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.

This tag is for questions about order statistics and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity of 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. Order statistics are widely used in non-parametric inference, methods which avoid assuming the form of distribution of the population from which a sample is drawn.

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Expectation of the maximum of gaussian random variables

Is there an exact or good approximate expression for the expectation, variance or other moments of the maximum of $n$ independent, identically distributed gaussian random variables where $n$ is large? If $F$ is the cumulative distribution function…
Chris Taylor
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Expectation of Minimum of $n$ i.i.d. uniform random variables.

$X_1, X_2, \ldots, X_n$ are $n$ i.i.d. uniform random variables. Let $Y = \min(X_1, X_2,\ldots, X_n)$. Then, what's the expectation of $Y$(i.e., $E(Y)$)? I have conducted some simulations by Matlab, and the results show that $E(Y)$ may equal to…
jet
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Order statistics of i.i.d. exponentially distributed sample

I have been trying to find the general formula for the $k$th order statistics of $n$ i.i.d exponential distribution random variables with mean $1$. And how to calculate the expectation and the variance of the $k$th order statistics. Can someone give…
geraldgreen
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Expected Value of Max of IID Variables

What is the expected value of the maximum of 500 IID random variables with uniform distribution between 0 and 1? I'm not quite sure of the technique to go about solving something like this. Could anyone point me the right direction? Thanks
Matt
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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 characteristics in a selected imaging window, we obtain…
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Trimmed mean: Take $n$ i.i.d. Gaussians and remove largest $m$ and smallest $m$ points. What is the variance of the mean of the remaining points?

Let $n,m\in\mathbb{Z}$ with $0 \le 2m < n$. Let $X_1, \cdots, X_n$ be i.i.d. standard Gaussians and let $X_{(1)} \le X_{(2)} \le \cdots \le X_{(n)}$ denote their order statistics (i.e., $\{X_1, X_2, \cdots, X_n\} = \{X_{(1)}, X_{(2)}, \cdots,…
Thomas
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Prove that the sample median is an unbiased estimator

My book says that sample median of a normal distribution is an unbiased estimator of its mean, by virtue of the symmetry of normal distribution. Please advice how can this be proved.
preeti
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Probability of getting into my favorite PhD

Suppose there are $n$ potential grad students and $n$ Universities. Each University has one scholarship to do a PhD. Each student has a strict ranking of Universities so that each University is comparable and she is not indifferent between any two…
fox
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Distribution of the maximum of a multivariate normal random variable

Suppose there is a vector of jointly normally distributed random variables $X \sim \mathcal{N}(\mu_X, \Sigma_X)$. What is the distribution of the maximum among them? In other words, I am interested in this probability $P(max(X_i) < x), \forall…
Ivan
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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. Then for $k\leq…
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the parametrization of a Gumbel in terms of a Gaussian

Extreme Value Distribution From a Gaussian. I was wondering how the parametrization of $\alpha$ and $\beta$ of a Gumbel $e^{-e^{-\frac{x-\alpha }{\beta }}}$ was done in terms of a cumulative Gaussian $F(x)^n$ (where $n$ is the number of…
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How to solve this confusing permutation problem related to arrangement of books?

Ms. Jones has 10 books that she is going to put on her bookshelf. Of these, 4 are mathematics books, 3 are chemistry books, 2 are history books, and 1 is a language book. Ms. Jones wants to arrange her books so that all the books dealing with…
RajSharma
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Conditional expectation to de maximum $E(X_1\mid X_{(n)})$

Let $X_1, \ldots, X_n$ a random sample of a Uniform(0,1): Which is $E(X_1\mid X_{(n)})$ ? where $X_{(n)}=\max\{X_1,\ldots,X_n\}$
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Upper bound of expected maximum of weighted sub-gaussian r.v.s

Let $X_1, X_2, \ldots$ be an infinite sequence of sub-Gaussian random variables which are not necessarily independent. My question is how to prove \begin{eqnarray} \mathbb{E}\max_i \frac{|X_i|}{\sqrt{1+\log i}} \leq C K, \end{eqnarray} where…
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Why is the maximum of i.i.d. Gaussians asymptotically $\sqrt{2 \log n}$?

Assuming that $\xi$ is bounded (as a function of $x$?), the claim is that given the equation: $$\xi \frac{\sqrt{2\pi}}{n} = \frac{1}{x} e^{-\frac{x^2}{2}} \left( 1 + O\left(\frac{1}{x^2} \right) \right) $$ one can solve ("after some calculation")…
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