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 which 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 \(X^{2}\) independent?

Short answer

For \(Q\) and \(\bar{X}\) to be independent, the elements in each row or column of the real symmetric matrix \(A\) must add up to zero. The same holds true for the independence of \(Q\) and \(X^{2}\).

Step-by-step solution

Understanding the Properties of Symmetric Matrix

Symmetric matrices have the property that their transpose is equal to the matrix itself, i.e, \(A=A^{T}\). Knowing A is symmetric helps when it comes to understanding the relationships between its elements and their implications on \(Q\) and \(\bar{X}\).

Relating the Elements of A to Q and \(\bar{X}\)

If \(Q\) and \(\bar{X}\) are independent, then all the elements of column (or row, due to symmetry) must add up to zero. This can be concluded because if \(Q\) is a quadratic form represented by matrix \(A\), then \(Q = X^{T}AX\) and for \(Q\) and \(\bar{X}\) to be independent, there should not be any linear term dependent on \(X\) in \(Q\) after expectation is subtracted, which means the elements of column (and row) of \(A\) should add up to zero.

Interpreting the Hint Provided

The hint asks to consider the independence of \(Q\) and \(X^{2}\). If \(Q\) and \(X^{2}\) are independent, it would further strengthen the claim made in the previous step, affirming that the sum of the elements in each row or column of \(A\) indeed needs to be zero for this independence to hold.

Key concepts

Symmetric Matrices

A matrix is termed symmetric if it is equal to its transpose. This means that for a symmetric matrix \(A\), each element located at position \((i, j)\) is equal to the element at position \((j, i)\), that is, \(A = A^{T}\). Symmetric matrices often arise in the study of quadratic forms like those in the given exercise. One key property of symmetric matrices is their real eigenvalues, making them particularly useful in various mathematical applications.

Why are symmetric matrices important in our context? Because they directly influence the form of quadratic expressions. For a quadratic form represented as \(Q = X^{T}AX\), knowing \(A = A^{T}\) implies any interactions between observations due to other terms in the matrix remain balanced. Thus, if the quadratic form \(Q\) is independent of another statistic, like the mean \(\bar{X}\), specific requirements on \(A\) must be met such as having non-zero row or column sums.

Symmetry constrains the structure of \(A\), and these constraints can be useful in proofs and verifications across statistical and algebraic calculations. By knowing \(A\) is symmetric, it becomes easier to verify independence conditions set forth in statistical problems.

Independence in Statistics

Independence is a crucial concept in statistics that refers to the lack of association between two random variables. When two variables are independent, knowing the value of one gives no information about the value of the other. In our original exercise, this independence applies to the quadratic form \(Q\) and the sample mean \(\bar{X}\).

To effectively establish independence, look for characteristics that imply no statistical relationship. For instance, if no linear transformations of the observations remain after the expectation is removed, it can suggest independence. Such requirements can often translate to specific requirements on matrices within quadratic forms, as seen in our problem where the row or column elements of matrix \(A\) must sum to zero.

Understanding independence allows statisticians to simplify complex problems. By recognizing that some data features do not influence others, unnecessary calculations can be avoided, focusing only on those relationships that do matter.

Random Samples

Random samples are selections made from a larger population where every possible sample of a particular size has an equal chance of being chosen. This randomness ensures that the statistics calculated from the sample can be generalized to the whole population with a known error margin.

In our exercise, the observations come from a random sample of size \(n\), drawn from a normal distribution \(N(0, \sigma^{2})\). When dealing with random samples in the context of quadratic forms, this randomness simplifies assumptions, such as the independence of \(Q\) and \(\bar{X}\). Because each observation is drawn independently, it further supports statistical claims of independence between derived statistics formed from the sample.

The utility of random samples extends well beyond the given problem, underpinning the fundamentals of inferential statistics. They allow for the estimation of population parameters and provide the basis for hypothesis testing, making them a vital tool in data analysis.