All Questions
Tagged with extreme-value self-study
32 questions
8
votes
4
answers
1k
views
Linearity of maximum function in expectation
I was solving an exercise for a probability theory course and stumbled upon the following problem.
Given a continuous random variable $X$, and $\max(a,b) = a$ if $a > b$ and $b$ otherwise, is
$$
E[\...
- 193
2
votes
1
answer
248
views
Extreme value theory for detrended series
I'm reading "An Introduction to Statistical Modeling of Extreme Values" by Stuart Coles, and using the pyextremes package for exploring the data which is time to return (in days). After ...
- 21
2
votes
0
answers
76
views
Tail-equivalence implying same domain of attraction
Suppose two distributions F and G that have the same extreme point ($x^F = x^G$) and
$$\lim_{x \to x^F}\frac{\bar{F}(x)}{\bar{G}(x)} = c \in (0, \infty)$$
Show that F and G belongs to the same domain ...
- 21
1
vote
0
answers
71
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Expected value of maximum of $n$ iid exponential random variables [duplicate]
I was recently playing around with this distribution. Let $Y_n \sim \max_i X_i$ where $X_i \sim \exp(\lambda)$. Then the well-known result
$$ f_{Y_n}(y) = \lambda n e^{-\lambda y}(1-e^{-\lambda y})^{n-...
- 141
5
votes
1
answer
171
views
$N \sim \text{Po}(\lambda)$ and $X_1,X_2,....,X_N$ are iid and independent of $N$, what is distribution of $Z_N = \max \{X_i\}_{i=1}^{N}$
I think the title covers most of my concerns. The distribution of the $X_i$ does not really matter, I am just experiencing difficulties in finding an expression for
$$\text{Pr}(Z_N \leq x) = F(x)^N$$
...
9
votes
1
answer
442
views
Intuition about the coupon collector problem approaching a Gumbel distribution
The coupon collector's problem
Let there be $n$ different types of coupons and we try to collect all of the types.
We do this by independent random draws of coupons in which each type of coupon has an ...
- 86.5k
1
vote
0
answers
125
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Distribution of maximum of sample means
Let $X_1, ..., X_n$ be a sample from $N(\mu, 1)$. Fix $1 \leq m<n$ and define $$T_i= \frac{1}{m}\sum\limits_{j=i}^{i+m-1} X_j,$$ for $i \in \lbrace 1, ..., n-m+1 \rbrace$. We have the test that ...
- 111
2
votes
1
answer
2k
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Proving the convergence of the maximum of Uniform Distribution
I have a random sample of size $X_1, X_2, .., X_n$ following $U(0,2)$. I need to prove that $X_{(n)}$ which is the maximum ordered statistics will converge to $2$ in probability and almost surely.
I ...
- 1,048
1
vote
1
answer
781
views
Asymptotic Distribution of Minimum Uniform Random Variables
I've been working on this problem for a while, and I've made some progress, but I'm still stuck on some parts. I was hoping to get some assistance with this!
Let $M_n = \min(X_1, ..., X_n)$ where $...
2
votes
1
answer
451
views
Show that $nX_{(1)}$ is not consistent
Consider a random sample from exponential distribution with mean $\frac{1}{\theta}$. I have to prove that $nX_{(1)}$ is not consistent for $\frac{1}{\theta}$ . A sufficient condition for consistency ...
- 1,397
1
vote
1
answer
151
views
Question regarding Extreme Value Theory and finding the distribution of X(n)
Hello stats stack exchange, I have a question regarding Order Statistics and the asymptotic distribution of $X_n$ which is the rv for max($X_1$, $X_2$,...,$X_n$) where $X_i$ are from some distribution....
- 41
2
votes
0
answers
300
views
Find the limiting distribution of $(bn)^{-\frac{1}{\alpha}} X_{(n)}$
let $\{X_n\}_{n\geq 1}$ be a sequence of i.i.d random variables with common distribution $F$, and write
$X_{(n)}=\max\{X_1,\cdots , X_n\}$ , $n=1,2,\cdots$
(a) for $\alpha >0$ , $\lim_{x\rightarrow ...
- 1,349
-1
votes
1
answer
210
views
Estimation of an exponential parameter
I´m trying to figure out the pdf $f_\min(X_i)$ of $\min(X_i)$, where the distribution of the sample $X_1,...,X_n$ is $\mathcal{E}xp(\lambda)$, where $\lambda$ is the unknown parameter.
I tried with ...
- 1
2
votes
1
answer
180
views
Maximum A Posteriori Estimate
The formula for calculating the MAP estimate of a particular parameter, $p$, is given by the following: $p^{MAP} =$ argmax $P(p)P(p|x)$.
Now I am trying to do a question where I am told the prior ...
- 571
5
votes
1
answer
487
views
Extreme value theory: show that $ \lim_{n\rightarrow \infty}a_n $ exists and is finite
Well known facts in extreme value theory:
Let $\{X_i\}_{\forall i \in \{1,...,n\}}$ be i.i.d. random variables with cdf $F$. If there exists $\{a_n\}_{n\in \mathbb{N}}>0$, and $\{b_n\}_{n\in \...
- 935
1
vote
0
answers
364
views
Normalising constant of the Gumbel in extreme value theory
Well known facts in extreme value theory:
Let $\{X_i\}_{\forall i \in \{1,...,n\}}$ be i.i.d. random variables with cdf $F$. If there exists $\{a_n\}_{n\in \mathbb{N}}>0$, and $\{b_n\}_{n\in \...
- 935
2
votes
2
answers
1k
views
Distribution of extreme values, case of uniform
Question: For $U_1 , \dots, U_n$ i.i.d. $U \sim \mathrm{unif}[0,1]$, we want to find the asymptotic distribution of $Z_n = n(1-U_{(n)})$ where $U_{(n)} = \max(U_1 , ... , U_n)$
I found this: ...
- 770
0
votes
1
answer
170
views
maximising a linear model function with unknowns
If i have this linear model
$$Y_{i,t}=\gamma_t(x_i)+v_{i,t}, v_{i,t} \stackrel{iid}{\sim}N(0,\sigma^2), i=1,\ldots,m.$$
$$\gamma_t(x)=\beta_{1,t}+\beta_{2,t}\frac{1-e^{-\lambda x}}{ \lambda x}+ \beta_{...
- 1
0
votes
0
answers
20
views
New question based on an existing question on Minimum and Maximum of N(0,1) [duplicate]
This question is an additional question to the given posted here: Variance of Minimum and Maximum of 2 iid Normal
Let $X, Y$ be independent $N(0,1)$ and let $M=Max(X,Y)$. In the previous problem, ...
- 1,347
8
votes
3
answers
1k
views
Variance of Minimum and Maximum of 2 iid Normal
Let $X$ and $Y$ be iid $\sim Normal(0,1)$
Let $A=max(X,Y)$ and $B=min(X,Y)$
What are $Var(A)$ and $Var(B)$?
From simulation, I get $Var(A)=Var(B)$ approximately 0.70.
How do I get this ...
- 1,347
0
votes
1
answer
4k
views
Finding the PDF of Y, where Y = min X
Have $ X_{1},X_{2},\cdots,X_{10}$ random sample from a distribution with PDF:
$$ f(x;\theta) = e^{ - (x- \theta) },\, \theta \leq x \lt \infty $$
Know that $ \hat{\theta}_{MLE} = Y = min(X_{i},\;i=...
- 39
6
votes
1
answer
3k
views
Maximum of a set of values from given mean and median
The arithmetic mean and median of $5$ distinct natural numbers are both $7$,what may be the maximum of the $5$ numbers?
Here we see that the median is the $3rd$ value.Also the sum of the numbers is $...
- 810
2
votes
2
answers
212
views
P(X<Y|Z=t) where Z=min(X,Y)
Lets X and Y be uniform random variable where $x \in [0,a]$ and $y \in [0,b]$ where a < b. We design $Z=\min(X,Y)$.
I know that the CDF of Z is $P(Z<z)=1-\frac{(a-z)(b-z)}{ab}$
And by ...
- 719
2
votes
1
answer
7k
views
Proof for the p.d.f of minimum and maximum of a sample
The following is a question from a past paper for one of my university statistical inference modules, and I know how to use the formula for each the max/min, but
Assume that the sample $X_1, X_2, ....
- 23
2
votes
1
answer
114
views
probability distribution of the maximum
Let T be a random variable giving the time to failure of led lights that follow exponential distribution with a mean value of 15 000 hours.
We put three new lights at the same time. Find the ...
- 61
10
votes
1
answer
3k
views
Maximum likelihood estimator for minimum of exponential distributions
I am stuck on how to solve this problem.
So, we have two sequences of random variables, $X_i$ and $Y_i$ for $i=1,...,n$. Now, $X$ and $Y$ are independent exponential distributions with parameters $\...
- 1,903
1
vote
1
answer
2k
views
Probability density function of the sample maximum of a random variable
According to my book, for a random sample $(X_1, \ldots, X_n)$ from a continuous distribution with p.d.f. $f(x)$ and c.d.f. $F(x)$, the p.d.f. of the maximum of the sample is $g(z)=nf(z)[F(z)]^{n-1}$, ...
- 13
3
votes
1
answer
158
views
Joint distribution of a random variable and the sample maximum
This is one necessary part of a slightly larger problem, but this part has me stumped.
We have that $X_1, X_2, ..., X_n\stackrel{iid}{\sim} U(0,\theta)$. What is the joint density of the first ...
- 39
2
votes
0
answers
589
views
Convergence in Probability of the minimum
This is a homework question. I think I have the correct answer, but I am not sure. Also, the wording sounds very awkward. Is there a better way to show this (or better way to word this)?
Let $X_1,\...
- 10.2k
4
votes
1
answer
1k
views
Convergence in probability of minimum
This is a homework problem. Suppose we have a random sample $X_1,\ldots,X_n \overset{iid}{\sim} F$ with density $f(x) = 2(x-\theta)$ for $x\in (\theta,\theta+1)$. Let $X_{(1)} = \min{\{X_1,\ldots,X_n\}...
- 10.2k
1
vote
0
answers
171
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What is the meaning of McFaddens Axiom: Irrelevance of Alternative Set Effect?
On page 110 of McFadden,1973 - Conditional logit analysis of Qualitative Choice Behavior, Frontiers in Economics, ed Zarembka, New York: Academic Press, pp. 105-142 the following three Axioms are ...
- 1,113
2
votes
2
answers
25k
views
Determine density of $\min(X,Y)$ and $\max(X,Y)$ for independently uniform distributed variables
Two independent random variables, $X$ and $Y$, are uniformly distributed on the unit interval $(-1,1)$.
Determine the density for $U=\min(X,Y)$ and for $W=\max(X,Y)$
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