g02dnf

Purpose

Synopsis

[est,stat,sestat,t,ifail] = g02dnf(irank,b,cov,p,f<,tol,ifail>)

Description

 
 This routine computes the estimates of an estimable function for 
 a general linear regression model which is not of full rank. It 
 is intended for use after a call to G02DAF. An estimable 
 function is a linear combination of the parameters such that it 
 has a unique estimate. For a full rank model all linear 
 combinations of parameters are estimable.
 
 In the case of a model not of full rank the routines use a 
 singular value decomposition (SVD) to find the the parameter 
            ^^^^^^                                             
 estimates, (beta), and their variance-covariance matrix. Given 
 the upper triangular matrix R obtained from the QR decomposition 
 of the independent variables the SVD gives:
 
                               (D 0) T
                           R=Q (0 0)P ,
                              *       
 
 where D is a k by k diagonal matrix with non-zero diagonal 
 elements, k being the rank of R, and Q  and P are p by p 
                                       *                 
 orthogonal matrices. This leads to a solution:
 
                         ^^^^^^    -1 T  
                         (beta)=P D  Q  c 
                                 1    *  1
                                       1 
 
 P  being the first k columns of P, i.e.,P=(P P ), Q   being the 
  1                                          1 0    *           
                                                     1          
 first k columns of Q  and c  being the first p elements of c.
                     *      1                                
 
 Details of the SVD, are made available, in the form of the matrix
  *
 P :
 
                               ( -1 T)
                               (D  P )
                               (    1)
                             * (  T  )
                            P =( P   )
                               (  0  )
 
 as given by G02DAF.
 
                                         T                      
 A linear function of the parameters, F=f (beta), can be tested to
                                             T               
 see if it is estimable by computing (zeta)=P f. If (zeta) is 
                                             0               
 zero, then the function is estimable, if not, the function is not
 estimable. In practice |(zeta)| is tested against some small 
 quantity (eta).
 
                                                   T^^^^^^     
 Given that F is estimable it can be estimated by f (beta) and its
 standard error calculated from the variance-covariance matrix of 
 ^^^^^^           
 (beta), C      , as
          (beta)  
 
                                 __________
                                / T       
                       se(F)=  / f C      f
                             \/     (beta)
 
 Also a t-statistic:
 
                               T^^^^^^
                              f (beta)
                           t= --------,
                               se(F)  
 
 can be computed. The t-statistic will have a Student's t-
 distribution with degrees of freedom as given by the degrees of 
 freedom for the residual sum of squares for the model.
 

Parameters

g02dnf

Required Input Arguments:

irank                                 integer
b (:)                                 real
cov (:)                               real
p (:)                                 real
f (:)                                 real

Optional Input Arguments:                       <Default>

tol                                   real     sqrt(eps)
ifail                                 integer  -1

Output Arguments:

est                                   logical
stat                                  real
sestat                                real
t                                     real
ifail                                 integer