Questions tagged [maximum]

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

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13 views

Biggest gap in symetric interval of the normal distribution

Suppose you have $n$ ordered points $x_1, \ldots, x_n$ sampled from the normal distribution $N(0,1)$ and bounded (all points are in $[-a, a]$ for a fixed $a>0$) What's the distribution of the ...
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78 views

Asymtotic distribution of the MLE of a Uniform

A property of the Maximum Likelihood Estimator is, that it asymptotically follows a normal distribution if the solution is unique. In case of a continuous Uniform distribution the Maximum Likelihood ...
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21 views

Finding maximum of quadratic function that depends on other variables

I am trying to fit a model of the following form in R: yield = solar_rad + I(solar_rad ^ 2) where each observation is a field and ...
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14 views

Process of the max of a gaussian process

I know how to calculate the distribution of the max of a Gaussian process. I am now wondering what's its process. Are its properties known? (For instance I guess that the length of constant parts ...
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19 views

Principles for choosing standard deviation of gaussian convolution filter

I have collected data from a chemical analysis, and I need to find the maximum of its first derivative; however it is too noisy to find this by simply taking the maximum. The noise can be smoothed out ...
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28 views

Exponential Inequality For Probability of Being Close to Maximum

Given $n$ independent identically distributed random variables $X_1, X_2, \ldots, X_n$ that have $|X_i| < \lambda$ for all $i$. Let $\max(X)$ be the maximum of these $n$ variables. Is there a ...
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12 views

Probability of random population value being higher than sample maximum

Considering a small sample size (n < 10) from a population, I'm trying to find how likely a random population value would be greater than the maximum of the sample. Hoping ye could help me with ...
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77 views

How to compute largest values of random variables? [closed]

Suppose we have two discrete random variables and we want perform maximum operation to obtain the max PDF. We know that max of two independent random variables is: if Z = max(X,Y) ...
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1answer
89 views

Distribution with 3 Modes, Find the 2 In-Between Minima

Suppose I have a dataset consisting of numbers drawn from three normal distributions $\mathcal N\!(\mu_{\rm left}, \sigma_{\rm left}^2),\ \mathcal N\!(\mu_{\rm center}, \sigma_{\rm center}^2),\ \...
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16 views

MAP versus Component-Wise Maximum Marginal

Suppose I have the joint distribution: \begin{align} p(\mathbf{x}) = p(x_1, ... , x_n) \end{align} The maximum a posteriori (MAP) solution is given by: \begin{align} \mathbf{x}_{MAP} = \arg \max p(\...
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42 views

Problem with two correlated random normals

Imagine you have a two-dimensional multivariate normal random variable with $\mu = [0, 0]$ and $\Sigma\ = \begin{bmatrix}1 & r\\r & 1\end{bmatrix}$. (Conceptually, you have two random normal ...
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18 views

Extreme Value Theory - Determining the positive normalising constant in the Extremal Types Theorem

I am working through the following question and cannot seem to work out how the final result is obtained from the last inequality involving $a_n$. Can someone shed some light?
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77 views

What is the distribution of a bivariate normal component conditional on the max of the other component?

Let $n$ be a large integer, and consider two independent multivariate Gaussian $n$-vectors $x, z$ with $x\sim\mathcal{N}\left(0,I\right),$ and $z\sim\mathcal{N}\left(0,\sigma^2 I\right)$. Let $y=x+z$. ...
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26 views

Normalizations: dividing by maximum [closed]

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 ...
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18 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 ...
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28 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}^...
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41 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 ...
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355 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 ...
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103 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 ...
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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 \...
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91 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 ...
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1answer
1k 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 $0$ ...
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582 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. ...
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130 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) ...
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134 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, ...
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23 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 ...
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178 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, ...
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678 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 ...
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41 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 ...
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226 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 ...
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30 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 ...
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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 ...
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54 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{\...
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112 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 ...
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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 ...
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1k 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 ...
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78 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 ...
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1answer
43 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 ...
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1answer
99 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 ...
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160 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 = $ { $...
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122 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 ...
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322 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 ...
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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 ...
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31 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 ...
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684 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 ...
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1answer
585 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 ...
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108 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_{...
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406 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}...
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1answer
102 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 ^{...
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1answer
94 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 ...