Some Notes on the Probability Space of Statistical Surveys

Transcription

1 Metodološk zvezk, Vol. 7, No., 200, 7-2 ome Notes o the Probablty pace of tatstcal urveys George Petrakos Abstract Ths paper troduces a formal presetato of samplg process usg prcples ad cocepts from Probablty Algebra ad Iformato Theory. Uder ths model, ay samplg scheme defes uquely a probablty measure, llustrated varous examples alog wth some applcatos survey desg ad maagemet. Itroducto basc deftos Let P be the target populato of a statstcal survey ad R = {r ν, ν =,2, N} a relevat regster had, cossts of N dvdual statstcal uts. Regardless of the parameters of the selecto process, all the possble outcomes cocerg the elemets of R comprse the set Ω R = {r +, r -, r 2 +, r 2 -,... r Ν +, r Ν - }, where r ν + deotes the presece of the ν th ut whle r ν - deotes ts absece (Kullback, 997). By B = {E Ω R } we determe the set of all subsets E of Ω R, called samples. Ay selecto process Ω R defes uquely a probablty measure p ad furthermore ay sample E ca ether have chaces to appear ( p(e) > 0) or ot ( p(e) = 0). Let us ow cosder a mappg o B ( φ : B E ) such that, φ (E) = E, p(e) > 0, p(e) = 0 (.) Dept. of Publc Admstrato, Pateo Uversty of ocal ad Poltcal ceces. Athes, GR ad Agls A, tatstcs ad Iformatcs, GR Acadmas 96-00, 0677 Athes, Greece; george.petrakos@agls-sa.gr

2 8 George Petrakos Thus we costruct a o-empty set E B whch, wth the basc Boolea operatos ad a probablty measure p whch s strctly postve, ormed ad addtve, form a probablty algebra (E, p) (Kappos, 969). Therefore for ay elemets E E () p(e) > 0 ad p(e) = 0 ff E= () p(e) =, where e s the ut E () p(e E 2 ) = p(e ) + p(e 2 ) f E E 2 = Ay elemet E dfferet tha ad e s called possble sample. We also cosder N+ classes B, =0,,2, N such that = { (r k, r 2 k, r N k ), I ( r k ) = 0, k, k = = + N ν = I( r k ν ) = }, where k = {+, -} ad whch cotas all subsets of B, where appearaces of statstcal uts occur., N, the class cotas N subsets, I ={, 2, By applyg φ o B, we costruct a o empty set N } φ : φ ( ) =, p(, p( ) > 0 ) = 0, I (.2) Uder the probablty algebra (E, p) defed by a chose samplg process, the class has the followg propertes (herted by E) () p( ) > 0, I (s) () p ( ) =, I (s) () p( j ) = p( ) + p( j ), (, j) I (s) x I (s) wth j where, I (s) I the subset of dces for whch p( ) > 0 ad e = I (s), the ut, wth p(e) =. Ths basc set of otos ad deftos troduces a more algebrac approach to measurable sample desgs tha the aalytcal oes (ärdal et all, 200) whch are focusg o the estmato of varous parameters. Ths algebrac approach

3 ome Notes o the Probablty pace of tatstcal urveys 9 seems to hadle multple samplg procedures, lke multple recapture desgs, more effcetly. 2 Applcato to varous samplg schemes I a sgle sample process t ca be show that j =, (, j) I (s) x I (s) wth j. The probablty that two dfferet samples wll be draw a sgle samplg process s zero, therefore p( j ) = 0 ad the oly evet (E, p) wth probablty 0 s the empty set,. There are samplg schemes where I (s) I (strctly),.e. stratfed radom samplg where oly the s that satsfy the proportoal to strata restrcto meet wth property (), whle for the rest t holds that p( ) = 0, I I (s). O the other had, a smple radom samplg I I (s), sce all, I satsfy property (). The above cocepts ca also be appled to multple samplg procedures. I ths type of samplg, both r ν + ad r ν - are preset the sample, dfferet stages of course. We wll exame the form of the evet space Ω R ad the class, for samplg wth replacemet ad multple recapture samplg. amplg wth replacemet. amplg from N statstcal uts by choosg oe ut each of the (sample sze) tmes ad put t back the populato before the ext tral s a process that correspods to a evet space Ω R such that: Ω R = {r k k ν } wth ν =,2,,N k={+,-} ad =,2, where I[ r ( )] =, ad a probablty algebra (E, p) s defed based o = x x x = where s the bass for a R of sze. Multple recapture. I a multple recapture expermet ru a populato of sze N (usually ukow), the sample space s expaded over the dscrete tme of trals (t=,2, T). If the populato s closed for ths tme perod, the sample space s: Ω R (T) = {r ν k (t)} wth ν =,2,,N, k={+,-} ad t =,2,,T. Whe the populato sze chages the dfferet pots of tme (ope populato), the sample space s: Ω R (T) = {r ν k (t)} wth ν =,2,,N(t), k={+,-} ad t =,2,,T. The basc class s T X = x x... varable wth elemets,2,,t. x = Χ X 2 X T X t, where X t {0,,,N(t)} a dscrete radom T t X X X N ν = 2 T = x x x 2 wth I T t =, 2,... ν X t X, N, t =

4 20 George Petrakos The probablty space Uder a pr. algebra (E, p) a class s uquely defed ad cotas all the possble samples ad oly them. Ths class forms a bass for the costructo of all evets E. Ay evet E E ca be costructed by usg oe or more basc samples ad expressed as a uo of these, based o the fact that ay possble evet related to the samplg process ca be realzed by uos of samples, I (s). It ca be easly show that E s closed uder the basc set operato. For that, let us cosder E, E 2 E as uos of some, such that: E E E = U, E 2 E E 2 = U j j, for some, j I (s),. The E E 2 = U U = j j U E where d s such that d belogs d d ether to U or U ad E E 2 = j j U E where g s such that g g g belogs both to U ad U. If there s o g such that g belogs to both j j of the uos above, the E E 2 = ad the two evets are mutually exclusve. These propertes ca be easly exteded for ay fte set of evets E. Moreover, the above defed possble evet E 2 cotas aother possble evet, oted as E E 2 whe U U, I, d D, or equvaletly I D. d d Let us ow llustrate the above wth a couple of examples: Example Let N = 4 ad =. The s a class of four basc sets, amely, = {r -, r 2 +, r +, r 4 + }, 2 = { r +, r 2 -, r +, r 4 + }, = { r +, r 2 +, r -, r 4 + }, 4 = { r +, r 2 +, r +, r 4 }. The evet of the presece of the frst two dvduals whch ca be oted by E 2 = {r +, r 2 + } ca be expressed as a uo of basc sets, E 2 = () 4 (). I other words, the evet E 2 occurs whe at least oe of the basc evets whch the frst two dvduals are preset occurs. Remark omeoe ca argue that the example above that B 2 = () 4 (), whch terms of pot set theory seems correct, sce { r +, r 2 +, r -, r 4 + } { r +, r 2 +, r +, r 4 - } = {r +, r 2 + }. However, our treatmet uder the gve samplg

5 ome Notes o the Probablty pace of tatstcal urveys 2 scheme, {r +, r 2 + } {r +, r 2 +, r k, r 4 k }, k=+,- whch explas why B 2 = () 4 () Example 2 Let N = so R = {r, r 2, r }. If they are placed a orthogoal space -D takg values of 0 ad for o-appearace ad appearace respectvely, we have the followg trasformato: [ (0) ] : (0,0,0) (r -, r 2 -, r - ) [ () ] : (,0,0) (r +, r 2 -, r - ), (0,,0) (r -, r 2 +, r - ), (0,0,) (r -, r 2 -, r + ) [ (2) ] : (,,0) (r +, r 2 +, r - ), (,0,) (r +, r 2 -, r + ), (0,,) (r -, r 2 +, r + ) [ () ] : (,,) (r +, r 2 +, r + ) whch produce all possble samples. For =2, we have orthogoal vectors (2), (2) (2) 2, whch are a bass for some samplg schemes where 2 out of are selected. r (2) (2) r (2) 2 r 2 Fgure : -D orthogoal space. Ay measure f appled to r ca assocate a measurable fucto φ =( f ) =,2, to a class ad therefore a value φ ( ) to each basc sample. Cosderg the probablty measure p mathematcal expectato

6 22 George Petrakos E(φ ) = p( ) φ ( ) (.) where φ = { f (r ), f 2 (r 2 ), f (r )} ad p( ), baryceter of the polytope formed by p( ) =, s located at the (Petrakos, 2000) ad expresses a mea value of φ before the sample s draw. A terestg applcato of ths approach s the determato ad applcato of a cost fucto C = {c (r ), c 2 (r 2 ), c (r )}, where c s varato s due to correspodg r s costly characterstcs (access, dstat locato, etc). The the cost of a sample s C = C I( ), where I( ) s a -dm vector wth oes for the correspodg r + - s ad zeros for the r s. Fally the expected cost of the samplg process estmated the desg phase wll be E(C ) = p( ) C I( ) (.2) 4 Coclusos A probablty algebra model has bee troduced order to descrbe the data collecto process a statstcal survey. Its suffcecy, effcecy ad smplcty was tested ad proved over dfferet samplg schemes. Future research ca adapt ths model to more complcated ad realstc samplg schemes, corporatg cost ad o-respose to the desg of a statstcal survey. From a theoretcal pot of vew, ths model ca be vewed ad further studed as a applcato of group theory. I both cases, ths paper aspres to provde some basc deas for substatal research. Ackowledgemets The author s grateful to the assocate edtor of the joural ad to the referees for ther costructve commets. The author would also lke to thak Mr. George Maats for revewg the fal verso of ths paper. Refereces [] Cochra, W. (977): amplg Techques. New York: J. Wley & os. [2] Kappos, D. (969): Probablty Algebras ad tochastc paces. Moograph Probablty ad Mathematcal tatstcs. Lodo: Academc Press.

7 ome Notes o the Probablty pace of tatstcal urveys 2 [] Kullback,. (997): Iformato Theory ad tatstcs. New York: Dover Publ. Ic. [4] Petrakos, G. (2000): The topologcal foudato ad some propertes of the mxed estmator. Computatoal tatstcs, 5, 09-4 [5] ärdal, C., wesso, B., ad Wretma, J. (200): Model Asssted urvey amplg. New York: prger.