MlNlMAX RESULTS IN LINEAR LEAST SQUARES PREDlCTlON ...

MlNlMAX RESULTS IN LINEAR LEAST SQUARES
PREDlCTlON APPROACH TO TWO-STAGE SAMPLING
THÈSE No 397 (1981)
PH. EICHENBERGER
PRÉSENTÉE AU DEPARTEMENT DE MATHÉMATIQUES
ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE
(Switzerland)

II.
III.
INTRODUCTION
-1-
C O N T E N T S
LINEAR PREDICTORS OF MINIMAL VARIANCE
I .1 Introduction 5
1.2 Particular cases 1 eading to equal i t y of the
estimators 9
LINEAR PREDICTORS FOR TWO-STAGE SAMPLING
11.1 Introduction
A 2
11.2 The predictor T(0, 1 1 v(x), p, 9 )
11.3 The predictor ?(l , 1 1 v(x), p, 9z)
2 2
II .4 The parti cul ar case of ?(O, 1 / x , p, 9 )
MINIMAX PREDICTORS FOR TWO-STAGE SAMPLING
111.1 Introduction
III .2 Relative precision of the predictors
III .3 Minimax predictors : one dimensional case
111.4 Minimax predictors : general case

IV. MINIMAX DESIGN
IV. 1 Introduction 72
IV.2 Unidimensional case : Q = p 1 73
IV.3 Minimax design for the mode1 : E(Y) = DO 7 7
IV.4 Minimax design for the mode1 : FIY) = B 1 x 86
IV.5 Minimax design for the mode1 : E(Y) = Do + BI x 89
V. NUMERICAL ILLUSTRATION
V.l Introduction 9 3
V.2 Type of system 93
V.3 Numeri cal eval uation of the minimax predictor 102
APPENDIX : PROGRAMS
A.l G~neral description
A.2 1 npu t
A.3 Output
A.4 Program for solving system number 1
A. 5 Program for solving system number 2
A.6 Program for sol ving system number 3
REFERENCES

O, INTRODUCTION
By the conventional assumption in the superpopulation approach
to fini te population sampl ing, the actual population vector
= (y, , y*,. . . , y#) is a real ized outcome of the random vector
Y w = (Y,, Y 2,... , YM) .
In order to estimate the total of the population T =
yi ,
i =l
a sample of m of the M components of Y - is chosen for observation
according to a nonrandom procedure and thus al1 probability statements
refer to the joint distribution of Y .
-
After the survey has been carried out, Yk
is known for the
uni ts in the sample and in order to estimate T , i t remains to predict
TI , the sum of the random variables corresponding to unsampled uni ts.
Suppose that the expected value of Yi
is a linear function
of p auxiliary variables, the values of which are known for each unit
of the population.
In Chapter 1 the following model is used :
E(!) = X 13 var (y) = V
For this mode1 , the best linear unbiased predictor iGLS
has
al ready been given expl ici tly.

1 t must be noted that the BLU predictor fGLS is based on the
knowl edge of the covariance matrix V . Therefore i t woul d be interesti ng
to find for which V matrices îGLS is identical to the ordinary least
2
squares predictor f , which is optimal when V = o 1 .
OLS
The greatest part of Chapter 1 is devoted to this question.
From now on i t is assumed that the fini te population of interest
consists of M elements arranged in N clusters. .The i-th cluster con-
N
tains Mi elements, so that Mi = M .
i =l
Moreover, we consider for Y ., the following mode1 :
where 6i = r
where Yij (resp. x ) denotes the variable Y (resp. x) associated
i j
with the j-th unit of the i-th cluster and ,v(x) is a positive function.
In Chapter II, the predictors of T and their error-variances
are given explicitly for p = 1 and 6 - = (y) or (;) .
1

The best linear unbiased predictor of T is normally of 1 imited
practical value because of i ts dependance on variances and correlation coef-
ficients which are usually unknown.
Two mi nimax approaches attempt to remedy these di fficul ti es.
Chapter III deals wi th minimax predictors.
2
Let us assume that v(x) and 0 , i = 1, 2 . . N are known.
Let i(p) - be the BLU predictor of T for the mode1 (M) . For any
r * in [O,
n
11 let
the relative increase of variance due to the use of ?(Y) - instead of the
,.
BLU T(p)
- when the true correlations are Q .
- r* is solution of
Then it can be shown that the minimax predictor T(c*) , where
is obtained under some weak conditions by solving a system of at most n
non 1 inear equations.
Minimax samples are studied in Chapter IV.

Let 7 be a 1 inear unbiased predictor of T . Then if the
function v(x) is known and if pi ~p for i = 1, 2 ,..., N , it is
shown how to find the minimax sample which satisfies
mi n
{sa:; es}
2 - T) .
max ~ ~ ( 7
rl s P s r2
This procedure i s then appl ied to the fol lowing mode1 s :
- E(Y..) = BO and x 1
1 J i j
In the last chapter, two numerical examples are given about the
results of chapter III. The first one concerns the kind of system to be
sol ved for the mi nimax predi ctor; the other one investi gates numerical 1 y
the qua1 i ty of the niinimax predictor on the basis of real data.