Estimating Population Proportions by Means of Calibration Estimators

Transcription

1 Revista Colombiaa de Estadística Jauary 2015, Volume 38, Issue 1, pp. 267 to 293 DOI: Estimatig Populatio Proportios by Meas of Calibratio Estimators Estimació de proporcioes poblacioales mediate estimadores de calibració Sergio Martíez 1,a, Atoio Arcos 2,b, Helea Martíez 1,c, Sarjider Sigh 3,d 1 Math Departmet, Uiversity of Almería, Almería, España 2 Departmet of Statistics ad Operatioal Research, Uiversity of Graada, Graada, España 3 Departmet of Mathematics, Texas A&M Uiversity-Kigsville, Kigsville Texas, Uited States Abstract This paper cosiders the problem of estimatig the populatio proportio of a categorical variable usig the calibratio framework. Differet situatios are explored accordig to the level of auxiliary iformatio available ad the theoretical properties are ivestigated. A ew class of estimator based upo the proposed calibratio estimators is also defied, ad the optimal estimator i the class, i the sese of miimal variace, is derived. Fially, a estimator of the populatio proportio, uder ew calibratio coditios, is defied. Simulatio studies are cosidered to evaluate the performace of the proposed calibratio estimators via the empirical relative bias ad the empirical relative efficiecy, ad favourable results are achieved. Key words: Auxiliary Iformatio, Calibratio, Estimators, Fiite Populatio, Samplig Desig. Resume El artículo cosidera el problema de la estimació de la proporció poblacioal de ua variable categórica usado como marco de trabajo la calibració. Se explora diferetes situacioes de acuerdo co la iformació auxiliar dispoible y se ivestiga las propiedades teóricas. Ua ueva clase de estimadores basada e los estimadores de calibració propuestos tambié a Professor. spuertas@ual.es b Professor. arcos@ugr.es c Ph.D. Research Assistat. hmartiez@ual.es d Associate Professor. sarjider.sigh@tamuk.edu 267

2 268 Sergio Martíez, Atoio Arcos, Helea Martíez & Sarjider Sigh es defiida y el estimador óptimo e la clase, e el setido de variaza míima, es obteido. Fialmete, u estimador de la proporció poblacioal, bajo uevas codicioes de calibració es tambié propuesto. Estudios de simulació para evaluar el comportamieto de los estimadores calibrados propuestos a través del sesgo relativo empírico y de la eficiecia relativa empírica so icluidos, obteiédose resultados satisfactorios. Palabras clave: calibració, diseño muestral, estimadores, iformació auxiliar, població fiita. 1. Itroductio I the presece of auxiliary iformatio, various approaches may be used to improve the precisio of estimators at the estimatio stage. The book of Sigh 2003 cotais several examples, icludig ratio, differece or calibratio estimators, followig the methodology proposed by Deville & Särdal 1992 ad Särdal 2007, or regressio estimators, as the papers of Arab, Shagodoyi & Sigh 2010 ad Sigh, Sigh & Kozak 2008 show. These techiques are geerally more efficiet tha other methods ot usig auxiliary iformatio. Usually social surveys are focused o categorical variables as sex, race, potetial voters, etc. Efficiet isertio of available auxiliary iformatio would improve the precisio the estimatios for the proportio of a categorical variable of iterest. Coceptually, it is difficult to justify usig a regressio estimator for estimatig proportios. Duchese 2003 cosidered estimators of a proportio uder differet samplig schemes ad preseted a estimator which used the logistic regressio estimator. The model calibratio techique proposed by Wu & Sitter 2001 ca be also used to estimate a proportio by usig a logistic regressio model. Based o logistic models, these estimators efficietly facilitate good modelig of survey data assumig that uit-specific auxiliary data i the populatio U are available. I this case it is assumed that the values of auxiliary variables are kow for the etire fiite populatio referred to as complete auxiliary iformatio but the values of mai variable are kow oly if the uit is selected i the sample. It is very commo for populatio data associated with auxiliary variables to be obtaied from cesus results, admiistrative files, etc., ad these sources ofte provide differet parameters for these auxiliary variables. For example, positio measures mea, media ad other momets are ormally provided, but there is o access to data for each idividual. I the preset study, it is assumed that the oly datum kow is the proportio of idividuals presetig oe or more characteristics related to the study variable. Uder this assumptio, Rueda, Muñoz, Arcos, Álvarez & Martíez 2011 defied a estimator ad various cofidece itervals for a proportio usig the ratio method. The results of their simulatio studies show that ratio estimators are more efficiet tha traditioal estimators. Cofidece itervals outperform alterative methods, especially i terms of iterval width. Revista Colombiaa de Estadística

3 Estimatig Proportios by Calibratio Estimators 269 Calibratio techiques were first employed by Deville & Särdal 1992 to estimate the total populatio, but this approach is also applicable to the estimatio more complex of parameters tha the total populatio. Relevat papers estimatig populatio variaces are Sigh 2001, Sigh, Hor, Chowdhury & Yu 1999 ad Farrell & Sigh The estimatio of fiite populatio distributio fuctios is studied i papers by Harms & Duchese 2006 or Rueda, Martíez, Martíez & Arcos 2007 ad the estimatio of quatiles i Rueda, Martíez- Puertas, Martíez-Puertas & Arcos I Sectio 2 we review a proportio estimator usig the calibratio techique. Sectio 3 describes alterative methods for derivig the calibratio estimator for the proposed parameter. I Sectio 4 we exted these methods to the multiple case. A simulatio study is performed i Sectio 5 ad our coclusios are reported i Sectio Calibratio Estimators for the Proportio 2.1. Defiitio of the Calibratio Estimator Assume a sample s with size from a fiite populatio U = {1, 2,..., } with size, selected by a specific samplig desig d, with iclusio probabilities π k ad π kl assumed to be strictly positive. Let A be a attribute of study i the populatio U, defiig A k = 1 whe a uit k of the populatio U has the attribute A ad A k = 0 otherwise. The populatio proportio of attribute A i the populatio U is give by: P A = 1 A k. 1 To estimate 1, the usual desig-weighted Horvitz-Thompso estimator is: where d k = 1/π k. P AH = 1 d k A k 2 If we cosider a auxiliary attribute B i which the value B k is kow for every uit k i the sample s ad is also kow, the above estimator caot icorporate the iformatio provided by the attribute B, i estimatig the populatio proportio of A. Oe way of icorporatig auxiliary iformatio i the parameter estimatio is via replacig the weights d k by ew weights ω k, usig calibratio techiques. Calibratio is a highly desirable property for survey weights, as Särdal 2007 argues, for the followig reasos: it provides a systematic way of takig auxiliary iformatio ito accout; it is a meas of obtaiig cosistet estimates, with kow aggregates; Revista Colombiaa de Estadística

4 270 Sergio Martíez, Atoio Arcos, Helea Martíez & Sarjider Sigh it is used by statistical agecies for estimatig differet fiite populatio parameters. Several atioal statistical agecies have developed software desiged to compute calibratio weights based o auxiliary iformatio available i populatio registers ad other sources. Such agecies iclude CLA Statistics Swede ad BASCULA Cetral Bureau of Statistics, The etherlads. Followig Deville & Särdal 1992 to obtai a calibratio estimator for the attribute A based o the attribute B, we calculate the weights ω k miimizig the chi-square distace χ 2 = ω k d k 2 3 d k q k subject to the coditio = 1 B k = 1 ω k B k 4 where q k are kow positive costats urelated to d k ad 0 < < 1. By miimizig 3 uder 4, the ew weights ω k are give by: ω k = d k + λd kq k B k where λ is the followig Lagrage multiplier λ = 2 H d k q k B k 5 ad H is the usual Horvitz-Thompso estimator for the attribute B. With the calibratio weights 5, assumig d k q k B k 0, the resultig estimator is: P AW = 1 ω k A k = P AH + H d k q k B k A k 6 d k q k B k By 4, whe the estimator is applied to estimate the populatio proportio of B, it coicides with Properties of the Calibratio Estimator Followig Deville & Särdal 1992, it ca be show that the estimator P AW is a asymptotically ubiased estimator for P A ad its asymptotic variace is give by AV P AW = 1 2 kl d k E k d l E l 7 l U Revista Colombiaa de Estadística

5 Estimatig Proportios by Calibratio Estimators 271 where kl = π kl π k π l ; E k = A k D B k ad D = q k B k A k q k B k A estimator for this variace is V P AW = kl d k e k d l e l 8 π kl l U d k q k B k A k with e k = A k B k D ad D = d k q k B k Example 1. Uder SRSWOR ad q k = 1 for all k U the estimator P AW is: where p A = 1 A k ; ad the asymptotic variace is P AW = p A + p B p p B p B = 1 B k ad p = 1 A k B k AV P AW = V P AV W = V p A + D 2 V p B 2DCov p A, p B [ 2 1 f = P A Q A P P A where Q A = 1 P A ; = 1, P = 1 ca be estimated by [ V P AW = 1 f p A q A + 1 p p B 2 p B q B 2 with q A = 1 1 A k ad q B = 1 1 B k 9 A k B k ad f =. This variace p 3. Alterative Calibratio Estimators p B p p A p B The usual estimator uder SRSWOR, p A has the followig shift ivariace property p A = 1 q A Revista Colombiaa de Estadística

6 272 Sergio Martíez, Atoio Arcos, Helea Martíez & Sarjider Sigh Hece p A has the same performace i the estimatio of P A as the performace of q A i the estimatio of Q A. I geeral, this property is ot satisfied by P AW. It is easy to see that this property is fulfilled if 1 = 1 ω k 12 Thus, we have two ways of obtaiig a estimator with the above property: i By cosiderig a calibratio estimator Q AW for Q A based o, ad determiig whe the estimator P AW has a smaller variace tha the estimator Q AW i order to defie a ew estimator based o these two. ii By cosiderig a calibratio estimator for P A based o ad because if we derive a calibratio estimator that provides perfect estimates for ad, the: 1 = + = 1 ω k B k + 1 ω k 1 B k = 1 ω k 3.1. A Estimator Based o the Complemetary: The P AT Estimator to The first alterative is developed oly uder SRSWOR, miimizig 3 subject = 1 = 1 ω k 1 B k 13 The resultig estimator, assumig q B 0, ca be expressed by with Q AW = q A + q B q q B 14 q = 1 1 B k 1 A k I the same way as with the estimator P AW i Example 1, the asymptotic variace of Q AW is give by: [ 2 AV Q 1 f Q AW = P A Q A Q 2 Q Q A where Q = 1 1 B k 1 A k. Revista Colombiaa de Estadística

7 Estimatig Proportios by Calibratio Estimators 273 A estimator V Q AW for 15 ca be easily defied by [ V Q AW = 1 f p A q A + 1 q q B 2 p B q B 2 q q B q q A q B 16 Let us ow compare the asymptotic variace of P AW with the followig estimator of P A, P AQ = 1 q. We have see Appedix A AV P AW < AV P AQ whe P AT = P < Q. 17 Hece, asymptotically, a more efficiet estimator for the populatio proportio P A is P AW if p < q or q B = 0 p B q B P AQ if p p B ote that the asymptotic variace of P AT is AV P AW AV P AT = AV P AQ q q B or p B = 0 if P otherwise < Q 18 To estimate the asymptotic variace of P AT 10 whe the followig coditio is satisfied the estimator ca be defied by p p B < q q B or q B = 0 19 ad it ca be defied by 16 if p p B q q B or p B = Whe i coditio 20 the equality is satisfied, we have P AW = P AQ, which implies that AV P AW = AV P AQ ad V P AW = V P AQ. The reaso for defiig P AT is to obtai a estimator with the property P AT = 1 Q AT. Accordigly, the estimator Q AT is defied by Q AT = Q AW if q q B 1 P AW f q q B < p p B or p B = 0 p p B or q B = 0 Revista Colombiaa de Estadística

8 274 Sergio Martíez, Atoio Arcos, Helea Martíez & Sarjider Sigh ad 1 Q AT = 1 Q AW if q q B P AW if q q B < p p B or p B = 0 p p B or q B = 0 Sice P AW = 1 Q AW whe q q B = p p B, we have 1 Q AT = P AT. Aother importat questio relative to the estimator P AT is: for low proportios, estimators such as P AW caot be calculated whe p B = 0. The estimator P AT does ot preset this problem, because if p B = 0 the estimator P AT = 1 Q AW ad sice q B = 1 we have P AT = 1 Q AW = 1 q A + q B q The the estimator P AT ca be obtaied for low proportios. Fially, the estimator P AT has the followig drawback: by 9; 15 ad 18, its asymptotic behaviour is worse tha that of the usual estimator whe P P A = Q Q A < 0 21 Thus, if coditio 21 occurs, we propose to use the attribute B = B c because P A B P A P B = 1 A k 1 B k P A 1 = = P A P P A + P A = P + P A > 0 Therefore, with attribute B, the above problem, whe 21 occurs, is solved A Optimal Estimator: The P Aα Estimator Aother way to improve the asymptotic behaviour is as follows: let us defie P Aα = α P AW + 1 α1 Q AW 22 ad take the value α that miimizes the variace. The miimum variace of P Aα is give by see Appedix B AV P Aα = G P A Q A φ2 with G = 1 f 1 ad φ = P P A = Q Q A Revista Colombiaa de Estadística

9 Estimatig Proportios by Calibratio Estimators 275 ad this is achieved where P α = P P A Q Q = The estimator P Aα has the desirable property because if we defie the P Aα = 1 Q Aα Q Aα = β Q AW + 1 β1 P AW Q Q Aα = 1 β P AW + β 1 Q AW = α P AW + 1 α1 Q AW with α = 1 β. Sice AV 1 Q Aα = AV Q Aα, we fid that miimizig the variace of Q Aα with respect to β is equal to miimizig the variace 1 Q Aα with respect to α, ad cosequetly α is agai give by 44 ad P Aα = 1 Q Aα. The P Aα estimator presets the followig disadvatages: It caot be calculated if p B = 0 or q B = 0 The optimum value α give by 44 depeds o theoretical variaces ad covariaces, which are geerally ukow. With respect to the secod questio, the value α ca be easily estimated whe the sample is draw by p p B α = p p B q p B q B Therefore, we obtai the followig estimator P Aα = α P AW + 1 α1 Q AW The estimator P Aα also has the desired property A estimator that Calibrate i ad at the Same Time: The P AR Estimator Whe usig the populatio size ad the populatio proportio of B, the auxiliary iformatio is the same as i the case of a post-stratified estimator. The secod way ii to obtai the property 11, based o a estimator for P A calibratio with ad, ca be developed i ay samplig desig. To do so, we must miimize the distace 3 uder the coditios Revista Colombiaa de Estadística

10 276 Sergio Martíez, Atoio Arcos, Helea Martíez & Sarjider Sigh = 1 ω k B k = 1 ω k 1 B k The calibratio weights with this ew calibratio process are 24 ω k = d k + H d k q k B k + H d k q k B k d k q k 1 B k d kq k 1 B k 25 ad the resultig estimator is where P AR = P AH + H B 1 + H B 2 26 B 1 = Therefore, whe d k q k B k A k ad B2 = d k q k B k d k q k B k = 0 d k q k 1 B k A k d k q k 1 B k or d k q k 1 B k = 0 the estimator 26 caot be obtaied. This problem is solved as follows: sice the two coditios are mutually exclusive, if d k q k B k = 0 we ca calibrate the estimator usig oly the attribute B. O the other had, if d k q k 1 B k = 0 the estimator P AR ca be developed oly with the attribute B. Thus, the estimator P AR is well defied. To prevet this article from becomig excessively log, i the same way as with the estimator P AW, the asymptotic variace of 26 is give by AV P AR = 1 2 l U 27 kl d k U k d l U l 28 Revista Colombiaa de Estadística

11 Estimatig Proportios by Calibratio Estimators 277 with U k = A k B 1 B K B 2 1 B k ad q k B k A k B 1 = ; B 2 = q k B k q k 1 B k A k q k 1 B k 28 is determied usig the followig estimator V P AR = 1 kl 2 d k u k d l u l 29 π kl l s where u k = A k B 1 B k B 2 1 B k Example 2. Uder SRSWOR ad q k = 1 for all k U, the estimator P AR ca be expressed by with P AR = p A + p B p p B p A B = 1 A k 1 B k. + q B p A B q B 30 I the same way as before with the estimator P AW, the asymptotic variace of P AR is see Appedix C. [ AV P 1 f P A 2 P A 2 AR = P A Q A [ P A 2 = G P A Q A = G P A Q A φ2 31 To estimate 45 the followig estimator is defied [ V P 1 f AR = p A q A φ 2 1 p B q b 32 with φ = p p A p B. Thus, the estimator P AR has the same asymptotic variace as the estimator P Aα. The, by 45, uder SRSWOR the estimator P AR is always more efficiet tha the estimators p A ad P AT. ote that the proposed estimator is essetially a post-stratified estimator ad, i this sese, is ot ew. Here, we look at it from a differet poit of view. I a practical situatio, the estimator P AR is preferred the estimator P Aα, sice P AR does ot eed estimatio of ay ukow populatio quatity. The case of estimator P Aα requires estimatig value α, which is geerally ukow. Revista Colombiaa de Estadística

12 278 Sergio Martíez, Atoio Arcos, Helea Martíez & Sarjider Sigh 4. Extesio to Multivariate Auxiliary Iformatio The previous sectio cosidered oly a auxiliary attribute B; let us ow assume that the study attribute A is related to J auxiliary attributes B 1,... B J. To develop the usual way of icorporatig the iformatio provided by J attributes i the estimatio of P A with calibratio techiques, we cosider a ew weight ω k subject to the followig coditios ext, we deote where P j = 1 B jk = 1 ω k B jk j = 1,..., J. 33 P = P 1,..., P J ; P H = P 1H,..., P JH ad B k = B 1k,..., B Jk P jh = 1 d k B jk j = 1,..., J. By T we deote the followig matrix T = d k q k B k B k. With the miimizatio of 3 uder the P coditios give by 33, the ew weights obtaied are: ω k = d k + d k q k P P H T 1 B k. 34 The calibratio estimator based o 34 is give by: with Ĥ = T 1 d k q k B k A k. P AW M = P AH + P P H Ĥ 35 ote that the weights 34 ad the estimator P AW M caot be obtaied if the matrix T is sigular. Followig Rueda et al. 2007a, the asymptotic variace of the estimator P AW M is AV P AW M = 1 2 kl d k Z k d l Z l 36 with Z k = A k B k H where l U 1 H = q k B k B k q k B k A k. Revista Colombiaa de Estadística

13 Estimatig Proportios by Calibratio Estimators 279 The asymptotic variace 36 ca be estimated by with z k = A k B k Ĥ V P AW M = 1 2 l s kl π kl d k z k d l z l 37 Example 3. Uder SRSWOR ad q k = 1, whe oly the two auxiliary attributes B ad C are cosidered, the matrix T ca be expressed by pb p T = BC p BC p C. Cosequetly, T = 2 p B p A p BC 2 ad pc p BC pc p BC T 1 = p BC T p B p BC p B = p B p C p BC 2. ext, we have p d k q k B k A k = p AC ad Ĥ = T 1 d k q k B k A k = pc p p p BC p B p AC p p BC p B p C p BC 2. Thus, the estimator P AW M, uder SRSWOR with two auxiliary attributes, is [ p C p p AC p BC P AW M = p A + p B p B p C p BC 2 [ 38 p B p AC p p BC + P C p C p B p C p BC 2 ow, if we deote A 1 = P CP P AC C P C C 2 ad A 2 = P AC P C P C C 2 Revista Colombiaa de Estadística

14 280 Sergio Martíez, Atoio Arcos, Helea Martíez & Sarjider Sigh The asymptotic variace of the estimator P AW M is AV P 1 f [ AW M = P A Q A + A A 2 2 P C Q C 1 + 2A 1 A 2 C P C 2A 1 P P A 1 f [ 2A 2 P AC P A P C = P A Q A A A 2 2 P C + 2A 1 A 2 C A 1 + A 2 P C 2 2A 1 P P A 2A 2 P AC P A P C. To determie 39 we use the followig estimator: V P 1 f [ AW M = p A q A + Â1 2 p B + Â2 2 p C + 2Â1Â2 p BC 1 Â1 p B + Â2 p C 2 2Â1 p p A p B 2Â2 p AC p A p C Simulatio study A limited study was carried out to ivestigate the desig-based fiite sample performace of the proposed estimators i compariso with that of covetioal estimators Simulated data The estimators were evaluated usig 15 simulated populatios with a populatio size = These populatios were geerated as a radom sample of 1000 uits from a Beroulli distributio with parameter P A = {0.5, 0.75, 0.9}, ad the attributes of iterest were thus achieved with the aforemetioed populatio proportios. Auxiliary attributes were also geerated, usig the same distributio, but a give proportio of values were radomly chaged so that Cramer s V coefficiet betwee the attribute of iterest ad the auxiliary attribute took the values 0.5, 0.6, 0.7, 0.8 ad 0.9. For each simulatio, 1000 samples with sizes = 50, 75, 100 ad 125, were selected uder SRSWOR to compare the estimators: 1 the Horvitz-Thompso estimator P AH 2 the ratio estimator P Aratio see Rueda et al., 2011, 3 P AW estimator, 4 P AT estimator, Revista Colombiaa de Estadística

15 Estimatig Proportios by Calibratio Estimators P Aα estimator with α estimated, 6 P AR estimator, 7 the multivariate ratio estimator P AratioM see Rueda et al., 2011 Multiple ratio, ad 8 the multivariate calibratio estimator P AW M. i terms of relative bias RB ad relative efficiecy with respect to the ratio estimator, where RB = E[ p A P A P A, = MSE[ p A MSE[ P Aratio, p A is a give estimator ad E[ ad MSE[ deote, respectively, the empirical mea ad the mea square error. Values of less tha 1 idicate that p A is more efficiet tha P Aratio. The results derived from this simulatio study gave values of RB withi a reasoable rage. All the calibratio estimators produced absolute relative bias values of less tha 1% except i case P A =0.9 ad φ=0.9. Uivariate ratio estimator produced the highest bias values, especially for small sample sizes. Figures 1, 2 ad 3 show the values of for the various populatios. These figures show: The ratio estimator performs poorly whe there is little associatio betwee the variables. Whe φ = 0.5 this estimator is worse tha the Horvitz- Thompso estimator. Eve whe φ = 0.6 as is sometimes the case P A = 0.75 ad P A = 0.9 the ratio estimator has a large MSE. I populatios with a large φ this problem does ot arise. With large φ values, all the estimators that use auxiliary iformatio produce good results: for φ 0.7 all calibratio ad ratio estimators are better tha the Horvitz-Thompso estimator. It is also see that as φ icreases, all the estimators achieve greater precisio, which is particularly marked for very high proportios. Of all the calibratio estimators, the first oe proposed P AW has the lowest degree of efficiecy. Although it performs better tha the Horvitz-Thompso estimator o most occasios except whe P A =0.9, φ=0.5 ad 0.6 the others produce a smaller MSE. The P AT, PAα ad P AR estimators perform very well i all cases. For high proportios P A =0.75 ad 0.9 the efficiecy of the estimators is fairly similar; oly i the case of P A =0.5 ad small values of φ is there a oticeable differece betwee them, i terms of efficiecy. I these cases, the best results are achieved by the P AR estimators that calibrate i ad at the same time. Revista Colombiaa de Estadística

16 282 Sergio Martíez, Atoio Arcos, Helea Martíez & Sarjider Sigh 1.0 φ = 0.5 Horvitz-Thompso 1.1 φ = 0.6 Horvitz-Thompso φ = 0.7 Horvitz-Thompso φ = 0.8 Horvitz-Thompso φ = 0.9 Horvitz-Thompso Figure 1: Empirical relative efficiecy of the differet estimators for the simulated populatios whe P A = 0.5 ad φ = 0.5, 0.6, 0.7, 0.8, 0.9. The sample size does ot produce a clear effect o the behaviour of the estimators; i some cases, as the sample size icreases, the efficiecy of the estimators icreases, while i others, it decreases as whe φ = 0.9 ad P A =0.9. ad calibratio estimators usig two auxiliary variables always have a lower tha those usig a sigle auxiliary variable. For P A =0.5 ad 0.5 φ 0.7 the multiple calibratio estimator is slightly more efficiet tha the multiple ratio estimator. For P A =0.75 ad 0.9 both estimators have similar levels of efficiecy. estimatio is usually kow to work well whe the variables auxiliary ad of iterest are positively correlated. I this case it is applied to 0-1 variables, so that a positive associatio is expected for the method to work higher freque- Revista Colombiaa de Estadística

17 Estimatig Proportios by Calibratio Estimators φ = 0.5 Horvitz-Thompso 1.1 φ = 0.6 Horvitz-Thompso φ = 0.7 Horvitz-Thompso 2.0 φ = 0.8 Horvitz-Thompso φ = Horvitz-Thompso Figure 2: Empirical relative efficiecy of the differet estimators for the simulated populatios whe P A=0.75 ad φ = 0.5, 0.6, 0.7, 0.8, 0.9. cies for the A=1; B=1 A=0; B=0 cases istead of the 0;1 1;0 cases. From this study we ca coclude that the associatio betwee the variables is the most importat factor ifluecig the behaviour of ratio ad also of calibratio estimators. As expected, as φ icreases the M SE of the calibrated estimators decreases. Eve for moderate values of φ the calibratio estimators improve cosiderably, i terms of efficiecy, o the Horvitz-Thompso estimator. The behaviour of calibratio estimators is similar for small proportios, whereas whe the proportio approaches 1 there are larger differeces amog the proposed calibratio estimators. Hovewer, it is ot a easy task to quatify how much associatio is eeded for a good improvemet i terms of efficiecy, or whe too small that it becomes harmful to itroduce extra variables i the calibratio costraits. Revista Colombiaa de Estadística

18 284 Sergio Martíez, Atoio Arcos, Helea Martíez & Sarjider Sigh 0.9 φ = 0.5 Horvitz-Thompso 0.9 φ = 0.6 Horvitz-Thompso φ = Horvitz-Thompso 2.0 φ = 0.8 Horvitz-Thompso φ = Horvitz-Thompso Figure 3: Empirical relative efficiecy of the differet estimators for the simulated populatios whe P A=0.9 ad φ = 0.5, 0.6, 0.7, 0.8, Real Data I this sectio we apply some proposed estimators to data obtaied i a survey o perceptios of immigratio i a certai regio i Spai. A sample of size = 1919 was selected from a populatio with size = , usig stratified radom samplig. Amog topics of iterest i the survey was estimatig the percetage of citizes who believe that the authorities should make immigratio more difficult by imposig stricter coditios. The auxiliary variable available is the respodet s geder. This variable was observed i the sample ad the totals are kow for each provice stratum. Revista Colombiaa de Estadística

19 Estimatig Proportios by Calibratio Estimators 285 Three mai variables are icluded i this study, related to goodess of immigratio ad amout of immigratio. The mai variables are the aswers to the followig questios: i geeral, do you thik that for Adalusia, immigratio is...? c1-very bad, c2 Bad, c3 either good or bad, c4 Good, c5 Very good, ad i relatio to the umber of immigrats curretly livig i Adalusia, do you thik there are...? c1-too may, c2-a reasoable umber, c3-too few. I this simulatio study, we use the sample as populatio ad we draw stratified radom samples of size = 240 with proportioal allocatio eight stratum. Relative efficiecy with respect to the ratio estimator is computed, as i the previous case, for compared estimators over 1000 simulatio rus. We computed this relative efficiecy for each category of the mai variable 5 categories i the first case, ad 3 i the secod case ad the average over categories is also computed. At the same time, cofidece itervals based o a ormal distributio ad usig proposed estimated variaces are computed for each proportio. Table 1 shows the average legth of the 1000 simulatio rus for each category ad the average over categories. I a similar way, the empirical coverage of the cofidece estimatio is computed. Tables 1 ad 2 show that, from a efficiecy stadpoit P Aα is best. Lookig the average legth of cofidece itervals for the proportio i each category, ad the average over categories, the best estimator is P AR, but the optimal PAα has very similar results. However, the empirical coverage of cofidece itervals is closer to the omial level whe the optimal estimator is used. 6. Applicatio IESA, the Istitute for Advaced Social Studies coducted a survey betwee Jauary 14th ad February 13th, 2011 o the perceptio of culture i the Spaish regio of Adalusia Barometer of Culture of Adalusia - BACU. It is based o a sample draw from a ladlie phoe frame = 5,064,304. From this frame a stratified radom sample without replacemet of dimesio = 641 was selected, where strata were made up by eight geographical regios. Strata populatio sizes are h = , , , , , , , ad the correspodig strata sample sizes are h = 53, 99, 66, 62, 38, 49, 131, 143. Amog several topics of iterest i the survey, is the iterest to estimate perceptio of their culture i relatio to Europea citizes. A auxiliary variable available is geder which totals are kow for each strata. From Table 3 we observe that the P Aα estimator produces the best cofidece itervals. Revista Colombiaa de Estadística

20 286 Sergio Martíez, Atoio Arcos, Helea Martíez & Sarjider Sigh Table 1: Relative efficiecy with respect to the ratio estimator; legth ad empirical coverage of 95% cofidece level estimatio of proportios. Mai variable: goodess of immigratio. Stratified radom samplig. Estimator c1 c2 c3 c4 c5 avg Relative Efficiecy P AW P AT P Aα P AR Legth P AW P AT P Aα P AR Coverage P AW P AT P Aα P AR Table 2: Relative efficiecy with respect to the ratio estimator; legth ad empirical coverage of 95% cofidece level estimatio of proportios. Mai variable: amout of immigratio. Stratified radom samplig. Estimator c1 c2 c3 avg Relative Efficiecy P AW P AT P Aα P AR Legth P AW P AT P Aα P AR Coverage P AW P AT P Aα P AR Revista Colombiaa de Estadística

21 Estimatig Proportios by Calibratio Estimators 287 Table 3: Estimated proportio ˆp, lower boud lb, upper boud ub ad legth l of a 95% cofidece iterval uder stratified radom samplig. Do you thik that i Adalusia the cultural level, compared to the Europea Uio, is...? Estimator prop lb ub le Much lower P AW P AT P AR P Aα lower P AW P AT P AR P Aα Equal P AW P AT P AR P Aα Higher P AW P AT P AR P Aα Much higher P AW P AT P AR P Aα Coclusios I practice, it is importat to make the best possible use of available auxiliary iformatio so as to obtai the most efficiet estimator possible. Whe a proportio ca be estimated i the case of complete auxiliary iformatio i.e., whe auxiliary iformatio is available at the populatio level for each uit it is possible to cosider estimators that use the logistic regressio model Duchese, 2003, Wu ad Sitter, 2001, as a improvemet o the simple estimator. Whe there is merely a rearragemet of the populatio proportio of a attribute with respect to the study variable, the traditioal idirect methods Revista Colombiaa de Estadística

22 288 Sergio Martíez, Atoio Arcos, Helea Martíez & Sarjider Sigh such as the ratio by Rueda et al or the calibratio studied i this work ca be applied. We have studied four calibratio estimators for the proportio which are simple to calculate from stadard calibratio packages ad ca give rise to cosiderable icreases i the precisio achieved, as illustrated by the theoretical results reported here ad by the simulatio performed. The proposed estimators P AW ad P AR ca be obtaied from ay arbitrary samplig desig, whereas P Aα ad P AT estimators are defied uder SRSWOR. However, the extesio to a stratified radom samplig is straightforward. Cofidece itervals based o the estimated variaces of the studied calibratio estimators is also ivestigated through a limited simulatio study, uder a more realistic survey stratified radom samplig usig real data. PAα ad P AR have good properties i cofidece estimatio, ad i some sese a balace betwee legth ad coverage the optimal estimator P Aα provides the best results. Ackowledgemets The authors thak the Editor ad the two aoymous referees of this joural for their helpful commets o a earlier versio of this paper. This study was partially supported by Miisterio de Educació y Ciecia grat MTM , Spai ad by Cosejería de Ecoomía, Iovació, Ciecia y Empleo grat SEJ2954, Juta de Adalucía. [ Received: October 2013 Accepted: ovember 2014 Refereces Arab, R., Shagodoyi, D. K. & Sigh, S. 2010, Variace estimatio of a geeralized regressio predictor, Joural of the Idia Society of Agricultural Statistics 642, Deville, J.-C. & Särdal, C.-E. 1992, Calibratio estimators i survey samplig, Joural of the America Statistical Associatio 87418, Duchese, P. 2003, Estimatio of a proportio with survey data, Joural of Statistics Educatio 113, Farrell, P. & Sigh, S. 2005, Model-assisted higher-order calibratio of estimators of variace, Australia & ew Zealad Joural of Statistics 473, Harms, T. & Duchese, P. 2006, O calibratio estimatio for quatiles, Survey Methodology 32, Rueda, M., Martíez-Puertas, S., Martíez-Puertas, H. & Arcos, A. 2007, Calibratio methods for estimatig quatiles, Metrika 663, Revista Colombiaa de Estadística

23 Estimatig Proportios by Calibratio Estimators 289 Rueda, M., Martíez, S., Martíez, H. & Arcos, A. 2007, Estimatio of the distributio fuctio with calibratio methods, Joural of Statistical Plaig ad Iferece 1372, Rueda, M., Muñoz, J. F., Arcos, A., Álvarez, E. & Martíez, S. 2011, Estimators ad cofidece itervals for the proportio usig biary auxiliary iformatio with applicatios to pharmaceutical studies, Joural of Biopharmaceutical Statistics 213, Särdal, C. 2007, The calibratio approach i survey theory ad practice, Survey Methodology 332, Sigh, H. P., Sigh, S. & Kozak, M. 2008, A family of estimators of fiitepopulatio distributio fuctios usig auxiliary iformatio, Acta Applicadae Mathematicae 1042, Sigh, S. 2001, Geeralized calibratio approach for estimatig variace i survey samplig, Aals of the Istitute of Statistical Mathematics 532, Sigh, S. 2003, Advaced Samplig Theory with Applicatios: How Michael selected Amy, Kluwer Academic Publishers. Sigh, S., Hor, S., Chowdhury, S. & Yu, F. 1999, Calibratio of the estimator of variace, Australia ad ew Zealad Joural of Statistics 41, Wu, C. & Sitter, Y. R. 2001, A model-calibratio approach to usig complete auxiliary iformatio from survey data, Jourals - America Statistical Associatio 96, Appedix A. Compariso betwee AV P AW ad AV P AQ Because AV P AQ = AV q we have AV P AW < AV P AQ whe [ P A Q A + or equivaletly P < [ P A Q A + Q P P A 2 2 P P A < Q2 2 Q Q Q Q A Q Q A. Revista Colombiaa de Estadística

24 290 Sergio Martíez, Atoio Arcos, Helea Martíez & Sarjider Sigh ow, we have Q Q A = 1 1 A k 1 B k 1 P A 1 = P P A. The AV P AW < AV P AQ whe 2 Q 2 2 Q P P A < 0 therefore Sice where Q [ + Q 2P P A = K 1 K 2 < 0. K 2 = P + P A = P ĀB + P A B 0 P ĀB = 1 1 A k B k ad P A B = 1 A k 1 B k, we deduce that AV P AW < AV P AQ whe k 1 < 0, that is P < Q. 41 Appedix B. Obtaiig the miimum variace of P Aα If we deote V 1 = AV P AW ; V 2 = AV Q AW ad C = Cov P AW, Q AW the miimum variace of P Aα is V 2 V 1 C 2 42 V 1 + V 2 + 2C ad this is achieved whe It is easy to see that α = V 2 + C V 1 + V 2 + 2C [ C = Cov P AW, Q 1 f AW = P A Q A Q P P A Q 43 Revista Colombiaa de Estadística

25 Estimatig Proportios by Calibratio Estimators 291 Therefore, we have: V 2 + C = 1 f [ P P A Q 1 Q 2 Q + = 1 f Q 1 [ P P A Q Similarly V 1 + V 2 + 2C = 1 f 1 [ 2 + = 1 f 1 Q 2 2 Q Q 2 By substitutig the values V 2 + C ad V 1 + V 2 2C i 43, the value of α is foud to be: P α = P P A Q Q = Q. 44 V 1 V 2 = G 2 [ PA Q A 2 + [ + Q Q 2 PB + 4 Q Q 2φ + Q P A Q A + P Q [ = G 2 P A Q A φ + Q 2 φ [ Q 2 + PB + Q + 2 Q P A Q A 2φ φ 2 Q 2 + Q P A Q A 2 P Q Revista Colombiaa de Estadística

26 292 Sergio Martíez, Atoio Arcos, Helea Martíez & Sarjider Sigh [ = G 2 P A Q A φ + Q 2 + P A Q A φ2 Q + 2P Q P A Q A φ + where K = P A Q A φ Q 2 PB = G 2 P A Q A + Q 2 2 Q 2. PB O the other had φ2 C 2 = G [φ 2 + Q P A Q A + Q 2 Q 2 + G 2 K 2 + 2P Q P A Q A φ + Q Q 2 = G 2 K Thus, by substitutig the values C 2 ; V 1 V 2 ad V 1 +V 2 +2C i 42, we have: AV P Aα = G P A Q A φ2 Appedix C. Obtaiig AV P AR [ AV P 1 f AR = P A Q A P P A 2 PA B P 2 A B P A B P A with P A B = 1 A k 1 B k. Revista Colombiaa de Estadística

27 Estimatig Proportios by Calibratio Estimators 293 ow, takig ito accout that P A B P A = P A P P A + P A = P A P the asymptotic variace is Sice AV P AR = P A B 1 f 1 [ P A Q A + PA B + 2P P A P. P = P A B + P P P 2 A B = P A P we have [ AV P 1 f P A 2 P A 2 AR = P AQ A [ P A 2 45 = G P AQ A = G P AQ A φ2 Revista Colombiaa de Estadística