Improved Ratio Estimators in Adaptive Cluster Sampling

Transcription

1 Section on Survey Rearch Methods JSM 28 Improved Ratio Estimators in Adaptive Cluster Sampling Chang-Tai Chao Feng-Min Lin Tzu-Ching Chiang Abstract For better inference of the population quantity of intert, ratio timators are often recommended when certain auxiliary variabl are available. Two typ of ratio timators, modified for adaptive cluster sampling via transformed population and initial intersection probability approach, have been studied in Dryver and Chao 27. Unfortunately, none of them are a function of a minimal sufficient statistic, and therefore can be improved with Rao-blackwellization procedure. The purpose of this paper is to obtain new ratio timators that are not only more efficient than the original ratio timators proposed by Dryver and Chao, but simple to calculate. Additionally, explicit formulas for the approximated variance of the easy-to-compute timators are derived. Key Words: Auxiliary Variable, Ratio Estimator, Adaptive Cluster Sampling, Rao-blackwell Theorem 1. Introduction First proposed by Thompson 199, adaptive cluster sampling is an alternative to timate the population quantity of intert pecially under rare or clustered populations. The basic idea behind this approach is to take a small initial sample by some conventional digns, and then to increase thampling efficiency in the neighborhoods of the sampling units satisfying a condition previously defined. Under the dign-based inferential approach, although the usual unbiased timators in adaptive cluster sampling arimple to calculate, they do not necsarily utilize all the information provided by the rultant final sample. More efficient timators, the Rao-blackwell timators, can be obtained by using Rao-blackwell idea of conditioning on a minimum sufficient statistic. However, Thompson did not prent analytical exprsions for any of the Rao-blackwell timators but he computed the value of a Rao-blackwell timator by averaging the valu of the timator over all the initial sampl giving rise to the observed final sample. Clearly, the excsive number of calculations is required and hence it is sential for ordinary applications to achieve simply analytical exprsions. A few papers have given some analytical exprsions for the Rao-blackwell timators. For example, Salehi 1999 and Félix-Medina 2 have provided the colsed-form exprsions for the Rao-blackwell timators based on the modified Hansen-Hurwitz timator and the modified Horvitz-Thompson timator. In addition, Dryver and Thompson 25 have prented alternative mathematical formulas for the two Rao-blackwell timators derived by taking the expected value of the usual timators conditional on a sufficient statistic. The alternative Rao-blackwell timators may not as efficient as the Rao-blackwell timators obtained by taking the expected value of the usual timators conditional on a minimum sufficient statistic, but they are rather simple. The timators mentioned previously utilize the information provided by the population variable of intert only. Neverthels, to improve the quality of the timate in sampling survey, one not only depends on the the information of the primary variable, but reasonably needs to take relevant aspects of data into account. For many sampling survey situations, certain auxiliary variabl are often available and it is suggted to make use of the the auxiliary information for better inference. Ratio timation is a popular and widely used method to take advantage of the data from the variable of intert along with available auxiliary variabl. Although dign-biased, ratio timators are more efficient because they can give lower mean-square errors when sufficient correlations between the variable of intert and auxiliary variable exist. Moreover, the performanc of the ratio timators become more apparent as the correlations increase e.g., Lohr, Dryver and Chao 27 used auxiliary information into timators in adaptive cluster sampling to obtain ratio timators, which is a straightforward extension of the ratio timator under unequal probability sampling. None of those ratio timators, however, is a function of a minimum sufficient statistic and therefore can be improved with the approach similar to that the univariate timators derived.in this paper the Rao-blackwell ratio timators are derived by taking expected value of the ordinary ratio timators conditional on thamufficient statistic that Dryver and Thompson 25 utilized. However, the formulas for the Rao-blackwell ratio timators are not easily computed as those univariate timators proposed by Dryver and Thompson, but are exprsed as an average over the ratios for all edge units in the observed final sample where edge units are as defined in Section 2. The computation becom much more intensive as the number of the edge units turns large. In the intert of simplicity, we therefore further propose new, efficient, and easy-to-compute ratio Department of Statistics, National Cheng-Kung University Department of Statistics, National Cheng-Kung University Institute of Statistical Science, Academia Sinica 321

2 timators. Furthermore, explicit formulas for the approximated variance of the easy-to-compute ratio timators are derived. The paper is organized as follows. In Section 2, we briefly dcribe the ordinary timators in adaptive cluster sampling, including the dign-unbiased timators and ratio timators. In Section 3 are two typ of the new ratio timators proposed in this article. The Rao-blackwell ratio timators, derived with Rao-blackwellization procedure by conditioning on a sufficient statistic, are dcribed in Subsection 3.1; the derivation is given in Appendix A. In Subsection 3.2, the easy-to-compute ratio timators are illustrated and the formulas for their approximated variance are derived in Appendix B. Section 4 prents concluding comments. 2. Ordinary timators in adaptive cluster sampling In adaptive cluster sampling, an initial sample of units can belected by different typ of conventional probability sampling. In this article thimplt form of adaptive cluster sampling, an initial sample is selected by simple random sampling without replacement SRSWOR, will be considered Thompson, 199. Neverthels, the rult can be extended to other various typ of adaptive cluster sampling associated with different initial sampling digns. In this section, we will briefly dcribe the methodology and concepts involved in adaptive cluster sampling, and we will also introduce the notation used throughout this paper. The ordinary timators in adaptive cluster sampling have been proposed, including the dign-unbiased and ratio timators, are addrsed in this section as well. 2.1 Dign-unbiased timators in adaptive cluster sampling In a basic sampling view, population is a finitet of units consisting of N units with labels 1,..., N, denoted as u = {u 1,..., u N }. Associated with each unit i, the valu of the population variable of intert is denoted as y i. Through this article, the population quantity of intert to be timated is the population mean of the y s, that is, µ y = N y i /N. 1 i=1 In adaptive cluster sampling, thampling procedure is selecting a small initial sample by some conventional digns, and whenever the variable of intert of a unit in thmall initial samplatisfi a given condition C, units in some predefined neighborhood will be included into thample and observed. C is typically a function of the population variable of intert based on the options and the experience of experts for various populations. Neighborhood can be defined by social or institutional relationships between units. The most prevalent, by far, is the neighborhood consisting of the unit itself and the four adjacent units, left, right, top and bottom. In this paper, consider an initial sampl = u 1,..., u n of size is selected from u via SRSWOR. If any of the units in s satisfy C, for example, y i c where c is a constant, their rpective neighborhoods are added to thample and observed. Furthermore, if any added units satisfy C, the units in the neighborhood are added to thample and observed as well, and so on. This procedure is iterated until no new units satisfy. Thet formed by the original unit in s and together with the units added as a consequence of selecting u i is called a cluster. The units adaptively selected but not meet C are called edge units. A cluster minus the edge units is called a network. Any unit not satisfying C is, by definition, a network of size 1. Let A k be the network containing unit i and m k denote the number of units in A k. Then the average of the y valu in the kth network is w yk = 1 m k i A k y i. 2 The population mean of the y s can be written in terms of networks and denote as µ y = K w y k /N, where K is the number of the distinct networks in the population Ordinary timators Section on Survey Rearch Methods JSM 28 Adaptive cluster sampling is a case of Unequal Probability Dign if networks are considered as sampling units. Thompson 199 developed two unbiased timators based on the modifications of the Hansen-Hurwitz and Horvitz- Thompson timators. With this dign, unfortunately, neither the draw-by-draw selection probability, nor the inclusion probability can be determined from the data for the units that do not satisfy C and are not included in the initial sample. Consequently, observations that do not satisfy C are ignore if they are not included in the initial sample. 3211

3 One of the unbiased timators in adaptive cluster sampling, the modified Hansen-Hurwitz timator, is based on the initial draw-by-draw selection probabiliti. Let I. denote an indicator function equalling 1 when the exprsion inside is true and otherwise. The number of units selected from the kth network in the initial sample is n k = i A k Ii s. 3 The modified Hansen-Hurwitz timator and its variance are where κ is the number of distinct networks intersected by the initial sample and varˆµ y hh = ˆµ y hh = 1 κ n k w yk, 4 N n NnN 1 Another unbiased timator using the partial inclusion probabiliti is ˆµ y ht = 1 N K m k w yk µ 2. 5 κ u yk, 6 [ ] N mk N where u yk = m k w yk and = 1 / is the initial intersection probability of the kth network. The joint initial intersection probabiliti is [ N mk h = 1 The variance of ˆµ y ht is Rao-Blackwell timators Section on Survey Rearch Methods JSM 28 varˆµ y hh = 1 N 2 K N mh K h=1 ] N mk m h N /. 7 αkh α h u yk u yh. 8 α h Under the dign-based inferential approach, the ordinary timators do not necsarily utilize all the information provided by the rultant final sample. Only the edge units in the initial sample are incorporated when computing them. The Rao-blackwell theorem can be used to improve the efficiency of the ordinary timators sincome variability can be reduced by making use of the observations of the edge units which are not in the initial sample. Dryver and Thompson 25 utilized a sufficient statistic instead of the minimum sufficient statistic, and obtained the easy-to-compute Rao-blackwell timators which were developed by considering only how many edge units were initial selected, but not which on. In this section, only the timators and their corrponding varianc will be introduced. More detailed proofs and dcriptions can be found in Dryver and Thompson s paper. A statistic d is defined as d = {i, y i, f i, j, y j ; i s c, j s c }. 9 For unit i, f i is the number of tim that the network to which unit i belongs is intersected by the initial sample. The union of a core part s c and the remaining part s c is the final sampl. The core part s c consists of all the distinct units in thample which satisfy the condition. The remaining part s c is thet of all the distinct units in thample for which the condition is not met. Thtatistic d is sufficient for µ so applying by the Rao-blackwell theorem to ˆµ y hh and ˆµ y ht, the easy-to-compute Rao-blackwell timators ˆµ y hh and ˆµ y ht are arrived. One of the Rao-blackwell timators, ˆµ y hh, is defined by where ˆµ y hh = E ˆµ y hh D = d = 1 κ n k w y n k. 1 w yk w y k = ȳ e = e i y i i s, if e i =,, if e i =

4 Section on Survey Rearch Methods JSM 28 Note ȳ e is the average y value for thample edge units in the final sample and is the number of sample edge units in s. For the ith unit in thample, the indicator variable e i is defined as { 1, if i do not meet the condition but is in the neighborhood, e i =, otherwise. 12 Additionally, for those units which are not in s, e i =. The variance of ˆµ y hh is varˆµ y hh = N n K m k w yk µ 2 NnN 1 1 n 2 Pd e s d D y 2 i 1 1 y i y j e 2 s ȳ 2 e, 13 where Pd is the probability that D = d and is the number of units picked in s. The other timators ˆµ y ht is defined by ˆµ y ht = E ˆµ y ht D = d = 1 N κ m k w yk 14 and has variance varˆµ y ht = 1 K K αkh α h N 2 u yk u yh α h=1 k α h 1 n 2 Pd e s y 2 i 1 e d D s 1 y i y j e 2 s ȳ 2 e Ordinary ratio timators in adaptive cluster sampling In many applied survey situations of adaptive cluster sampling, auxiliary variable is often collected together with the population variable of intert. Dryver and Chao 27 utilized the auxiliary information into the timation, and proposed two ratio timators in adaptive cluster sampling by taking advantage of the correlation between the variable of intert and the auxiliary variable. In this section the two ordinary ratio timators and their varianc will be briefly dcribed. Let µ x be the population mean of the auxiliary variable x. The ordinary ratio timator related to the modified Hansen-Hurwitz timator is ˆµ r hh = ˆµ y hh ˆµ x hh µ x, 16 where ˆµ x hh is the modified Hansen-Hurwitz timator for µ x. The approximated variance of ˆµ r hh is Avarˆµ r hh = N n NnN 1 K m k w yk Rw xk 2, 17 where R is the population ratio between w yk and w xk. The other ratio timator of µ y can be constructed based on the modified Horvitz-Thompson timator and is defined by ˆµ r ht = ˆµ y ht ˆµ x ht µ x, 18 where ˆµ y ht and ˆµ x ht are the modified Horvitz-Thompson timators for µ y and µ x, rpectively. The approximated variance is the variance of the modified Horvitz-Thompson timator of the variable u k = u y k Ru xk. Avarˆµ r ht = 1 N 2 K h=1 K u ku h αkh α h α h

5 3. New ratio timators in adaptive cluster sampling In subsection 3.1 the real Rao-blackwell ratio timators, derived with Rao-blackwellization procedure by conditioning on thufficient statistic d, are arrived. The derivation of the Rao-blackwell ratio timators is given in Appendix A. The formulas for them are not as easy as those univariate timators proposed by Dryver and Thompson 25, and the computation becom much more intensive as the number of the edge units turns large. In the intert of simplicity, we therefore further propose two efficient and easy-to-compute ratio timators and the formulas for them are prented in Subsection 3.2. Furthermore, explicit formulas for the approximated variance of the easy-to-compute ratio timators are derived and the derivation is given in Appendix B. 3.1 Rao-Blackwell ratio timators Mentioned in the previous section, thtatistic d is sufficient for µ. So the Rao-blackwell ratio timators ˆµ r hh and ˆµ r ht are able to be arrived at by applying the Rao-blackwell theorem to ˆµ r hh and ˆµ r ht. The Rao-blackwell ratio timator ˆµ r hh is defined by ˆµ r hh = E ˆµ r hh D = d. 2 The timator is not easily computed as shown by the formula κ ˆµ r hh = e n kw yk 1 e i e i y i s i Ψ k e κ s n kw xk 1 µ x. 21 e i e i x i The other Rao-blackwell ratio timator ˆµ r ht is not easily computed as well and the formula is given as ˆµ r ht =E ˆµ r ht D = d 22 κ u yk 1 N e iy i = κ e i n u xk 1 e i n N e µ x. 23 ix i The proofs that equations 2 and 21, and 22 and 23 are rpectively equivalent are given in Appendix A. 3.2 Easy-to-compute ratio timators The formulas for the real Rao-blackwell ratio timators are too complicated to be calculated in practice. To simplify the calculation, we therefore further construct other improved timators via a ratio of the Rao-blackwellized univariate timators conditioning on thufficient statistic d. The new ratio timators are very easily computed and their approximated varianc are ls than or equal to the varianc of the original ratio timators proposed by Dryver and Chao 27. The two easy-to-compute ratio timators and their varianc will be briefly dcribed. One of the easy-to-compute timators ˆµ r hh is defined to be κ ˆµ r hh = ˆµ n kw yk 1 e i ȳ e y hh i Ψ ˆµ µ x = k κ x hh n kw xk 1 µ x, 24 e i x e where ˆµ y hh and ˆµ x hh are the Hansen-Hurwitz type Rao-blackwell timator for µ y and µ x. x e is the average x value for thample edge units in the final sample. The approximated variance of ˆµ r hh is Avar ˆµ r hh = Avar ˆµr hh 1 n The other easy-to-compute timator ˆµ r ht is Section on Survey Rearch Methods JSM 28 ˆµ r ht = ˆµ y ht ˆµ µ x = x ht Pd d D κ κ y i Rx i 2 y i Rx i y j Rx j e 2 s ȳ e R x e 2. u yk 1 1 u xk e i n e i n N ȳe N x e 25 µ x

6 Section on Survey Rearch Methods JSM 28 The approximated variance of ˆµ r ht is Avar ˆµ r ht = Avar ˆµr ht 1 n 2 Pd e s y i Rx i 2 d D 1 y i Rx i y j Rx j e 2 s 1 ȳ e R x e Conclusions In order to make the bt use of survey data in adaptive cluster sampling, we discuss how to utilize auxiliary information into timation. Improving ratio timators with Rao-blackwellization is the main object in this study. We derive the real Rao-blackwell ratio timators by taking expected value of the ordinary ratio timators conditional on thamufficient statistic that Dryver and Thompson 25 utilized. However, the formulas for the Raoblackwell ratio timators are too complicated to be calculated in practice. In the intert of simplicity, we therefore further construct other improved timators via a ratio of the Rao-blackwellized univariate timators conditioning on thufficient statistic. Furthermore, we have been able to obtain the explicit formulas for the approximated variance of those easy-to-compute ratio timators and therefore guarantee their approximated mean square errors are lower than those of the unimproved ratio timators proposed by Dryver and Chao 27. From the model-based perspective population valu are considered to be random variabl, and reprent just one outcome of many possible outcom under a specific model. This probability model can be constructed by detailed surveys or experience and may offer more efficient inferenc than dign-based approach. However, validity of inference depends on the correctns of this assumed model. We did not discuss the inferenc via model-based perspective in this rearch. Similar study under the model-based point of view will be invtigated in the future. Appendix A. Derivation of the Rao-blackwell ratio timators We only derive ˆµ r hh and leave the derivation of ˆµ r ht to the reader because the approach to obtain the formula for ˆµ r ht is not much different from the approach for ˆµ r hh. Let ˆµ r hh s reprent ˆµ r hh as a function of the initial sampl. Let S be a random variable taking on valu from thamplpace S and PS = s D = d is the probability of that initial sample given d. Thus the Hansen-Hurwitz type Rao-blackwell ratio timator is ˆµ r hh = E ˆµ r hh D = d = s S ˆµ r hh s PS = s D = d. The conditional probability PS = s D = d can be written as I{gs = d }/L, where I{.} is an indicator function and L = s I{gs S = d } is the total number of combinations compatible with d. And out of the L combinations any single unit appears 1 1 L. Thus the timator can be written as ˆµ r hh = 1 I{gs = d }ˆµ r hh L s s S 1 = 1 ˆµ r hh s s S = κ n kw yk 1 e i e i y i κ n kw xk 1 e i e i x i µ x. 3215

7 B. Derivation of the approximated variance of the easy-to-compute ratio timators The improved ratio timator is hence the timator can be written ˆµ r = ˆµ y ˆµ µ x = ˆRµx ; x ˆµ r = ˆµ y ˆRµx ˆµ x. The first term in Taylor s formula, expanding about the point µ x, µ giv the approximation Consequently, The approximation for the variance is Avar ˆµ r Hence, Avar ˆµ r =var Eˆµy Rˆµ x D =var ˆµ y Rˆµ x E varˆµ y Rˆµ x D ˆµ r ˆµ y Rµ x ˆµ x. ˆµ r µ ˆµ y Rµ x ˆµ x µ = ˆµ y Rˆµ x. = E ˆµ y Rˆµ 2 x = var ˆµ y Rˆµ x = var Eˆµy Rˆµ x D. =Avar ˆµ r E ˆµ y Rˆµ x ˆµ y Rˆµ x 2 =Avar ˆµ r 1 Pd n 2 Ld I{gs = d } 2 y i Rx i ȳ e R x e d D s S i s,e i =1 =Avar ˆµ r 1 Pd n 2 Ld I{gs = d } y i Rx i 2 e 2 s ȳ e R x e 2 d D s S i s,e i =1 =Avar ˆµ r 1 Pd n 2 Ld I{gs = d } d D s S y i Rx i 2 y i Rx i y j Rx j e 2 s ȳ e R x e 2 i s,e i =1 i s,e i =1 j i =Avar ˆµ r 1 n 2 Pd d D y i Rx i 2 2 =Avar ˆµ r 1 n 2 Pd d D e s y i Rx i Section on Survey Rearch Methods JSM 28 REFERENCES y i Rx i y j Rx j e 2 s ȳ e R x e 2 y i Rx i y j Rx j e 2 s ȳ e R x e 2 Dryver, A. L., and Thompson, S. K. 25, Improved unbiased timators in adaptive cluster sampling, Journal of the Royal Statistical Society B, 67, Dryver, A. L., and Chao, C. T. 27, Ratio timators in adaptive cluster sampling, Environmetrics, 18, Félix Medina, M. H. 2, Analytical exprsions for Rao-Blackwell timators in adaptive cluster sampling, Journal of Statistical Planning and Inference, 84, Salehi, M. M. 1999, Rao-Blackwell versions of the Hansen-Hurwitz and Horvitz-Thompson timators in adaptive cluster sampling, Ecological and Environmental Statistics, 6, Thompson, S. K.199, Adaptive cluster sampling, Journal of the American Statistical Association, 77, Thompson, S. K., and Seber, G.A.F. 1996, Adaptive Sampling, New York: Wiley. 3216