Linked Questions
19 questions linked to/from Distribution of a quadratic form, non-central chi-squared distribution
2
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0
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475
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Proving the given quadratic form is chi-squared $k$ [duplicate]
Suppose $\underline{X}$ is an $m$-dimensional vector following multivariate Normal distribution i.e. $\underline{X}$~$N_m(\underline{\mu},\Sigma)$ where $\Sigma$ is positive definite. Let $B$ be a ...
- 1,565
34
votes
3
answers
16k
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Prove that F-statistic follows F-distribution
In light of this question : Proof that the coefficients in an OLS model follow a t-distribution with (n-k) degrees of freedom
I would love to understand why
$$ F = \frac{(\text{TSS}-\text{RSS})/(p-1)...
- 443
12
votes
2
answers
8k
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Quadratic form and Chi-squared distribution
It's about the demostration of the quadratic forms and chi-squared distribution.
Let's split the problem:
We have a $n$ vector with n standardized normal distribution called $z={[z_1,z_2...z_n]}$.
...
- 413
8
votes
2
answers
8k
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General linear hypothesis test statistic: equivalence of two expressions
Assume a general linear model $y = X \beta + \epsilon$ with observations in an $n$-vector $y$, a $(n \times p)$-design matrix $X$ of rank $p$ for $p$ parameters in a $p$-vector $\beta$. A general ...
- 12.1k
5
votes
3
answers
18k
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Multiplication of chi-square distribution by constant
Is it true that multiplication of a chi-square random variable by a real constant remains chi-square? I tried to check this using a change of variables, but it didn't look promising.
- 3,750
6
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1
answer
2k
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Prove that $\frac{(n-2)s^2}{\sigma^2}\sim \chi^{2}_{n-2}$
Consider the following simple linear regression model involving the $\epsilon_i$ error term,
$$y_i = \alpha + \beta x_i + \epsilon_i$$
such that,
$$\epsilon_i \sim \mathcal N(0,\sigma^2)$$
we know ...
- 750
2
votes
3
answers
1k
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If $X \sim \mathcal{N}(\mu,\sigma^2)$, then how is $X^2$ distributed?
If $X \sim \mathcal{N}(0,\sigma^2)$, then $X^2$ is distributed according to a scaled chi-square distribution.
If $X \sim \mathcal{N}(\mu,1)$, then $X^2$ is distributed according to a noncentral chi-...
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3
votes
1
answer
1k
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Distribution of the sample variance of values from a multivariate normal distribution
What is the sampling distribution of the variance of a collection of variables that follow a multivariate normal distribution? Specifically, assume that the $n-$dimensional vector $\boldsymbol{x} \sim ...
- 45
2
votes
3
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431
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Prove that the distribution of $Q$ is chi-squared with $p_2$ degrees of freedom
Suppose $X$ is a $p$-dimensional vector following $N_p(\mu,\Sigma)$ distribution, where $\mu$ is $p$-dimensional and $\Sigma$ is $p\times p$. Let $X=\left(\begin{array}{ccc}X_1\\X_2\end{array}
\right)$...
- 1,565
1
vote
1
answer
580
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Applying Cochran's Theorem to prove $\sum_{i = 1}^{n}\left(\frac{X_{i} - \bar{X}}{\sigma}\right)^{2} \sim \chi_{n - 1}^{2}$? [duplicate]
Apply Cochran's theorem to show that if $X_{1}, \dots ,X_{n} \stackrel{i.i.d.}{\sim} \mathcal{N}(\mu, \sigma)$, then
\begin{align*}
\sum_{i = 1}^{n}\left(\frac{X_{i} - \bar{X}}{\sigma}\right)^{2} \...
- 133
4
votes
1
answer
416
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How can I show that $\frac{1}{\sigma^2}\sum^k_{i=1}n_i[(\bar{Y}_{i.}-\bar{\bar{Y}})-(\theta_i-\bar{\theta)}]^2 \sim \chi^2_{k-1}$?
Define $\bar{\bar{Y}}=\sum n_i \bar{Y}_{i.}/\sum n_i$ and $\bar{\theta}=\sum n_i\theta_i / \sum n_i$, where $Y_i \sim N(\theta,\sigma^2)$. How does can I show that $\frac{1}{\sigma^2}\sum^k_{i=1}n_i[(\...
- 2,043
2
votes
1
answer
399
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Mean and variance of $Y'\Sigma^{-1}Y-Y_1^2/\sigma_1^2$ when $Y\sim N_2(0,\Sigma)$
Let $\underline Y=(Y_1,Y_2)'$ have the bivariate normal distribution $N_2(\underline0,\Sigma)$, where
$$\Sigma=\begin{pmatrix}\sigma_1^2 & \rho\sigma_1\sigma_2 \\[1em] \rho\sigma_1\sigma_2 & \...
- 43
1
vote
2
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327
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Scaled sample variance as sum of squares of normal variables
I want to prove that $(n-1)S^2 = \sum_{i=1}^{n} (X_i - \bar{X})^2$ can be written as $\sum_{i=2}^{n} Y_i^2$, with $Y_i = N(0,\sigma^2)$, $X_i$ and $Y_i$ $i.i.d$.
I managed to do it by taking a big ...
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1
vote
1
answer
242
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Proof on $SS_{AB} / \sigma^2 \sim \chi^2_{(I-1)(J-1)}$ under the null hypothesis $H_{AB}: \delta_{ij} = 0$ for $i=1,\dotsc,I$ and $j=1,\dotsc,J$
First time to ask a question. I am now reading a textbook "Mathematical Statistics and Data Analysis, 3ed" by Rice. On Page 495, there is a Theorem B for the two-way layout ANOVA.
Let me ...
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