moreover, let $f\in\mathcal{F}$ such that $$... 1 vote 0 answers 38 views ### Conditional expectation involving Order Statistics [duplicate] Let X_1,X_2,...,X_n be a random sample of size n \geq 2 from \exp(1/\theta), where \theta \in (0,\infty). Let$$Y = \min\{X_1,X_2,...,X_n\} \qquad \textrm{and} \qquad T = \sum_{i=1}^{n} X_i .$$... 1 vote 1 answer 43 views ### Are the probability distributions of order statistics log concave? If some distribution is log concave, then would the distribution of order statistics sampled from that distribution also be log concave? Specifically, I'm thinking of a case where I have a random ... 1 vote 1 answer 59 views ### finding the marginal pdf of second order statistics Suppose that X_1, X_2, X_3 is random sample (iid) from a population with pdf f(x) = 3x^2 if 0<x<1 I would like to compute g_2(y_2) which is the marginal pdf of order statistics Y_2 ... 0 votes 0 answers 33 views ### Expectations of Max Order Statistic on Exponential distribution Let X_1,...,X_n be iid exponential with scale parameter \theta. That is for each i \in \{ 1,2,...,n\}, X_i \sim f_{X_i}(x) = \frac{1}{\theta}\exp(-x/\theta) . I am interested in the max order ... -1 votes 1 answer 89 views ### What is the closed form solution for this infinite sum? [closed] We are working on a problem related to order statistics. This requires the computation of the following infinite sum. Let i be a positive integer. Let real numbers$$\alpha >0$$and$$1>\beta>...
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Let $X_i$ be i.i.d uniform variables between 0 and 1. Define $Min_n = \min(X_1, \ldots, X_n)$ $Max_n = \max(X_1, \ldots, X_n)$ How can we show that the variance of $Min_n$ and $Max_n$ are the same? ...