In a linear model, we defined residuals as: e = y − y ^ = ( I − H ) y where H is the hat matrix X ( X T X ) − 1 X T and we defined standardized residuals as: r i = e i s 1 − h i i , i = 1 , . . . , nwhere s 2 is the usual estimate of σ 2 , v a r ( e i ) = σ 2 h i i , and h i i is the diagonal entry of H at the i t h row and i t h columnWhy r i and e i are functions of h i i rather than the whole row h i ?

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In a linear model, we defined residuals as:  e  =  y  −      y    ^    =  (  I  −  H  )  y where   H is the hat matrix   X  (      X    T    X      )          −      1            X    T  and we defined standardized residuals as:      r    i    =            e      i              s              1        −                  h                      i            i                              ,   i  =  1  ,  .  .  .  ,  nwhere       s    2   is the usual estimate of       σ    2  ,   v  a  r  (      e    i    )  =      σ    2        h          i      i      , and       h          i      i       is the diagonal entry of   H at the       i          t      h       row and       i          t      h       columnWhy       r    i   and       e    i   are functions of       h          i      i       rather than the whole row       h    i  ?

bigfreakystargl · Accepted Answer

Let&#039;s look at the variance of the residuals vector   e,  V  a  r  (  e  )  =  V  a  r  (  (  I  −  H  )  y  )  =  (  I  −  H      )    T    V  a  r  (  y  )  (  I  −  H  )  =      σ    2    (  I  −  H  )  .The main diagonal of       σ    2    (  I  −  H  ) are the variance terms of   e. Particularly, the variance of       e    i   is the   ith diagonal element of       σ    2    (  I  −  H  ), that is       σ    2    (  1  −      h          i      i        ). Since you don&#039;t know       σ    2   and estimate it with       s    2  , thus the estimated variance of       e    i   is       s    2    (  1  −      h          i      i        ).

In a linear model, we defined residuals as: e=y−y=(I−H)y where H is the hat matrix X(X^TX)^−1 X^T and we defined standardized residuals as: r_i=e_i/(s sqrt(1-h_ii), i=1,...n

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