Question

Let \(A\) be the real symmetric matrix of a quadratic form \(Q\) in the observations of a random sample of size \(n\) from a distribution that is \(N\left(0, \sigma^{2}\right) .\) Given that \(Q\) and the mean \(\bar{X}\) of the sample are independent, what can be said of the elements of each row (column) of \(\boldsymbol{A}\) ? Hint: Are \(Q\) and \(\bar{X}^{2}\) independent?

Short answer

Under the conditions given, namely that \(Q\) and the sample mean \(\bar{X}\) are independent and given \(Q\) and \(\bar{X}^{2}\) are independent, the sum of the elements in each row (or each column) of the matrix \(A\) of the quadratic form \(Q\) must be zero.

Step-by-step solution

Understand the problem

The problem tells us that the mean (\(\bar{X}\)) of the sample and the quadratic form \(Q\) are independent. We are asked to determine what can be said about the elements of each row (or column) of the real symmetric matrix \(A\). This will require understanding the conditions for independence of the quadratic form and the sample mean in the scenario presented.

Recall the condition

One important condition is that for \(Q\) and \(\bar{X}\) to be independent, and since we know \(Q\) and \(\bar{X}^{2}\) are independent, the sum of the values in each row or column of the matrix \(A\) that defines the quadratic form \(Q\) should be equal to zero. This is because in this case, the quadratic form \(Q\) does not bear any information about the mean of the sample, and vice versa.

The conclusion

Therefore, given the conditions of the problem, we can conclude that in order for \(Q\) and \(\bar{X}\) to be independent, the sum of the elements in every row or every column of the matrix \(A\) must be equal to zero.