Var½b

i

^

b

i

¼½Z

0

i

R

1

i

Z

0

i

þ D

1

i

1

ð17Þ

The expressions given for

^

b and

^

b

i

in (15)and(16)

assume that V

i

is known, e.g., D and R

i

are known.

However in normal practice a consistent estimator given

by

^

V

i

¼ Z

i

^

DZ

0

i

þ

^

R

i

must be used instead. Likelihood-

based methods are used for estimat ing D and R

i

based on

the assumptions that b

i

and e

i

are normally distributed

(see Littell et al. 1996; Schabenberge r and Pierce 2002).

Appendix 2

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DATA Example ;

INPUT h d ;

one = 1 ;

lnd = LOG(d) ;

RES = LOG(h-1.37)-(1.4027+0.4386*lnd) ;

CARDS ;

21.0 25.9

17.7 16.5

17.7 18.5

;

RUN ;

PROC IML ;

USE Example ;

READ ALL VAR {one lnd} INTO Z ;

READ ALL VAR {RES} INTO RES ;

D = { 0.2243 -0.0499, -0.0499 0.0127 } ;

R = 0.00575 * I(3) ;

b = D*Z`*INV(Z * D * Z` + R)*RES ;

PRINT b ;

QUIT ;

Fig. 4 SAS program for computing random parameters

261