Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [non-central]

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

83 questions
24
votes
1 answer

sum of noncentral Chi-square random variables

I need to find the distribution of the random variable $$Y=\sum_{i=1}^{n}(X_i)^2$$ where $X_i\sim{\cal{N}}(\mu_i,\sigma^2_i)$ and all $X_i$s are independent. I know that it is possible to first find the product of all moment generating functions…
pitfall
  • 427
  • 1
  • 3
  • 7
14
votes
1 answer

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. Anyway I found a tool called G*Power which seems to…
Deepend
  • 405
  • 1
  • 5
  • 19
13
votes
1 answer

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? Lehr's formula for t-tests is $n = 16 / \Delta^2$, where $\Delta$ is the effect size (e.g. $\Delta = (\mu_1 - \mu_2) / \sigma$). This can be generalized…
shabbychef
  • 10,388
  • 7
  • 50
  • 93
11
votes
1 answer

Distribution of a quadratic form, non-central chi-squared distribution

Definition. Suppose $\mathbf{y} \sim \mathcal{N}(\boldsymbol{\mu}, I_{n \times n})$. Then $$w = \mathbf{y}^{T}\mathbf{y} = \|\mathbf{y}\|^2 \sim \chi^{2}_{n}\left(\theta = \|\boldsymbol{\mu}\|^2/2 =\boldsymbol{\mu}^{T}\boldsymbol{\mu}/2…
Clarinetist
  • 3,761
  • 3
  • 25
  • 70
11
votes
2 answers

What is the power of the regression F test?

The classical F-test for subsets of variables in multilinear regression has the form $$ F = \frac{(\mbox{SSE}(R) - \mbox{SSE}(B))/(df_R - df_B)}{\mbox{SSE}(B)/df_B}, $$ where $\mbox{SSE}(R)$ is the sum of squared errors under the 'reduced' model,…
shabbychef
  • 10,388
  • 7
  • 50
  • 93
10
votes
2 answers

What is the median of a non-central t distribution?

What is the median of the non-central t distribution with non-centrality parameter $\delta \ne 0$? This may be a hopeless question because the CDF appears to be expressed as an infinite sum, and I cannot find any information about the inverse CDF…
shabbychef
  • 10,388
  • 7
  • 50
  • 93
8
votes
5 answers

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 $\vec{b}^{\top} \vec{a}$, based only on the observed $\vec{b}$ and…
shabbychef
  • 10,388
  • 7
  • 50
  • 93
7
votes
1 answer

Moment-generating function (MGF) of non-central chi-squared distribution

I need to compute the moment-generating function of the non-central chi-squared distribution, but I have no idea where to begin.
CT.
  • 73
  • 1
  • 6
6
votes
4 answers

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. Many of the distributions folks learn about at…
Alexis
  • 26,219
  • 5
  • 78
  • 131
6
votes
1 answer

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…
user179309
6
votes
1 answer

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):49–63. doi:10.1080/03610927608827331 In this paper…
Vicent
  • 639
  • 1
  • 5
  • 18
5
votes
0 answers

Estimation After Selection on Non-central F Random Variables

Suppose that you observe $F_1,F_2,\ldots,F_k$ each independently. drawn from non-central F distributions with common, known, d.f. $\nu_1, \nu_2$, and with (unknown) non-centrality parameters $\lambda_1,\lambda_2,\ldots,\lambda_k$. Suppose that the…
shabbychef
  • 10,388
  • 7
  • 50
  • 93
5
votes
1 answer

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: $$ x_1 \sim \mathrm{Bin}(n_1, p_1), \quad x_2 \sim \mathrm{Bin}(n_2,p_2), $$ and the null hypothesis is…
Stéphane Laurent
  • 17,425
  • 5
  • 59
  • 101
5
votes
1 answer

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 distributed according to the noncentral chi-squared…
Tim
  • 1
  • 29
  • 102
  • 189
5
votes
1 answer

Is there a PDF for a generalized non-central chi-squared distribution

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 writing the PDF of such a distribution as a…
Andrei Kucharavy
  • 397
  • 2
  • 10
1
2 3 4 5 6