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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4 views

What is asymptotic variance and how do I find it?

I don't understand the concept of asymptotic variance. Given the mle of a probability function, the likelihood function and the random variables how do I find asymptotic variance? What exactly is ...
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35 views

CDF of minimum of correlated and iid random variables

Consider two random variables $X_1=\min (W_1, W_2)$ and $ X_2=\min (W_3, W_4),$ where $W_1$, $W_2$,$W_3$ and $W_4$ are positive, identically distributed random variables. While $W_1$, $W_2$ are ...
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36 views

Probability distribution of $Y_1=min(X_i)$

Let $Y_1<Y_2<\ldots<Y_n$ denote the order statistics of a random sample of size $n$ from the distribution with pdf : $$f(x;\theta)=e^{-(x-\theta)}I_{(\theta,\infty)}(x)$$ Here we use the ...
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23 views

Obtain distribution of mid-range in uniform

I want to obtain distribution of mid-range, $(x_{(1)} + x_{(n)})/2$, of an uniform(a, b) random variable. One can use the following transformation. $M = \frac{X_{(1)} + X_{(n)}}{2}$ and $W = ...
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16 views

iid and correlated order statistics - A comparison

This is an extension of my previous question from the post iid and correlated order statistics a comparison Consider the two random variables $X_1$ and $X_2$ defined via non-negative random ...
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21 views

iid and correlated order statistics a comparison

Consider the two random variables $X_1$ and $X_2$ defined via i.i.d, non-negative random variables $Y_1$ and $Y_2$ as $ X_1=\min(Y_1,Y_2)\\$ $ X_2=\min(Y_1,Y_1)$ - (maximally correlated) The ...
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1answer
19 views

Expectation of maximum of Binomial RVs

Given an iid sample $X_{1}, \ldots, X_{n} \sim Bin(n, p)$ I'm trying to find $$E(X_{(n)})$$ that is the expectation of the sample maximum. Unfortunately I don't know where to start. It seems that ...
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26 views

how to create unbiased estimator in uniform distribution

$X_{1}, X_{2},...X_{n},n\geqslant 2, $ is a random sample from unif[$\theta -1, \theta +1$] Followed with the problem, I got T(X)=($X_{(1)}, X_{(n)} $) is sufficient but not complete, But I got ...
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31 views

Expected value of power of exponential order statistics

Finding the expected value of the $\gamma$:th power of the $k$:th standard exponential order statistic in a sample of $n$ implies evaluating the integral: ...
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42 views

PDF of the r-th Order Statistic (essentially need help with a derivative or combinatorics)

So I get why it is that the CDF of the $r$th order statistic is $$ F_{X_{(r)}}(x) = \sum_{i=r}^{n} \binom{n}{i}[F_{X}(x)]^{i}[1-F_{X}(x)]^{n-i} $$ but I'm not seeing how to prove that the PDF is ...
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58 views

Xmas Special: 25 identical sweets shared between two indivuduals and…

Ready?.. 25 identical sweets must be shared bewteen 1 boy & 1 girl. Each of the children MUST recieve at least 10 sweets each. all sweets must be distrubuted. order not important although I will ...
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31 views

Statistics: Odd Moments

Need help with this stat question. I know you start by integrating $z^k f(z)$ from $-\infty$ to $0$ + integral of $z^k f(z)$ from $0$ to $\infty$. After that I'm stuck.
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22 views

Evenly selected order statistic?

Recently, I'm studying Nonparametric methods, especially Order Statistics. And I saw a sentence which says that Order statistic has to be evenly selected. And now I'm wondering what 'evenly ...
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29 views

probability that at least one observation of a random sample

What is the probability that at least one observation of a random sample of size n=5 from a continuous type distribution exceeds the 90th percentile. Let W~Binomial n=5, P=?. how can i get the p of ...
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1answer
19 views

Bayesian Statistics: Estimators and Posterior Probability

If I let $M ∼ Γ(α,β)$ (where $α, β$ are known) Let $X_1,...,X_n$ be discrete random variables such that $X_i$|$θ$ ∼ i.i.d. Poisson with parameter $θ$, where $θ$ is a realization of $M$. I have two ...
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11 views

Bayesian Statistics … Γ(α,β) Posterior Probability and Estimators

If I let $M ∼ Γ(α,β)$ (where $α, β$ are known) Let $X_1,...,X_n$ be discrete random variables such that $X_i$|$θ$ ∼ i.i.d. Poisson with parameter $θ$, where $θ$ is a realization of $M$. I have two ...
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15 views

Use condition to make r.v.s independent and then use Laplace transform to sum them up?

Let $X_1,X_2$ be two nonnegative i.i.d. random variables. Let $k_1 = \text{argmax} (X_1, X_2)$. This means, $k_1$ is also a random variable. Now I define $c_k$ as $c_k = A$ if $k = k_1$, and $c_k = ...
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1answer
26 views

Difference of two largest values from a set of i.i.d. random variables

I have a set of i.i.d. random variables $(E_1, \ldots, E_N)$ that are exponentially distributed with parameter $\lambda$. I guess, if $E_1$ lives in $(S,\mathcal{S},\mathbb{P})$, then the whole set ...
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27 views

Order statistics of mixed (iid as well as non iid) random variables

Does anyone know if there are results (PDF or the CDF) on the order statistics (at least minimum or maximum) of $n$ random variables in which a few of them are i.i.d. and the rest of them are ...
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90 views

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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35 views

Joint distribution $(X_1,X_{(n)})$ order statistics

Let $X_1, \ldots, X_n$ a random sample of a Uniform(0,1), I want to show which the joint distribution of $(X_1,X_{(n)})$ is. I do the following: $$ P(X_1\leq x, X_{(n)}\leq y)=P(X_1\leq x, X_1\leq y, ...
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55 views

Question about piecewise exponential distribution

This is an excerpt from the following paper. I am particularly interested in knowing how the authors got the displayed equations. We let [Z] denote the distribution of a generic random variable ...
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54 views

Difficulty to compute an integral

Have somebody ideas to evaluate the following integral ? $$J_n=\int_{-\infty}^{+\infty} \left(\frac{\pi^2}{4}-\arctan(x)^2\right)^n\,dx$$ I'm trying this because I have shown that the empiric ...
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53 views

Squence of order statistic converges to median?

Suppose that $(X_1,...,X_N)$ be $N$ iid samples from uniform distribution. Let $X_{(n)}$ be $n$-th order statistic. It sounds natural that $X_{N/2}$ converges to the median, $1/2$. (interpret $N/2$ as ...
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19 views

orderstatistics of uniform distributions on different ranges

During a simulation I discovered an interesting phenomenon: Given you have 3 agents. 2 are uniformly distributed between [0,1] and one between [0,2]. The question is how often do the smaller agents ...
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61 views

Order Statistics Expected Value

Can someone help me with this question? I have arrived with the distribution function equal to $g(y) = \frac{n}{\theta }\left(1-\frac{y}{\theta }\right)^{n-1}$. But couldn't solve the integral for the ...
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1answer
36 views

Common distribution of order statistics

Let $X_1,\dots,X_n$ be $iid$ with distribution function F and $Y_1,\dots,Y_n$ $iid$ with distribution function G. I've proved for some function $g$ that $X_1$ is equal in distribution to $g(Y_1)$, ...
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18 views

Order Statistics interval sizes

Suppose an i.i.d. sample of size $n \geq 2$ drawn from a known distribution with density $g$. Let us note the associated order statistics as $(X_{(1)}, \ldots ,X_{(n)})$. I am interested in the number ...
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42 views

Maximum Likelihood Estimator where x>= theta

Let X = X1, X2, . . . , Xn be i.i.d. random variables from a population with a density if x ≥ θ f(x;θ) = 2θ^2/x^3, if x ≥ θ, 0 elsewhere. What is the Maximum Likelihood Estimator of θ? Is it unbiased? ...
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28 views

Cumulative minimum of an Ornstein-Uhlenbeck process

Assume we generate a sample path $X_t$ from an Ornstein-Uhlenbeck distribution (i.e. a mean-reverting random walk), where $dX_t = −\rho(X_t − \mu)dt + \sigma dW_t$. For concreteness, take $\mu = 0$, ...
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1answer
60 views

The probability of observing a value as extreme, or more extreme, than the sample maximum.

Consider a random sample of continuous random variables $X_1, X_2, ..., X_n$ with CDF $F$. Define the sample maximum: $$ M = \max(X_1, X_2, ..., X_n) $$ Some simulations I've done (see the R code ...
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1answer
24 views

Rigorous Order Statistics with Indicator Functions

If three people are randomly placed along a 1 mile road, the probability that no two of them are less than $m$ miles apart for $m \leq \frac{1}{2}$ could be solved by using the density for the order ...
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20 views

How do I describe probabilities of realised variables in terms of CDFs and PDFs?

I have a game theory problem here where the realisation of the random variable $Y₂$ (termed $y₂$ is observed in the first stage of the game. $Y₂$ is the second highest order statistic for $n-1$ ...
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1answer
144 views

Joint distribution of range $(R=X_n-X_1)$ and mid-range $(V=\frac{1}{2}(X_1+X_n)$order statistics

Let $X_1,X_2, · · · , X_n$ be independent and identically distributed Uniform random variables on the interval (0, a) for a > 0, each having a density function $f(x) = \frac{1}{a}$, $0<x<a$. Let ...
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1answer
21 views

Order statistics, no distribution, probability of percentile

$Let \ Y_1<...<Y_8 \\ be \ the \ order \ statistics \ of \ n \ independent \ observations\ from \ a \ continuous \ type \ distribution \ with \\ 70th \ percentile \ \pi_0._7 = 27.3$ The ...
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1answer
41 views

Find the Distribution of an Order Statistic

I am working on a practice prelim question. It states: Let $Y_1 < Y_2 < Y_3 < Y_4$ denote the order statistics of a random sample of size 4 from a distribution which is uniform on ...
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13 views

$Y_{(n)} = X_{(n)}/\mu$?

If $ X_1, ...,X_n$ are iid random variables such that $ X_i \sim U(0, \mu)$, is that true that if $Y_i = X_i/\mu$, then $Y_{(n)} = X_{(n)}/\mu?$ I am sorry if the question looks so simple and I am nt ...
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1answer
31 views

Given a set of partial orderings of samples from a set of distributions, can we estimate the (relative) mean of the distributions?

This problem is motivated by attempting to construct a total ordering out of an arbitrarily large set of potentially contradictory partial orderings. Let's assume we have some set of items I for ...
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30 views

probability that a variable, as a function of choice variables, is among the top k out of n when ordered

Suppose $(h_1,h_2,...,h_n)'$ is an $n\times 1$ vector. Let $h_i=g_iX_i$, where $g_i$ is a choice variable which can vary across $i$ and $X_i$ is a random shock with Pareto Type I distribution. ...
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38 views

Order statistics of a max-min problem.

$X_{A,1}, X_{B,1}, X_{A,2}, X_{B,2}$ are independent and identically distributed (i.i.d.) exponential random variables with parameter $\lambda=1$, i.e., the means of the four random variables are all ...
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1answer
51 views

Probability that a random variable is among the top k out of n when ordered

Suppose $X_1,X_2,\ldots,X_n $ are $n$ i.i.d. random variables with a continuous distribution $F(x)$ and density function $f(x)$. What is the probability distribution that any given $X_i$ is among the ...
3
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1answer
103 views

Expectation of first and second order statistics in a random distribution

Let $E(f_{i}^{n})$ and $E(s_{i}^{n})$ denote the expected first and second order statistics for $n$ draws from the distribution $V_i$ .i.e set $X_{i}^{n}=\{x^1,.....,x^n | x^j \sim V_i \}$ and let ...
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1answer
26 views

Testing a hypothesys about a survey?

A survey of $61, 647$ people including questions about office relationships. Of the respondents, $26$% reported that bosses scream at employees. Use a $.05$ significance level to test the claim that ...
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1answer
28 views

Proving some properties about the expected first order statistic (maximum) with respect to sample size.

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as ...
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108 views

Threshold calculation

I need to calculate a threshold $[T]$ for a data set. The elements are a total count $[t]$ of items, and of those total items there will be variable counts of hot$[h]$ and Unknown$[U]$. So the ...
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39 views

Definite integral of product of regularized incomplete beta functions

I'm trying to find $$\int_{0}^{1} \! {I_x'(\alpha_1,\beta_1) I_x(\alpha_2,\beta_2) \dots I_x(\alpha_n,\beta_n)} \,\mathrm{d} x$$ where $\alpha_i, \beta_i$ are constant. Note the first term is ...
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21 views

Distribution maximum with small sample related to large sample

Suppose the random variables $X_i$, $i=1,\cdots,n$ and $Y_j$, $j=1,\cdots,m$ all have distribution $F(x)$, with order statistics denoted by $X_{(i)}$ and $Y_{(j)}$. Assuming $n<m$ (e.g. $n=m/100$), ...
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1answer
51 views

Gamma distribution Norming constant for extreme minima

the norming constants for extreme maxima of Gamma distribution is known and is give in link.springer.com/article/10.1007/s10687-010-0125-3. I would like to know is there reference or paper that states ...
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1answer
38 views

How to calculate $\int\limits_{-\infty}^{+\infty} \frac{n-1}{\alpha}\Phi(\frac{x}{\alpha})^{n-2}\phi(\frac{x}{\alpha})^2dx$?

I was working on a research project that involves taking the integral of $$\frac{n-1}{\alpha}\int\limits_{-\infty}^{+\infty} ...
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1answer
91 views

Normalizing constants for Extreme value distributions

I have a question regarding the normalizing constants $\mu$ and $\sigma$ that appear in the following problem. Let the random variable $Y_n$ be $Y_n=max(a_1,a_{2},\cdots, a_n)$ and $X_{n}$ be ...