Constrained Inverse Adaptive Cluster Sampling

Transcription

1 Joural of Of cial Statistics, Vol. 19, No. 1, 2003, pp. 45±57 Costraied Iverse Adaptive Cluster Samplig Emilia Rocco 1 Adaptive cluster samplig ca be a useful desig for samplig rare ad clustered populatios. I this article a ew adaptive cluster samplig, which is a extesio of the classical oe, is suggested. It is deomiated costraied iverse adaptive cluster samplig ad its distictive characteristic is to make sure that the iitial sample cotais at least oe uit satisfyig the coditio for extra samplig. This is achieved by meas of a sequetial selectio of the iitial sample. This sort of selectio of the iitial uits itroduces a bias ito the estimators of the mea of the populatio usually used i the adaptive cluster samplig. To overcome this dif culty two ew ubiased estimators of the mea of the populatio are suggested i the article. The expressios of their variace ad of their sample variace estimators are also proposed. To study the properties of the proposed strategies a simulatio study is carried out. Key words: Rare ad clustered populatios; sequetial selectio. 1. Itroductio Iformative samplig (also kow as adaptive samplig) desigs are those i which the procedure for selectig uits may deped o values of the variable of iterest or o values of ay other variable observed durig the survey. For rare ad clustered populatios, such as populatios examied i may evirometal ad atural resources studies, they ca produce gais i precisio compared to covetioal desigs. For studies of hidde huma populatios, such as ijectio drug users ad others at risk of HIV trasmissio, adaptive lik-tracig desigs ofte provide the oly practical way to obtai a large eough sample. A particular iformative desig, which is applicable i either case is the adaptive cluster samplig rst proposed by Thompso (1990, 1991a, 1991b, 1992). I adaptive cluster samplig a iitial sample of uits is selected ad, wheever the value of the variable of iterest satis es a speci ed coditio, eighbourig uits are added to the sample. The coditio for extra samplig might be the presece of rare aimal or plat species, detectio of ``hot spots'' i a evirometal pollutio study, ifectio with a rare disease i a epidemiology study or observatio of a rare characteristic of iterest i a household or busiess survey. The eighbourhood of a uit may be de ed by spatial proximity or, i the case of huma populatios, by social or geetic liks or other coectios. Differet types of adaptive cluster samplig have bee proposed (Thompso 1990, 1991a, 1991b, 1992, 1993, 1996, 1997, 1998; Thompso ad Seber 1994, 1996; 1 Uiversity of Florece, Viale Morgagi, 59, I Fireze, Italy. rocco@ds.uifi.it q Statistics Swede

2 46 Joural of Of cial Statistics Thompso ad Frak 1998; Brow 1994; Brow ad Maly 1998; Smith, Coroy ad Brakhage 1995; Chao ad Thompso 1997; Dryer ad Thompso 1998; Roesh 1993; Salehi ad Seber 1997a, 1997b) ad their advatages have bee poited out. However, the possibility that o uit i the iitial sample would satisfy the coditio for extra samplig is, accordig to our research, a problem that is, for the most part, utouched. If this were to happe, the adaptive cluster samplig would coicide with the iitial sample ad o iformatio o the distributio of the relevat values of the variable of iterest would be gathered. Either takig a larger iitial sample or settig a less restrictive coditio for extra samplig (but suf ciet to discover the relevat cluster i the populatio, for example to locate the areas i which a pollutat exceeds a dagerous threshold) would reduce this drawback but ot always avoid it. I this article, a ew adaptive cluster desig, which is a extesio of the adaptive cluster samplig of Thompso, is itroduced. We deomiated it costraied iverse adaptive cluster samplig (CIAC) ad its aim is to esure the presece i the iitial sample of at least oe uit satisfyig the coditio for extra samplig. This is achieved by a sequetial selectio of the iitial sample. This kid of selectio of the iitial uits, explaied i detail i the ext sectio, itroduces a bias ito the estimators of the populatio mea usually used i the adaptive cluster samplig. To overcome this dif culty two ew ubiased estimators of the populatio mea are suggested i the article. I order to obtai ubiased estimators, however, it is ot suf ciet to iclude i the iitial sample oe uit satisfyig the coditio for extra samplig. Rather it is ecessary to iclude at least two uits. Thus, i practice, i the CIAC procedure the iitial selectio process does ot stop util the secod relevat uit is icluded i the sample. The expressios of the variace of the two estimators ad of their sample variace estimators are also proposed. Fially, the relative ef ciecy of the ew strategy, compared to simple radom samplig ad to iverse samplig, is empirically evaluated. 2. Samplig Scheme The cluster adaptive samplig desig proposed by Thompso ca be brie y described as follows. A iitial probability sample of xed size is selected. For each selected uit the variable of iterest y is observed ad if the observed value y i i ˆ 1; 2;...; satis es a coditio of iterest C 0 (speci ed a priori), additioal uits i the eighbourhood of the ith uit are sampled. If the coditio is met i ay uits of the i th eighbourhood, the their eighbourhoods will be also sampled. This is repeated util the coditio is ot met for ay adaptively sampled uits. The result is a sample of clusters. Each cluster has a core of uits satisfyig the coditio C 0 called etwork ad a boudary of uits called ``edge uits'' which do ot satisfy C 0. The uits of the iitial sample which do ot satisfy the coditio C 0 are size-oe clusters. Fially, ay uit that does ot satisfy C 0 is de ed as a etwork of size oe (this meas that ay cluster of size oe is a etwork of size oe ad that ay edge uit is also a etwork of size oe). The proposed samplig scheme is differet from Thompso's i the samplig desig used for the selectio of the iitial sample. It ca be described as follows. Let y be the study variable, let l deote a miimal size for the iitial sample ad let C 0 be the coditio for extra samplig. Assume l uits are selected by simple radom samplig. If amog these uits at least two satisfy C 0, the the procedure for selectig the iitial samplig is stopped

3 Rocco: Costraied Iverse Adaptive Cluster Samplig 47 ad what follows is idetical to what happes i the adaptive cluster strategy proposed by Thompso. Otherwise, the samplig is carried o i a sequetial way ± that is, oe uit is added to the iitial sample at each step, util at least two uits satisfyig C 0 are selected. The last uit, that is the secod satisfyig C 0, may either be retaied or rejected from the sample. Therefore, if the umber of uits selected i the iitial sample is larger tha l, we ca have two possible samples: the sample s which does ot iclude the last selected uit ad the sample s 1 which icludes all the selected uits. From these iitial samples, usig the same mechaism of adaptive additio of uits used by Thompso, we ca obtai respectively the al samples s F ad s 1 F. 3. Estimators Let us cosider the possible CIAC samples: i) s F obtaied from the iitial sample s ii) s 1 F obtaied from the iitial sample s 1 For each of them we shall de e a ubiased estimator of the populatio mea. These will be described respectively i Subsectios 3.1 ad Estimator related to sample s The iitial sample s ca be thought of as a simple radom sample without replacemet of l uits (the rst l selected uits), with the possible additio of other uits sequetially selected. No additioal uits are selected if two or more uits amog the rst l satisfy C 0. Otherwise, the additioal uits are all the uits selected before the secod oe that satis es C 0. Let s l deote the rst l selected uits ad sf l the adaptive cluster sample with iitial sample s l.ift 0 is a ubiased estimator of the populatio mea for a adaptive cluster sample obtaied from a iitial simple radom sample, the T 0 s F ˆT 0 s l F 1 is a ubiased estimator of the mea of the populatio for our sample s F. However, whe the size of s is larger tha l, T 0 (beig calculated o the al sample obtaied from the iitial sample formed by oly the rst l uits selected) does ot take ito accout the oly ouitary etwork i s F if this etwork is ot itersected from oe of the rst l selected uits. Therefore, we propose a ew estimator that depeds o the complete iformatio i s F. This ew estimator is obtaied takig the expected value of T 0 coditioal o a suitable suf ciet statistic. I other words, it is obtaied by a applicatio of the Rao-Blackwell theorem to T 0 (Rao 1945, Blackwell 1947). Let d ˆf i; y i ; i [ s g 2 be the set of distict data i the iitial sample. This set is a fuctio of the set of reduced data associated to sf ad sice this last set is a miimal suf ciet statistic for v ˆ y 1 ; y 2 ;...; y N, d is a suf ciet statistic for the same parameter. The ew ubiased

4 48 Joural of Of cial Statistics estimator is T s F ˆEbT 0 s l F jd c 3 Let deote the size of the iitial sample, s, ad T 0 sfp l the value of T 0 whe the iitial sample cosists of the permutatio p of the uits i s (ad the T 0 is applied to the rst l uits of this permutatio). The umber of permutatios is! ad coditioally o d each of these is equally likely. Hece a explicit expressio of T sf is T sf ˆ 1 X T! 0 sfp l 4 p [ where is the set of all the permutatios of s.ift 0 is ivariat for the permutatio (i.e., its value does ot deped o the positio of the iitial sample uits) the Expressio (4) ca be writte as follows: T sf ˆ 1 X T 0 sfc l 5 c [ C l where C is the set of all possible combiatios of l uits from the i the iitial sample, c is ay of the possible combiatios ad T 0 sfc l the value of T 0 whe the iitial sample is the combiatio c. If we de e I ˆ l ˆ 1 if ˆ l ad I > l ˆ 1 I ˆ l 6 0 if > l Expressio (5) becomes T sf ˆI ˆ l T 0 sf I C T 0 sfc l > l l Also the expressio of varbt sf c ca be obtaied by the Rao-Blackwell theorem. If var T 0 Š is the variace of T 0, we have " # var T sf Š T s ˆ var T 0 Š E F T 0 sfp Š l 2 8! ad if T 0 is ivariat for the permutatio 2 3 var T sf Š 6 C T s ˆ var T 0 Š E F T 0 sfc Š l l To d a ubiased estimator of varbt sf c, we eed ubiased estimators of the two terms i (9). If var T à 0 Š deotes a ubiased estimator of var T 0 Š, a ubiased estimator of varbt sf c is 7 var T s à F Š T s ˆ var T à 0 Š F T 0 sfp Š l 2! 10

5 Rocco: Costraied Iverse Adaptive Cluster Samplig 49 ad if T 0 is ivariat for the permutatio var T s à F Š C T s ˆ var T à 0 Š F T 0 sfc Š l 2 l Sice var T 0 Š is also a fuctio of v ad var T à 0 Š deotes a ubiased estimator, we ca apply the Rao-Blackwell theorem oce agai to obtai aother ubiased estimator of varbt sf c with a smaller variace, amely var à 0 T sf Š ˆ var à 0 T sf Š ˆ var T à 0 sfp Š l T sf T 0 sfp Š l 2! ad if T 0 is ivariat for the permutatio C à var T 0 sfc Š l T sf T 0 sfc Š l 2 l It should be oted that both the estimators of the variace could produce egative estimates with some samples i some data sets. T ca be calculated from ay ubiased estimator of the populatio mea i the adaptive cluster samplig with iitial radom sample without replacemet. Two possible estimators of this type are the modi ed versio of the Horvitz-Thompso estimator ad the modi ed versio of the Hase-Hurvitz estimator proposed by S. K. Thompso (1990). For both these estimators Thompso has proposed a expressio of their variace ad a ubiased estimator of their sample estimate. The estimator T 0 used i the applicatios of Sectio 4 is the modi ed versio of the Horvitz-Thompso estimator proposed by S. K. Thompso Estimator related to the sample sf 1 I cotrast to s, which does ot iclude the last uit selected i the iitial sample whe these are more tha l, s 1 always icludes all the selected uits. Note that whe the uits i s 1 are more tha l, two ad oly two uits satisfy the coditio C 0 ad oe of the two is ecessarily the last uit. So, i cotrast to the uits i s that are permutable i all ways, the uits i s 1 are ot permutable i all ways. The last uit ca be chaged oly if the other satis es C 0. But, after havig made this chage, we ca agai cosider all the possible permutatios of the rst uits. This observatio ca be used to de e aother estimator of the populatio mea which is based o all the iformatio i s 1 ad takes ito accout the ouitary etwork itersected by the secod selected uit satisfyig C 0 eve whe this uit is the last of the iitial sample ad is selected after the rst l. This estimator, as well as T, is obtaied by takig the expected value of T 0 coditioal o a suf ciet statistic. The subset of uits i s 1 satisfyig C 0 is deoted by sc 1 0 ad the statistic cosidered is ( d 1 ˆ f i; y i ; i [ s 1 g if 1 ˆ l ff i 1 ; y i 1 ;...; i ; y i 1 g; i 1 ; y i 1 1 with i 1 [ sc 1 0 g if 1 > l that is, the set of distict ad uordered data i s 1 if the size of s 1 is l, ad the set of

6 50 Joural of Of cial Statistics the distict ad uordered data i s 1 but so that the last uit satis es C 0 if the size of s 1 is larger tha l. Furthermore d 1 is a fuctio of the reduced data associated with. So it is also a suf ciet statistic for v as well as beig d. The, the ew estimator is s 1 F T M s 1 F ˆE T 0 s l F jd 1 Š 15 Whe the size of s 1 is larger tha l, the umber of samples compatible with d 1 is 2!, that is, all the permutatios of the rst values icludig i tur oe of the two uits satisfyig C 0. Whe the size of s 1 is equal to l, s 1 is equal to s ad d 1 is equal to d, ad thus the umber of possible permutatios of data is l!. I either case each possible permutatio is equally likely. It follows that a explicit expressio for T M is T M sf 1 ˆI 1 ˆ l T 0 sfp l I 1 T 0 s l Fp 1 2 T 0 s l Fp 2 l! 1 > l 2! where, if 1 is equal to l, is the set of all permutatios of s 1. If, istead, 1is larger tha l, 1 ad 2 are the sets of the permutatios of the rst terms of s 1 amog which there are, respectively, either the rst or the secod uit satisfyig C 0. If T 0 is ivariat for the permutatio, the T M sf 1 ˆI 1 ˆ l T 0 sfc I l C 1 T0 s l Fc1 C2 T0 s l FC2 1 > l 2 17 l where C 1 ad C 2 are the sets of combiatios correspodig to those of permutatios 1 ad 2. T M is obviously ubiased ad its variace is " var T M sf 1 Š ˆ var T 0 Š 1 # E X # T M sf 1 T 0 sfp l where deotes the set of possible permutatios, ˆ if 1 ˆ l 1 È 2 if 1 > l If T 0 is ivariat for the permutatio " var T M sf 1 Š ˆ var T 0 Š 1 # C E X # T M sf 1 T 0 sfc l 2 C 19 where C is the set of possible combiatios. Two ubiased estimators of var T M Š are var T Ã M sf 1 Š ˆ var T Ã 0 Š 1 X # T M s 1 F T 0 sfp l 2 20 var T Ã M sf 1 Š ˆ 1 X # var T Ã 0 sfp l Š T M sf 1 T 0 sfp l 2 21 To obtai their expressios i the case i which T 0 is uchageable for permutatio, it is suf ciet to cosider C istead of. It is easy to verify that T M is oly the uweighted mea of the two estimators that we deote with T 1 ad T 2 ad that they are equal to T if the uits selected i the iitial sample

7 Rocco: Costraied Iverse Adaptive Cluster Samplig 51 are l. Otherwise, if the uits selected i the iitial phase are more tha l, T 1 is the estimator T applied to the part of sf 1 obtaied from the rst selected uits ad T 2 is the estimator T applied to the part of sf 1 obtaied from the rst uits of the iitial sample after the last uit (the secod uit selected satisfyig C 0 ) has bee substituted with the rst uit selected satisfyig C A Simulatio Study A simulatio was used to study the properties of the CIAC. Te patchy populatios were simulated usig a oisso cluster process model (Neyma 1939; Neyma ad Scott 1958; Cressie 1991) withi a de ed study site divided ito equal sized quadrats. The umber of clusters i the study site was a radom variable from a oisso distributio with a mea equal to 4. Cluster cetres were radomly located i the study site. The umber of idividuals per cluster was a radom variable from a oisso distributio with a mea equal to 90. Each idividual was located at a radial uitary distace from the cetre of the cluster selected from a ormal distributio uiformly distributed betwee 08 ad For each populatio, three Mote Carlo experimets were carried out i order to compare the estimators related to the CIAC samplig with the sample mea related to the simple radom samplig ad two other estimators related to the iverse samplig. What we did is described i detail i the followig three items: i) Each populatio was sampled 10,000 times usig costraied iverse adaptive samplig. The coditio for extra samplig was C 0 : y > 0...Š ad the miimal size of the iitial sample was 50. I detail, from each populatio 10,000 iitial samples were selected usig the desig described i Sectio 2. Not icludig i the iitial sample the last selected uit whe the selected uits were more tha 50, we obtaied the samples s from which we had the al samples sf. From sf we estimated the mea usig the estimator T. Icludig i the iitial sample all the selected uits also whe these were more tha l we obtaied the samples s 1 from which we had the al samples sf 1. From sf 1 we estimated the mea usig the estimator T M. The modi ed versio of the Horvitz-Thompso estimator was used as the startig estimator to calculate T M as well as T. At the ed we had 10,000 samples sf ad 10,000 samples sf 1. The sample size ad the umber of sampled uits satisfyig the coditio for extra samplig were also calculated for each sample sf ad for each sample sf 1. ii) Each populatio was sampled 20,000 times usig simple radom samplig without replacemet. 10,000 simple radom samples equal i size to the expected size (empirically evaluated from the data described i the previous item ad obviously icludig all the uits selected, ot oly those satisfyig C 0 but also the edge uits) of the samples sf, ad a further 10,000 simple radom samples equal i size to the expected size of the samples sf 1 were selected. The sample mea of the rst 10,000 simple radom samples was compared to the estimator T, whilst the sample mea of the other 10,000 simple radom samples was compared to the estimator T M. iii) Each populatio was sampled 20,000 times usig the iverse samplig without

8 52 Joural of Of cial Statistics replacemet. 10,000 iverse samples cotaiig a umber of uits with relevat values of the variable of iterest (values satisfyig the coditio for extra samplig) equal to the expected umber of relevat values of the variable of iterest i sf (empirically evaluated from the data described i the rst item). I additio 10,000 iverse samples cotaiig a umber of uits with relevat values of the variable of iterest equal to the expected umber of relevat values of the variable of iterest i sf 1 were selected. For each sample of the rst 10,000 ad for each sample of the additioal 10,000 the followig two ubiased estimators of the meas were calculated: T iv ˆ 1 X y i 22 i ˆ 1 ad T ivm ˆ 1 k X k k ˆ 1 T iv s h where 1 deotes the umber of uits selected ad the umber of uits cosidered i order to estimate the mea ( just as the costraied iverse adaptive samplig is ot used to calculate T, the last selected uit is ot used to calculate T iv ). T ivm works with the same logic as T M. The last uit ca be ay of the uits satisfyig the coditio selected. Let k deote the umber of uits satisfyig the coditio selected i the iverse samplig, ad let s h deote the iverse sample mius the h th uit from which satisfyig the coditio for extra samplig. The T ivm is othig else tha the uweighted mea of the k possible values of T iv. Table 1 gives some properties of the te populatios, deoted by roma umerals from 1 to X. It gives the mea m, the size N, the umber of ouitary etworks : et, the umber of uits satisfyig the coditio for extra samplig : y i > 0, the total variace V TOT, the ratio betwee the variace withi ad the total variace V W =V TOT. Table 2 provides some results of the Mote Carlo experimets for the populatio IV which is less rare ad that where the ratio betwee the variace withi ad the total variace is the highest. For the other populatios, to avoid prolixity oly the Mote Carlo-derived ef ciecy 23 Table 1. Some key characteristics of simulated populatios opulatios m N : et : y i > 0 V TOT V W =V TOT I II III IV V VI VII VIII IX X

9 Rocco: Costraied Iverse Adaptive Cluster Samplig 53 Table 2. opulatio IV: empirical ad theoretical results p p MSE : Š MSE : Š : y i > 0 % irrelevat eff T Š eff T M Š T T M y ± y ± T iv ± T ivm ± Tiv ± TivM ± idexes for the two proposed estimators T ad T M are provided i Table 3. Before studyig these tables, let us explai the otatios used i them: T: mea estimator related to the sample sf T M : mea estimator related to the sample sf 1 y: sample mea of the simple radom sample of size equal to the expected size of sf y : sample mea of the simple radom sample of size equal to the expected size of s 1 F T iv : mea estimator that does ot take ito accout the last uit selected related to the iverse sample cotaiig a umber of relevat uits equal to the expected umber of relevat uits i s F T ivm : mea estimator that takes ito accout all the uits selected related to the iverse sample cotaiig a umber of relevat uits equal to the expected umber of relevat uits i s F T iv: mea estimator that does ot take ito accout the last uit selected related to the iverse sample cotaiig a umber of relevat uits equal to the expected umber of relevat uits i s 1 F T ivm : mea estimator that takes ito accout all the uits selected related to the iverse sample cotaiig a umber of relevat uits equal to the expected umber of relevat uits p i sf 1 MSE : Š: root square of the mea squared error of the correspodig estimator : p expected size of the sample to which the estimator is related MSE : Š : product of the mea squared error ad expected size of the correspodig sample : y i > 0: expected umber of uits satisfyig the coditio for extra samplig preset i the sample % [... ] irrelevat sample: percet of sample that do ot cotai ay uit satisfyig the coditio for extra samplig p eff T p Š: ef ciecy of the estimator T evaluated as a ratio betwee MSE T Š ad MSE : Š, where : Š is i tur oe of the estimators to which T is compared

10 54 Joural of Of cial Statistics Table 3. Ef ciecy idexes for the two estimators, T ad T M, empirically evaluated ad compared with all the estimators used i the experimets opulatios T T M y y T iv T ivm T iv T ivm I eff T Š ± ± ± eff T M Š 1 1 ± ± ± II eff T Š ± ± ± eff T M Š ± ± ± III eff T Š ± ± ± eff T M Š ± ± ± V eff T Š ± ± ± eff T M Š ± ± ± VI eff T Š ± ± ± eff T M Š ± ± ± VII eff T Š ± ± ± eff T M Š ± ± ± VIII eff T Š ± ± ± eff T M Š ± ± ± IX eff T Š ± ± ± eff T M Š 1 1 ± ± ± X eff T Š ± ± ± eff T M Š ± ± ± p eff T p M Š: ef ciecy of the estimator T M evaluated as a ratio betwee MSE T M Š ad MSE : Š, where : Š is i tur oe of the estimators to which T M is compared. Notice that Tables 2 ad 3 do ot iclude ay referece to the expected values of the estimators because they are all ubiased such that their expected values are approximately equal to the mea of the populatio, which is give for each populatio i Table 1. Two other importat cosideratios regardig Tables 2 ad 3 are the followig: i) for quatities for which it was possible we have cosidered the theoretical calculus apart from the empirical. This ca be foud i italics uder the correspodig theoretical oe; ii) the estimator T is compared oly to y which shares its expected size, to T iv ad T ivm which share its expected umber of uits satisfyig the coditio for extra samplig, ad T M which shares its iitial sample. Equivaletly, T M is compared oly to y which shares its expected size, to Tiv ad TivM which share its expected umber of uits satisfyig the coditio for extra samplig, ad to T which shares its iitial sample. Table 4. Some characteristics of a populatio for which T M is more ef ciet tha T m Size Variability opulatio Network Network

11 Rocco: Costraied Iverse Adaptive Cluster Samplig Discussio From Tables 2 ad 3 it should be oted that, apart from populatios I ad IX, the most ef ciet estimator is T. The relative ef ciecy of a estimator i compariso to aother is evaluated usig the ratio betwee the relative mea squared error multiplied by the expected size of the correspodig sample (through variability for observatio cost). T is also more ef ciet tha T M despite the fact that it is based o less iformatio: it does ot cosider the last uit selected i the iitial sample ad the correspodig etwork. But the icrease of the sample size as a result of their cosideratio is larger tha the decrease i the estimator variability obtaied by usig more iformatio. It should be oted, however, that T is ot more ef ciet tha T M for all possible populatios. Table 4 gives some characteristics of a populatio for which T M is more ef ciet tha T. For this populatio the Mote Carlo-derived ef ciecy idex of T M compared to T is A factor which could always make T M more ef ciet tha T is the itroductio of a cost fuctio which assigs a lower cost of observatio to the uits belogig to the same etwork. I relatio to the compariso betwee T ad y it is clear that T is more ef ciet tha y for all the populatios cosidered. This result is more evidet for some populatios tha for others because it is strictly related to the patchy structure of the cosidered populatio. Apart from the variability by uit of observatio aother factor which makes the costraied iverse adaptive cluster samplig preferable to the simple radom samplig is the larger expected umber of the relevat uits selected (see Table 2). I relatio to the compariso betwee the CIAC samplig ad the iverse samplig, it should be oted that, apart from populatios I ad IX, the rst is more ef ciet. opulatios I ad IX iclude oly oe etwork: sice i the iverse samplig we wat to select the same umber of relevat uits as i the CIAC samplig, we are selectig almost all the populatios i order to capture the uits of this etwork. As a cosequece, the variability of the two mea estimators is almost zero. The relative ef ciecy of the estimators related to the CIAC samplig i compariso to the sample mea related to the simple radom samplig ad to the estimators related to the iverse samplig icreases if we cosider a cost fuctio which assigs a lower cost of observatio to the uits belogig to the same etwork. We do ot have to cosider the compariso betwee the estimators associated to the CIAC samplig ad the estimators of the mea associated to the adaptive cluster samplig of Thompso. The last are of course more ef ciet because the same elemets work o the variability of both with the exceptio that also the variability of the size of the iitial sample works o the estimators related to the CIAC samplig. But the aim of this work was ot to d a strategy more ef ciet tha the adaptive oe proposed by Thompso, but to d a strategy that permits us i every case to say somethig about the study variable. We deem that the method proposed here ful ls this objective. 6. Refereces Blackwell, D. (1947). Coditioal Expectatio ad Ubiased Sequetial Estimatio. Aals of Mathematical Statistics, 18, 105±110.

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