Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [maximum]

A maximum is the largest value in a set, function, variable, distribution etc.

1
vote
1answer
17 views

Normalizations: dividing by maximum [on hold]

I'd like to know what are the reasons and benefits of dividing all the values of a dataset by the maximum of the dataset. Are they referred by authors? This normalization is well known in gene ...
0
votes
0answers
14 views

Maximum likelihood second derivative test

Can anyone explain what to do if the maximum likelihood second derivative test comes back positive such that the M is a saddle point instead of a maximum value? What do I do after that to figure out ...
0
votes
0answers
22 views

Likelihood ratio statistic lognormal

I want to determine the LRS lambda of a lognormal distribution under H0: the variances are equal. What I have so far is the following: $H_0: \sigma^2 = \sigma_{o}^{2}$ $H_1: \sigma^2 \neq \sigma_{o}^...
0
votes
0answers
36 views

Estimate true mean of the maximum of N sample means

Let's say we have N distributions $\mathcal N(\mu_i, \sigma_i)$, each with unknown mean $\mu_i$ and unknown standard deviation $\sigma_i$, $i=0,...,N-1$. For each $i$, $M$ independent random samples ...
2
votes
2answers
97 views

What is element-wise max pooling?

I came across this term in the VoxelNet paper in relation to point cloud based object detection using machine learning. It is mentioned in figures 2&3 and in 2.2.1 I am familiar with 2d max ...
4
votes
1answer
82 views

distribution for scaled Maximum of n independent Weibulls for $n \to \infty$

Assume that $X_1, X_2,...\sim Weibull(\lambda, k) \quad iid.$, i.e. $F(X_1\leq x) = 1-e^{-(\lambda x)^k}$ define $M_n:= \max\{X_1, ..., X_n\}$ and $\tilde{M}_n:=\frac{M_n-b_n}{a_n}$ according to ...
0
votes
0answers
31 views

Confidence Intervals of not Gaussian functions

Is anybody know a good tutorial about how we calculate Confidence Intervals of not Gaussian functions? I give some example of what I kind of function I think about: 1st example: Let be $ X_1, X_2 \...
1
vote
1answer
52 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 ...
1
vote
1answer
275 views

cross entropy loss max value

The cross entropy loss function for multiclass can be computed as: $$-\sum\limits_{i=1}^N y_i log \hat{y}_i$$ where $y_i$ is a class and $\hat{y}_i$ the estimated probability. The minimum value is $0$ ...
1
vote
1answer
261 views

Expectation of max of two normal random variables

I have been reading this paper about the maximum and minimum of two normal distributed variables. Inside the paper there is the formula for the expectation of this the maximum of the two variables. ...
1
vote
0answers
55 views

m out of n bootstrap implementation in R

I am wishing to estimate the sampling distribution of an extreme order statistic (the sample maximum). The usual nonparametric (n-out-of-n) bootstrap fails miserably in this case. Chernick (2011) ...
1
vote
2answers
79 views

Estimating the min and max of a distribution

I have a measurement problem where I am attempting to measure the minimum and maximum height of a surface by taking point samples of heights. If I then look at the distribution of all height values, ...
0
votes
0answers
22 views

Variability metric for Top 3 sports teams in a league

**I am a bit unsure what to mark this as/title this as. We refer to this type of phenomena as 'volatility' but this apparently has a specific context with regards to statistical phenomena so any ...
1
vote
1answer
100 views

Mean and variance of maximum of normal random variables

I'm trying to find the mean and variance of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. Note that the $X_i$ are independent, but not identically distributed. That is, ...
1
vote
1answer
401 views

Computing the Hessian of maximum log likelihood function

I am trying to find the Hessian matrix for the maximum log likelihood function given training data {(xi, yi)} for i=1:N with yi ∈ {+1, −1} for each i = 1, . . . , N for the function: When I try to ...
0
votes
0answers
36 views

Name for (maximum+minimum)/2 and relationship to average?

Is there a common name for $c := \frac{max(X)+min(X)}{2}$? What is the relationship between $\tilde{x} := Avg(X)$ and $c$? What metrics or information can I derive from $\tilde{x}$ and $c$? If I ...
0
votes
0answers
70 views

Unable to solve using lagrangian multipliers

Suppose $$K(x,z) = \theta(x)^T \theta(z) = \left\{ \begin{array}{ll} 1 & \text{if } x = z \\ 0 & \text{otherwise} \end{array} \right. $$ and $y_1=+1$ or $-1$. I ...
4
votes
2answers
224 views

Is it possible to obtain more accurate annual extremes predictions from sub-annual data?

I'm looking at various extreme climate variables, such as 50-year or 500-year maximum daily precipitation, using a generalized extreme value (GEV) distribution. The problem with this is that there are ...
0
votes
1answer
26 views

Cumulative Probability Distribution of Maximum and 2nd from Maximum of 4 Variables

I understand that the cumulative probability distribution cum(x) of the maximum of 2 variables x1 and x2 with probability distribution p1(x1) and p2(x2) is the product of the two cumulative ...
0
votes
0answers
14 views

Distribution of maximum variance explained by 1 variable

Say I do principal component analysis on $n$ variables, and I sort the fractions of variance explained to find the largest. What is the probability distribution for this figure? For context I just ...
4
votes
1answer
46 views

Integral from the Adversarial Spheres paper (maximum of the difference between a constant and a normal random variable)

I'm trying to follow a proof in the Adversarial Spheres preprint on arXiv. The proof requires the computation of the integral in Appendix F, page 14: $$\mathbf{E}\left[\max\left(\sqrt{2}\left(\frac{\...
9
votes
2answers
109 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 these ...
1
vote
0answers
47 views

Determine maxima and minima of fitted GAMM smooth

I have been using gamm4 to model the daily activity pattern of a certain behavior as a binomial response (whether the behavior occurs or not for each hour of the day). I am comparing the daily ...
5
votes
1answer
705 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 ...
0
votes
0answers
54 views

Expectation of Maximum Value

I'm trying to understand the basic statistics involved in trading. Suppose I'm trying to decide whether to buy a stock whose current price is $V_0$. Suppose I have some fancy statistical model from ...
0
votes
1answer
50 views

Estimating max value from statistical data

Assuming that you have the following values for a data set: Median Mean First quartile Third quartile Standard deviation Number of elements Minimum value , would it be possible to somewhat ...
1
vote
1answer
39 views

Algorithm for selecting largest possible value, when observing online sequence of unknown distribution?

I have been trying to devise an algorithm for a problem that's been bugging me for a while. For some weird reason I haven't been able to find any mention of this problem in the literature, so far. I ...
5
votes
1answer
82 views

Effect of adding and removing data on variance

Consider a set of distinct numbers. After removing both the max and the min from the set and adding the median to the set, the set of numbers obviously becomes less dispersed and the variance should ...
0
votes
0answers
10 views

Measuring the cost of the prediction error when picking the maximum value in a sequence

I have several entities denoted as a set $A=\{1, \ldots, K\}$. They form a sequence $(y_n)_{n\in A}$ with $y_i\in \mathbb{R}$. The values of $y_i$ are unobservable and get estimated as $\hat{y}_i$. ...
0
votes
2answers
107 views

Maximization of Output based on Input

What I want to do is find the values for $X = $ { $x_j$ } that will produce the maximum $y$. I'm currently trying to maximize my output $y$, based on my inputs $X$. Say there are inputs, $X = $ { $...
4
votes
2answers
119 views

Need handy formula for $Var[\max(V, K)]$

In Appendix 12A, p. 262 of this book, the author Hull derives a handy, tractable formula for the expression $E[\max(V-K, 0)]$, where $V$ is a lognormally distributed random variable and $K$ is a ...
3
votes
1answer
198 views

Mean of maximum of exponential random variables (independent but not identical)

I am looking for the the mean of the maximum of N independent but not identical exponential random variables. I found the CDF and the pdf but I couldn't compute the integral to find the mean of the ...
1
vote
0answers
14 views

Error estimate for the position of a maximum given data

I posted the following question on Physics SE (here), but was told it might be better placed on Cross Validated. Alright, so I am not sure what terminology easily describes this, but I have an excel ...
1
vote
0answers
30 views

Compare maxima of two Bernouilli experiments

I am looking at the following question -- which has already been solved for the case of Gaussian samples Compare maxima of two Gaussian samples but I am unable to find a similar answer for the ...
2
votes
0answers
458 views

Upper bound on KL divergence

Is there a maximum (unique?) to the KL divergence between discrete distributions p & q, with the restriction that q is a proper probability distribution? I know KL is unbounded from above when q ...
0
votes
1answer
441 views

maximization of a function with nlminb in R

I know nlminb () takes a function, objective, and finds values for the parameters of this function at which the objective function achieves its minimum value and ...
0
votes
1answer
105 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_{...
3
votes
2answers
330 views

Generalized Pareto distribution (GPD)

I would like to understand the functional form of the Generalized Pareto distribution (GPD) presented in Wikipedia. My questions are: what is the rationale for replacing $z$ with $\frac{x-\mu}{\sigma}...
2
votes
1answer
96 views

Interpretation of Constraint in Maximum Entropy Derivation of Cauchy distribution

As per Wikipedia: The Cauchy distribution is the maximum entropy probability distribution for a random variate $X$ for which $$ {\displaystyle \operatorname {E} [\log(1+(X-x_{0})^{2}/\gamma ^{...
3
votes
1answer
82 views

Finding mode using mean and skewness (and higher moments)?

I have a pdf that doesn't yield trivial derivatives, so I cannot differentiate it and find the root to determine where its max exactly occurs. However, I have a general formula to express all its ...
0
votes
1answer
80 views

3-level hierarchical model and ferquentist approach

Could I use maximum likelihood method or any other frequenist method to estimate parameters for 3-level hierarchical model? Is there any references help me in this case? Thank you
1
vote
0answers
52 views

Maximise the probability of a linear combination of random variables

I have a data set representing a random vector $\mathbf{X}=(X_1,\ldots, X_p)'$. Define $Z=\alpha' X $, where $\alpha \in \mathbb{R}^p$ and $\alpha'\mathbf{1}_p =1$. I would like to find the $\alpha$ ...
1
vote
0answers
141 views

Why does this sequence of random variables converge in distribution?

Given iid random variables $X_1, \dots, X_n$ with common density: $$ f(x) = 1\{ x > 0 \} \cdot \frac{1}{(x+1)^2} $$ it is supposed to be the case that $\frac{\max_i X_i}{n}$ converges in ...
1
vote
0answers
31 views

How can I show that R^2 in multiple linear regression is maximum of corr(y,Xbeta)?

let $Y=X\beta +\epsilon, \ \epsilon \sim N(0,\sigma^2 I_n)$, (Y: nx1 vector, X: nxp matrix, beta:px1 vector) assume that both $Y$ and $X$ are centered, so that the sum of them becomes 0. How can I ...
1
vote
0answers
42 views

Modeling relationship between one variable and maximum values of another variable

I'm having trouble with this because I suspect I'm missing some key terminology in how to ask this question. I have data that shows a relationship shown below: Notice how as js_avg increases, the ...
2
votes
1answer
256 views

Boundaries on correlation coefficient given five other correlations

Is there a general formula for the boundaries of a correlation coefficient given a set of other correlation coefficients? I have seen the formula for three random variables where two correlations are ...
12
votes
3answers
529 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 ...
1
vote
0answers
52 views

Why “softmax” is called “softmax”? How does it related to “max”? [duplicate]

Why "softmax" is called "softmax"? How does it related to "max"? I am trying following code and they are not look like each other. ...
0
votes
0answers
16 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
vote
0answers
96 views

Expected value of maxima of dependent random variables

I don't know if such theorem exists, but what I am looking for is a closed-form solution for $$E[\max(X_1, ..., X_N)]$$ where $X_1, ..., X_N$ is a sequence of dependent identically distributed ...