American Journal of Mathematics and Statistics 2012, 2(6): 199-205 DOI: 10.5923/j.ajms.20120206.06
Alternative Estimation Method for a Three-Stage Cluster Sampling in Finite Population
L. A. Nafiu1,*, I. O. Oshungade 2, A. A. Adewara
1Department of Mathematics and Statistics, Federal University of Technology, Minna, +234, Nigeria 2Department of Statistics, University of Ilorin, Ilorin, +234, Nigeria
Abstract This research investigates the use of a three-stage cluster sampling design in estimating population total. We focus on a special design where certain number of visits is being considered for estimating the population size and a weighted factor of / is introduced. In particular, attempt was made at deriving new method for a three-stage sampling design. In this study, we compared the newly proposed estimator with some of the existing estimators in a three-stage sampling design. Eight (8) data sets were used to justify this paper. The first four (4) data sets were obtained from[1],[2],[3] and[4] respectively while the second four (4) data sets represent the number of diabetic patients in Niger state, Nigeria for the years 2005, 2006, 2007 and 2008 respectively. The computation was done with software developed in Microsoft Visual C++ programming language. All the estimates obtained show that our newly proposed three-stage cluster sampling design estimator performs better. Ke ywo rds Sampling, Three-stage, Cluster Design, Estimator, Bias and Variance
Sub sampling has a great variety of applications[3] and the 1. Introduction reason for mu ltistage samp ling is ad ministrative convenienc e[10]. The process of sub sampling can be carried to a third In a census, each unit (such as person, household or local stage by sampling the subunits instead of enumerating them government area) is enumerated, whereas in a sample survey, complete ly[11]. Comparing multistage cluster sampling with only a sample of units is enumerated and information simp le rando m samp ling, it was observes that multistage provided by the sample is used to make estimates relating to cluster sampling is better in terms of efficiency[12]. all units[5] and[6]. In designing a study, it can be Multistage sampling makes fieldwork and supervision advantageous to sample units in more than one-stage. The relatively easy[4]. Multistage sampling is more efficient than criteria for selecting a unit at a given stage typically depend single stage cluster sampling[13] and references had been on attributes observed in the previous stages[7]. Mu ltistage made to the use of three or more stages sampling[9]. sampling is where the researcher divides the population into clusters, samples the clusters, and then resample, repeating Let N denote the number of primary units in the the process until the ultimate sampling units are selected at population and n the number of primary units in the sample. the last of the hierarchical levels[8]. If, after selecting a Let M i be the number of secondary units in the primary unit. sample of primary units, a sample of secondary units is The total number of secondary units in the population is selected from each of the selected primary units, the design is N referred to as two-stage sampling. If in turn a sample of MM= ∑ i (1) tertiary units is selected from each selected secondary unit, i=1 the design is three-stage sampling[9]. Let yij denote the value of the variable of interest of the The aim of this paper is to model a new estimator for jth secondary unit in the ith primary unit. The total of the three-stage cluster sampling scheme which is to be compared y-values in the ith primary unit is with the other existing seven conventional estimators. M i Yyi= ∑ ij (2) j=1 2. Methodology Accordingly, the population total for over-all sa mple in a two-stage is given as
N M i * Corresponding author: = (3) [email protected] (L. A. Nafiu) Yy∑∑ ij = = Published online at http://journal.sapub.org/ajms ij11 Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved For a three-stage sampling, the population contains N
200 L. A. Nafiu et al.: Alternative Estimation Method for a Three-Stage Cluster Sampling in Finite Population
first-stage units, each with M second-stage units, each of estimator of the population total at secondary unit in the which has K third-stage units. The corresponding numbers primary unit in the sample is: 𝑗𝑗𝑗𝑗ℎ for the sample are n, m and k respectively. Let y be the 1 iju 𝑖𝑖𝑖𝑖ℎ = 𝑘𝑘𝑖𝑖𝑖𝑖 th th 2 value obtained for the u third-stage units in the j 𝑖𝑖 =1 𝑁𝑁 second-stage units drawn from the ith primary units. The 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖 𝑦𝑦� 𝑖𝑖𝑖𝑖 � 𝑖𝑖 𝑦𝑦 = 𝛾𝛾 𝑙𝑙=1 𝑛𝑛2 (8) relevant population total for over-a ll samp le in a three-s tage 𝐾𝐾𝑖𝑖𝑖𝑖 𝑘𝑘𝑖𝑖𝑖𝑖 𝑁𝑁𝑖𝑖 is given as follows: 𝑖𝑖𝑖𝑖𝑖𝑖 where = is the known𝑘𝑘𝑖𝑖𝑖𝑖 ∑ 𝑙𝑙sampling𝑛𝑛𝑖𝑖 𝑦𝑦 fraction for tertiary NMK 𝑘𝑘𝑖𝑖𝑖𝑖 = (4) 𝑖𝑖𝑖𝑖 Yy∑∑∑ iju units in𝛾𝛾 the 𝐾𝐾𝑖𝑖𝑖𝑖 secondary unit of the primary unit. i=111 ju = = Also, let denote the number of individuals (tertiary ^ units) in the𝑗𝑗𝑗𝑗 sampleℎ from the secondary𝑖𝑖𝑖𝑖ℎ unit of the th 𝑖𝑖𝑖𝑖𝑖𝑖 For any estimation Θ h in the h cell based on primary unit𝑦𝑦 who engage in the treatment of diabetes. An completely arbitrary probabilities of selection, the total unbiased estimator of the population𝑗𝑗𝑗𝑗ℎ total in the primary𝑖𝑖𝑖𝑖ℎ variance is then the sum of the variances for all strata. The unit in the sample is: symbol E is used for the operator of expectation, V for the 𝑖𝑖𝑖𝑖ℎ = =1 (9) ^ 𝑀𝑀𝑖𝑖 𝑚𝑚𝑖𝑖 𝑖𝑖 𝑖𝑖𝑖𝑖 variance, and V for the unbiased estimate of V. We may Finally, an unbiased𝑦𝑦� estimator𝑚𝑚𝑖𝑖 ∑𝑗𝑗 𝑦𝑦�of the population total of then write the diabetic patients undergoing treatment in all the hospitals ^ ^ ^ at the secondary unit (city) in the primary unit V (Θh ) = V (E(Θh )) + E(V (Θh )) (5) (local government area) is: 1 >1 1 >1 𝑗𝑗𝑗𝑗ℎ 𝑖𝑖𝑖𝑖ℎ The expression (5) may be written into three components 3 = 𝑛𝑛 as: =1 𝑁𝑁𝑁𝑁𝑁𝑁 𝑁𝑁 𝑖𝑖 ^ ^ ^ ^ 𝑌𝑌� � 𝑦𝑦� = =1{ 𝑛𝑛 𝑖𝑖 =1( =1 2 )} (10) V (Θ h ) = V (E(E(Θ h ))) + E(V (E(Θ h ))) + E(E(V (Θ h ))) (6) 𝑁𝑁 𝑀𝑀𝑖𝑖 𝐾𝐾𝑖𝑖𝑖𝑖 𝑘𝑘𝑖𝑖𝑖𝑖 𝑁𝑁𝑖𝑖 1 2 >2 1 2 >2 1 2 >2 𝑛𝑛 𝑚𝑚𝑖𝑖 𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖 𝑛𝑛 ∑ 𝑚𝑚𝑖𝑖 ∑𝑗𝑗 𝑘𝑘𝑖𝑖𝑖𝑖 ∑𝑙𝑙 𝑛𝑛𝑖𝑖 where “>1” is the symbol to represent all stages of sampling 𝑦𝑦 after the first[3]. 4. Theorems
3. Proposed Three-Stage Cluster 4.1. Theorem 1: is Unbi ased for the Popul ati on total Y Sampling Design 𝒀𝒀�𝟑𝟑𝟑𝟑𝟑𝟑𝟑𝟑 To estimate the population size at different hospitals using Proof: three-stage sampling, the unbiased estimator of population We know that expectation of given by equation (8) total can be derived as follows. In a three-stage sampling conditional on samples 1 and 𝑖𝑖𝑖𝑖2 of primary units and without replacement design supported by[3],[4] and[14]; a secondary units respectively equals𝑦𝑦� of engaging in the sample of primary units is selected, then a sample of 𝑠𝑠 𝑠𝑠 variable of interest in each primary unit𝑖𝑖𝑖𝑖 and each secondary secondary units is chosen from each of the selected primary unit [15]. That is; 𝑦𝑦 units and finally, a sample of tertiary units is chosen from 1, 2 = (11) each selected secondary unit. For instance, the state consists Also, the expectation of given by equation (9) of number of local government areas out of which a � 𝑖𝑖𝑖𝑖 � � 𝑖𝑖𝑖𝑖 conditional on sample𝐸𝐸 𝑦𝑦� 𝑠𝑠of 𝑠𝑠primary𝑦𝑦 units equals of simple random sampling of n number of local government 1 𝑖𝑖 engaging in the variable of interest𝑦𝑦� in each primary unit. areas𝑁𝑁 is selected. Each local government area consists of 𝑖𝑖 That is; 𝑠𝑠 𝑦𝑦 number of cities out of which a simple random sampling 𝑖𝑖 ( | ) = (12) without replacement of number of cities is selected.𝑀𝑀 1 To obtain the expected value of given by equation Finally, from the selected sample of city containing 𝑖𝑖 3𝑖𝑖 𝑖𝑖 (10) over all possible samples𝐸𝐸 𝑦𝑦� 𝑠𝑠 of primary𝑦𝑦 units. number of hospitals, 𝑚𝑚number of hospitals is selected at 𝑁𝑁𝑁𝑁𝑁𝑁 𝑖𝑖𝑖𝑖 Then, the expectation of is𝑌𝑌� : random without replacement and the number of diabetic𝐾𝐾 3 𝑘𝑘𝑖𝑖𝑖𝑖 3 = 1 [ 2 3 3 1, 2 1 ] (13) patients in this hospital is collected. 𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁 where 1 and 2 denote the samples of primary units and Then; 𝐸𝐸�𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁 � 𝐸𝐸 𝐸𝐸 �𝐸𝐸 �𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁 �𝑠𝑠 𝑠𝑠 ��𝑠𝑠 � = secondary units respectively. =1 =1 (7) 𝑠𝑠 𝑠𝑠 𝑛𝑛 𝑚𝑚𝑖𝑖 3 = [ =1 1, 2 1 ] Again, let be the numbe𝑖𝑖 r𝑗𝑗 of 𝑖𝑖𝑖𝑖primary units (local 𝑦𝑦 ∑ ∑ 𝑦𝑦 𝑁𝑁 𝑛𝑛 government areas) sampled without replacement, be the 𝑁𝑁𝑁𝑁𝑁𝑁 𝑖𝑖 𝐸𝐸�𝑌𝑌� � 𝐸𝐸 𝐸𝐸 �𝐸𝐸 �𝑛𝑛 ∑ 𝑦𝑦�𝑖𝑖𝑖𝑖 �𝑠𝑠 𝑠𝑠 ��𝑠𝑠 � number of secondary𝑛𝑛 units (cities) selected without = [ { 𝑛𝑛 | 1}] 𝑖𝑖 replacement from the sampled primary unit𝑚𝑚 (local =1 𝑁𝑁 𝑖𝑖 government area) and be the number of tertiary units 𝐸𝐸 𝐸𝐸 � 𝑦𝑦� 𝑠𝑠 𝑛𝑛 𝑖𝑖 𝑖𝑖𝑖𝑖ℎ = { 𝑛𝑛 } (hospitals) selected from 𝑖𝑖𝑖𝑖the secondary unit (city) in the primary unit (local𝑘𝑘 government area). An unbiased 𝑁𝑁 =1 𝐸𝐸 � 𝑦𝑦𝑖𝑖 𝑗𝑗𝑗𝑗ℎ 𝑛𝑛 𝑖𝑖 𝑖𝑖𝑖𝑖ℎ American Journal of Mathematics and Statistics 2012, 2(6): 199-205 201
known for = 1,2, , and = 1,2, , . The third term = 𝑁𝑁 contains variance due to estimating the ’s fro m a 𝑖𝑖 =1 𝑖𝑖 ⋯ 𝑛𝑛 𝑗𝑗 ⋯ 𝑚𝑚 𝑁𝑁 𝑛𝑛 𝑖𝑖 subsample of tertiary units within the selected secondary = � 𝑦𝑦 (14) 𝑦𝑦𝑖𝑖𝑖𝑖 𝑛𝑛 𝑁𝑁 𝑖𝑖 units. An unbiased estimator of the variance of 3 given This implies that the proposed estimator 3 is in equation (18) is obtained by replacing the population unbiased. 𝑌𝑌 𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁𝑁𝑁 variances with the sample variances as follows: � 2 2 Hence, the variance of the newly proposed esti𝑌𝑌 mator 1 3 = ( ) + =1 ( ) + 3 of the population total is derived as follows: 𝑠𝑠 𝑁𝑁 𝑛𝑛 𝑠𝑠𝑖𝑖 𝑁𝑁𝑁𝑁𝑁𝑁 =1 =1 𝑖𝑖 𝑖𝑖 𝑖𝑖 2 𝑖𝑖 (22) In line with[3] and[14], we use 𝑉𝑉��𝑌𝑌� � 𝑁𝑁 𝑁𝑁 − 𝑛𝑛 𝑛𝑛 𝑛𝑛 ∑ 𝑀𝑀 𝑀𝑀 − 𝑚𝑚 𝑚𝑚𝑖𝑖 𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁 3 = 3 1, 2 | 1 where 𝑁𝑁𝑛𝑛𝑖𝑖 𝑛𝑛𝑀𝑀𝑖𝑖𝑚𝑚𝑖𝑖𝑗𝑗 𝑚𝑚𝑖𝑖𝐾𝐾𝑖𝑖𝑗𝑗𝐾𝐾𝑖𝑖𝑗𝑗−𝑘𝑘𝑖𝑖𝑗𝑗𝑠𝑠𝑖𝑖𝑗𝑗 𝑘𝑘𝑖𝑖𝑗𝑗 𝑁𝑁𝑁𝑁𝑁𝑁 + 𝑁𝑁𝑁𝑁𝑁𝑁 3 1, 2 1 = � � � � � � � � � 3 2 𝑉𝑉 𝑌𝑌 𝑉𝑉 𝐸𝐸 𝑌𝑌 𝑠𝑠 𝑠𝑠 𝑠𝑠 =1( ) 3 + 3 +𝑁𝑁𝑁𝑁𝑁𝑁 { 3 } (15) 2 � �� � � �� �� 1 = 𝑛𝑛 𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁 (23) 𝐸𝐸 𝑉𝑉 𝑌𝑌 𝑠𝑠 𝑠𝑠 𝑠𝑠 ∑𝑖𝑖 𝑦𝑦𝑖𝑖 − 1 Because of the simple random sampling of primary units 𝑛𝑛 2 𝑁𝑁𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁𝑁𝑁 =1( ) 𝑉𝑉�𝐸𝐸� 𝑌𝑌� �� 𝐸𝐸�𝑉𝑉�𝐸𝐸� 𝑌𝑌� ��� 𝐸𝐸 𝐸𝐸�𝑉𝑉� 𝑌𝑌� �� 2 𝑛𝑛− and secondary units without replacement at the first stage 𝑠𝑠 = 𝑚𝑚 𝑖𝑖 𝑦𝑦 𝑖𝑖 (24) ∑ 𝑦𝑦𝑖𝑖𝑖𝑖 −1 𝑗𝑗 𝑚𝑚 𝑖𝑖 and second stage respectively, the first term to the right of the 2 2 𝑖𝑖 2 𝑠𝑠 𝑚𝑚𝑖𝑖 − 2 equality in equation (15) is: = 2 =1( 4 2 )( ) (25) ( ) 2 𝐾𝐾𝑖𝑖𝑖𝑖 𝑘𝑘𝑖𝑖𝑖𝑖 𝑁𝑁𝑖𝑖 𝑁𝑁𝑖𝑖 3 1, 2 | 1 = { =1 } = 1 (16) 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑘𝑘𝑖𝑖𝑖𝑖 ∑𝑙𝑙 𝑛𝑛𝑖𝑖 𝑛𝑛𝑖𝑖 𝑁𝑁 𝑛𝑛 𝑁𝑁 𝑁𝑁−𝑛𝑛 𝑠𝑠 − 𝑦𝑦 − 𝑦𝑦� The second𝑁𝑁𝑁𝑁𝑁𝑁 term to the right of the𝑖𝑖 equality𝑖𝑖 in equation (15) 4.2. Theorem 2: is Un bi as ed for 𝑉𝑉�𝐸𝐸�𝑌𝑌� �𝑠𝑠 𝑠𝑠 � 𝑠𝑠 � 𝑉𝑉 𝑛𝑛 ∑ 𝑦𝑦 𝑛𝑛 𝜎𝜎 is: Proof: 𝑽𝑽��𝒀𝒀�𝟑𝟑𝑵𝑵𝑵𝑵𝑵𝑵 � 𝑽𝑽�𝒀𝒀�𝟑𝟑𝑵𝑵𝑵𝑵𝑵𝑵 � { 3 1, 2 | 1} = { 𝑛𝑛 | 1} We note that 2 =1 2 1 2 3 𝑁𝑁𝑁𝑁𝑁𝑁 𝑁𝑁 𝑖𝑖 � 1 = ( =1 2 ) (26) 𝑉𝑉 �𝑌𝑌 �𝑠𝑠 𝑠𝑠 � 𝑠𝑠 𝑉𝑉 � 𝑦𝑦� 𝑠𝑠 1 � 2 𝑛𝑛 𝑖𝑖 𝑛𝑛 𝑛𝑛𝑌𝑌 𝑁𝑁𝑁𝑁𝑁𝑁 = 𝑛𝑛 ( | ) + ( ) 𝑛𝑛 ( ) Next, we note that 1 3 𝑠𝑠 𝑛𝑛− ∑𝑖𝑖 𝑦𝑦𝑖𝑖 − 𝑁𝑁 =1 =1 𝑁𝑁 𝑖𝑖 𝑁𝑁 𝑁𝑁𝑁𝑁𝑁𝑁 2 2 � �� 𝑉𝑉 𝑦𝑦� 𝑠𝑠 � 𝑉𝑉 𝑌𝑌� ( 𝑛𝑛 ) = { ( 𝑛𝑛 | 1)} 𝑛𝑛 𝑖𝑖 2 𝑛𝑛 2𝑖𝑖 2 = 𝑛𝑛 + ( ) 𝑛𝑛 =1 =1 𝑖𝑖 𝑖𝑖 𝑁𝑁 =1 𝑁𝑁 =1 𝐸𝐸 � 𝑦𝑦 𝐸𝐸 𝐸𝐸 � 𝑦𝑦 𝑠𝑠 𝑖𝑖𝑖𝑖 𝑖𝑖 𝑖𝑖 ( | ) 𝑖𝑖 ( | ) 2 �, � � 𝜎𝜎 � 𝜎𝜎 = ( 𝑛𝑛 [ 1 + { 1 } ]) 3 1𝑛𝑛 2 𝑖𝑖 1 𝑛𝑛 𝑖𝑖 =1 2 𝑖𝑖 𝑖𝑖 𝐸𝐸�𝑉𝑉��𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁 �𝑠𝑠 𝑠𝑠 ��𝑠𝑠 �� 𝐸𝐸 � 𝑉𝑉 𝑦𝑦 𝑠𝑠 𝐸𝐸 𝑦𝑦 𝑠𝑠 = 𝑁𝑁 𝑀𝑀𝑖𝑖 ( ) 𝑖𝑖 2 2 𝑛𝑛 𝑛𝑛 𝑖𝑖𝑗𝑗 = ( + ) =1 𝑖𝑖 =1 𝜎𝜎 𝑁𝑁 𝑀𝑀 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 =1 =1 � � 𝐾𝐾 𝐾𝐾 − 𝑘𝑘 2 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑗𝑗 𝑖𝑖𝑖𝑖 𝐸𝐸 � 𝜎𝜎 � 𝑌𝑌 𝑛𝑛 2 𝑚𝑚 𝑘𝑘 + ( ) 𝑁𝑁 ( ) 𝑖𝑖 2 𝑖𝑖 2 = ( 𝑁𝑁 + 𝑁𝑁 ) =1 𝑖𝑖 𝑁𝑁 𝑛𝑛 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝜎𝜎 =1 =1 2 � 𝑀𝑀 𝑀𝑀 − 𝑚𝑚 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝐸𝐸 � 𝑧𝑧 𝜎𝜎2 � 𝑧𝑧 𝑌𝑌2 = =1 ( 𝑛𝑛 ) 𝑁𝑁 + =1 =1𝑚𝑚 ( = (𝑖𝑖 =1 + 𝑖𝑖 =1 ) (27) 𝑁𝑁 𝜎𝜎𝑖𝑖 𝑁𝑁 𝑀𝑀𝑖𝑖 𝑀𝑀𝑖𝑖 𝑛𝑛 𝑁𝑁 ) 𝑁𝑁 2 𝑁𝑁 𝑁𝑁 𝑛𝑛 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑚𝑚𝑖𝑖 𝑛𝑛 𝑖𝑖 𝑚𝑚𝑖𝑖 𝑗𝑗 𝑖𝑖𝑖𝑖 (17)𝑖𝑖𝑖𝑖 In addition; 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 ∑ 𝑀𝑀 𝑀𝑀 − 𝑚𝑚 ∑ ∑ 𝐾𝐾 𝐾𝐾 − 2 𝑁𝑁 ∑ 𝜎𝜎 ∑ 𝑌𝑌 2 3 = 3 + { 3 } 𝑘𝑘𝑖𝑖𝑗𝑗 𝜎𝜎𝑖𝑖𝑗𝑗 𝑘𝑘𝑖𝑖𝑗𝑗 ( ) Equations (16) and (17) give; = 2 + 2 + 2 (28) 2 2 𝐸𝐸�𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁 � 𝑉𝑉�𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁1� 𝐸𝐸�𝑌𝑌=�1𝑁𝑁𝑁𝑁𝑁𝑁 � ( ) 1 ( ) 𝑁𝑁 𝑁𝑁−𝑛𝑛 𝑁𝑁 𝑁𝑁 3 = + =1 + Combining equations (27) and (28), we have: 𝑛𝑛 𝑛𝑛 ∑𝑖𝑖 𝑖𝑖 𝜎𝜎 𝑁𝑁 𝑁𝑁 𝜎𝜎𝑖𝑖 ( 2)𝜎𝜎= 𝜎𝜎 𝑌𝑌 𝑁𝑁𝑁𝑁𝑁𝑁=1 =1 𝑖𝑖 𝑖𝑖 2𝑖𝑖 (18)𝑖𝑖 1 𝑉𝑉�𝑌𝑌� � 𝑁𝑁 𝑁𝑁 − 𝑛𝑛 𝑛𝑛 𝑛𝑛 ∑ 𝑀𝑀 𝑀𝑀 − 𝑚𝑚 𝑚𝑚𝑖𝑖 ( ) ( 2 + 2) ( 2 + ( 1) =1 =1 2 ( 1) 1 𝑁𝑁𝑛𝑛𝑖𝑖 𝑁𝑁𝑀𝑀𝑖𝑖𝑚𝑚𝑖𝑖𝑗𝑗 𝑀𝑀𝑖𝑖𝐾𝐾𝑖𝑖𝑗𝑗𝐾𝐾𝑖𝑖𝑗𝑗−𝑘𝑘𝑖𝑖𝑗𝑗𝜎𝜎𝑖𝑖𝑗𝑗 𝑘𝑘𝑖𝑖𝑗𝑗 𝑛𝑛 𝐸𝐸 𝑠𝑠 𝑛𝑛 𝑁𝑁 𝑁𝑁−𝑛𝑛 where 𝑁𝑁 𝑁𝑁=1 2+ 2) 2 𝑁𝑁 𝑛𝑛− ∑𝑖𝑖 𝑖𝑖 ∑𝑖𝑖 𝑖𝑖 𝑁𝑁 𝑛𝑛− 𝑛𝑛 =1( ) 𝜎𝜎 𝑌𝑌 − 𝜎𝜎 2 ( ) 2 1 = 𝑁𝑁 (19) 1 2 1 1 2 ( ) 2 ∑ 𝑌𝑌𝑖𝑖1−𝑌𝑌� = + + (29) 𝑖𝑖 2 =1 ( 1) 𝑁𝑁𝑛𝑛𝑖𝑖 1 𝑁𝑁𝜎𝜎=𝑖𝑖1 𝑌𝑌 ( 1) 1 =1( ) 𝑁𝑁− 𝑛𝑛 𝑌𝑌 𝑁𝑁−𝑛𝑛 2 𝑁𝑁− 𝑁𝑁 𝑁𝑁 𝜎𝜎= 𝑀𝑀 𝑖𝑖 𝑌𝑌 𝑖𝑖 (20) Using𝑖𝑖 the𝑖𝑖 fact that: 𝑖𝑖 𝑖𝑖 ∑𝑗𝑗 𝑌𝑌𝑖𝑖𝑖𝑖 −1 𝑁𝑁 ∑ 𝜎𝜎 𝑛𝑛− 𝑁𝑁 �𝑁𝑁− �∑ 𝑌𝑌 − 𝑁𝑁 �� 𝑁𝑁 𝑛𝑛− 𝜎𝜎 𝑀𝑀 𝑖𝑖 2 2 2 1 𝑖𝑖 2 2 2 𝜎𝜎 𝑀𝑀𝑖𝑖− 2 = 𝑁𝑁 = 2 =1( 4 2 )( ) (21) 1 𝐾𝐾𝑖𝑖𝑖𝑖 1 𝑘𝑘𝑖𝑖𝑖𝑖 𝑁𝑁𝑖𝑖 𝑁𝑁𝑖𝑖 =1 𝑌𝑌 We note that𝑖𝑖𝑖𝑖 the first𝑙𝑙 term to the right𝑖𝑖𝑖𝑖𝑖𝑖 of 𝑖𝑖𝑖𝑖the equality in 𝑖𝑖 𝜎𝜎 𝑘𝑘𝑖𝑖𝑖𝑖 ∑ 𝑛𝑛𝑖𝑖 − 𝑛𝑛𝑖𝑖 𝑦𝑦 − 𝑦𝑦� Equation (29) 𝜎𝜎becomes; �� 𝑌𝑌 − � equation (18) is the variance that would be obtained if every 𝑁𝑁 − 𝑖𝑖 𝑁𝑁 1 ( 1) ( ) tertiary unit in a selected secondary unit and every secondary 2 2 2 2 ( ) = 𝑁𝑁 + unit in a selected primary unit was observed, that is, if ’s 1 ( 1) 1 ( 1) 1 =1 𝑁𝑁 − 𝑛𝑛 𝑁𝑁 − 𝑛𝑛 were known for = 1,2, , . The second term contains 𝑖𝑖 𝑖𝑖 𝐸𝐸 𝑠𝑠 �1 𝜎𝜎 2 ( ) 𝜎𝜎 2 − 𝜎𝜎 variance that would be obtained if every tertiary unit in𝑦𝑦 a =𝑁𝑁 𝑖𝑖 =1 +𝑛𝑛 − 𝑁𝑁 1 𝑁𝑁 𝑛𝑛 − (30) 𝑁𝑁 𝑁𝑁−𝑛𝑛 selected secondary𝑖𝑖 unit was⋯ observed,𝑛𝑛 that is, if ’s were Next, we note that;𝑁𝑁 𝑁𝑁 ∑𝑖𝑖 𝜎𝜎𝑖𝑖 𝑛𝑛 𝜎𝜎
𝑦𝑦𝑖𝑖𝑖𝑖 202 L. A. Nafiu et al.: Alternative Estimation Method for a Three-Stage Cluster Sampling in Finite Population
2 2 2 1 ( ) = { ( 𝑛𝑛 )} { 3 } = ( ) =1 𝜎𝜎 2 𝐸𝐸 𝑠𝑠𝑖𝑖 𝐸𝐸 𝐸𝐸 � 𝑠𝑠𝑖𝑖 𝐸𝐸 𝑉𝑉��𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁 � 𝑁𝑁 𝑁𝑁 − 𝑛𝑛 𝑖𝑖 2 + 𝑁𝑁 𝑛𝑛 ( ) = { ( 𝑛𝑛 )} 𝑁𝑁 =1 𝜎𝜎𝑖𝑖 =1 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 � 𝑀𝑀 𝑀𝑀 − 𝑚𝑚 2 𝐸𝐸 𝐸𝐸 � 𝑠𝑠 𝑛𝑛 𝑖𝑖 𝑚𝑚𝑖𝑖 𝑖𝑖 2 + 𝑁𝑁 𝑀𝑀𝑖𝑖 = { ( 𝑛𝑛 | 1)} 𝑖𝑖 𝑖𝑖𝑖𝑖 =1 𝑁𝑁 =1 𝑀𝑀 =1 𝜎𝜎 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 � 2 � 𝑖𝑖 � 𝐾𝐾 �𝐾𝐾 − 𝑘𝑘 � 𝑖𝑖𝑖𝑖 𝐸𝐸 𝐸𝐸 𝑠𝑠 𝑠𝑠 𝑛𝑛 𝑖𝑖 𝑚𝑚 𝑗𝑗 𝑘𝑘 𝑖𝑖 2 = 3 (33) = { 𝑛𝑛 2 ) 𝑖𝑖 =1 𝑀𝑀 That is; 𝑁𝑁𝑃𝑃𝑃𝑃 𝑖𝑖 𝑉𝑉�𝑌𝑌� � 𝐸𝐸 � 𝑖𝑖 2𝜎𝜎 3 = 3 𝑖𝑖 𝑚𝑚 2 = { 𝑁𝑁 ) 2 Hence, ( 3 ) is an𝑁𝑁𝑁𝑁𝑁𝑁 unbiased sample𝑁𝑁𝑁𝑁𝑁𝑁 estimator of the 𝑖𝑖 𝐸𝐸�𝑉𝑉��𝑌𝑌� �� 𝑉𝑉�𝑌𝑌� � =1 𝑀𝑀 proposed estimator ( 3 ) in three-stage cluster sampling � 𝑖𝑖 𝑖𝑖 � � 𝑁𝑁𝑁𝑁𝑁𝑁 2 𝐸𝐸 𝑧𝑧2 𝑖𝑖 𝜎𝜎 2 design. 𝑉𝑉 𝑌𝑌 𝑖𝑖 𝑚𝑚 1 𝑁𝑁𝑁𝑁𝑁𝑁 𝑁𝑁 ( ) 𝑁𝑁 ( ) 𝑌𝑌� = { 2 } This estimator, 3 , is then compared with these seven 𝑛𝑛 𝑁𝑁 =1 𝜎𝜎𝑖𝑖 =1 𝜎𝜎𝑖𝑖 conventional three stage cluster sampling design estimators: 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑁𝑁𝑁𝑁𝑁𝑁 � 𝑀𝑀 𝑀𝑀 − 𝑚𝑚 2 − � 𝑀𝑀 𝑀𝑀 − 𝑚𝑚 2 𝑌𝑌� 1 .. 𝑁𝑁 𝑛𝑛 𝑖𝑖 𝑚𝑚𝑖𝑖 1 𝑛𝑛 𝑖𝑖 𝑚𝑚𝑖𝑖 i. = (34) ( ) ( ) =1 = 𝑁𝑁 𝑁𝑁 𝑛𝑛 𝑌𝑌�𝑖𝑖 𝑖𝑖 𝑖𝑖 𝐻𝐻𝐻𝐻 𝑖𝑖 =1 =1 � 𝑛𝑛 ∑ =𝑃𝑃𝑖𝑖 =1 𝑁𝑁 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝜎𝜎 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝜎𝜎 𝑌𝑌 � � ii. = 0 𝑚𝑚 𝑖𝑖 𝑛𝑛 (35) Therefore; 𝑀𝑀 𝑀𝑀 − 𝑚𝑚 𝑖𝑖 − 𝑀𝑀 𝑀𝑀 − 𝑚𝑚 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑛𝑛 𝑖𝑖 𝑚𝑚 𝑁𝑁 𝑖𝑖 2 𝑚𝑚 ∑𝑗𝑗 ∑=1𝑖𝑖 𝑦𝑦� 1 1 ( 2) ( ) ( 𝐻𝐻𝐻𝐻𝐻𝐻 𝑛𝑛.. = =1 =1 iii. � = ∑𝑖𝑖 𝑀𝑀 𝑖𝑖 (36) 𝜎𝜎 𝑌𝑌 𝑀𝑀 =1 𝑁𝑁 𝑁𝑁 𝑖𝑖 𝑁𝑁 �𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 2 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑛𝑛 𝑌𝑌 𝑛𝑛 ∑ 𝑚𝑚𝑖𝑖 𝑁𝑁 ∑ (31) 𝐸𝐸 𝑠𝑠 𝑀𝑀 𝑀𝑀 − 𝑚𝑚 − 𝑀𝑀 𝑀𝑀 − iv. 𝐻𝐻𝐻𝐻 = 𝑖𝑖 =1 𝑖𝑖 .. (37) 𝑌𝑌� 𝑛𝑛 ∑ 𝜋𝜋𝑍𝑍𝑔𝑔 𝑛𝑛 𝑚𝑚𝑖𝑖𝜎𝜎𝑖𝑖 𝑚𝑚𝑖𝑖 𝑅𝑅𝑅𝑅𝑅𝑅 𝑔𝑔 𝑔𝑔 𝑔𝑔 Also; 𝑌𝑌� ∑ 𝑧𝑧𝑔𝑔 𝑀𝑀 𝑌𝑌� 2 2 v. = =1 =1 =1 (38) ( ) = | 1, 2 𝑖𝑖𝑖𝑖 𝑛𝑛 𝑚𝑚𝑖𝑖 𝑘𝑘 vi. = (39) 𝑖𝑖𝑖𝑖 2 𝑖𝑖𝑖𝑖 �𝐶𝐶 =𝑖𝑖 1 =𝑗𝑗 1 𝑢𝑢=1 𝑖𝑖𝑖𝑖𝑖𝑖 � � � ��� 𝑌𝑌 ∑ ∑ ∑ 𝑖𝑖𝑖𝑖 𝑦𝑦 𝐸𝐸 𝑠𝑠 𝐸𝐸 𝐸𝐸𝑘𝑘𝑖𝑖𝑖𝑖𝐸𝐸 𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑛𝑛 𝑚𝑚𝑖𝑖 𝑘𝑘 = { [ 2 ( | 1)]} and 𝑖𝑖𝑖𝑖 �𝑇𝑇 𝑖𝑖 𝑗𝑗 𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖 𝐾𝐾 =1 𝑌𝑌 ∑ ∑ ∑ 𝑦𝑦 � 𝑖𝑖𝑖𝑖𝑖𝑖 vii. = =1 =1 =1 (40) 𝐸𝐸 𝐸𝐸 𝑖𝑖𝑖𝑖 𝐸𝐸 𝑦𝑦 𝑠𝑠 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 2 𝑘𝑘 𝑙𝑙 2 𝑁𝑁 𝑛𝑛 𝑀𝑀𝑖𝑖 𝑚𝑚𝑖𝑖 𝐾𝐾 𝑘𝑘 𝑚𝑚𝑖𝑖 𝑘𝑘𝑖𝑖𝑖𝑖 𝑂𝑂 𝑖𝑖 𝑖𝑖 𝑗𝑗 𝑖𝑖𝑖𝑖 𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖 = { 2 [ 2 ( | 1)]} 𝑌𝑌� 𝑛𝑛 ∑ 𝑚𝑚 ∑ 𝑘𝑘 ∑ 𝑦𝑦 𝑖𝑖𝑖𝑖 𝑀𝑀𝑖𝑖 =1 𝐾𝐾 =1 𝐸𝐸 𝐸𝐸 � � 𝐸𝐸 𝑦𝑦𝑖𝑖𝑖𝑖𝑖𝑖 𝑠𝑠 𝑚𝑚𝑖𝑖 𝑗𝑗 2 𝑘𝑘𝑖𝑖𝑖𝑖 𝑙𝑙 2 5. Data Used for this Study 𝑀𝑀𝑖𝑖 2 = { 2 [ 2 ]} 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑀𝑀 =1 𝐾𝐾 There are eight (8) categories of data sets used in this 𝐸𝐸 𝐸𝐸 � 𝜎𝜎𝑖𝑖𝑖𝑖 𝑚𝑚𝑖𝑖 2 𝑗𝑗 𝑘𝑘𝑖𝑖𝑖𝑖 2 paper. The first four (4) data sets were obtained 𝑀𝑀𝑖𝑖 2 = { 𝑁𝑁 2 [ 2 ]} fro m[1],[2],[3] and[4] respectively. The second four (4) data 𝑖𝑖𝑖𝑖 =1 𝑖𝑖 =1 𝐾𝐾 sets used are of secondary type and were collected from[16] 𝑀𝑀 𝑖𝑖𝑖𝑖 𝐸𝐸 � 𝐸𝐸 � 𝜎𝜎 and[17] where we constructed a samp ling frame fro m all 𝑖𝑖 2𝑚𝑚𝑖𝑖 𝑗𝑗 𝑘𝑘𝑖𝑖𝑖𝑖 2 𝑀𝑀𝑖𝑖 diabetic patients with chronic eye disease (Glaucoma and = { 𝑁𝑁 2 } 𝑖𝑖𝑖𝑖 =1 𝑖𝑖 =1 𝜎𝜎 Retinopathy) in the twenty-five (25) Local Government 𝑖𝑖 𝑀𝑀 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝐸𝐸 �2 𝑧𝑧 � 𝐾𝐾 �𝐾𝐾 − 𝑘𝑘 � 2 Areas of the state between 2005 and 2008. 𝑖𝑖 𝑚𝑚𝑖𝑖 𝑗𝑗 𝑘𝑘𝑖𝑖𝑖𝑖 = { 2 =1 =1 } 𝜎𝜎𝑖𝑖𝑖𝑖 𝑛𝑛 𝑁𝑁 𝑁𝑁 𝑀𝑀𝑖𝑖 𝑀𝑀𝑖𝑖 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑁𝑁 𝑛𝑛 ∑ 𝑚𝑚𝑖𝑖 ∑𝑖𝑖 𝐾𝐾 �𝐾𝐾 − 𝑘𝑘 �2𝑘𝑘𝑖𝑖𝑖𝑖 = 𝑁𝑁 𝑀𝑀𝑖𝑖 6. Results 𝑖𝑖𝑖𝑖 𝑁𝑁 =1 𝑀𝑀𝑖𝑖 =1 𝜎𝜎 � � 𝐾𝐾𝑖𝑖𝑖𝑖 �𝐾𝐾𝑖𝑖𝑖𝑖 − 𝑘𝑘𝑖𝑖𝑖𝑖 � Therefore; 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑛𝑛 𝑚𝑚 𝑘𝑘 2 The estimates obtained with the aid of software developed 2 using Visual Basic C++ Programming Language[18] are = =1 =1 (32) 𝜎𝜎𝑖𝑖𝑖𝑖 𝑁𝑁 𝑁𝑁 𝑀𝑀𝑖𝑖 𝑀𝑀𝑖𝑖 given in tables 1 - 12 for the illustrated and the real-life data 𝑖𝑖𝑖𝑖 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 Combining𝐸𝐸�𝑠𝑠 equations� 𝑛𝑛 ∑ (30),𝑚𝑚𝑖𝑖 ∑ (31)𝑖𝑖 𝐾𝐾and� 𝐾𝐾(32),− we𝑘𝑘 �have;𝑘𝑘𝑖𝑖𝑖𝑖 respectively.
American Journal of Mathematics and Statistics 2012, 2(6): 199-205 203
Ta b l e 1 . Estimat ed Populat ion Totals using Illustrat ed Dat a Ta b l e 5 . Variances of the Estimat ed Populat ion Tot als using Illustrat ed Dat a Cases Es t ima to Estimator I II III IV Case I Case II Case III Case IV r 400 98,983 36 13,898 584.3718 163,955.9954 3.0753 2,012.5838 𝑌𝑌�𝐻𝐻𝐻𝐻 440 111,310 22 14,000 𝑉𝑉�(�𝑌𝑌�𝐻𝐻𝐻𝐻�) 498.0637 146,797.7757 2.9402 1,958.2391 𝑌𝑌�𝐻𝐻𝐻𝐻𝐻𝐻 479 139,527 27 14,791 𝑉𝑉� 𝑌𝑌�𝐻𝐻𝐻𝐻𝐻𝐻 444.7838 131,942.6350 1.6489 1,861.7476 𝑌𝑌�𝐻𝐻𝐻𝐻 437 98,830 31 15,214 𝑉𝑉��𝑌𝑌�𝐻𝐻𝐻𝐻� 425.8846 109,764.4333 1.6226 1,844.8793 𝑌𝑌�𝑅𝑅𝑅𝑅 𝑅𝑅 385 131,675 39 13,963 𝑉𝑉��𝑌𝑌�𝑅𝑅𝑅𝑅𝑅𝑅 � 418.4156 107,030.5504 1.6201 1,842.4048 𝑌𝑌�𝐶𝐶 456 102,635 34 14,576 𝑉𝑉��𝑌𝑌�𝐶𝐶� 315.7266 96,800.8501 1.5117 1,834.4763 𝑌𝑌�𝑇𝑇 397 141,194 24 13,950 𝑉𝑉��𝑌𝑌�𝑇𝑇� 296.8283 94,624.7532 1.1674 1,810.5586 3𝑌𝑌�𝑂𝑂 492 99,136 42 15,016 𝑉𝑉�(�𝑌𝑌3�𝑂𝑂� ) 274.6806 91,963.5764 1.1105 1,619.5559 � 𝑁𝑁𝑁𝑁𝑁𝑁 𝑌𝑌Ta b l e 2 . Estimat ed Populat ion Totals using Real Life Dat a Ta b𝑉𝑉� l e 𝑌𝑌6�. 𝑁𝑁𝑁𝑁𝑁𝑁 Variances of the Est imated Populat ion Tot als using Real Life Data Estimator POP1 POP2 POP3 POP4 Estimator POP1 POP2 POP3 POP4 26,022 26,541 28,428 29,356 17,464 16,636 15,360 15,146
�𝐻𝐻𝐻𝐻 24,355 25,019 25,162 28,610 𝐻𝐻𝐻𝐻 𝑌𝑌 𝑉𝑉�(�𝑌𝑌� �) 15,714 16,286 14,139 15,100 �𝐻𝐻𝐻𝐻𝐻𝐻 25,514 27,197 28,731 29,096 𝐻𝐻𝐻𝐻𝐻𝐻 𝑌𝑌 𝑉𝑉� 𝑌𝑌� 13,419 14,626 11,493 13,315 �𝐻𝐻𝐻𝐻 26,043 26,428 27,301 27,451 𝐻𝐻𝐻𝐻 𝑌𝑌 𝑉𝑉��𝑌𝑌� � 11,684 11,788 11,398 12,396 �𝑅𝑅𝑅𝑅 𝑅𝑅 24,420 25,321 26,851 28,365 𝑅𝑅𝑅𝑅𝑅𝑅 𝑌𝑌 𝑉𝑉��𝑌𝑌� � 9,985 11,532 10,938 12,239 �𝐶𝐶 25,804 27,197 27,609 29,472 𝐶𝐶 𝑌𝑌 𝑉𝑉��𝑌𝑌� � 9,749 11,441 10,532 12,069 �𝑇𝑇 27,204 28,124 28,631 29,251 𝑇𝑇 𝑌𝑌 𝑉𝑉��𝑌𝑌� � 9,568 10,507 10,330 11,087 �𝑂𝑂 𝑂𝑂 3𝑌𝑌 26,151 26,625 27,511 28,090 𝑉𝑉(��3𝑌𝑌� � ) 9,118 9,726 9,884 10,612
𝑁𝑁𝑁𝑁𝑁𝑁 𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁 𝑉𝑉� 𝑌𝑌� Ta b l e 3 . Biases of Estimat ed Populat ion Totals using Illustrat ed Dat a Ta b l e 7 . St andard Error for Estimat ed Populat ion Tot al using Illustrated Dat a Estimator Cases I Cases II Cases III Cases IV Estimator Case I Case II Case III Case IV 29 516 2 446 24.1738 404.9148 1.7537 44.8618 �𝐻𝐻𝐻𝐻 19 337 8 540 𝑌𝑌 𝑌𝑌�𝐻𝐻𝐻𝐻 22.3173 383.1420 1.7147 44.2520 𝑌𝑌�𝐻𝐻𝐻𝐻𝐻𝐻 29 427 3 133 𝑌𝑌�𝐻𝐻𝐻𝐻𝐻𝐻 21.0899 363.2391 1.2841 43.1480 �𝐻𝐻𝐻𝐻 15 364 2 461 𝑌𝑌 𝑌𝑌�𝐻𝐻𝐻𝐻 20.6370 331.3072 1.2738 42.9521 𝑌𝑌�𝑅𝑅𝑅𝑅 𝑅𝑅 25 309 1 265 𝑌𝑌�𝑅𝑅𝑅𝑅 𝑅𝑅 20.4552 327.1552 1.2728 42.9232 𝑌𝑌�𝐶𝐶 33 463 9 369 𝑌𝑌�𝐶𝐶 17.7687 311.1283 1.2295 42.8308 𝑌𝑌�𝑇𝑇 13 261 3 476 𝑌𝑌�𝑇𝑇 17.2287 307.6114 1.0805 42.5507 3𝑌𝑌�𝑂𝑂 11 219 1 112 3𝑌𝑌�𝑂𝑂 16.5785 302.2597 1.0538 40.2437 𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁 Ta b l e 8𝑌𝑌�. 𝑁𝑁𝑁𝑁𝑁𝑁 St andard Error for Estimat ed Populat ion Tot al using Real Life Ta b l e 4 . Biases of Estimat ed Populat ion Totals using Real Life Dat a Dat a Estimator POP1 POP2 POP3 POP4 Estimator POP1 POP2 POP3 POP4 196 136 136 125 132.1502 128.9710 130.6399 123.0702 𝑌𝑌�𝐻𝐻𝐻𝐻 147 176 127 168 125.3573 127.6163 125.3413 122.8823 𝑌𝑌�𝐻𝐻𝐻𝐻 𝑌𝑌�𝐻𝐻𝐻𝐻𝐻𝐻 122 117 114 128 115.8417 120.9390 113.0051 115.3810 𝑌𝑌�𝐻𝐻𝐻𝐻𝐻𝐻 𝑌𝑌�𝐻𝐻𝐻𝐻 155 153 134 118 108.0937 108.5724 112.5340 111.3351 𝑌𝑌�𝐻𝐻𝐻𝐻 𝑌𝑌�𝑅𝑅𝑅𝑅 𝑅𝑅 127 146 151 120 99.9287 107.3872 110.2437 110.6305 𝑌𝑌�𝑅𝑅𝑅𝑅 𝑅𝑅 𝑌𝑌�𝐶𝐶 164 156 124 195 98.7379 106.9641 108.1752 109.8593 𝑌𝑌�𝐶𝐶 𝑌𝑌�𝑇𝑇 137 137 137 158 97.8150 102.5056 107.1344 105.2964 𝑌𝑌�𝑇𝑇 3𝑌𝑌�𝑂𝑂 112 104 103 107 95.4893 98.6229 104.7944 103.0137 3𝑌𝑌�𝑂𝑂 𝑁𝑁𝑁𝑁𝑁𝑁 𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁 𝑌𝑌� 204 L. A. Nafiu et al.: Alternative Estimation Method for a Three-Stage Cluster Sampling in Finite Population
Ta b l e 9 . 95% Confident Int ervals for Estimated Populat ion using Illustrat ed Dat a Estimator Case I Case II Case III Case IV (353,447) (98189,99777) (33,39) (13810,13986) (396,484) (110559,112061) (19,25) (13913,14087) 𝑌𝑌�𝐻𝐻𝐻𝐻 (438,520) (138815,140239) (24,30) (14706,14876) 𝑌𝑌�𝐻𝐻𝐻𝐻𝐻𝐻 (397,477) (98181,99479) (29,33) (15130,15298) 𝑌𝑌�𝐻𝐻𝐻𝐻 (345,425) (131034,132316) (37,41) (13879,14047) 𝑌𝑌�𝑅𝑅𝑅𝑅𝑅𝑅 (421,491) (102025,103245) (32,36) (14492,14660) 𝑌𝑌�𝐶𝐶 (363,431) (140591,141797) (22,26) (13867,14033) 𝑌𝑌�𝑇𝑇 (460,524) (98542,99730) (40,44) (14939,15095) 3𝑌𝑌�𝑂𝑂 𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁 Ta b l e 1 0 . 95 % Co nfident Int ervals for Estimat ed Populat ion Totals using Real Life Dat a Estimator Population 1 Population 2 Population 3 Population 4 (25760,26290) (26290,26790) (28170,28680) (29110,29600) (24110,24600) (24770,25270) (24920,25410) (28370,28850) 𝑌𝑌�𝐻𝐻𝐻𝐻 (25290,25740) (26960,27430) (28510,28950) (26870,27320) 𝑌𝑌�𝐻𝐻𝐻𝐻𝐻𝐻 (25830,26250) (26220,26640) (27080,27520) (27230,27670) 𝑌𝑌�𝐻𝐻𝐻𝐻 (24220,24620) (25110,25530) (26630,27070) (28150,28580) 𝑌𝑌�𝑅𝑅𝑅𝑅𝑅𝑅 (25610,26000) (26990,27410) (27400,27820) (29260,29690) 𝑌𝑌�𝐶𝐶 (27010,27400) (27920,28320) (28420,28840) (29040,29460) 𝑌𝑌�𝑇𝑇 (25960,26340) (26410,26800) (27310,27720) (27890,28290) 3𝑌𝑌�𝑂𝑂
� 𝑁𝑁𝑁𝑁𝑁𝑁 Ta b l e 11 . Coefficient𝑌𝑌 of Variat ion for Estimated Populat ion Tot als using Illustrated Data 7. Discussion of Results Estimator Case I Case II Case III Case IV The estimation methods given in equation (10) was 6.04% 0.41% 4.87% 0.32% applied to four different illustrated data (Cases I – IV) and 5.07% 0.34% 7.79% 0.32% 𝑌𝑌�𝐻𝐻𝐻𝐻 four real life data (Populations 1 - 4). The population totals obtained for illustrated data are given in table 1 while the �𝐻𝐻𝐻𝐻𝐻𝐻 4.40% 0.26% 4.76% 0.29% 𝑌𝑌 population totals obtained for real life data are given in table 4.72% 0.34% 4.11% 0.28% 𝑌𝑌�𝐻𝐻𝐻𝐻 2. Table 3 give the biases of the estimated population totals 5.31% 0.25% 3.26% 0.31% 𝑌𝑌�𝑅𝑅𝑅𝑅 𝑅𝑅 for illustrated data for our own estimator as 11, 219, 1, and 112 fo r cases I – IV respectively while table 4 gives that of 𝐶𝐶 3.90% 0.30% 3.62% 0.29% 𝑌𝑌� the four life data sets as 112, 104, 103, and 107 respectively. 4.34% 0.22% 4.50% 0.31% 𝑌𝑌�𝑇𝑇 This implies that our own estimator has the least biases using 3.98% 0.31% 2.51% 0.27% both data sets. Table 5 shows the variances obtained using 3𝑌𝑌�𝑂𝑂 illustrated data for our own estimator as 274.6806, Ta b l e 1 2 . 𝑁𝑁𝑁𝑁𝑁𝑁Coefficient of Variation for Estimat ed Populat ion Tot als using Real Life𝑌𝑌� Dat a 91963.5764, 1.1105 and 1619.5559 for cases I – IV Populations respectively while table 6 shows that of life data sets as Estimator 1 2 3 4 9118.2037, 9726.4809, 9883.6215 and 10611.8216 0.44% 0.43% 0.42% 0.42% respectively meaning that our own estimator has the least variances using both data sets. Table 7 shows the obtained 0.51% 0.51% 0.44% 0.43% 𝑌𝑌�𝐻𝐻𝐻𝐻 standard errors for the estimated population totals using 𝑌𝑌�𝐻𝐻𝐻𝐻𝐻𝐻 0.45% 0.44% 0.37% 0.34% illustrated data for our own estimator as 16.5785, 302.2597, 1.0538 and 40.2437 for cases I – IV respectively while table 𝑌𝑌�𝐻𝐻𝐻𝐻 0.42% 0.41% 0.37% 0.41% 8 shows that of life data sets as 95.4893, 98.6229, 104.7944 0.37% 0.42% 0.38% 0.33% 𝑌𝑌�𝑅𝑅𝑅𝑅 𝑅𝑅 and 103.0137 respectively meaning that our own estimator 𝑌𝑌�𝐶𝐶 0.38% 0.33% 0.36% 0.37% have the least standard errors using both data sets. The confidence intervals of the estimated populations in 𝑌𝑌�𝑇𝑇 0.36% 0.36% 0.36% 0.36% table 1 are given in table 9 for α=, 5%. The confidence 3𝑌𝑌�𝑂𝑂 0.34% 0.33% 0.35% 0.37% intervals of the estimated populations in table 2 are given in 𝑌𝑌� 𝑁𝑁𝑁𝑁𝑁𝑁
American Journal of Mathematics and Statistics 2012, 2(6): 199-205 205
table 10 for α =5 % which shows that all the estimated [5] Kish, L., Survey Sampling, John Wiley and Sons, New York, population totals fall within the computed intervals as 1967. expected. For our own estimator, table 11 gives the [6] Adams, J. L., S. L. Wickstrom, M. J. Burgess, P. P. Lee, and J. coefficients of variations for the estimated population totals J. Escarce, “Sampling Patients within Physician Practices and using illustrated data as 3.98%, 0.31%, 2.51% and 0.27% for Health Plans: M ultistage Cluster Samples in Health Services cases I – IV respectively while table 12 gives that of life data Research”, The Global Journal for Improving Health Care Delivery and Policy, vol. 38: 1625-1640, 2003. sets as 0.34%, 0.33%, 0.35% and 0.37% respectively which means that our newly proposed three stage cluster estimator [7] Kuk, A., “Estimation of Distribution Functions and Medians has the least coefficient of variation, hence it is preferred. under Sampling with Unequal Probabilities”, Biometrika, vol. 75, 97-103, 1988.
[8] Goldstein, H., Multilevel Statistical Models, Halstead Press, 8. Conclusions New York, 1995. The alternative estimation method of population allows [9] Tate, J.E. and M.G. Hudgens, “Estimating Population Size the use of certain number of visits to the venues (hospitals) with Two-stage and Three-stage sampling designs”, American Journal of Epidemiology,vol. 165 no.11, within the clusters (cities) and a more precise (minimum 1314-1320, 2007. mean square error) estimate was obtained and the estimates presented indicate that substantial reduction in the variances [10] Kalton, G., Introduction to Survey Sampling, Sage was obtained through the use of newly proposed estimator. Publications, Thousand Oaks, C.A., 1983. We also observed that irrespective of the data considered, the [11] Henry, G. T., Practical Sampling, Thousand Oaks, C.A.: variance of newly proposed estimator is always less than Sage Publications, 1990. those of already existing estimators in three-stage cluster [12] Fink, A., How To Sample In Survey, C.A. Sage Publications, sampling designs. The newly proposed estimator ( 3 ) is Thousand Oaks, 2002. preferred to the already existing estimators considered in 𝑁𝑁𝑁𝑁𝑁𝑁 this study and is therefore recommended. 𝑌𝑌� [13] Kendall, M. G. and A. S. Stuart, The Advanced Theory of Statistics, Hafner Publishing Company, New York, 1968.
[14] Thompson, S. K., Sampling, John Wiley and Sons, New York, 1992. REFERENCES [15] Nafiu, L. A., An Alternate Estimation Method for Multistage Cluster Sampling in Finite Population, Unpublished Ph.D [1] Horvitz, D. G. and D. J. Thompson, “A Generalization of Thesis, University of Ilorin, Ilorin, Nigeria, 2012. Sampling without Replacement from a Finite Universe”, Journal of American Statistical Association, vol. 47 663-685, [16] Niger State, Niger State Statistical Year Book, Niger Press 1952. Printing & Publishing, Nigeria, 2009.
[2] Raj, D., The Design of Sample Survey, M cGraw-Hill, Inc., [17] National Bureau of Statistics, Directory of Health New York, USA, 1972. Establishments in Nigeria, Abuja Printing Press, Nigeria, 2007. [3] Cochran, W.G., Sampling Techniques, Third Edition, John Wiley and Sons , New York, 1977. [18] Hubbard, J. R., Programming with C++, Second Edition. Schaum’s Outlines, Tata McGraw-Hill Publishing Company [4] Okafor, F., Sample Survey Theory with Applications, Limited, New Delhi, 2000. Afro-Orbis Publications, Nigeria, 2002.