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Can we find the closed-form solution to this optimization problem?
I have a simple optimization problem:\begin{equation}\max\limits_{k}\overline{Q}_{k}^{-1}(10^{-3})-k^2\\s.t. k>0\end{equation}where $\overline{Q}_{k}(b)$ is the complementary CDF of the chi-square...
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Approximation to the CDF of chi-squared random variable
I am trying to simulate the approximation of the CDF of a chi-squared random variable, using the method proposed By Luisa Canal2006.In Luisa Canal2006, the CDF of the $\chi_n^2$, denoted as $F_n(x,n)$,...
View Articledegrees of freedom should be 'k-p-1'... why??
In the goodness of fit test of Poisson distribution, the degrees of freedom should be$$k - p - 1$$which means,ν = (number of categories after pooling) − (number of parameters estimated) − 1.for...
View ArticleDistribution of the square of the euclidean norm of a gaussian vector with...
Let $X\sim\mathcal{N}_d(0,\Sigma)$ be a $d$-dimensional gaussian vector, where $0\in\mathbb{R}^{d}$ and $\Sigma\in\mathbb{R}^{d\times d}$ is diagonal. I'm interested on the distribution of:$$||X||^2$$I...
View ArticleProof and precise formulation of Welch-Satterthwaite equation
In my statistics course notes the Welch-Satterthwaite equation, as used in the derivation of the Welch test, is formulated as follows:Suppose $S_1^2, \ldots, S_n^2$ are sample variances of $n$ samples,...
View ArticleGiven the L2 norm of a Gaussian matrix, what distribution does the Gaussian...
Given a random gaussian matrix X with zero mean matrix and covariance matrix Σ, and two deterministic matrices A and B. If I know the value of $||{\bf{AX}}||_F^2$, how could I get the pdf of...
View ArticleTransform Wishart distribution to Chi-square distribution
This's actually what I'm trying to prove:$$ \frac {a^{'}\Sigma^{-1}a}{a^{'}W^{-1}a} \sim \chi^{2}_{n-p+1} $$$a$ is any P-dimensional nonzero constant vector, and $W \sim W_{p}(n,\Sigma)$, $\Sigma$ is a...
View ArticleWhat test can we use to compare the sample proportion of multiple dependent...
I am currently studying hypothesis testing for dependent two-sample (proportion). The crux for my question is this, what test does one use to compare the proportion of multiple samples for...
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How to explain light reduction in Helmert demonstration (Chi square...
I search to understand how Helmert find his second equation (2) in his article that explain Chi Square distribution function.You can find Helmert article from 1876 on Goettingen University Site (PDF...
View ArticleSimple(r) way to derive the expectation of an inverse Wishart?
I am looking for a simple way to derive the expectation of an inverse Wishart matrix., with distribution $W^{-1}$ where $W=\sum_{i=1}^n \Sigma^{1/2} g_i g_i^T \Sigma^{1/2}$ for a covariance $\Sigma\in...
View Articlepdf of Linear Combination of the same random variable
Let's say that a random variable X has a probability p to be Gamma($\alpha,\beta$) and a 1-p probability to be $\chi^2$(r). How do I prove that $f_x(x) = p...
View ArticleStatistical test for null hypothesis $\|p-q\|\le \epsilon$ for...
Is there any known (asymptotic) statistical test for the null hypothesis $$\|p-q\|\le \epsilon$$ for $k$-dimensional categorical data independently taken from two societies for some given norm...
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sum of two independent scaled noncentral $\chi$-squared random variables
I want to analyze or approximate a random variable that is a sum of two scaled independent non central $\chi$-squared random variables with the same degrees of freedom.For example,$$X = X_1 + a...
View Articlechi2 probability
I am trying to use this equation$Prob_d(\chi^2 > \chi_0^2) = \frac{2} {2^{d/2} \Gamma(d/2)} \int_{\chi_0}^{\inf} x^{d-1}e^{-x^2/2} dx$to compute probabilities of an empirical distribution $\chi^2_0$...
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suitable statistical method for variables dependency testing
I made a survey among small and medium enterprises and I asked two questions:Q1: How many employees does Your enterprise have ?none, 2. 1 to 9, 3. 10 to 19, 4. 20 to 49, 5. at least 50.Q2: How...
View ArticleAsymptotic convergence of sampling distribution of the sample variance
Let's consider a set $\{X_i\}_{i=1}^N$ of $N$ i.i.d. random variables drawn from the distribution $P_X(x) = \mathcal{N}(\mu, \sigma^2)$. Define the variable$$\hat{\sigma}^2 = \frac{1}{N} \sum_i (X_i -...
View ArticleFrobeinus norm of multiplication of two complex Gaussian distributed matrices
There are two complex Gaussian distributed matrices, $\mathbf{A}\in \mathbb{C}^{L\times M}$ and $\mathbf{B}\in \mathbb{C}^{N\times M}$.The elements of $\mathbf{A}$ and $\mathbf{B}$ are followed i.i.d....
View ArticleWhen to use chi square law for confidence intervals with mahalanobis distance?
So right now i'm reading this paper: Distance-based detection of out-of-distribution silent failures for Covid-19 lung lesion segmentation, available here: https://arxiv.org/abs/2208.03217In brief,...
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Distributing the error in a frequency table so that the $\chi^2$ statistics...
When using the $\chi^2$ statistic, if the errors (difference between observed and expected) are too low, the resulting statistic will be low. If we repeat the experiment several times with similar...
View ArticleSquare of a Chi-squared Random Variable
I have a Chi-squared RV with 2NL degrees of freedom and I am interested in the distribution of its square ($Y=X^2$). I have tried the transformation method to get the expression for the pdf of Y...
View ArticleChi Square Contingency Table - Formula Derivation
A chi-square distribution is constructed from normal random variables $X_i i=1,...n$ , each with normal distribution and mean $\mu$ and variance $\sigma^2$. Transforming to standard normal and...
View Articlethe expectation of linear combination of chi-squared random variables with 1...
I'm calculating the problem descripted in the title, and found it a little bit hard, here is the problem:Suppose $X_i\sim\mathcal{N}(0,1)$ is standard normal random variables, now we need to calculate...
View ArticleA integer calculation problem whose integrand is $\frac1{A \chi^2+B}$
I'm calculating an expectation of the form $$\mathbb E\left[1\over aX+b\right]$$where $X\sim\chi^2_1(0)$ obeys an central chi squared distribution with 1 degree of freedom. The integral formula is...
View ArticleAre sum and ratio of two independent chi-squared random variables independent?
Suppose that $X \sim \mathcal{X}^2_n$ and $Y \sim \mathcal{X}^2_m$ are independent. Can we say that $\frac{X}{Y}$ is independent of $X+Y$? For example, can we show that $$p(X/Y|X+Y) = p(X/Y)?$$We know...
View Article$X$ normally distributed, then $X^T \Sigma ^{-1} X$ follows chi square...
Suppose $X\sim \mathcal{N} _p (0, \Sigma )$. I am not sure why $X^{\top} \Sigma ^{-1} X$ follows a $\chi ^2$-distribution with $p$ degrees of freedom.I think it has something to do with the square root...
View ArticleFinding quantiles of chi square ditributions
I really need help on this exercise i honestly havent got a scooby on how to do this, any help would be much appreciatedExercise 17. Let $X_1, \ldots, X_n \sim \mathcal{N}\left(\mu, \sigma^2\right)$ be...
View ArticleEstimate the integral $\int_{4}^{\infty} x^{2}\exp(-x^{2}/2)/2\pi\ dx$ in...
I know that if we had $\int_{-\infty}^{\infty} x^2\exp(-x^{2}/2)/2\pi dx$, we would be talking about the mean of $\chi^{2}$ with 1 freedom degree, that is 1. However, as the bounds of integration are...
View ArticleFinding α-Quantiles of χ2 Distribution for Variance Estimation
I posted this same question yesterday but it got closed because i hadn't met the guidlines for questions, my apologies guys, so i'm going to re-write it better this time.Exercise 17. Let $X_1, \ldots,...
View ArticleHow can we write a non-central chi-squared distribution as gamma distribution?
Consider a random variable that has a non-central chi-squared distribution\begin{eqnarray*}L & = & \chi_{1}^{2}(b^{2}),\end{eqnarray*}where$\chi_{1}^{2}(b^{2})$ represents a non-central...
View ArticleWhat is the distribution of the observed counts of the Chi-squared test?
I was reading this answer, trying to get some intuition for how Pearson's chi-squared test works:https://math.stackexchange.com/a/2074074/1226290Everything makes sense from this answer, except for the...
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