All Questions
Tagged with minimum or extreme-value
606 questions
83
votes
3
answers
105k
views
How is the minimum of a set of IID random variables distributed?
If $X_1, ..., X_n$ are independent identically-distributed random variables, what can be said about the distribution of $\min(X_1, ..., X_n)$ in general?
- 941
73
votes
4
answers
191k
views
How do you calculate the probability density function of the maximum of a sample of IID uniform random variables?
Given the random variable
$$Y = \max(X_1, X_2, \ldots, X_n)$$
where $X_i$ are IID uniform variables, how do I calculate the PDF of $Y$?
- 871
69
votes
9
answers
8k
views
Taleb and the Black Swan
Taleb's book "The Black Swan" was a New York Times best seller when it came out several years ago. The book is now in its second edition. After meeting with statisticians at a JSM (an annual ...
- 43.2k
32
votes
3
answers
17k
views
Extreme Value Theory - Show: Normal to Gumbel
The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory.
How can we show that?
We have
$$P(\max X_i \leq x) = P(...
- 1,271
26
votes
6
answers
53k
views
Can mean plus one standard deviation exceed maximum value?
I have mean 74.10 and standard deviation 33.44 for a sample that has minimum 0 and maximum 94.33.
My professor asks me how can mean plus one standard deviation exceed the maximum.
I showed her ...
- 261
25
votes
2
answers
2k
views
Which distribution has its maximum uniformly distributed?
Let's consider $Y_n$ the max of $n$ iid samples $X_i$ of the same distribution:
$Y_n = max(X_1, X_2, ..., X_n)$
Do we know some common distributions for $X$ such that $Y$ is uniformly distributed $U(a,...
- 403
25
votes
2
answers
1k
views
Fitting custom distributions by MLE
My question relates to fitting custom distributions in R but I feel it has enough of a probability element to remain on CV.
I have an interesting set of data which has the following characteristics:
...
- 2,911
24
votes
2
answers
11k
views
Distribution of the maximum of two correlated normal variables
Say I have two standard normal random variables $X_1$ and $X_2$ that are jointly
normal with correlation coefficient $r$.
What is the distribution function of $\max(X_1, X_2)$?
- 2,295
24
votes
5
answers
5k
views
Why use extreme value theory?
I'm coming from Civil Engineering, in which we use Extreme Value Theory, like GEV distribution to predict the value of certain events, like The biggest wind speed, i.e the value that 98.5% of the wind ...
- 1,285
24
votes
6
answers
48k
views
Why doesn't k-means give the global minimum?
I read that the k-means algorithm only converges to a local minimum and not to a global minimum. Why is this? I can logically think of how initialization could affect the final clustering and there is ...
- 589
23
votes
3
answers
3k
views
Distribution of the largest fragment of a broken stick (spacings)
Let a stick of length 1 be broken in $k+1$ fragments uniformly at random. What is the distribution of the length of the longest fragment?
More formally, let $(U_1, \ldots U_k)$ be IID $U(0,1)$, and ...
- 14.9k
21
votes
5
answers
2k
views
Let X,Y be 2 r.v. with infinite expectations, are there possibilities where min(X,Y) have finite expectation?
If it is impossible, what is the proof?
- 542
21
votes
2
answers
2k
views
How can we bound the probability that a random variable is maximal?
$\newcommand{\P}{\mathbb{P}}$Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_n$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$....
- 738
19
votes
2
answers
12k
views
What is the variance of the maximum of a sample?
I'm looking for bounds on the variance of the maximum of a set of random variables. In other words, I'm looking for closed-form formulas for $B$, such that
$$
\mbox{Var}(\max_i X_i) \leq B \enspace,
$$...
- 273
15
votes
2
answers
21k
views
What is the distribution for the maximum (minimum) of two independent normal random variables?
Specifically, suppose $X$ and $Y$ are normal random variables (independent but not necessarily identically distributed). Given any particular $a$, is there a nice formula for $P(\max(X,Y)\leq x)$ or ...
- 355
15
votes
1
answer
10k
views
Finding local extrema of a density function using splines
I am trying to find the local maxima for a probability density function (found using R's density method). I cannot do a simple "look around neighbors" method (where ...
- 373
14
votes
1
answer
1k
views
Any example of (roughly) independent variables that are dependent at extreme values?
I am looking for an example of 2 random variables $X$, $Y$ such that
$$\newcommand{\cor}{{\rm cor}}|\cor(X,Y)| \approx 0 $$
but when consider the tail part of the distributions, they are highly ...
- 143
14
votes
4
answers
1k
views
Unbiased estimator for the smaller of two random variables
Suppose $X \sim \mathcal{N}(\mu_x, \sigma^2_x)$ and $Y \sim \mathcal{N}(\mu_y, \sigma^2_y)$
I am interested in $z = \min(\mu_x, \mu_y)$. Is there an unbiased estimator for $z$?
The simple estimator ...
- 141
13
votes
2
answers
9k
views
Markov chain Monte Carlo (MCMC) for Maximum Likelihood Estimation (MLE)
I am reading a 1991 conference paper by Geyer which is linked below. In it he seems to elude to a method that can use MCMC for MLE parameter estimation
This excites me since, I have coded BFGS ...
13
votes
3
answers
1k
views
Does there exist someone faster than Usain Bolt today?
EDIT: I am more interested in the technical issues and methodology of determining the likelihood of a "true" maximum in a given population given a sample statistic. There are problems with estimating ...
- 283
12
votes
3
answers
32k
views
Calculating distribution from min, mean, and max
Suppose I have the minimum, mean, and maximum of some data set, say, 10, 20, and 25. Is there a way to:
create a distribution from these data, and
know what percentage of the population likely lies ...
- 123
12
votes
3
answers
1k
views
Classes of distributions closed under maximum
Let $Q_p$ be a class of probability distributions on non-negative reals parametrized by $p$, so that
$$
Q_p([0,\infty)) = 1.
$$
I wonder which known classes of distributions are closed under ...
- 473
12
votes
1
answer
1k
views
Card game: If I draw four cards randomly and you draw six, what is the probability that my highest card is higher than your highest?
As stated in the title, say if I draw randomly 4 cards and you draw 6 from the same deck, what is the probability that my highest card beats your highest card?
How will this change if we draw from ...
- 611
12
votes
2
answers
3k
views
Order statistics (e.g., minimum) of infinite collection of chi-square variates?
This is my first time here, so please let me know if I can clarify my question in any way (incl. formatting, tags, etc.). (And hopefully I can edit later!) I tried to find references, and tried to ...
- 696
12
votes
2
answers
489
views
Expected value of spurious correlation
We draw $N$ samples, each of size $n$, independently from a Normal $(\mu,\sigma^2)$ distribution.
From the $N$ samples we then choose the 2 samples which have the highest (absolute) Pearson ...
- 2,020
12
votes
1
answer
494
views
Male and Female Chess Players - Expected Discrepancies at Tails of Distributions
I'm interested in the findings of this paper from 2009:
Why are (the best) women so good at chess?
Participation rates and gender differences
in intellectual domains
This paper attempts to explain ...
- 223
11
votes
2
answers
8k
views
Decision trees, Gradient boosting and normality of predictors
I have a question regarding the normality of predictors. I have 100,000 observations in my data. The problem I am analysing is a classification problem so 5% of the data is assigned to class 1, 95,000 ...
- 365
11
votes
2
answers
4k
views
Asymptotic distribution of maximum order statistic of IID random normals
Is there a nice limiting distribution of $\max( X_1,X_2,...,X_n) $ as $n$ goes to $\infty$, assuming that they are iid normal distributions with variance $\sigma^2$.
This is almost certainly a well ...
- 1,511
11
votes
1
answer
4k
views
Using bootstrap to obtain sampling distribution of 1st-percentile
I have a sample (of size 250) from a population. I do not know the distribution of the population.
The main question: I want a point estimate of the 1st-percentile of the population, and then I want ...
- 69.5k
11
votes
1
answer
236
views
Distribution of $\min_{j\ge 1}(X_1+X_2+\cdots+X_j)/j$ when $X_i$'s are i.i.d $\text{Exp}(1)$
Suppose $(X_n)_{n\ge 1}$ is a sequence of independent Exponential random variables with mean $1$. I am trying to find the distribution of $\min_{j\ge 1}(X_1+X_2+\cdots+X_j)/j$.
Simulation suggests the ...
- 11.6k
11
votes
2
answers
547
views
Tail bounds on Euclidean norm for uniform distribution on $\{-n,-(n-1),...,n-1,n\}^d$
What are known upper bounds on how often the Euclidean norm of a uniformly chosen
element of $\:\{-n,~-(n-1),~...,~n-1,~n\}^d\:$ will be larger than a given threshold?
I'm mainly interested in bounds ...
- 111
10
votes
1
answer
642
views
Distribution of argmax of beta-distributed random variables
Let $x_i \sim \text{Beta}(\alpha_i, \beta_i)$ for $i \in I$. Let $j = \operatorname*{argmax}_{i \in I} x_i$ (ties broken arbitrarily). What is the distribution of $j$ in terms of $\alpha$ and $\beta$? ...
- 1,033
10
votes
1
answer
4k
views
What's the difference between Bayesian Optimization (Gaussian Processes) and Simulated Annealing in practice
Both processes seem to be used to estimate the maximum value of an unknown function, and both obviously have different ways of doing so.
But in practice is either method essentially interchangeable? ...
- 459
10
votes
3
answers
433
views
If $Z_i =\min \{k_i, X_i\}$, $X_i \sim U[a_i, b_i]$, what is the distribution of $\sum_iZ_i$?
Assume the following set up:
Let $Z_i = \min\{k_i, X_i\}, i=1,...,n$. Also $X_i \sim U[a_i, b_i], \; a_i, b_i >0$. Moreover $k_i = ca_i + (1-c)b_i,\;\; 0<c<1$ i.e. $k_i$ is a convex ...
- 60.8k
10
votes
2
answers
575
views
Improving the minimum estimator
Suppose that I have $n$ positive parameters to estimate $\mu_1,\mu_2,...,\mu_n$ and their corresponding $n$ unbiased estimates produced by the estimators $\hat{\mu_1},\hat{\mu_2},...,\hat{\mu_n}$, i.e....
- 4,504
10
votes
1
answer
539
views
Expected value of maximum ratio of n iid normal variables
Suppose $X_1,...,X_n$ are iid from $N(\mu,\sigma^2)$ and let $X_{(i)}$ denote the $i$'th smallest element from $X_1,...,X_n$.
How would one be able to upper bound the expected maximum of the ratio ...
- 103
10
votes
2
answers
11k
views
How to find when a graph reaches a peak and plateaus?
This may sound very basic, but I have this problem:
I've got a queue of data with a window size of 300. New data is added at one end, old values are removed from the other end.
I expect the queue ...
- 203
10
votes
1
answer
2k
views
Extreme Value Theory: Lognormal GEV parameters
Lognormal distribution belongs to the Gumbel maximum domain of attraction, where:
$F^{logN}(x; \mu,\sigma)=\Phi\left(\frac{\ln x - \mu}{\sigma}\right)$,
$F^{Gum}(x;\mu,\beta) = e^{-\exp\left({-\frac{...
- 1,271
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
9
votes
4
answers
855
views
Why is the average of the highest value from 100 draws from a normal distribution different from the 98th percentile of the normal distribution?
Why is the average of the highest value from 100 draws from a normal distribution different from the 98% percentile of the normal distribution? It seems that by definition that they should be the ...
- 19k
9
votes
3
answers
10k
views
What is the expected MINIMUM value drawn from a uniform distribution between 0 and 1 after n trials?
Assume you draw a uniformly distributed random number between 0 and 1 n times. How would one go about calculating the expected minimum number drawn after n trials?
In addition, how would one go ...
- 597
9
votes
1
answer
6k
views
MAP estimation as regularisation of MLE
Going through the Wikipedia article on Maximum a posteriori estimation, it got confusing after reading this:
It is closely related to the method of maximum likelihood (ML) estimation, but employs ...
- 1,059
9
votes
3
answers
2k
views
Extreme value theory for count data
I am aware of extreme value theory for continuous distributions. I need to fit an extreme value distribution to the maximum observation of number of events on a day, per month. This seems to be the ...
- 205
9
votes
1
answer
3k
views
The code variable in the nlm() function
In R there is a function nlm() which carries out a minimization of a function f using the Newton-Raphson algorithm. In particular, that function outputs the value of the variable code defined as ...
- 22.4k
9
votes
1
answer
9k
views
Expected value of minimum order statistic from a normal sample
UPDATE Jan 25th 2014: the mistake is now corrected. Please ignore the calculated values of the Expected Value in the image uploaded - they are wrong- I don't delete the image because it has generated ...
- 60.8k
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
9
votes
2
answers
189
views
What is the most powerful result about the maximum of i.i.d. Gaussians? The most used in practice?
Given $X_1, \ldots, X_n, \ldots \sim \mathscr{N}(0,1)$ i.i.d., consider the random variables
$$ Z_n := \max_{1 \le i \le n} X_i\,. $$
Question: What is the most "important" result about ...
- 6,479
9
votes
2
answers
932
views
What is the distribution of maximum of a pair of iid draws, where the minimum is an order statistic of other minima?
Consider $n\cdot m$ independent draws from cdf $F(x)$, which is defined over 0-1, where $n$ and $m$ are integers. Arbitrarily group the draws into $n$ groups with m values in each group. Look at the ...
- 1,049
9
votes
1
answer
2k
views
Approximating the mathematical expectation of the argmax of a Gaussian random vector
Let $X = \left( {{X_1},...,{X_n}} \right) \sim \mathcal{N}\left( {{\mathbf{\mu }},{\mathbf{\Sigma }}} \right)$ be a Gaussian random vector and $I = \mathop {\arg \max }\limits_{i = 1,n} {X_i}$.
$I$ ...
user193726
8
votes
4
answers
1k
views
Linearity of maximum function in expectation
I was solving an exercise for a probability theory course and stumbled upon the following problem.
Given a continuous random variable $X$, and $\max(a,b) = a$ if $a > b$ and $b$ otherwise, is
$$
E[\...
- 193