Consider the leverages for a linear model with full-rank model matrix and p parameters. a. Prove...

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Consider the leverages for a linear model with full-rank model matrix and p parameters. a. Prove that the leverages fall between 0 and 1 and have a mean of p/n. b. Show how expression XXXXXXXXXXfor h ii simplifies when each pair of explanatory variables is uncorrelated.

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Ans. 
A)
Let say Y = Xß + e = Y cap +n
Where X has full rank p,
E€ = 0 and Cov(e) = σ 2 I,
Assume, P = PX is the projection matrix on 
C(X) so Y cap = PX, n = Y – capY = (I-P)Y,
And PX = X 
Therefore, X^T *P = X^T
So, the predictor variables and residuals are orthogonal. Therefore,

Consider the leverages for a linear model with full-rank model matrix and p parameters. a. Prove that the leverages fall between 0 and 1 and have a mean of p/n. b. Show how expression XXXXXXXXXXfor h...

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