11
votes
Accepted
Does scaling a central $\chi^2$ distribution produce a non-central $\chi^2$ distribution?
Unfortunately, the Wikipedia article on "F-test of equality of variances" is incorrect. When the variances are unequal, the distribution of $F$ is neither $F$ nor non-central $F$, it is simply scaled $...
- 13.5k
9
votes
Accepted
Expected value of Rayleigh quotient, non-centered Gaussian vector
I found the answer for the case where the covariance of $X$ above is the identity ($\Sigma=I$). This was enough to solve my problem of interest, so I'm posting here and voting provisionally as the ...
- 1,405
7
votes
Accepted
Power Analysis and the non central t distribution: what is the non-centrality parameter?
Below is a geometric view of the $t$-test (a similar view is also expressed here).
The $t$-statistic, which is a ratio of the sample mean and the sample standard deviation, follows a ratio ...
- 86.5k
7
votes
Accepted
Random number generator for non-central chi-squared with non-integer dimension
I am assuming you know how to generate random draws from a central chi-squared distribution, or from its equivalent gamma version; see below for the details. I also suggest possible readings on ...
- 12.1k
6
votes
Accepted
Possible to use moment generating function of standard normal to find variance of noncentral $\chi^{2}$?
Here's an inelegant but straightforward, elementary solution. It begins with the case $n=1$.
Generally, a non-central chi-squared distribution arises as the sum of squares of independent Normal ...
- 334k
6
votes
PDF of $x_1^2+x_2^2-x_3^2-x_4^2$ with $x_i \sim N(\mu_i,1)$
As noted by the OP, this problem can be simplified to the difference in two independent noncentral $\chi^2$ random variables:
$$X=X_1^2+X_2^2-X_3^2-X_4^2=Z_1-Z_2$$
where $Z_i\sim \chi^2(2,\lambda_i)$,...
- 4,505
6
votes
Accepted
Can power decrease despite an increase in sample size?
To answer the question in the title (besides all the problems raised in the comment section), power can decrease while the sample size increases, if simultaneously you reduce enough the effect size ...
- 5,848
5
votes
Is there a generalized concept of noncentrality of a distribution?
It's hard to understand how to answer this question.
For any given hypothesis and any given test statistic, the distribution under an alternative hypothesis is considered a "non-central" ...
- 64.8k
5
votes
Confidence Interval on a random quantity?
Geometric view of the problem and distributions of $\vec{b}\cdot \vec{a}$ and $|\vec{b}|^2$
Below is geometrical view of the problem. The direction of $\vec{a}$ doesn't really matter and we can just ...
- 86.5k
4
votes
Accepted
Distribution of the step size of diffusion in 3-dimensional space
Write $Z_i = \dfrac{X_i}{\sigma} \sim \mathcal{N}(0, 1)$.
Then, $$Y = \sigma\sqrt{Z_1^2+Z_2^2+Z_3^2}\text{.}$$
Using a standard result, $Z_1^2+Z_2^2+Z_3^2 \sim \chi^2_3$.
The square root of a $\chi^...
- 5,147
4
votes
Why doesn't the TOST equivalence testing procedure use non-central $t$ distribution to determine the $p$ value?
What you are proposing does not appear to be any different from what actually occurs in this test. Remember that in a classical hypothesis test, the p-value is calculated using the null distribution ...
- 133k
4
votes
Accepted
Expected value of $X^{-1}$, $X$ being a noncentral $\chi^2$. Cannot understand a step of a equation in a paper
Yes, $EY^{−r}$ stands for $E[Y^{−r}]$. (I dislike not making it explicit because it leaves too many opportunities for misunderstandings and errors.)
With respect to the later part, consider:
$Y=\...
- 290k
4
votes
Accepted
How to obtain histograms of non-central t distributions from a normal distribution?
As I mentioned in my comment, matching a histogram to a density in general requires some scaling considerations, as explained in this answer.
However, there are several issues with your code.
$Z$ is $...
- 2,338
3
votes
Accepted
Derive the distribution of the ANOVA F-statistic under the alternative hypothesis
Consider $Y_i \sim N(\mu_i, \sigma^2)$ (independently) as a random vector with a multivariate normal distribution, $\vec Y \sim N(\vec\mu, \sigma^2 I)$.
The main idea is to consider this distribution ...
- 1,203
3
votes
Relationship between the gamma distribution and Non-central chi squared distribution?
You can put this into the form of a non-central chi-squared distribution by scaling the random variables to get unit variance. Since $X_i / \sigma \sim \text{IID N}(\mu / \sigma, 1)$ you have:
$$\...
- 133k
3
votes
Accepted
Why doesn't the TOST equivalence testing procedure use non-central $t$ distribution to determine the $p$ value?
You are correct, in the sense that the uniformly most powerful (UMP) test for equivalence for the one-sample, paired, and two-sample $T$-test does involve a non-central $t$-distribution.
The two one-...
- 66
3
votes
Is there a generalized concept of noncentrality of a distribution?
I think a simple way to think about noncentral distributions is to consider how they're built from normal distribution, e.g., non central t variable is $\frac{Z+\mu}{\sqrt{V/\nu}}$, where $Z$ is ...
- 62.3k
3
votes
How can we centralise a non-central chi squared random variable?
The generic answer is to use the cdf and the inverse cdf, namely
$$Y=F^{-1}_{k,0}(F_{k,\lambda}(X))$$
where $F_{k,\lambda}(\cdot)$ is the $\chi^2_k(\lambda)$ cdf.
- 108k
3
votes
Accepted
Different results for noncentrality parameter (formula vs. G*Power)
You're right. My answer was missing a divisor of the degrees of freedom. In this case, there are two degrees of freedom for either of the means. The non-centrality parameter in a power calculation is ...
- 64.8k
3
votes
Difference of two non-central chi squared random variables
You can write
$P(Z>z) = P\left(\frac{|X|^2}{|Y|^2+z}>1\right)$
with the assumption $|Y|^2+z \ne 0$. Then the problem turns to studying the ratio of two non-central chi-squared distributed ...
- 31
3
votes
Accepted
ttest where the difference in the null hypothesis is not 0: non-centrality parameter?
Apart from not being centred around zero, a non-central T is also non-symmetric about its mean. As @Scortchi mentions in the comments, the null hypothesis can be expressed in another way to make it ...
- 577
3
votes
Sample size formula for an F-test?
I am wondering if there is a sample size formula like Lehr's formula that applies to an F-test?
The webpage "Power Tools for Epidemiologists" explains:
Difference Between Two Means (Lehr):
Say, ...
- 2,110
3
votes
PDF of $x_1^2+x_2^2-x_3^2-x_4^2$ with $x_i \sim N(\mu_i,1)$
Here is the inverse Fourier transform approach you've labeled as equation (3). (Again, I've used Mathematica as your profile indicates you have access to it and it does all of the algebraic ...
- 4,505
3
votes
Accepted
Distribution of $X'\Sigma^{-1}X$ for $X$ following a multivariate $t$ distribution
Suppose the random variable $\textbf{X}$ has a multivariate t-distribution with mean vector $\mu$, covariance matrix $\boldsymbol{\Sigma}$ and degrees of freedom $\nu$, then $\textbf{X}^\prime \...
- 86.5k
3
votes
Accepted
Lower Tail Bound for Noncentral chi-squared distribution
The noncentrality parameter of the described non-central chisquared distribution will be the sum of squares of the means: $\lambda^2=\mu_1^2+\mu_2^2+\cdots+\mu_k^2$ so if $ce^{-c' t^2}$ can be the ...
- 4,505
2
votes
Significant support of non-central chi-quared distribution
You do it the same way you could do it for a normal distribution, by using the cdf (cumulative distribution function) $F$ or better, its inverse, the quantile function $Q=F^{-1}$. There is only one ...
- 82.8k
2
votes
Confidence Interval on a random quantity?
For the case $p=1$, we can find a two sided interval.
In this case we can assume that $0 < a$ is the population parameter,
and we observe $b=\mathcal{N}\left(a,1\right).$ We wish to
bound $ab$ in ...
- 15k
2
votes
What is the median of the non-central F ratio distribution
Here is an approach (as opposed to a complete answer unfortunately) to obtain a good approximation with maybe a limited amount of programming.
It appears that for any fixed values of the two degrees ...
- 4,505
2
votes
Is there a generalized concept of noncentrality of a distribution?
I agree with Aksakal and AdamO, the non-central varieties are a result of investigating the power of a test. The test itself assumes a particular null hypothesis for the purposes of argument and ...
- 3,000
2
votes
Accepted
The distribution of the difference between two correlated non-central t distribution
Here is an approach that shows how to obtain a numerical approximation to the probability density function of $Z=X/S-Y/S$. (I haven't been successful in finding an analytic solution.)
Using the joint ...
- 4,505
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