23. Derivation of the Bivariate Projected Asymptotic Distributionb

Settings

  1. Consider the case where there are unconstrained parameters (corresponding to the block ) and parameters constrained by a lower bound (corresponding to the block ).
  1. Let and the unconstrained normal random vector be .
  1. Let the symmetric information matrix be , where is , is , and is .
  1. Objective: Find the constrained minimizer by projecting onto the constrained parameter space . This requires minimizing the quadratic distance function .

Proof

Step 1: Block Expansion of the Quadratic Form

Define the difference vector with . The quadratic form can be expanded using block matrix multiplication: Multiply the block matrix by the column block vector: Multiply the row block vector by the resulting column vector: Since (a scalar equals its transpose), combine the symmetric cross-terms and substitute back:

Step 2: First-Order Condition for the Unconstrained Block

Since is unconstrained, take the gradient of with respect to and set it to zero: Since is positive definite (hence invertible), solve for to obtain its optimal value conditional on :

Step 3: Boundary Truncation for the Constrained Block

Substitute equation (6) into the objective to concentrate out . From (6), . Let and , so . Substitute each term of equation (4):
  • Term 1: (using by symmetry of ).
  • Term 2: .
  • Term 3: . Summing the three terms cancels one copy of the cross-quadratic and leaves: Restoring yields the concentrated quadratic in : The bracketed matrix is the Schur complement (舒尔补) of in , and inherits positive definiteness from . Because of the componentwise hard boundary , the optimal solution is the unconstrained minimizer truncated at the lower bound : where the max is applied component-wise.

Step 4: Substitution and Simplification

Substitute the truncated solution back into equation (6) to find the final estimator for : Apply the componentwise identity (letting ): Substitute this identity back into equation (10):

Conclusion

By minimizing the generalized quadratic distance subject to the parameter space constraints, the asymptotic distributions of the estimators (parameterized by the true values ) are precisely given by:
Q.E.D.
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