Questions tagged [minimum]

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

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1answer
56 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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0answers
33 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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81 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 ...
2
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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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0answers
10 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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0answers
44 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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1answer
37 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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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 \...
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0answers
29 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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0answers
34 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
82 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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0answers
47 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 ...
2
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2answers
76 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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0answers
37 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
217 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
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 ...
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0answers
70 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
45 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 ...
2
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1answer
51 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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2answers
305 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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0answers
86 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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0answers
370 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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0answers
16 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
113 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
113 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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0answers
40 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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0answers
48 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 ...
3
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1answer
58 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,...
4
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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
44 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. ...
2
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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 ...
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0answers
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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0answers
165 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 ...
4
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1answer
497 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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0answers
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
54 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
96 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
706 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 ...
2
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1answer
111 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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0answers
74 views

Estimate minimum values of the dependent variable [closed]

We know that linear regression estimates the expected mean value of a dependent variable, as a function of the independent variables. I would like to know, however, if there is any theory about some ...
3
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1answer
167 views

What are bounds for the covariance of x and min(x,y)?

Suppose $x$ and $y$ are independently distributed according to CDFs $F_x$ and $F_y$ respectively over compact support $[0,1]$. Under what condition is $\operatorname{Cov}(x, \min\{x,y\})\gt 0$?
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2answers
249 views

Find distribution of two i.i.d. variables when their minimum is known

If I know the CDF of $X$ where $X=\min(X_1,X_2)$, is it possible to find the CDFs of $X_1$ and $X_2$? I know that $X_1$ and $X_2$ are i.i.d. such that $$F_{X}(x)=2F_{X_i}(x)-F_{X_i}(x)^2$$ where $i=$ ...
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3answers
1k views

Why is empirical risk minimization prone to overfitting?

According to Chapter 8 of the book Deep Learning, "..empirical risk minimization is prone to overfitting. models with high capacity can simply memorize the training se." My question why is it so? ...
2
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0answers
599 views

glmnet returning lambda that gives all-zero coefficients as optimal lambda

Before I start, I have already looked at the answers for related questions: How to interpret all zero coefficients in the results of cv.glmnet? Why is cv.glmnet giving a lambda.min that is ...
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1answer
1k views

Calculating a confidence interval for the min/max of a distribution when sample may not reflect the underlying distribution

I'm interested in finding the minimum and maximum of a particular distribution. However, I don't have a model/function of the distribution, but I can generate at random samples from this distribution....
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1answer
20 views

Assign each sample in B to each element of a matrix A

I've measured the distance between 100 brain regions and 5 "core" brain regions. This led me to a 100x5 matrix (A) of empirical distances. Now, I have a second 100x5 matrix (B), where the distances ...
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1answer
57 views

Reference request: parameter inference on simulated distribution

Say I have some data $(x_{1},\ldots,x_{n})$ which I believe to be drawn from some distribution $\nu_{\theta}(x)$. I'm moderately familiar with estimation techniques for $\theta$ when I have some ...
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0answers
1k views

Normalization when Max and Min Values are Reversed

I'm running an experiment where I'm continuously sampling from a dial hooked into a physiological recorder as a hack because the dial won't interface with the equipment we're using any other way. By ...
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0answers
120 views

Distribution of the minimum of the squared Euclidean norm of a $N(\mu,\Sigma)$ random variable

Suppose that $X^n := \{x_1, x_2, \ldots, x_n\}$ is a sample of $n$ i.i.d $p$-dimensional points, where $X \sim N(\mu, \Sigma)$. What is known about the distribution of $\min_{x_i \in X^n} \|x_i\|^2_2$?...