# Questions tagged [non-central]

use for questions about socalled non-centrality parameters, in distributions such as t, F, chisquare, wishart and others.

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### Does scaling a central $\chi^2$ distribution produce a non-central $\chi^2$ distribution?

For independent samples from two normal populations, $X_1,\dotsc,X_n\sim N(\mu_X, \sigma_X^2)$ and $Y_1,\dotsc,Y_m\sim N(\mu_Y,\sigma^2_Y)$, the $F$ test for equality of variances uses the statistic \...
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### Is there a PDF for a generalized non-central chi-squared distribution [duplicate]

Is there a PDF for a distribution defined as a sum of squares of random variables pulled from a family of normal distributions with different standard deviation? Is there a way of analytically ...
103 views

### Expected value of $X^{-1}$, $X$ being a noncentral $\chi^2$. Cannot understand a step of a equation in a paper

I am reading the following paper: Mudholkar GS, Chaubey YP, Ching-Chuong L (1976). Approximations for the doubly noncentral-$F$ distribution. Communications in Statistics - Theory and Methods, 5(1):...
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### Relationship between the gamma distribution and Non-central chi squared distribution?

If $Y = \sum_{i=1}^N X_i^2$, where $X_i \sim \mathcal{N}(\mu,\sigma^2)$, i.e. all $X_i$ are i.i.d gaussian random variables of same mean and variance, then what is the resultant PDF of $Y$? How the ...
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### ttest where the difference in the null hypothesis is not 0: non-centrality parameter?

I have problems to understand when a non-centrality parameter is relevant. As far as I understood, if the t distribution is not centered on 0, a non-central t distribution is used. For example, in ...
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### Bayesian inference on noncentrality parameters of t- and F-distributions

Suppose I observe a random variable $x$ drawn from a non-central t- or F- distribution, and would like to perform inference on the non-centrality parameter. How would a Bayesian approach this problem? ...
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### Why doesn't the TOST equivalence testing procedure use non-central $t$ distribution to determine the $p$ value?

I'm looking at Lakens' (2017) primer and he tests the hypothesis that there is a difference between two groups $X_1$and $X_2$of magnitude $\Delta=E[X_1]-E[X_2] \neq0$ by subtracting $\Delta$ from the ...
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### Is there standard nomenclature for Confidence Distributions?

I am wondering if there are standard naming conventions in general for confidence distributions or for the particular cases of the non-centrality parameters of (non-central) $t$- and $F$-distributions....
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### Efficient fitting of noncentral chi-squared distribution to data?

I am looking for the most efficient way to fit a noncentral chi-squared distribution with fixed d.o.f. to a given data set. So the inputs are d.o.f. and the data and the output should be the ...
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### Expansion of Non-central Chi-square density function

About non central chi-square, I have read paper from Castano-Martinez and Lopez-Blazquez(2005). They made expansion of $\chi_n^2(\delta)$ using inverse-Laplace transform and used Laguerre polynomial. ...
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### Expected value of statistic based on sample correlations

Let there be $i = 1,...,m$ random variables, for simplicity assume that each of these random variables follows a normal distribution: $x_{i} \sim N(\mu_{i}, \sigma_{i})$. Let $\hat{\rho_{i,j}}$ be the ...
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### Where is the error: doubly noncentral $F$-distribution

Let $F=(X/n_1)/(Y/n_2)$ where $X$ and $Y$ are independent, $X$ is $\chi^2$ with noncentrality parameter $\lambda_1\geq 0$, and $Y$ $\chi^2$ with NCP $\lambda_2\geq 0$. According to this Wolfram ...
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### expectation of Log Noncentral Chi

Let $X$ follow non central chi distribution, the formula of $\mathbb{E}[\log(X)]$ (or equivalently $\mathbb{E}[\log(X^2)]$ where $X^2$ is Non central chi square) can be found here and here. However, ...
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### Which method to choose for constructing a confidence interval around Cohen's w?

There are two methods to calculate a confidence interval (CI) around a measure of effect size: you can either construct a CI around the noncentrality parameter of the corresponding noncentral ...
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### How to calculate non centrality parameter?

Suppose $Y\sim N_p(\mu,\sigma^2I_p)$. Let $A$ be a symmetric idempotent matrix. I want to show $Y^{T}AY$ follows a chi-sq distribution with some non centrality parameter. Suppose $tr(A)=r(A)=k$. ...
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### Is there a known generalization of the doubly noncentral t-distribution?

Given $p$-variate normal $X \sim \mathcal{N}\left(\mu,\Sigma\right)$, consider the random variable $$t_* = \frac{X_1}{\sqrt{\sum_{2\le i \le p} X_i^2}}.$$ For some values of $\Sigma$, this ...
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### Distribution of the step size of diffusion in 3-dimensional space

I need to find the distribution of the random variable $$Y=\sqrt{X_1^2+X_2^2+X_3^2}$$ where $X_i\sim{\cal{N}}(0,\sigma^2)$ and $\sigma^2$ is related to diffusion coefficient. All $X_i$s are ...
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### Different results for noncentrality parameter (formula vs. G*Power)

This post states the formula to calculate the noncentrality parameter of a test statistic that has a standard normal distribution under the null hypothesis. For a t-test under an equal variance ...
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### How do I get the alpha and beta of a non-central beta distribution from mean and variance in R?

I need to fit a beta-distribution a real data, with a mean of 0 and a standard deviation of 0.17. I have read that this is possible by using the non-central beta distribution, and I would wlike to ...
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### Noncentral t-distribution — relationship to shifted/scaled normal distribution

Let $x$ be 100 random samples from a $N(10,4)$ distribution. Suppose that I want to calculate the likelihood of these data, given my knowledge that $\mu=10,\sigma=4$. For the normal distribution, this ...
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### Why does G*power calculate the noncentrality parameter in this way?(different from formula)

I'm calculating the power of some F-family tests and I found the key to the power was the noncentrality parameter (ncp). So I'm trying to calculate the ncp $\lambda$. However, I got different results ...
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### Power Analysis with Noncentral F distribution

I'm looking to replicate a power analysis for binary data using an MRMC analysis (details found here). The data looks like this, where Score is a binary variable. ...
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### How to calculate mean and variance of non central chi distribution of the problem?

If $Y = \sqrt{\sum_{i=1}^N X_i^2}$, where $X_i \sim \mathcal{N}(\mu,\sigma^2)$, i.e. all $X_i$ are i.i.d gaussian random variables of same mean and variance, then what is the resultant PDF of $Y$? ...
I am trying to implement few equations for the distribution of Chi square from the paper Castaño-Martínez and López-Blázquez (2005). In equation (5.1), I don't understand what $(p)_j$ means. Could you ...
Suppose I have a sample of 10 $t$ variates which I think has come from a non-central $t-distribution$. I was wondering how I can estimate the non-centrality parameter $(ncp)$ of the mother non-...