# Tagged Questions

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 ...

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### Conditional expected value of mutlitple draws from uniform distribution

There are $m$ i.i.d. draws of $x$ made from a uniform distribution on $[0,1]$. The $n$ ($n\leq m$) lowest draws are "winners", i.e. if we write $x_1\leq\ldots\leq x_n\ldots\leq x_m$, the draws $x_1$ ...
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### Order Statistics Intuition

I'm having trouble understanding the intuition behind the answer to a practice question. An internet company has three redundant servers for its web site. Thus, the site functions properly as long as ...
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### Operations on two normal distributions using order statistics

$G(x)$ is a Normal distribution with mean $\mu$ and standard deviation $\sigma$. I observe realization of $X$ which are a function of $s$. The distribution $F(s)$ is found as the root (between 0 and ...
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### Compute completion time using order statistics

I am trying to figure out how order statics work, I need to estime the completion time of a computation. The computation consists of n stages and runs at N threads (parallel computing). Assuming that ...
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### Relation between estimator's consistency and biasedness

I have two quick question: If an estimator is consistent, does that imply it is unbiased? If an estimator is biased, does that imply it is not consistent? we know that consistency means ...
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### Covariance of order statistics

I'm a researcher in social science and I have encountered the following math formulation of a problem in my field. Let $x_1,x_2,...,x_n,x_{n+1}$ be $n+1$ i.i.d. random variable with non-negative ...
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### Density of maximum of order statistics

Let $X_{1},X_{2},\ldots,X_{n}$ be independently drawn from a distribution $F$ and let $Y_{k}^{(n)}$ be the $k$-th order statistic (Convention: $Y_{1}^{(n)}>Y_{2}^{(n)}>\cdots>Y_{n}^{(n)}$). ...
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### Marginal Order Statistics

I'm trying to find the marginal statistic of 2 order stats from the full joint and I can't figure out the bounds on the following problem. Consider $0\leq Y_1\leq Y_2\leq Y_3\leq Y_4 < 1$ from the ...
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### An Estimator Based on Exponential RVs

Let $X_1$, $X_2$, $\cdots$, $X_n$ be $n$ random variables independently sampled from the exponential distribution $\text{exp}(1)$. Suppose $k \leq n$, and $X_{(k)}$ is the $k$-th order statistic, i.e.,...
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### The expected value of the $k$th order statistic of iid geometrically distributed rvs, and its asympotic expansion.

I have read the paper Combinatorics of geometrically distributed random variables: Left-to-right maxima. In the paper, the largest order statistic $X_{n:n}$ (i.e., $\max\{X_1,X_2,\ldots,X_n\}$) is ...
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### expected value of order statistic

I am having troubles with solving this task 7.4.5. Show that the first order statistic $Y_1$ of a random sample of size $n$ from the distribution having pdf f(x;\theta) = e^{-(x-\theta)}, \...
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### What is the asymptotic behaviour of $\max_{n \in \{1, 2,\, \dots N\}} \tan n$?

What is the asymptotic behaviour of $f(N) = \max_{n \in \{1, 2,\, \dots N\}} \tan n$? Any non-trivial bounds above or below would be of interest. Some quick numerical experimentation shows: ...
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### For $n \geq 2$, let $X_1,X_2,\ldots,X_n$ be independent samples from $P_{\theta}$, the uniform distribution $U(\theta,\theta +1),\theta \in \mathbb R$

For $n \geq 2$, let $X_1,X_2,\ldots,X_n$ be independent samples from $P_{\theta}$, the uniform distribution $U(\theta,\theta +1),\theta \in \mathbb R$. Let $X_{(1)},X_{(2)},\ldots,X_{(n)}$ be order ...
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### Given $X_i \sim Geom(p)$, what is the PDF of $R := \min \{ r:\sum_{j=1}^n \mathbb{1}_{\{ X_j \leq r\}} \geq m \}$?

I am considering a file transfer problem that given a file divided into $n$ blocks, the number of transmission rounds of each block $X_i$ satisfies $X_i \sim Geom(p)$. Thus the number of transmission ...
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### Show that $(\frac{S_1}{S_n+1},\frac{S_2}{S_n+1},…\frac{S_n}{S_n+1})=_d (U_{(1)},U_{(2)},…,U_{(n)})$.

Let $(X_1, X_2,...,X_n) \in \mathbb R^n$ have density function $p(x)$. (1) Find the density of $(U_{(1)},U_{(2)},...,U_{(n)})$, the order statistics from a sample of iid $\mathbb U[0,1]$ (uniform ...
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### subsample order statistics

I am interested in characteristics depending of the $r$-th order statistics $X_r$ of a distribution with unknown pdf. For example, I would like to estimate the Gini mean difference $E(|X_r - X_r'|)$, ...
Under symmetry, $F^{−1} (0.5) = E(X)$. Compare the sample mean as an estimator of µ to that of the sample median ($F^{−1} (0.5)$) for n sufficiently large, assuming that $X_i$ ∼ Z = N(0, 1), i = 1,...
### Distribution of Interquartile Range on $X_i$ ∼ U(0, 1), i = 1, . . . , 20, iid.
Let Xi ∼ U(0, 1), i = 1, . . . , 20, iid. IQR = $F^{−1}(.75)−F^{−1}(.25)$ = $X_{(15)}−X_{(5)}$ in this example as n = 20. a. Find the distribution of the random variable W = IQR. b. Devise ...