A C T A U N I V E R S I T A T I S L O D Z I E N S I S
F O L IA O E C O N O M IC A 175, 2004
M a r c i n S k i b i c k i *
O N M A X IM IZ A T IO N O F E S T IM A T IO N A C C U R A C Y IN M U L T IP A R A M E T E R T W O S T A G E S A M P L IN G
Abstract. In the paper the problem o f sample allocation for both stages in such a way, that the accuracy o f an estim ation o f the m eans o f many variables is maximal and the survey cost is restricted is considered. A s the measure o f accuracy, value o f the spectral radius o f covariance matrix o f m eans estim ators vector is taken. It w as proved that the spectral radius is a convex function o f sam ple sizes. T hat allow s effective solving the problem using know n m ethods adapted to this issue.
Key words: tw o stage sam pling, sam ple allocation, covariance matrix, spectral radius.
1. IN T R O D U C T IO N
Let us co nsider the problem o f sam ple allocation on b o th sam pling stages in the case o f estim atio n o f m any variables m eans. A ssum in g lim ited financial resources, we w ant to m axim ize accuracy o f the estim atio n . As the m easure o f accuracy we take value o f the spectral radiu s o f co variance m atrix o f m ean s estim ato rs vector (see W y w i a ł 1995). T h e p ro blem o f m inim izing o f the spectral radius when the survey co st is restricted in the stratified sam pling case was considered am o n g o th ers by J. W y w i a ł (1988), J. W y w i a ł and G. K o ń c z a k (1994). T h e solutio n p ro p o se d here is based on convexity o f the m axim um eigenvalue o f the co varian ce m atrix as a functio n o f sam ple sizes.
2. N O T A T IO N S
Let U = {1, ..., N} be a p o p u la tio n o f N units p artitio n ed into G g ro up s ..., UG (G > 2), so th a t U j n U k = ę, for j ^ k . T here sizes are ad eq u ately
* P hD , D epartm en t o f Statistics, University o f E conom ics, K atow ice, e-m ail: ski- bi@ ae.katow ice.pl.
N x, N a ( Nj^ 2 fo r j = 1, G). Suppose th a t m variables are observed in the p o p u latio n (m js 2). C onsidering th e tw o-stage sam pling as the first stage sam pling units let tak e groups U v, U c . O n the first sam pling stage
a sam ple S o f size g is d raw n w ithout replacem ent from th e g rou ps. O n the second sam pling stage form each group, a sam ple S h o f size nh is d raw n w ithout replacem ent fo r h e S. T h e vector o f m eans values in th e p o p u latio n is o f the form :
A n unbiased e stim a to r o f the m eans vector у is y , = LVis, —, )>ms]r with its co-ordinates:
У = [Уи ý J T
where
fc= 1
T h e co variance m a trix o f vector у , is o f the form :
C . is the betw een-groups covariance m atrix with its elem ents: 1 с
C'h(yi,yj) = £ t o * ~ У|)(Уд - У})>
h = 1
w here
" h k e V„
C . is the w ith in -g ro u p s covariance m atrix w ith its elem ents: Ул = т г И Уш, i = 1, m, h= 1, H.
3. FRO M O F T H E PRO BLEM
Let us define the to tal cost o f the survey as
(3)
where k 0 is the u n it cost o f surveying the first stage elem ents (e.g. co st o f organizing the survey o f each group, area). H ow ever kh is th e unit co st o f surveying the second stage elem ents from /i-th group. D efined in this way cost o f the survey is th e expected costs o f sam ple o bservation .
T he problem consist in calculating sam ple sizes on b o th sam pling stages in such a way th a t the spectral radius (the m axim al ab so lu te eigenvalue) o f the covariance m atrix V (y s) is m inim al and th e survey cost k ( g, n{, nG)
do n o t exceed specified value. Let us denote the co varian ce m a trix as the function o f sam ple sizes V ( g , n it nH) = V (y,). T h e prob lem m ay be
w ritten d ow n as:
T he value o f objective fu n ctio n can n o t be calculated an aly tical, how ever if it is a convex function o f sam ple sizes, searching for so lu tio n is sim plified. In this case the problem is a non-linear constrained p ro g ram m in g pro blem with convex objective fu nction (see W i t 1986).
Theoreme 1. M axim um eigenvalue o f the covariance m atrix \ ( g , n l , nH)
is a convex fu n ctio n o n set:
(
m a x |A |: A e S p e c tiV ig ,« !, ..., nG))} = m in k(3, n i. .... nG) ^ K 2 ś g ś G 2 ^ n h ^ N h, / i = l , G (4) ( 2 ^ g ^ G [ 22 ^ n h ^ N h, < nh ^ I\ / i = l , ..., G (5) Proof: Let n = [g,nm 1, ..., nH]r be the vector o f sam ple sizes. T h e fu n ction s defined as follow:G N h l ( N „ N 2 g \ n h
/ * ( » ) = * » = 1... G,
arc convex on the set given by inequalities (5). F o r n o ta tio n sim plification let us w rite C. 0 = C .. T h e covarian cc m atrix V (g , n lt ..., n„ ) m ay be w ritten in the form:
V (и) = £ / » ( * ) C .h h = о
Sym bol A^V (л)) d en o tes the m axim um eigenvalue o f the m atrix V (n). M axim um ab so lu te value o f the eigenvalues is a norm on the space o f real m atrices o f fixed size. M oreo v er, the eigenvalues o f th e covariance m atrix are non-negative, th en fo r all covariance m atrices V , i V2 and every
t e (0, 1) we have
Xl ( t \ 1 + (1 - t)V 2) < tA .ÍV J + (1 - t)^ i(V 2).
Because the functions АЛ(п) are convex and the m atrices С .Л are non-negative definite we obtain: A [(V (tn a ) + (1 — t)n (2))) = sup x £ / * ( in(1) + 0 - 0 n (2,)C .AJ x *eR" XXT 1 Л ('П (1) + (1 - 1)n(2))xTC./,x I [tf„(nU) + (1 - t)fkn(2))]xTC.*x = sup — --- =--- < sup — --- =---= ,e r- x x * £r- X X x 7[ t Z / * ( n (1,)C .A + (1 - 1) Z / * ( n (2>)C J x = sup L *=t> T - - --- *L = A1( iK n (1)) + ( l - 0 K n (2))) x e R " x*x
for all vectors n(1) and n(2) which fulfils (5) and every ie ( 0 , 1). Finally we have A i(V (tn(1) + (1 - i)n (2>)) < t A ^ n ' 1»)) + (1 - OAjCVCn'2»)).
which m eans convexity o f the m axim um eigenvalue as a fu n ctio n o f sam ple sizes g , n {, ..., rtG w hich fulfils (5).
I he problem (4) can be solved using know n n o n -lin ear p ro g ram m in g m ethods and the m axim um eigenvalue m ay be calculated fo r exam ple with the pow er m eth o d . Since we use a non-integer p ro g ram m in g m e th o d , there is a problem o f ro u n d in g obtained sam ple sizes. T h e sim plest m eth o d is to ro u n d the sam ple sizes to lower integers, witch allow s to fulfil the survey cost restrictio n but also increases the spectral rad iu s value. In practice, ro u n d in g size g o f the first stage sam ple m ay indeed affect the solution. Because o f th a t, if non-integer value o f g was o b tain ed , we m ay for exam ple solve the problem tw o m ore tim es for g ro u n d e d respectively to lower and higher integer. F inally b etter from these tw o so lu tio n s is selected.
4. EXA M PLE
We have the p o p u latio n o f 7523 farm s from adm inistrative unit D ą b ro w a T arn o w sk a. T h e d a ta used fo r calculating optim al sam ple sizes com e from the general ag ricu ltu ral census for 1996.
F o u r variables have been selected to survey: 1) arab le land (in hectares),
2) stock o f cattle (in heads), 3) stock o f pigs (in heads),
4) value o f sale (in th o u sa n d s o f zlotys).
T h e p o p u la tio n was divided into 6 g ro ups on the basis o f territo ria l division. G ro u p sizes, unit costs and m ean values o f the variables are presented in T ab . 1. T h e value 30 was taken as th e unit cost o f surveying first stage elem ents.
T a b l e 1
Sizes o f groups
U n ite costs o f survey
M ean values o f variables
1 11 III IV 1046 1.0 3.864 2.41 5.42 3.732 1312 1.0 4.574 1.97 3.44 2.247 695 1.5 5.141 3.66 9.87 6.607 1459 2.0 4.671 2.63 5.19 3.181 1163 2.0 4.847 1.98 3.98 2.559 1848 2.5 3.714 1.82 5.26 2.751
T h e betw een-groups covariance m atrix is o f the form : C . = c.(y„y j) 0.326 0.239 0.499 0.425 0.239 0.491 1.478 1.092 0.499 1.478 5.315 3.699 0.425 1.092 3.699 2.675
R estriction o f the survey cost am o u n ts К = 800. S olutions ob tained for each value o f the first stage sam ple size presents T ab . 2. T h e first row o f results in the tab le co n tain s solu tio n optim al in the sense o f considered problem .
T a b I e 2
First stage, sam ple size
Second stage sam ple size
Spectral radius, value 2 236, 140, 131, 144, 87, 474 0.2300 3 150, 90, 84, 92, 56, 303 0.2617 4 106, 64, 60, 66, 41, 218 0.2850 5 87, 49, 46, 50, 30, 166 0.3052 6 69, 39, 36, 40, 24, 132 0.3249 R EFEREN CES
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W y w i a t J. (1988), L okalizacja p ró b y и> warstwach m inim alizująca prom ień sp e ktra ln y m acierzy wariancji i kow ariancji w ektora średnich z próby, “ Prace N au kow e A kadem ii Ekonom icznej we W rocław iu” , 404, s. 195-200.
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spektralny m acierzy wew nątrzw arstw ow ej m acierzy wariancji i kowariancji, [w:] X I Sem inarium E kon o m etryczn e im. P rofesora Zbigniew a Pawłowskiego, T rzem ieśnia 2 4 -2 6 III, A kadem ia Ekonom iczna, K raków , 85-92.
M a r c in S k i h i c k i
O M A K SY M A L IZ A C JI D O K Ł A D N O ŚC I EST Y M A C JI W W IE L O P A R A M E T R O W Y M L O SO W A N IU D W U S T O P N IO W Y M
W pracy rozw ażano zadanie ustalenia liczebności prób losow anych na obydw u stopniach losow ania tak, aby d ok ładn ość estymacji średnich wielu cech populacji była m aksym alna przy kosztach obserwacji próby nieprzekraczających zadanego poziom u. Za miarę dok ładności estymacji przyjęto w artość prom ienia spektralnego macierzy kowariancji w ektora estym atorów wartości średnich cech. W ykazano, że prom ień spektralny tej macierzy dla przekształconego zadania jego minimalizacji jest wypukłą funkcją odpow iedniego wektora. Pozw ala to na efektyw ne poszukiw anie optym alnego rozw iązania przy użyciu znanych m etod adaptow anych do tego problemu.
