How Can the Variance of a Quadratic Form Be Simplified?

In summary, the variance of quadratic form is a statistical measure that quantifies the spread or variability of a quadratic expression or function around its mean value. It is commonly used in hypothesis testing and regression analysis to assess the significance of a relationship between variables and can also be used to determine the precision of estimates in statistical models. The variance of quadratic form cannot be negative and is used to measure the variability of a quadratic expression or function, while the variance of linear form is used for linear expressions. A larger variance indicates a higher degree of variability and a smaller variance indicates a lower degree of variability.
  • #1
zli034
107
0
In the Searle's 1971 book Linear Model, page 57, has a formula for the Variance of Quadratic form:

var(Y[tex]^{T}[/tex]AY)=2tr(A[tex]\Sigma[/tex]A[tex]\Sigma[/tex])+4[tex]\mu[/tex][tex]^{T}[/tex]A[tex]\Sigma[/tex]A[tex]\mu[/tex]

The proof of this showed on page 55 was based on MGF. I'm looking for proofs are less complicated. Some thing that is similar to show the expectation of a quadratic form.

Anyone has read about quadratic form please help.

Thanks
 
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  • #2
If you're talking about a constant matrix A and a random vector Y that is jointly gaussian, one way is to write the quadratic form as a double sum.
 

Related to How Can the Variance of a Quadratic Form Be Simplified?

1. What is the definition of variance of quadratic form?

The variance of quadratic form is a statistical measure that quantifies the spread or variability of a quadratic expression or function around its mean value. It is calculated by taking the expected value of the squared deviation from the mean.

2. How is the variance of quadratic form used in statistical analysis?

The variance of quadratic form is commonly used in hypothesis testing and regression analysis to assess the significance of a relationship between variables. It can also be used to determine the precision of estimates in statistical models.

3. Can the variance of quadratic form be negative?

No, the variance of quadratic form cannot be negative as it is a measure of variability and cannot have a negative value. It will always be equal to or greater than zero.

4. What is the difference between variance of quadratic form and variance of linear form?

The variance of quadratic form is used to measure the variability of a quadratic expression or function, while the variance of linear form is used to measure the variability of a linear expression or function. The formula for calculating the variance also differs between the two.

5. How can the variance of quadratic form be interpreted?

The variance of quadratic form can be interpreted as the average squared distance of the values of a quadratic expression or function from its mean. A larger variance indicates a higher degree of variability, while a smaller variance indicates a lower degree of variability.

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