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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Distribution of $X'\Sigma^{-1}X$ for $X$ following a multivariate $t$ distribution

According to Golam Kibria & Joarder (2006, p.7) available here and Kotz & Nadarajah (2004, p. 19) visible in google, the distribution of $X'\Sigma^{-1}X /p$, for a known correlation matrix $\...
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Which is the noncentral multivariate t distribuiton

In various documentations, I found out two definitions of the multivariate noncentral student $t$ distribution. The most commonly density (PDF) found is (e.g., wikipedia): $$\mathcal{T}(x;\mu, \sigma^...
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The distribution of the difference between two correlated non-central t distribution

Suppose a binormal population $\{X, Y\}$ with means $\mathbf{\mu} = \{\mu_1,\mu_2\} \ne \{0,0\}$ and covariance $\Sigma= \sigma^2\begin{bmatrix}1 & \rho\\ \rho &1 \end{bmatrix}$. Let $S^2$ be ...
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How to get rid of a nuisance parameter when the two parameters are multiplied

I have the distribution of a sample statistics which is given by a noncentral t distribution. This distribution is dependent on the population correlation ($\rho$) which is unknown. However I have the ...
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3 votes
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PDF of $x_1^2+x_2^2-x_3^2-x_4^2$ with $x_i \sim N(\mu_i,1)$

What is the probability distribution function of the random variable $X$ $X=x_1^2+x_2^2-x_3^2-x_4^2, \tag{1}$ where $x_i$ are independent normally distributed variables $\mathcal{N}(\mu_i,1)$? What I ...
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The average distance in a 3D noncentral-chi-distribution random walk process after N steps

In the Noncentral Chi Distribution Wikipedia page, the calculated Mean is: $$ {\sqrt{{\pi\over2}}L_{1/2}^{(k/2-1)} \large( {\small{-\lambda^2\over2}})}$$ I am calculating the average distance after N ...
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Show that noncentral $X_2$ is $\chi^2(r-r_1,\theta -\theta_1)$

As I read a book named 'Introduction to mathematical statistics' written by Hogg et al, I stuck with the below question. $X_1$ and $X_2$ be two independent random variables. And Let $X_1$and $Y=X_1+...
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How to estimate a parameter from a unknown model as follow?

Recently I have met a question. I have derived a important indicator in my research on radar signal processing. The indicator $y$ can be calculated by another measured value $x$, and their ...
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How can we centralise a non-central chi squared random variable?

Say that $X\sim {\chi '}_{k}^{2}(\lambda)$ and $Y \sim \chi_k$. What transformation of $X$ will produce $Y$? If we also let $Z \sim N(\mu, 1)$, $\lambda = \mu^2$, and $k=1$, then I understand that $(Z ...
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Is there a generalized concept of noncentrality of a distribution?

The theory of probability distributions forms one of the pillars of statistics, and is a foundation for statistical inference. There are more than a few probability distributions, and they are neat-O. ...
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Non-central correlated normal ratio - distribution of the ratio of two dependent normally distributed variables

I am currently trying to solve a problem in the context of a Bayesian analysis that concerns normal distributions. The situation is as follows. I have an equation that looks like this, where I know ...
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Confusion with definition of Non-Centrality Parameter

So part of my research involves working with non-central distributions. One of the most important but at the moment more confusing ones is the non-central $\chi^2$, whose ncp is the one used in the ...
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How to obtain histograms of non-central t distributions from a normal distribution?

My Question; I'd like to know how to generate random numbers that follows a non-central t distribution using the normal random numbers. I made a calculation code for this using R (See Box2, below), ...
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Power Analysis and the non central t distribution: what is the non-centrality parameter?

I want to investigate the power of t test in detecting shift so I have a sample $X_1, \dots,X_n \sim N(\mu_x,\sigma)$ and other sample $Y_1, \dots,Y_n \sim N(\mu_y,\sigma)$. My null hypothesis is $H_0:...
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Convergence of function of central limit theorem

I have an estimator of the form: $$\frac{\hat{\mu}-c}{\sqrt{n\hat{\sigma}^2}}$$ Where $c$ is constant, $\hat{\mu}$, is an estimate, $n$ is number of observations and $\hat{\sigma}^2$ is a variance ...
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Mean of non-central Poisson deviance distribution is lower than that of the associated central distribution

We have observed some strange behaviour regarding distributions of Poisson deviance statistics. In short, we find for certain Poisson parameters, that the non-central deviances are smaller than the ...
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How to test signifcance of a sharpe ratio

Let say you have measured a Sharpe Ratio of $S^*$. What is the simplest way (ie no fancy distributions) to do a hypothesis that this is different from $0$? So $H_0: \text{ The sharpe ratio is equal to ...
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Rice $\sim R(\nu,\sigma)$ to Noncentral $\chi^2$

Statistics beginner here. I have a sample data set which is Rice distributed, $R \sim R(\nu,\sigma)$. However, I'm interested in fitting $R^2$. According to Wikipedia, If $R ∼ Rice ⁡ ( ν , 1 )$ ...
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Is the ratio of two noncentral exponential distribution the logistic distribution?

Is the ratio of two noncentral exponential distribution the logistic distribution? If yes, How to set the parameter of the logistic distribution using the parameters of noncentral exponential ...
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likelihood view of F-test

Suppose I observe independent $X, Y$ where $X/\sigma^2 \sim \chi^2(\nu_1,\delta)$ and $Y/\sigma^2 \sim \chi^2(\nu_2,0)$, where $\sigma^2$ is some nuisance parameter. (The notation $\chi^2(\nu,\delta)$ ...
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Distribution of noncentrality parameter in noncentral t-distribution

Let $X$ be a random vector $X \sim N(0, \Sigma)$, where $\Sigma = \frac{1}{n} Id$. Find probability $$ P(|T| > a)$$ where $T$ has noncentral t-distribution with noncentrality parameter $\delta = \...
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Different results for simulated and closed form distribution of T-statistic under the alternative hypothesis

While running some simulations to get a better grasp of the concept of statistical power I stumbled upon an unexpected result. I was trying to simulate the sampling distribution of test statistic $T$ ...
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Power Analysis and Non-Central t Distribution

I'm a bit confused about power analysis and the non-central t-distribution. If I am in a situation where $H_0$ is $\mu = 0$, and $H_1$ is $\mu > 0$, I calculate for what value $\mu$ would have a ...
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2 votes
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172 views

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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3 votes
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Derive the distribution of the ANOVA F-statistic under the alternative hypothesis

Say we have $k$ samples of data, where sample $i$ is of size $n_i$ and we write it as $x_{i1}, ... , x_{in_i}$. Let the total sample size be $N$. The ANOVA model is $X_{ij} \sim N(\mu_i, \sigma^2)$ ...
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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 $Y$? ...
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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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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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0 votes
1 answer
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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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non-central Chi-square distribution [duplicate]

If $X \sim \mathcal{N}(\mu,\sigma^2)$ where $\mu > 0$ and $\sigma > 2$, how do I find the PDF of $X^2$, and will it correspond to non-central Chi-square distribution?
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2 votes
0 answers
381 views

t-test: understanding alternative hypothesis, power, and noncentral distributions

While the alternative hypothesis $H_a: \mu_c \ne \mu_t$ covers an infinite number of possible values for $\mu_t$, the power of the test can only be defined based on a point value of $\mu_t$ - is this ...
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4 votes
2 answers
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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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4 votes
2 answers
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What is the median of the non-central F ratio distribution

I am looking for a simple approximation to the median of the (simply) non-central F distribution with parameters dlnum, dldenominator, and ncp, the non-centrality parameter. Clearly, there is no ...
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6 votes
1 answer
214 views

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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4 votes
3 answers
633 views

Possible to use moment generating function of standard normal to find variance of noncentral $\chi^{2}$?

In a homework problem, I have been asked to find the variance of the noncentral $\chi_{n,\delta}^{2}$ distribution with degrees of freedom $n$ and noncentral parameter $\delta >0$ by using the ...
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4 votes
0 answers
111 views

What is a reasonable alternative when the mean does not exist?

Consider a continuous random variable $F''$ following a doubly noncentral $F$ distribution with $8$ and $2$ degrees of freedom, and noncentrality parameters $0$ and $10$; which is to say: $$ F'' \sim ...
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1 answer
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(p)_j notation in the paper Castaño-Martínez and López-Blázquez (2005)

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 ...
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1 vote
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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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6 votes
1 answer
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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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2 votes
0 answers
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How can i express non central chi square random variable in terms of gamma function?

I know the relation between central chi square and gamma random variables.But i am not able to get relation between gamma and non central chi-square distribution.
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1 answer
684 views

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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2 answers
181 views

Estimating non-centrality parameter from some obtained sample of t variates

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-...
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3 votes
1 answer
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Power of the $\chi^2$ test constraint?

I understand that the distribution of the $\chi^2$ goodness of fit test ($\sum\dfrac{(x_i-m_i)^2}{m_i}$) is a non-central chi-squared with non centrality parameter $\lambda=\sum\dfrac{(m_i-m'_i)^2}{m'...
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4 votes
1 answer
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Difference of two non-central chi squared random variables

I have a problem where I want to find the distribution of the difference of two non-central chi squared random variables (RV), both independent. Given $$ X=a+A\\ Y=b+B $$ where $a=a_r+ja_i$, $b=b_r+...
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4 votes
1 answer
474 views

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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3 votes
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147 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 $\min_{x_i \in X^n} \|x_i\|^2_2$?...
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279 views

Sum of non-central chi-square using Laguerre expansion

I moved this question from my answer on this link. From Castano-Martinez and Lopez-Blazquez(2005), They explained pdf of sum of noncentral Chi-square on equation 3.2, that is: $f(y)= \frac{e^{-\frac{...
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1 vote
1 answer
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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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3 votes
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In inferential statistics, is the noncentrality parameter of the F-distribution simply the F-value 'in the population'?

I'm writing functions (in R) that allow working with the $\omega^2$ distribution (the effect size for analysis of variance; the less biased alternative to $\eta^2$)....
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