Questions tagged [order-statistics]

Question 1

For the median ($m$), there is a simple formula that it falls within the range of a random sample of size $n$ from a continuous distribution: $$ \mathbb{P}(X_{(1)} \leqslant m \leqslant X_{(n)}) = 2^{...

Question 2

I am currently trying to find out the pdf and cdf of the $r$-th order statistic with $X_1,\ldots,X_n$ having the independence but not identical distribution. First I tried it using the usual methods ...

Question 3

Let $X=(X_1, X_2, \dots, X_n)$ a centered gaussian vector with a positive covariance matrix and $X_{(1)} < X_{(2)} < \dots < X_{(n)}$ his increasing rearrangement. Let $k\in \{2, 3, \dots, n-...

Question 4

It's known that in some cases we can infer distributions on order statistics, given the distributions of samples. (Perhaps relatedly, it's also known that there are cases where we can achieve ...

Question 5

I'm taking an inference course via Casella/Berger where students are encouraged to look up "stumper" questions in the online solutions or beyond. (Yes, really.) I'm treating exercise 8.5, ...

Question 6

Let $\mathbf q\in[0,1]^n$ be a random vector following a symmetric Dirichlet distribution with parameter $\alpha$: $$ \mathbf{q}\sim \operatorname{Dir}(\alpha,\cdots,\alpha). $$ Define $q_{(i)}$ as ...

Question 7

Given two samples $x_1, x_2, ... x_n$ and $y_1, y_2, ... y_n$, whose values are in [0, 1], what is the distribution of $Z$ = min({$y_1, y_2, ... y_n$}) $-$ max($x_1, x_2, ... x_n$)? It is usually the ...

Question 8

Suppose two independent ordered draws from some continuous and bounded distribution $F_{[0,\bar{x}]}$ with $f>0$ everywhere, represented by the order statistics $$X_{(1:2)}=\min(X_1,X_2)\text{, and ...

Question 9

Given $X_{1},\ldots,X_{n}$ an iid sample of a standard normal distribution, let $M=M(X_1,\ldots,X_n)$ be the median of this sample. What is the value of$$\mathbb E\left[M(X_1,\ldots,X_n)\times\sum_{i=...

Question 10

Suppose, $n$ units are placed on a life test. The time-to-failure follows a continuous probability distribution with non-existing finite moments(like a lower-truncated Cauchy or inverse Lomax). Let, $...

Question 11

Suppose $X_1, X_2,\cdots, X_n$ is a random sample of size $n$ from a continuous probability distribution with cumulative distribution function (CDF) $F(x)$ and probability density function (PDF) $f(x)$...

Question 12

Show that $P(M_n > t) \leq n(1 - \Phi(t))$ My work: \begin{align} & P(M_n > t) \leq P\left(\bigcup_{i=1}^n (Z_i > t) \right) \\ \leq {} & \sum_{i=1}^n P(Z_i > t)= n(1 - \Phi(t)) \...

Question 13

If $X_i \overset{\textrm{iid}}{\sim} \text{Lognormal}(0, \sigma^2)$ for $i=1,\ldots,n$ and $Y_1 = X_1 / \sum_{j=1}^n X_j$, then I would expect that a particular* limiting distribution of $Y_1$, ...

Question 14

For random variables $X_1, \cdots, X_n$, we denote the order statistics by \begin{align} X_{(1)} & = \min (X_1,\ldots, X_n) \\[6pt] X_{(2)} & = \text{second-smallest of } X_1,\ldots, X_n \\ &...

Question 15

I am working through "Computer Age Statistical Inference" as a self-study and am stuck on the follow exercise (1.3): The details of equation 1.6 are unimportant for the exercise, so far as ...

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Questions tagged [order-statistics]

Probability that the range of a random sample of size $n$ from a continuous distribution includes the population mean?

The pdf of $r$-th Order statistics with independent but not identical distribution

Gaussian order statistics

Inverse problem of inferring sample distribution from order statistic distribution

Independent random exponential variables, minus their minimum, are independent (proof?)

The first moment of symmetric Dirichlet order statistics

Distribution of $Z$ = min($y_1, y_2, ... y_n$) $-$ max($x_1, x_2, ... x_n$) [closed]

Conditional expectation of order statistics when observing ratio/difference

What is $\mathbb E[M\times\sum{X_i}]$ in normal distribution, $M$ being the sample median?

What are the alternative summary measures of a maximum order statistic when the expectation of the underlying distribution is not finite?

Can median or mode of the order statistics be obtained?

Suppose $Z_i$ are i.i.d. $N(0, 1).$ And let $M_n = \max\{Z_1, \ldots, Z_n\}$, show that $P(M_n > t) \leq n(1 - \Phi(t))$

In a sum of high-variance lognormals, what fraction comes from the first term?

Approximation of the expected value of the $i$-th standard normal order statistic in a sample of size n

Rough answer for the maximum of absolute value of $n$ standard gaussians (Computer Age Statistical Inference Problem 1.3)

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