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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49 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 ...
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1answer
30 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 ...
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28 views

What is the difference between a non-central limit theorem and the usual central limit theorems?

I'm reading a paper where the authors prove the following theorem. They then say that this constitutes a non-central limit theorem for the variables in question. Since I have never heard this term (...
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1answer
35 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 $R ∼ Rice ⁡ ( ν , 1 )$ ...
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22 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 ...
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93 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 $\chi^2(\nu,\delta)$ ...
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0answers
18 views

G*Power: Sensitivity Z-test, Compute non centrality parameter given alpha and power

I am using the Z test: Generic Z test, sensitivity tool. I have a distribution of values of the normal population sample with a mean and SD so I thought the non centrality parameter would give me the ...
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41 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 = \...
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173 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 $T$ ...
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135 views

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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63 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 ...
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65 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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1answer
522 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 $X_{ij} \sim N(\mu_i, \sigma^2)$ ...
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243 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 $Y$? ...
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1answer
1k 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 ...
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51 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 ...
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245 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, ...
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1answer
52 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 ...
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238 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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2answers
204 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 ...
3
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2answers
103 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 ...
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1answer
118 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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3answers
304 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 ...
4
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0answers
85 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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1answer
21 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 ...
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45 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 ...
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1answer
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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219 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.
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1answer
433 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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2answers
54 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-...
2
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1answer
64 views

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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1answer
260 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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0answers
224 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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1answer
132 views

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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0answers
301 views

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 $\eta^2$)....
3
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1answer
203 views

Non-central t-distributions with different degrees of freedom

I have two non-central t distributions with CDF given by $t_{n,a}$ and $t_{m,a}$ with $n>m$ degrees of freedom respectively and the same non-centrality parameter $a$. The question is: for which ...
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1answer
167 views

Why is the noncentral chi-squared approximate of the deviance bad?

The statistical model under consideration is given by the independent realizations of two independent binomial distributions: $$ x_1 \sim \mathrm{Bin}(n_1, p_1), \quad x_2 \sim \mathrm{Bin}(n_2,p_2), $...
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1answer
252 views

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 ...
3
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0answers
89 views

A question concerning distribution of $\mathbf{Y}/\|\mathbf{Y}\|_2$ where $\mathbf{Y}\sim \mathcal{N}(\boldsymbol{\mu},\mathbf{I})$

I know that when $\mathbf{Y}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$, $\mathbf{Y}/\|\mathbf{Y}\|_2$ is distributed uniformly on the unit sphere. But to my surprise, I failed to find a simple closed ...
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0answers
494 views

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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1answer
92 views

Which distribution do I get?

Be $X\sim N(\mu,1)$ and $Y\sim Inverse-Gamma(\alpha,\beta)$. For the Inverse-Gamma, I usually use the parameterization which leads to the following probability distribution function for Y: $f(y;\...
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0answers
38 views

Histogram of noncentral chi-square variable shoots up when the number of histogram cells increases

Consider the random variable defined by: $$y=-8x+108$$ where $x$ follow a non-central chi-square distribution with 1 degree of freedom and a non-centrality parameter equal to 3.5, $x \sim \chi_{1}^{2}(...
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0answers
84 views

which random variable is a rescaled non-central $\chi^2$ random variable?

Probably simple question. Consider the Cox-Ingersoll-Ross model (1985) model for interest rates $$ dr = k(\theta - r)dt + \sigma \sqrt{r}dz $$ Then it is known in closed form the conditional pdf $f(r(...
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1answer
450 views

Why are these two random variables identically distributed?

Let $(X_1, X_2, \ldots, X_i, \ldots, X_k)$ be $k$ independent, normally distributed random variables with means $\mu_i$ and variances $1$. Then the random variable $$ \sum_{i=1}^k X_i^2$$ is ...
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5answers
588 views

Confidence Interval on a random quantity?

Suppose $\vec{a}$ is an unknown $p$-vector, and one observes $\vec{b} \sim \mathcal{N}\left(\vec{a}, I\right)$. I would like to compute confidence intervals on the random quantity $\vec{b}^{\top} \...
5
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1answer
2k views

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 ...
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1answer
8k views

Noncentrality Parameter - what is it, what does it do, what would be a suggested value?

I have been trying to brush up on my stats knowledge, especially in relation to Sample size determination and Statistical Power Analysis. But it seems that the more I read the more I need to read. ...
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0answers
108 views

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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0answers
1k views

Generate data from a t-distribution with specified mean and standard deviation [duplicate]

How does one randomly sample from a T-distribution in R. From what I've found, the function rt in R doesn't let you specify the mean and standard deviation. For a ...
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1answer
1k views

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 ...