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Questions tagged [maximum]

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

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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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18 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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74 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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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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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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44 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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99 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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33 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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35 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 ...
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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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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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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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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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307 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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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 ...
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48 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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32 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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77 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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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 $\hat{y}_i$. ...
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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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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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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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Add a sample to Maximum correlation?

Assume we have two random variables X and Y. After taking 100 samples of $X$ and $Y$ as $\{x_1, x_2, ... , x_{100}\}$ and $\{y_1, y_2, ... , y_{100}\}$, the correlation between these two series of ...
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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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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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279 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
288 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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103 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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280 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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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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74 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 ...
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73 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
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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 $\alpha$ ...
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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 ...
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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 ...
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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 ...
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1answer
216 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 ...
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447 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 ...
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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. ...
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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, ...
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80 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 ...
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1answer
54 views

Simple probability question (similar to birthday paradox)

If $x$ objects are randomly distributed to $n$ groups, what is the formula for working out how big $x$ needs to be for the probability that at least one of the groups gets an amount $y$ (or larger) to ...
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Unbiased estimator for top-k bernoullis

Supposed I have $n$ coins and I'm interested in finding the $k < n$ coins which have the highest odds of coming up heads and I want to know $p(heads)$ for each of these $k$ coins. Assume that I'm ...
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Find “best” range of one value based on correlated value

Scenario: I want to give advice on how long a post should be for maximum number of interactions (lets call "Likes"). I have Description Length and Likes for each post. (I've excluded posts with no ...
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534 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 ...
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1answer
86 views

Hypothesis test for minimum/maximum

I need some kind of hypothesis test (or at least a reasonable rule of thumb) that will enable me to validate if observed minimum and/or maximum is "close enough" to the theoretical minimum/maximum. I ...
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What are some of the methods for constrained maximization?

This question is related to a question that I asked earlier. Since I am not able to decipher the algorithm in the original question, I would like to know whether there are other algorithms that can ...
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Constrained Maximization Algorithm with Linear Contours

The Original Problem Let $M(X)$ be a function of a multivariate random variable $X$ with probability density function $f_X(X)$. Given a small positive real number $\alpha$, find $m_{\alpha}$ such ...