Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Invariance of the constrained least squares quadratic form

  1. Jan 31, 2010 #1
    Hi everyone,

    I'm trying to show that the quantity

    [tex](L'\hat{\beta}-\eta)'(L'(X'X)^{-1}L)^{-1}(L'\hat{\beta}-\eta)[/tex]

    where [itex]\beta[/itex] is a (p+1)x1 vector, [itex]L'[/itex] is a q x (p+1) vector, and [itex]\eta[/itex] is a q x 1 vector, is invariant under transformations of L which are row and column operations.

    This quantity arises in the test statistic for the constrained (restricted) least squares process, where the objective is to minimize

    [tex]S(\beta,\Lambda) = (Y-X\beta)'(Y-X\beta) + 2\Lambda'(L'\beta-\eta)[/tex]

    where [itex]\Lambda[/itex] is treated as a Lagrangian multiplier and [itex]L'\beta=\eta[/itex] is the restriction to be imposed on the unrestricted objective function [itex]S(\beta,0)[/itex].

    The point is that there is no unique choice of the matrix L', and this should not affect the form of the test statistic.

    Here, [itex]\beta' = (\beta_0, \beta_1, \ldots, \beta_p)[/itex].

    For instance if one wants to impose

    [tex]\beta_1 = \beta_2 = 0 [/tex]

    two possible choices of the L' matrix are

    [tex]\left(\begin{array}{ccc}0 & 1 & -1 \ldots\\0 & 0 & 1\end{array}\right)[/tex]

    and

    [tex]\left(\begin{array}{ccc}0 & 1 & 0 \ldots\\0 & 0 & 1\end{array}\right)[/tex]

    with [itex]\eta = 0[/itex] in both cases.


    How do I show that for some such nonsingular transformation T which gives L* = TL, the quantity [itex](L'\hat{\beta}-\eta)'(L'(X'X)^{-1}L)^{-1}(L'\hat{\beta}-\eta)[/itex] is invariant?

    Thanks in advance.
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Invariance of the constrained least squares quadratic form
  1. Quadratic forms (Replies: 2)

  2. Quadratic forms (Replies: 1)

  3. Quadratic Forms (Replies: 5)

Loading...