# 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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### if $f(x) = a exp(-a (x-b))$ . Find sufficient statistic for b

Intuitively I feel that the answer should be $\min(X_1,X_2,\ldots,X_n)$ where $X_i$'s are iid, but I don't know how to prove it.
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### Finding more information after finding MLE with Indicator functions.

Ex: $X_1 , X_2 , ... , X_n$ ~ $U(-\theta, \theta); f(x; \theta) = \frac{1}{2\theta}; -\theta \leq X \leq \theta; \theta > 0$ I believe this is the correct approach to finding the MLE in this ...
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### Show that $X_{(1)}$ and $X_{(2)}−X_{(1)}$ are independent, and determine their distributions.

Let $X_1$ and $X_2$ be independent, $\text{Exp}(a)$-distributed random variables. Show that $X_{(1)}$ and $X_{(2)}−X_{(1)}$ are independent, and determine their distributions. Although it looks like ...
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### Compute the correlation coefficient $r(X_{(1)},X_{(3)})$ .

I got this problem where: The random variables $X_1, X_2,$ and $X_3$ are independent and $Exp(1)-$ distributed. Compute the correlation coefficient $r(X_{(1)},X_{(3)})$ . I know through research ...
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### Compute $P(X_{(1)} + X_{(3)} \le 1),$

Any Idea why I keep getting $3/4$ ? Let $X_{1}, X_{2}, X_{3},$ be independent, $U(0, 1)$-distributed random variables and $X_{(1)}, X_{(2)}, X_{(3)}$ be the corresponding order variables. Compute (a)...
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### probability that a random variable is greater than a limit in given ordering of random variables

I am currently working on a modified version of the classic greedy algorithm for the 0/1 knapsack problem. Suppose that one has $N$ items with given weights and profits that are iid random variables ...
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### Finding PDF of Range of Order Statistic

Let $Y_1, Y_2,..., Y_n$ denote a random sample from the uniform distribution $f (y) = 1, 0 ≤ y ≤ 1.$ Find the probability density function for the range $R = Y_{(n)} − Y_{(1)}$. First Attempt: ...
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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|}{\...
Dear Statisticians and Mathematicians, I am interested in proving the following lemma by induction I have shown that it holds true for $n=2$ which I don't provide its proof here. We assume it is true ...