# 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.

489 questions
Filter by
Sorted by
Tagged with
24 views

### Which statistical test is suitable to compare order statistics of two independent samples?

Say I want to compare two order statistics (say the 2nd largest value or min value) of two samples. Let's not make any distributional assumptions except that the variance is finite? Is something like ...
36 views

52 views

81 views

1 vote
44 views

11k views

### Why does R say 'cannot compute exact p-values with ties' when I can do it with pen and paper?

Suppose I have two sets of three numbers: $x_1, x_2, x_3$ and $y_1, y_2, y_3$ and I want to test the Null hypothesis that they are drawn from the same distribution using the Wilcoxon-Mann-Whitney test....
Let $n \geq 1$ be an integer. Let $X \sim \operatorname{Beta}(i, n - i + 1)$ where $i \in \{1, ..., n\}$. Therefore: $$X = \frac{A_n}{A_n + B_n}$$ where A_n = \sum_{r = 1}^i Z_r, \qquad B_n = \... 2 votes 0 answers 147 views ### Lower Bound on Expected Maximum and Upper Bound on Expected Minimum of Order Statistics This question relates to bounds on expectations of order statistics, elaborated upon in the Book by Arnold and Balakrishnan (1989). Let X_1,\ldots,X_n be i.i.d. continuous random variables ... 3 votes 0 answers 65 views ### Proof that \sqrt{n}\left( \hat{F}_n(x_1)-F(x_1),\dots,\hat{F}_n(x_k)-F(x_k)\right) \rightarrow \mathcal{N}_k(0,\Sigma) By definition \begin{align*} \Sigma = F(\min(x_i,x_j))-F(x_i)F(x_j)\end{align*} Note: \hat{F}_n(x) = \frac{1}{n} \sum_{j=1}^n 1_{\{X_j \leq x\}} I think that \mathbb{E}\left[\hat{F}_n(x_i)\right] =... 3 votes 1 answer 40 views ### Jacobian of function returning m evenly-spaced order statistics of an n-dimensional vector Let y\in\mathbb{R}^n, and let f:\mathbb{R}^n\to\mathbb{R}^m be the transformation that outputs m evenly-spaced order statistics (including the extremes) of y. What is the Jacobian of this ... 0 votes 0 answers 36 views ### Probability that a normal RV is greater than multiple other normal RVs Let X_1, X_2, ... X_n be independently drawn from different normal distributions, such that  X_i \sim N(\mu_i, \sigma^2_i)  For any j what is the probability that X_j is the greater than all ... 1 vote 2 answers 76 views ### Probability random variable is less or equal to k-th out of two samples when ordered Given the random variable X, \{X_{i}\}_{i=2}^{n}, \{Y_{i}\}_{i=2}^{n} all iid and lets denote X_{(k)} as the k-th statistic of \{X\} \cup \{X_{i}\}_{i=2}^{n} and Y_{(k)} for \{X\} \cup \{... 0 votes 2 answers 64 views ### Estimator for Range (Length of Stay ) I have sample data ad I want to test the claim that the mean length of stay is 7 days. Data is give as pairs ( Arrival Date, Departure Date) , each date given as the nth day of the year, e.g., Jan 3 ... 13 votes 1 answer 1k views ### QR interview problem Guessing order of draws from iid U(0,1) This is for QR at two well know trading firms (think jane street, HRT, Citadel, Jump ...)(not BB bank). Question prompt: Given n iid Uniform distributed r.v.s. x_i ~ U(0,1). x_1 is drawn first, ... 0 votes 1 answer 26 views ### Summing ordered samples Let's pick a random number between 0 and 1 uniformly 1000 times and put the results in an array for example : [0.176,0.765,0.879,0.234,0.152,0.765,0.645,0.897,0.762,0.087...] and sort this sample from ... 0 votes 0 answers 103 views ### PDF of the linear combination of two order statistics I have just started to learn order statistics so please feel free to correct my notation/terminology. In my field is common to provide data as the median of some sample (for example several ... 3 votes 3 answers 492 views ### Nonparametric Order Statistics - Does this Exist? I was reading about order statistics on Wikipedia [retrieved 29 June 2022]: Apparently, if we have a sample with k elements (e.g., x_1, x_2, ..., x_k) and assume a probability distribution for ... 3 votes 1 answer 49 views ### Boolean order statistic Inspired from this LeetCode question. I authored this question myself however. Please review my answer for accuracy. Let's sort an array (n \ge 1) of 1s and 0s, so that all 0s come before 1s.... 3 votes 1 answer 44 views ### Calculate how many defective parts with p=0.95 After a 2 month test, there is a result of 5 defective parts out of the total sample size of 50 tested parts. How many defective parts can be expected for an annual production of 70000 parts (p=0.95)? ... 2 votes 1 answer 186 views ### Distribution/estimation of maximum change of a stationary time series Any help on this would be much appreciated. Let x_{t} = b x_{t-1} + u_{t}, where u_{t} \sim N(0,1) and \lvert{b}\rvert < 1. What can we say about the distribution of y_{t} = \max(x_{t+2},x_{... 4 votes 3 answers 168 views ### Estimate normal distribution parameters from smallest N samples I have a bunch of small datasets (billions of sets of 7 samples). Each dataset represents the smallest 7 samples of a larger set of 15 values which are normally distributed. Given just the smallest 7 ... 1 vote 2 answers 83 views ### Probability that a given number falls between the minimum and the maximum of a sample Let X be a real random variable with absolutely continuous cumulative distribution function F. Let x_{(1)}, ..., x_{(n)} be a i.i.d. ordered sample of size n of X: x_{(1)} \leq x_{(2)} \...
Assume I have $n$ points sampled independently from the uniform distribution on the unit interval. After ordering the sample I get the points $X_1, X_2, \dots X_n$ such that \$X_1 \leq X_2 \leq \dots \...