Ug 
out by applying the weighted least squares method after introducing 
appropriate fictitious weighted observations for the unknowns/s. Now 
still assume model (1) to be true. And suppose that zero observa- . 
tions with weight matrix Pg are introduced for the additional 
parameter vector S and then the weighted least squares method is 
conducted. Thus, the following error equations can be established: 
vq A H 1 
= X + S - 
Y 0 T 0 
with the weight matrix of ond being IT oss 
0 o 2. 
After adjustment the estimators of the unknowns is obtained as 
follows: 
€ - (AAA I-GEGS £jEQ 
§ = (ag + RS 
in which Q, = I- AAA) a, Qas = (H'Q,,H) ; 
are respectively the cofactor matrice of v and 3 obtained from the 
adjustment treating s as free parameters. X and S are obviously 
biased. Their biases are respectively 
EX) u (ala A ut SD RSS) (14) 
and = 4 
BS) - 8 = ({I + QsP,) - I) s ; (12) 
Assume that Ps = diag(Ps, » Psy» +++» Psmp) is a diagonal matrix 
and s has been transformed so that Qaa = diag(as, » Agrees dsm5)* 
Then the ith component of Eq.(12) is 
-- Is; Ps. 
E(S;) - 8; = anor Tio Si (13) 
1 + Qs Psy 
From section 2 we know that to assess the reliability is to evaluate 
the biases as in Eq.(11),(12) or (13). In the application of 
robust methods, the least squares estimation with all the unknowns 
being free is usually also carried out as the preliminary 
adjustment. So its result can be used to evaluate the biases. By 
using 81, we acquired the maximum of the absolute values of the 
bias for S, which are possible in the sense of probability 8, the 
posteriori internal reliability for S; . It is expressed as 
follows: 
^ 
VSi 
imu 
Sj6{S;: 8 probable given 8] 
ds, Ps; 
= [Si] + 18/2 S IO. (14) 
1 As; Ps; 
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