Final answer:
The Hat Matrix H = [hij] in a full rank linear model with an intercept has diagonal elements hii that satisfy 1/n ≤ hii ≤ 1.
Step-by-step explanation:
The task is to prove that the diagonal elements of the Hat Matrix must satisfy the inequality 1/n ≤ hii ≤ 1, assuming that the design matrix Xn×(p+1) is of full rank and includes a first column of 1's (an intercept term).
To prove this, we first note that the Hat Matrix is defined as H = X(XTX)-1XT where X is the design matrix of the linear model.
It has the property that HX = X, meaning that it 'projects' the design matrix onto the space spanned by its own column vectors.
Because the first column of X is composed of all 1's (representing the intercept), the sum of the elements in each row of H must be 1, since the product of H and the first column of X (all 1's) must give back the first column of X.
As a result, the diagonal element hii must be less than or equal to 1 because it cannot be larger than the sum of the positive elements in its row.
The other inequality, 1/n ≤ hii, can be proved by using the properties of projection matrices and the fact that X is of full rank, meaning its columns are linearly independent.
This ensures that the diagonal elements of H, which represent the 'leverage' of each observation in the model, must have a minimum value that is related to the number of leverage points (observations) available (n), hence 1/n.