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 the number of iterations to achieve success

Let $Z=X+jY$ ($j$ is the imaginary unit), with $X\sim\mathcal{N}(\mu,1)$ and $Y\sim\mathcal{N}(0,1)$. I'm running an algorithm that at every iteration $k$ samples a complex number $z_k$ that follows $...
mateusgl's user avatar
3 votes
2 answers
90 views

non-central F-statistic confidence intervals seem inconsistent with ANOVA p-values

I have done a series of one-way ANOVA tests. For each test I have calculated the corresponding alternative hypothesis F-statistic and confidence intervals, based on the degrees of freedom involved and ...
treemake's user avatar
5 votes
1 answer
169 views

Can power decrease despite an increase in sample size?

I've analyzed fish density data (log(x+1) transformed) to see what power I'll have to detect a 30% increase and decrease in density from future surveys, and for which species of interest. Using ...
Nate's user avatar
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CDF for squared sum of Rayleigh random variables

In short, I am looking to estimate the distribution of $ \eta = \sum_{i=1}^N (X_i - z_i)^2$, for each $X_i \sim \text{Rayleigh}(1)$ and constants $z_i$. If $X_i$ were Gaussian, then this could be ...
mirrormere's user avatar
5 votes
0 answers
139 views

sum of noncentral Chi random variables

if $X_1,...,X_n$ are independent random variables with noncentral chi distributions (same $df$ but different $\lambda$), What is the distribution of $\sum_{i=1}^{n}{X_i}$ Just wondering if it can be ...
Nika Tsereteliii's user avatar
4 votes
1 answer
316 views

How to approximate non-central chisquare distribution to Poisson weighted sum of central chi-square distribution in case of non-unit variances?

According to Statistics libre texts Equation 5.9.20, a non-central chi square distribution can be approximated as sum of Poisson weighted central chi square distributions. $\tag{1}g(y) = \sum_{k=0}^\...
amitha's user avatar
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3 votes
2 answers
263 views

Expected value of Rayleigh quotient, non-centered Gaussian vector

Let $X \sim \mathcal{N}\left(\mu, \Sigma \right)$, and let $A$ be a symmetric matrix. My understanding is that the Rayleigh quotient of vector $X$ is given by: $$R=\frac{X^T A X}{X^T X}$$ I've been ...
dherrera's user avatar
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6 votes
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225 views

Random number generator for non-central chi-squared with non-integer dimension

Does someone know of a random number generation algorithm for a non-central chi-squared distribution with a non-integer dimension? PS By algorithm, I am interested in the detailed procedure or ...
user378619's user avatar
1 vote
1 answer
190 views

Noncentrality parameter Repeated-Measures ANOVA - formula vs. G*Power

I am trying to figure out the correct expression for the noncentrality parameter $\lambda_{ws}$ for the within-subjects effect in a one-way Repeated-Measures ANOVA with $k$ trials/groups. Comparing ...
Barlon Mrando's user avatar
2 votes
1 answer
133 views

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 $\...
Denis Cousineau's user avatar
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118 views

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^...
Denis Cousineau's user avatar
1 vote
1 answer
120 views

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 ...
Denis Cousineau's user avatar
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83 views

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 ...
Denis Cousineau's user avatar
5 votes
2 answers
231 views

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 ...
granular_bastard's user avatar
2 votes
1 answer
66 views

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+...
Minho Kang's user avatar
1 vote
0 answers
48 views

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 ...
Wannier's user avatar
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3 votes
1 answer
159 views

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 ...
Migwell's user avatar
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7 votes
4 answers
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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. ...
Alexis's user avatar
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384 views

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 ...
Masel's user avatar
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2 votes
0 answers
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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 ...
Matthew Ferrell's user avatar
1 vote
1 answer
424 views

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), ...
Blue Various's user avatar
2 votes
1 answer
967 views

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:...
Sextus Empiricus's user avatar
0 votes
0 answers
30 views

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 ...
JDoe2's user avatar
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3 votes
0 answers
85 views

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 ...
Jayson Vavrek's user avatar
1 vote
1 answer
1k views

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 ...
Trajan's user avatar
  • 449
0 votes
1 answer
75 views

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 $R ∼ Rice ⁡ ( ν , 1 )$ ...
Krlngc's user avatar
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71 views

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 ...
user53154's user avatar
2 votes
0 answers
108 views

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 $\chi^2(\nu,\delta)$ ...
shabbychef's user avatar
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1 vote
0 answers
301 views

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 = \...
Berto's user avatar
  • 13
1 vote
0 answers
229 views

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 $T$ ...
Barlon Mrando's user avatar
2 votes
0 answers
150 views

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 ...
MathStudent's user avatar
2 votes
0 answers
304 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 ...
RAND's user avatar
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4 votes
1 answer
2k views

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 $X_{ij} \sim N(\mu_i, \sigma^2)$ ...
Denziloe's user avatar
  • 1,093
0 votes
0 answers
405 views

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 $Y$? ...
D Satya Ganesh's user avatar
4 votes
1 answer
2k views

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 ...
D Satya Ganesh's user avatar
1 vote
0 answers
198 views

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 ...
yurnero's user avatar
  • 205
1 vote
0 answers
356 views

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, ...
Cath Maillon's user avatar
0 votes
1 answer
101 views

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 ...
fermat4214's user avatar
0 votes
0 answers
83 views

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?
Ahmed Cheema's user avatar
2 votes
0 answers
467 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 ...
Ben's user avatar
  • 451
5 votes
3 answers
527 views

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 ...
matus's user avatar
  • 578
4 votes
2 answers
193 views

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 ...
Denis Cousineau's user avatar
6 votes
1 answer
395 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 \...
user avatar
5 votes
1 answer
781 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 ...
NotThatGrumpyAnymore's user avatar
4 votes
0 answers
139 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 ...
Vicent's user avatar
  • 799
0 votes
1 answer
27 views

(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 ...
assadabbasi's user avatar
2 votes
0 answers
114 views

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 ...
Joe's user avatar
  • 277
6 votes
1 answer
130 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):...
Vicent's user avatar
  • 799
2 votes
0 answers
385 views

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.
Akhil Singh's user avatar
0 votes
1 answer
864 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 ...
Hendrik's user avatar
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