Questions tagged [minimum]

A minimum is the smallest value in a set, function, variable, distribution etc.

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17 views

Rewards in reinforcement learning for minimization problem

I am new to ML/DL/RL. I am looking to solve the classic travelling salesman problem (TSP), where the salesman has to visit all cities only once and finding the smallest path to do that (minimize ...
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6 views

Sample size for uniform multinomial to control minimum

I'm interested in controlling the minimum for a uniform multinomial distribution. Specifically, What would be the sample size to have the following inequality regarding the minimum? Do you know of ...
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1answer
52 views

Min, max and average value of a multiple regression line [closed]

Let's suppose that we have a theoretical MLR as followed: sqrt(number_of_divorces_per_country) = income +0.2*income^2 -years_married +frequency_conflicts^1/4 + ε with a,b and c being the coefficients....
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31 views

How to explain local minima found between two trained Neural networks?

I have trained 2 neural networks with SGD and then I have taken a linear path between their weights. Say W_0 and W_1 are the weight matrices of network 1 and network 2, respectively. Then I compute ...
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1answer
62 views

How to solve for the minimum KL Divergence when the distribution is discrete?

Say we have a simple case of $p(x,y)$ is a 3x3 matrix: $$\begin{bmatrix} 1/6 & 0 & 0 \\ 1/6 & 3/6 & 0 \\ 0 & 0 & 1/6 \end{bmatrix}$$ And $q(x,y)=...
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19 views

Back-calculation of the minimum sample size for an mechanical experiment

I want to back-calculate the minimum sample size for an experiment. I have a known mean and standard variance of a statistical population. From the statistical population I chose two samples with ...
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1answer
54 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 $...
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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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26 views

Energy based learning for HMMs: Viterbi training

I understand why we want to maximise the posterior probability to find the most likely sequence of hidden variables but I've read that this is equivalent to minimising some concept of free energy. I'...
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18 views

How to minimise and expectation with respect to a parameter?

Suppose that $X$ is a random variable with distribution $G$. Let $H(X;\theta)$ be a parametric function with $\theta \in \Theta \subset {\mathbb R}^p$. I want to maximize the function $$\varphi(\...
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1answer
70 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 ...
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45 views

Limiting Distribution of $n\left[Y_n\right]$ where $Y_n$ is the minimum of a sample of size n from Uniform$\left(0,\theta\right)$ distribution

Suppose $X_1,X_2,\dots,X_n$ is a random sample from Uniform$(0,\theta)$ for some unknown $\theta > 0$. Let $Y_n$ be the minimum of $X_1,X_2,\dots,X_n$. (a) Suppose $F_n$ is the CDF of $nY_n$. Show ...
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178 views

Initial simplex for the Nelder-Mead algorithm in R using the “optim” function

How does R create the initial simplex when using the "optim" function? Is it different to the method applied in Matlab's "fminsearch"? Are there any other differences in the implementation? I have ...
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1answer
15 views

What is the variance of the least of several series?

Given a number of series (or images) that are independent and have a Poisson distribution with the same mean, what is the variance of the series generated from taking the point-by-point minimum? i.e. ...
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16 views

'Shrink' Step occuring often in Nelder Mead optimization?

I have an implementation of Nelder Mead, which is giving me good results. I had a bug, though, and while I managed to fix that bug, I noticed during my debugging that the 'shrink' step is occurring ...
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57 views

How is the minimum logarithmic loss calculated when initializing the XGBoost algorithm?

Suppose there are $5$ sample units, $2$ of which carry the feature $y=1$ to be predicted and three of which carry the feature $y=0$. So, $2$ are positive. The XGBoost algorithm initializes with $\...
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75 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 ...
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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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43 views

How to prove that the prior for which Bayes rule is also the minimax rule, is the least favorable prior?

I have read in the book Mathematical Statistics: A Decision Theoretic Approach by Thomas Ferguson that The prior for which the Bayes rule is also minimax rule, then that prior is Least favorable prior....
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35 views

Normal equations issue

Let $A\mathbf{x}=\mathbf{b}$ be an overdetermined system, with $A$ being an $n \times m $ full-column rank rectangular matrix. Are these minimization problems equivalent? $$ 1) \;\underset{\mathbf{x}...
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56 views

Identity on expectation of the minimum of two iid random variables with bounded support

I am reading the 2008 annals of statistics paper "Ranking and empirical minimisation of U-statistics" by Clémençon et. al, and read a statement which I do not know why is true. In order to accurately ...
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2answers
138 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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52 views

Minimum of n independent, but not identically distributed inverse Gaussians

I would like to find the probability distribution of the minimum of of n independent, but not identically distributed, i.e. differently parametrized inverse Gaussians. I would prefer an analytical ...
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2answers
77 views

Measure that takes samples that is minimized in expectation for a uniformly-distributed random variable?

I am having trouble thinking of a function that operates on a set of samples, that is, single-valued random variables between zero and one, $x_i \in (0,1), i\in\{1,2,...I\}$, and provides a measure of ...
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44 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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1answer
352 views

How to minimize Chi-Square using the CDF instead of the PDF?

Suppose one has data that is suspected to obey a normal distribution. One computes a histogram of the data, and performs Pearson's Chi-Squared Test. To perform this test, one must compare the observed ...
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1answer
103 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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72 views

Computing the CDF of the minimum of particular dependent random variables

For each $i=1,\dots,n$ let $Z_i\sim\text{Poisson}(\lambda_i)$, and suppose $\{Z_i\}$ are independent. Also for each $i=1,\dots,n$, let $\{Y_{ij}\}_{j\in\mathbb{N}}$ be an infinite sequence of iid ...
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1answer
48 views

Minimizing a least square [duplicate]

I'm a bit confused with matrix calculus. Given is $$f(x) = \frac{1}{2}||Ax-b||^2_2$$ and the derivate of it is in my book $$\nabla_xf(x) = A^T(Ax-b). $$ I don't see how this works. My plan was to ...
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1answer
65 views

Distribution of distance of N-1 gamma distributed iid random variables from minimum

I have the minimum value of N iid random variables that are gamma-distributed. The parameters of the gamma distribution are known. What would be the distribution of the distance of the remaining N - 1 ...
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317 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....
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113 views

What is the most efficient implementation of the min-k-cut algorithm?

Simple question, I'd like to apply min-k-cut algorithm on a graph to partition the graph in k clusters by minimizing the sum of edges cut during the formation of clusters. In my case I already know k ...
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677 views

Calculating minimum detectable effect from sample size and conversion rate

I have a function for calculating the required sample size based on four inputs: baseline conversion rate, minimum detectable effect, confidence and statistical power: ...
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19 views

Calculating minimum sum of ordered statistics

Can someone help me with this problem? Thanks and apologies in advance! The following sample of observations of X is given: 1,2,2,3,4,4,5. Calculate
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1answer
127 views

Expected minimum distance from a point with varying density

I'm looking at how the expected minimum Euclidean distance between randomly uniform points and the origin changes as we increase the density of random points (points per unit square) around the origin....
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1answer
273 views

ADAM Gradient descent oscillates close to minimum

I am using ADAM as an optimization algorithm to minimize some black box function $f(x,y)$. I know this function is convex and has a minimum $f(5,5) = 0$. Initially, the algorithm proceeds as expected:...
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61 views

What is the optimal gradient descent variation for a smooth function with clear minimum?

I can implement a vanilla gradient descent alogrithm, to minimize some function $f(x,y)$. Now if I know a priori that this function is smooth with a single global minimum (i.e. no local minima for a ...
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52 views

Fitting a distribution to random variable in R when the data is available for minimum of those random variables

I am new in R. I have the following problem: I have a dataset which presents the minimum of a set of n random variables (x1, x2,..., xn). The formula is Min(x1, x2,...,xn) = 1-(1-F(x))^n. It can be ...
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1answer
60 views

Minimum of Poissons

Let $X_i\sim\text{Pois}(\lambda_i)$ for $i=1,2,\ldots,n$ and $Y = \min X_i$. Can we show that, for example $\mathbb{E}[Y] \leq f(\lambda,n)\min\lambda_i$ for some $f : (\mathbb{R}^n,\mathbb{N}) \to [0,...
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1answer
2k views

Cdf of minimum of two iid random variables

I am struggling with the following sentence: Using the fact that the cumulative distribution of the minimum of two i.i.d. random variables can be expressed as $1 - (1 - F(x))^2$.... Can anyone ...
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1answer
31 views

Minimum of Unit-Exponentials Plus Constants

Define $e_i$ to be iid random variables drawn from an exponential distribution with parameter $\lambda=1$. $a_i$ are numerical constants. I am interested in the probability that $a_1 + e_1 < a_2 ...
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1answer
63 views

Linear regression: Why derviative = 0 is the minimum for OLS

I am new in this field, but I wanna advance fast and wan't to have a complete picture of everything. I have a very simple question, but I guess I am missing something in imagining the whole picture. ...
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1answer
312 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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33 views

Relationship between expected minimum as sample size increased

Suppose I have $N$ random variables $X_i$, $i = 1, \dots, N$. I am interested in the quantity $$ A = \mathbb E \left[\min_{i=1, \dots, N} X_i \right]. $$ Now suppose I take a subset $S \subset \{1, \...
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210 views

Gradient-informed global optimization

I am looking for a review or comparison of global optimization techniques where the gradient of the function is available and utilized to speed up search, like the following: A hybrid descent method ...
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1answer
665 views

What is the population minimizer for Huber loss

Suppose we make predictions for a continuous outcome $Y$ conditional on a vector of covariates $X$. If we use mean squared error loss (MSE), the population minimizer is $\mathbf{E}[Y \;|\; X]$. If ...
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40 views

Drawing the smallest value from a set of distributions

I have a specific number of normal distributions $N_D$ all with their own mean $\mu$ and std $\sigma$. Now I obtain a sample from all distributions which results in a set of $N_D$ samples. What is the ...
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1answer
64 views

How to regularize parameters across a 2D array

I'm attempting to fit a parameter (which will be a 2D array) to an array of data which corresponds to spatial locations (i.e. longitude/latitude). The parameter can vary from point to point but I want ...
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1answer
126 views

How to find coefficient that will minimize the distance between few times series

I have 3 time series X1, X2, X3. I want to find the coefficient (c1, c2) that will minimize the distance between them as follow: $$MIN\sum\sqrt{(X1-(c1*X2+c2*X3))^2}$$ The constrains are: $$-1< ...
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3answers
951 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 ...