Domain Mean Estimation Using Double Sampling with Non-Linear Cost Function in the Presence of Non Response

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1 ciece Joural of Applie Mathematics a tatistics 08; 6(: oi: 0.648j.sjams I: (Prit; I: (Olie Domai Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o Respose Alila Dai Aekeya,, Ouma Christopher Oyago, yogesa Keey Departmet of Mathematics Masie Muliro Uiersity of ciece a Techology, Kakamega, Keya Departmet of tatistics a Actuarial ciece Keyatta Uiersity, airobi, Keya aress: Correspoig author To cite this article: Alila Dai Aekeya, Ouma Christopher Oyago, yogesa Keey. Domai Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o Respose. ciece Joural of Applie Mathematics a tatistics. Vol. 6, o., 08, pp oi: 0.648j.sjams Receie: December 7, 07; Accepte: Jauary 5, 08; Publishe: February 5, 08 Abstract: This paper escribes theoretical estimatio of omais mea usig ouble samplig with a o-liear cost fuctio i the presece of o-respose. The estimatio of omai mea is propose usig auiliary iformatio i which the stuy a auiliary ariable suffers from o-respose i the seco phase samplig. The epressio of the biases a mea square errors of the propose estimators are obtaie. The optimal stratum sample sizes for gie set of o-liear cost fuctio are eelope. Keywors: Double amplig for Ratio Estimatio, Domai Mea, Auiliary Variable, o-liear Cost Fuctio a o-respose. Itrouctio.. Domais Domai is a subgroup of the whole target populatio of the surey for which specific estimates are eee. I samplig, estimates are mae i each of the class ito which the populatio is subiie; for istace, the focus may ot oly be the uemploymet rate of the etire populatio but also the break-ow by age, geer a eucatio leel. Uits of omais may sometimes be ietifie prior to samplig. I such cases, the omais ca be treate as separate stratum from a specific sample take. tratificatio esures a satisfactory leel of represetatieess of the omais i the fial sample. These omais are calle plae omais... Domai Estimatio Cosier a fiite populatio uer stuy U of size iie ito D omais; U, U,..., U D respectiely. Domai membership of ay populatio uit is ukow before samplig. Its assume that the omais are quite large th a for a typical omai U seeral characteristics maybe efie as escribe by Gamrot [4]. This iclues; Domai total; U yk Domai mea; U u y u Domai ariace; ( ( U u k k u Domai Coariace betwee two characters a is gie by; ( ( Co y U k u k u k u Accorig to Meee [7] omai ca be estimate by use

2 ciece Joural of Applie Mathematics a tatistics 08; 6(: of a o-iformatie Bayesia approach where a polya posterior is use o fiite populatio that has little or o prior iformatio about the populatio. Although a prior istributio is ot specifie there is a posterior istributio which may be use to make ifereces. Uofia [] propose estimate of omais usig ouble samplig for probabilities proportioal to size (PP with kow costituet omai. The assumptios propose by Uofia [] are; (i The size of auiliary ariable is ot kow. Z that efies the omai is the ot kow prior a therefore the populatio (ii The istributio of the ariable ( size ( h j of the omai is also kow. (iii The cost of measurig the ariable a Z i each stratum is much lower tha that of measurig of the stuy ariable. Aitya et al. [] eelope a metho of estimatig omai total for ukow omai size i the presece of orespose with a liear cost fuctio usig two-stage samplig esig. I this metho the respose mechaism is assume to be etermiistic..3. Double amplig i the Presece of Auiliary Iformatio I may samplig proceures the prior kowlege about the populatio mea of the auiliary ariable is require. If there is o such iformatio, it s easier a cheaper to take o the large iitial sample from which the auiliary ariable is measure a from which the estimatio of the populatio parameters like the total, mea or the frequecy istributio of the auiliary ariable is mae. riastaa [] propose a large class of ratio a prouct estimators i ouble samplig. It was fou that the asymptotic miimum ariace for ay estimator of this class is equal to that which is geerally beliee to be liear regressio estimators. Accorig to ahoo a Paa [0] if a eperimeter kows the populatio mea of a aitioal auiliary ariable, say, Z whereas the populatio mea of a auiliary ariable is ukow a ca be estimate usig ouble samplig scheme, it is possible to come up with a class of estimators for the fiite populatio mea µ..4. Double amplig for the Ratio Estimator i the Presece of o-respose Hase a Hurwitz [5] propose a way of ealig with o-respose to aress the bias problem. I this case, whe ealig with o-respose, a sub-sample is take from the o-respoets to get a estimate of the sub-populatios represete by the o-respoets. Cochra [] employe Hase a Hurwitz [5] techique a propose ratio a regressio estimatio of the populatio mea of the stuy ariables where the auiliary ariable iformatio is obtaie from all the sample uits with some of the sample uits failig to supply iformatio o the stuy ariable. Accorig to Oh a cheure [8] a Kalto a Karsprzyk [6], o-respose is ofte compesate by weightig ajustmet a imputatio respectiely. I these methos it was argue that the proceure use i weightig ajustmet a imputatio aime at elimiatig the bias ue to orespose. Okafor a Lee [9] employe the ouble samplig metho to estimate the mea of the auiliary ariable a wet ahea to estimate the mea of the stuy ariable i a similar way as Cochra []. I this metho ouble samplig for ratio a regressio estimatio was cosiere. The istributio of the auiliary iformatio was ot kow a hece the the first phase sample was use to estimate the populatio istributio of the auiliary ariable while the seco phase was use to obtai the require iformatio o the ariable of the iterest. The optimum samplig fractio for the estimators for a fie cost was erie. Performaces of the propose estimators were compute a compare with those of Hase a Hurwitz [5] estimators without cosierig the cost. It was ote that for the results for which cost compoet was ot cosiere, regressio estimator fuctios were cosistet tha the Hase a Hurwitz [5] estimator. Chauhary a Kumar [3] propose a metho of estimatig mea of a fiite populatio usig ouble samplig scheme uer o-respose. The propose moel was base o the fact that both the stuy a auiliary ariable suffere from the o-respose with the iformatio of ot aailable. Hece the estimate of at first phase is gie by, + h With the correspoig ariace of, Where uits a ( ( V W a L + h are meas from the o-respoig uits respectiely. ( respoig a are mea square errors of the etire group a orespoig respectiely with L as the ierse samplig rate at first phase of the samplig. From the preious stuies, a umber of researchers hae cosiere a liear cost fuctio whe estimatig omais. I ealig with o-respose most of them hae cosiere subsamplig while holig to the iea that the respose mechaism is etermiistic. This stuy therefore focuses o the estimatio of omai mea usig ouble samplig for ratio estimatio with o-liear cost fuctio with a raom respose mechaism. I this stuy we therefore establish a efficiet a cost effectie metho of estimatig omais whe the trael cost compoet is iclusie a it is ot liear. The problem of miimum ariace a cost is aresse while cosierig o-liear cost fuctio a optimal sample size.

3 30 Alila Dai Aekeya et al.: Domai Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o Respose. Estimatio of Domai Mea a Variace i the Presece of o-respose.. Deelopig Domai Cocept Theory with o-respose The problem of o-respose is iheret i may sureys. It always persists ee after call-backs. The estimates obtaie from icomplete ata will be biase especially whe the respoets are ifferet from the o-respoets. The o-respose error is ot so importat if the characteristics of the o-respoig uits are similar to those of the respoig uits. Howeer, such similarity of characteristics betwee two types of uits (respoig a o-respoig is ot always attaiable i practice. I ouble samplig whe the problem of o-respose is preset, the strata are irtually iie ito two isjoit a ehaustie groups of respoets a o-respoets. A sub-sample from orespoig group is the selecte a a seco more etesie attempt is mae to the group so as to obtai the require iformatio. Hase a Hurwitz [5] propose a techique of ajustig the o-respose to aress the problem of bias. The techique cosists of selectig a subsample of the o-respoets through specialize efforts so as to obtai a estimate of o-respoig uits i the populatio. This sub-samplig proceure albeit costly, it s free from ay assumptio hece, oe oes ot hae to go for a hure percet respose which ca be substatially more epesie. I eelopig the cocept of omai theory with orespose the followig assumptios are mae; i. Both the omai stuy a auiliary ariables suffers from o-respose. ii. The respoig a o-respoig uits are the same for the stuy a auiliary characters. iii. The iformatio o the omai auiliary ariable is ot kow a hece is ot aailable. i. The omai auiliary ariables o ot suffer from orespose i the first phase samplig but suffers from o-respose i the seco phase of samplig... Propose Domai Estimators Let U be a fiite populatio with kow first stage uits. The fiite populatio is iie ito D omais; U, U,..., U D of sizes,,...,,..., D respectiely. Further, let U be the omai costituets of ay populatio size which is assume to be large a kow. LetU a be efie as, U a respectiely. D U D Let a be the omai stuy a auiliary ariables respectiely. Further, let a be their respectie omai populatio meas a auiliary meas with y ( i,,3,..., a ( i,,..., i th obseratios o the i uit. I estimatig the omai auiliary populatio mea ouble samplig esig is use. A large first phase sample of size is selecte from uits of the populatio by simple raom samplig without replacemet (RWOR esig from which sample uits fallig i the i out first th omai. The assumptio here is that all the uits supply iformatio of the auiliary ariable at first phase. A smaller seco phase sample of size is selecte from by RWOR from which out th of seco phase sample uits fall i the omai. For estimatig the omai populatio mea of the auiliary ariables from a large first phase sample of size, alues of the obseratios i ( i,,3,..., are obtaie a a sample auiliary omai mea compute. From the seco sample of size, let is y a i be the omai stuy a auiliary obseratios with ( i,,3,...,. Let y i a i respoets while i uits supply the iformatio o be the o-respoets for both the stuy a the auiliary omai ariables respectiely such that, +. For the o-respoet group at the seco phase samplig, a RWOR of r uits is selecte with a ierse samplig rate of r such that,, With > All the r uits respo after makig etra efforts of subsamplig o-respoig uits. I eelopig the framework of ouble samplig there are two strata that are o-oerlappig a isjoit. tratum oe cosist of those uits that will respo i the first attempt of the seco phase populatio mae up of uits a stratum two cosist of those uits that woul ot respo i the first attempt of phase two with omai populatio uits. Both a uits are ot kow i aace. The stratum weights of the respoig a o- respoig groups are efie by W their estimators efie by respectiely. a W W i respectiely with w a W w

4 ciece Joural of Applie Mathematics a tatistics 08; 6(: Followig the Hase a Hurwitz [5] techiques, the ubiase estimator for estimatig the omai populatio mea usig ( r + obseratios o character is gie by; y y + y r w y r y i omai stuy + w y (3 imilarly the estimate for omai auiliary ariable is gie by; + r w r + w (4 Where y a are the sample omai meas for the obseratio y i a i respectiely. The followig sample characteristics are efie whe estimatig omai mea, i ii y y i i from the respose group base o y iii r r r j y j for the o-respoig group of i i -omai mea of the stuy character uits -omai mea of the stuy character r respoet uits -omai mea of the auiliary character from the respose group base o uits i r r r j j -omai mea of the auiliary character for the o-respoig group of respoet uits I estimatig the oerall omai populatio mea i the presece of o-respose, ouble samplig ratio estimatio of the omai mea is use. Defie; y y a. r. R r With the assumptio that, E E, E ( y r (5 3. Bias a Mea quare Error of the Ratio Estimator a The epressio for the Mea square error (ME of R R are erie by the use of the Taylor's series approimatio. Let ε ε y 0 0 ( ε y + ( ε + ε 0 ( ε + (6 With the assumptio that E ( ε E ( ε E ( ε Further efie; 0 0 E ( ε0 y E ( y E Ey Var ( y VE E3 ( y + EV E3 ( y + EEV 3 ( y r y + y + W y (7

5 3 Alila Dai Aekeya et al.: Domai Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o Respose C + C + W C y y y y Variace of the whole omai populatio mea of the stuy ariable Variace of the omai populatio mea for the y stratum of o-respoets for tratum of o-respoets for the stuy ariable Cosier also Eε E E Var ( V E ( + E V ( r + W (8 the auiliary ariable The ierse samplig rate Variace of the omai populatio mea for the stratum of o-respoets for tratum of o-respoets for the auiliary ariable et cosier ( ε Cosier, E E E Var ( (9 Variace of the whole omai populatio mea of E y ε ε 0 E E( y ( Co( y CoE ( y E ( E Co( y ECo( y r + + ECo y ECo y r + ( ( y y ρ y + ρ y W (0 et, y Eε ε 0

6 ciece Joural of Applie Mathematics a tatistics 08; 6(: E( y ( Co( y ( ( ( ( CoE y, E + ECo y + E Co yr r Cosier, y ρ y Eε ε E ( ( E E E ( 3.. The Bias of the Ratio Estimator a R R The ratio estimator of. R r a R R ( ca be efie as; y y a. r respectiely Defie as R y. R r ( + ε ( + ε 0 ( + ε ( + ε ( + ε 0 ( + ε ( ( + ε + ε + ε ε ε ε ε ε ε + ε ( Bias of Ratio Estimator R Propositio The bias of the ratio estimator is gie by, R ( ρ ( ρ C C c + W C C C y y y C, C y y, C y a Cy y Variace of the whole omai populatio mea of the stuy ariable Variace of the omai populatio mea for the stratum of o-respoets for tratum of o-respoets for the y stuy ariable Variace of the whole omai populatio mea of the auiliary ariable The ierse samplig rate Variace of the omai populatio mea for the stratum of o-respoets for tratum of o-respoets for the auiliary ariable Proof R ( + ε + ε 0 + ε ε 0 ε ε ε 0 ε ε + ε

7 34 Alila Dai Aekeya et al.: Domai Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o Respose ( R ( E ( ( ( ( ( ( ( + E ε + E ε 0 + E ε ε 0 E ε E ε ε 0 E ε ε + E ε ( ( ε ε ( ε ε ( ε ε 0 0 ( ε + E E E + E + C C C C W C C C + C + W C ρ y y ρ y y ρ y y + C C C + W C C C ( ρ ( ρ y y y y ( R E ( C ρ y C C y + ( W C ρ y C C y Hece Bias of R ( C ρ y C C y + ( W C ρ y C C y 3... Bias of Ratio Estimator R Propositio y y The bias of the ratio estimator R is gie by, ρ y + W ρ y Proof Defie y R as. R r ( + ε ( + ε 0 ( + ε ( + ε ( + ε 0 ( + ε ( ε + ε + ε ε ε ε ε ε ε + ε R ( + ε + ε 0 + ε ε 0 ε ε ε 0 ε ε + ε ( R ( E ( ( ( ( ( ( ( + E ε + E ε 0 + E ε ε 0 E ε E ε ε 0 E ε ε + E ε ( ( ε ε ( ε ε ( ε ε 0 0 ( ε + E E E + E + + y y y ρ y ρ y W ρ y +

8 ciece Joural of Applie Mathematics a tatistics 08; 6(: Hece Bias of R y y ρ y + W ρ y + R ρ R 3.. Mea quare Error (ME of the Ratio Estimator R a R The ratio estimator of a ca be efie as;. R r R R y y a. r respectiely Propositio 3 The mea square error (ME of the estimator efie by y. R r is gie by; + + y W R R ( + ε ( + ε 0 ( + ε E ( + ε ( + ε 0 ( + ε E y y y R y + ρ y y R R R With the otatios efie as i prepositio aboe a R Populatio ratio of to Proof By efiitio, ( R ME E R y E. ubstitutig the alues of equatios (5 we obtai ( ( E ε + ε + ε ε ε ε + ε + E ( ε + ε ε 0 E ( ε + E ( ε + E ( ε + E ( ε ε E ( ε ε E ( ε ε y y y W W y + ρ y y ρ y ρ y y W ] + + y y R

9 36 Alila Dai Aekeya et al.: Domai Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o Respose R + W + R ρ R y y y + ρ R y y ρ R y y + + { ρ } y y R y R y + W + R ρ R y y y y + + R W R Propositio 4 y The mea square error (ME of the ratio estimator. R is gie by; y + + R W y + + ρ y y R R R R With the otatios as efie i propositio aboe Proof ME of y + + ρ y y R R R R ME ( E R R y E E. ( + ε 0 ( + ε ( + ε E( ε + ε + ε 0 ε ε 0 ( ε + ε... + E ( ε + ε ε 0 E ( ε + E ( ε + E ( ε + E ( ε ε E ( ε ε ( ε ε y y y W W

10 ciece Joural of Applie Mathematics a tatistics 08; 6(: y y + ρ y + W ρ y y ρ y y y W y R R W R + ρ R + W ρ R y y y y ρ y R y R ( ρ y y R y R y + W ( y + R + ρ y R y y + + R W R 4. Estimatio of ample ize i the Presece of o-respose Estimatio of omai mea is eelope usig ouble samplig esig base o the techique of sub-samplig of both the stuy a auiliary ariable of the o-respose with ukow omai size. A stuy of cost sureys is therefore cosiere where a o-liear cost fuctio is employe i obtaiig the optimal sample sizes by miimizig ariace for a fie cost 4.. Optimal Allocatio i Double amplig for the Estimatio of Domai A optimum size of a sample is require so as to balace the precisio a cost iole i the surey. The optimum allocatio of a sample size is attaie either by miimizig the precisio agaist a gie cost or miimizig cost agaist a gie precisio. I this stuy, a o-liear cost fuctio has bee cosiere. Deote the cost fuctio for the ratio estimatio by ( c c c r 0 C c (4 size c The cost of measurig a uit i the first sample of c The cost of measurig a uit of the first attempt o 0 y with seco phase sample size. c The uit cost for processig the respoe ata ofy at the first attempt of size. c The uit cost associate with the sub-sample of size r from o-respoets of size Howeer the first sample of size a sub-samples of size r are ot kow util the first attempt is carrie out. The cost will therefore be use i the plaig for the surey. Hece the epecte cost alues of sizes a r will be gie by; W a r W.. Hece the epecte cost fuctio is; C c ( c 0 c c W E C + + +

11 38 Alila Dai Aekeya et al.: Domai Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o Respose W c c c W c C ( 0 (5 4.. Results for Double amplig for Domai Estimatio i the Presece of o-respose Propositio 5 The ariace for the estimate omai mea for the y estimate omai mea. R is miimum for a specifie cost C whe, + ω c c ( W R R ( c + cw 0 ( W R R. R 0 ( + c c W Let R ( c ( R y c y R ( y R y + c ω > 0, thus, ( + ω c + ω + ω c c et the partial eriatie with respect to ( W R G W c W 0 obtaie as; y ω > 0 R y + ± ρ y y R R R y + ± ρ y y R R R c R W W W R c W Cosier the equatio c (7 R Proof To etermie the optimum alues of, a miimizes ariace at a fie cost, efie that G( W + + W y R R W + c ( + c + c. 0 W + c C (6 To obtai the ormal equatios, the epressio of Equatio (6 is ifferetiate partially with respect to, a, a the partial eriaties are equate to zero ( G W y ( R + + c 0 G ( W + + y W R R W + c ( + c + c W + c. C 0 (8 a But from (7, c R R (9 c ubstitutig this i Equatios (8 we obtai

12 ciece Joural of Applie Mathematics a tatistics 08; 6(: G ( W c + + y W R R R ( R + c + c + c W + c W C 0 c (0 The partial eriatie of the equatio (0 with respect to is obtaie as ( G W W ( c cw R R ( + W + c + c W 0 R R 0 ( c + c W W 0 R R W R R ( c + cw 0 0 W R R c c 0 + W Where Thus, But c. from equatio (9 R c ( W R R. R 0 ( + c c W To obtai the alues of, a uppose the cost fuctio is gie by The, are substitute i the cost fuctio equatio (6 a the sole for the alue of. W c + c + c. 0 W + c C ( + C W R R W R ω c W R c + c + c 0 W + c c + c 0 W c c + c 0 W + + ω W R R C c W ( c c W R 0 c c c 0 W + ( Let,

13 40 Alila Dai Aekeya et al.: Domai Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o Respose The equatio (0 becomes; + + ω ( A c R ( B c + c W + W c W R 0 R c + c 0 W C C A B C ( If a substitutig this alue i the equatio (0 we obtai a liear equatio of the form With the alues of A a B efie as; + 0 A B C ω ( A c B c + c W + W c W R 0 R c + c 0 W R, ( a C C olig the liear equatio solutio obtaie is, A+ B C Whe a substitutig this alue i the equatio ( 3 we obtai a liear equatio of the form, A B C With the alues of A efie as; ω ( A c While B a C remais as earlier efie olig the equatio (3 solutio obtaie is, A ( B + 4AC B 4 (3 Propositio 6 If the epecte cost fuctio is of the form C W c log + c + c. 0 W + c the the ariace of the estimate omai mea y is miimum for a specifie cost C if; ω c Proof The proof for a propositio 5 aboe. For techique is use. Let, G ( W c ( W R R ( c + cw 0 ( W R R. R 0 ( + c c W is the same as the oe i the Lagragia multiplier + + y W R R

14 ciece Joural of Applie Mathematics a tatistics 08; 6(: W + c log + c + c. 0 W + c C (4 To obtai the ormal equatios for the epressio (4 the equatio is ifferetiate partially with respect to a the partial eriaties are equate to zero ( y R c G W y + + c 0 y c R R But, y R y c R ω > 0, thus, ω ω c c To sole for, let the ariace be gie as V 0 the substitute the alues of, a ito the equatio, G( W + + y W R V R 0 ubstitute the alues of, a y W R V R ( ( y y R W R + R W + V (5 R ito the equatio (5 a simplify to obtai, ( ( c + W 0 R c R + c 0 W W + c R V (6 Let, V y + V 0, A c R R 0 R B W c + c W + W c a ( ( Thus equatio (6 becomes, 0 A + B V (7 olig for i equatio (7 the solutio becomes, 5. Coclusio ( B 4 AC B + A From the results it is ote that as alues of first sample omai size ( tes to ( omai populatio size (, seco sample size ( samplig rate ( tes to ( tes to ( a ierse the the ME tes asymptotically to 0. From theoretical aalysis it is obsere that the Mea quare Error of the propose estimator will ecrease as the sub-samplig fractio together with the umber of auiliary characters is icrease. As the subsamplig fractio also icreases a the alue of icreases the the alues of a are miimize with the reuctio i the alue of Lagragia multiplier ( which miimizes the cost fuctio. Refereces [] Aitya, K., u U., a Chara H., (04. Estimatio of Domai Mea Usig Two-tage amplig with ub-amplig o-respose. Joural of the Iia ociety of Agricultural tatistics 68 ( pp [] Cochra W. G., (977 amplig techiques. ew ork: Joh Wiley a os, (977.

15 4 Alila Dai Aekeya et al.: Domai Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o Respose [3] Chauhary M. K, a Kumar A., (06. Estimatio of Mea of Fiite Populatio usig Double samplig cheme uer o-respose, Joural of Mathematical cieces 5 (, 4 pp [4] Gamrot, W., (006. Estimatio of Domai Total uer orespose usig Double amplig, tatistics i Trasitio, 7 (4 pp [5] Hase M. H. a Hurwirtz W. W, (946. The problem of o-respose i ample ureys, The Joural of the America tatistical Associatio, [6] Kalto, G., a Kasprzyk, D (986. The treatmet of Missig urey Data, urey Methoology, pp. -6. [7] Meee, G., (005. A o-iformatio Bayesia Approach to Domai Estimatio. Joural of tatistical Plaig a Iferece, 9 ( pp [8] Oh, H. L, a cheure F. J., (983. Weightig ajustmets for uit o-respose, I W. G Maow, I, Olki A B Rui (Es, Icomplete ata i sample sureys ew, ork Acaemic press, pp [9] Okafor F. C, (00. Treatmet of o-respose i uccessie amplig, tatistica, 6 ( [0] ahoo, L. a Paa, P., (99. Estimatio Usig Auiliary Iformatio i Two-tage amplig, Austrialia a ew Zeala Joural of tatistics 4 (4 pp [] riastaa,. K., Jhajj, H.., (983. A Class of Estimators of Populatio Meas Usig Multi-Auiliary Iformatio. Cal. tat. Assoc. Bull. 0 ( pp [] Uofia G. A., (00. Estimatio for Domais i Double amplig for Probabilities Proportioal to ize, the Iia Joural of tatistics 63 pp