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1、Research article Determining the Efficiency of Ranked Set Sampling techniques over Cluster Sampling Using Water Pollution Data Badmus N. IdowuDepartment of Statistics, Abraham Adesanya Polytechnic Ijebu Igbo, Ogun SIkegwu, Emmanuel M.Department of Statistics,Yaba College of Technology, Yaba, LagosOd
2、etunde, O.S.Department of Mathematical Science, Olabisi Onabanjo University, Ago-Iwoye, NigeriaAbstractThis paper review the ultimate goal of the researcher in sampling design techniques to arrive at the best possible design and knowledge of efficiency of various designs will help in selecting a par
3、ticular design from among different designs. The comparison of the variance were made between the ranked set sampling design and cluster sampling design, therefore the variance of and .(C) is established to known the most efficient one. It is established in this paper that both the variance and the
4、relative efficiency of ranked set sampling are smaller than that of cluster sampling. Therefore, ranked set sampling is more efficient than cluster sampling design. Copyright WJSTR, all rights reserved. Key words: Cluster Sampling, Efficiency, Pollution, Ranked Set Sampling, 1. IntroductionRanked se
5、t sampling (RSS) is a procedure which combines random sampling and the ability to rank the sampling units, with respect to the characteristic of interest, without making the actual measurements. See Ref: 6. The RSS procedure can be described as follows: Randomly Sample a group of sampling units from
6、 the target population. Then, randomly partition the group into disjoint subsets each having a pre-assigned size r. In most practical situations, the size r will be 2, 3, or 4. Rank the elements in each subset by a suitable method of ranking such as prior information, visual inspection or by the sub
7、ject-matter experimenter himself etc. Then the ith order statistic from the i-th subset, X(i), i = 1,., r , will be quantified (actual measurement). Therefore, X1(1), X2(2), Xr(r) constitutes the RSS. This represent one cycle. The whole procedure can be repeated m-times as needed, to get a RSS of th
8、e size n = mr for the theoretical aspect of RRS see Refs: 4, 6, 7. The research in the literature has been focusing on the comparison between procedures based on RRS and their counter parts based on SRS.The earliest such research is on the relative precision of ranked set sample means to simple rand
9、om sample means as estimators of the population mean see Refs: 3 . The optimal ranked-set sampling scheme for inference on population quantiles see Ref 2 where he has derive the asymptotic properties of the unbalanced ranked set sample quantiles for any unbalanced ranked-set sampling scheme. see Ref
10、 7 that considered both maximum likelihood estimates and best linear unbiased estimates for the parameters of the underlying parametric family. They developed methodology for determining optimal unbalanced schemes according to certain optimality criteria based on the asymptotic variance-covariance m
11、atrix of the estimates when is large.The introduction of the matched pairs sign test using bivariate ranked-set sampling, where numerical comparisons between the performance of the BVRSS sign test and the performing of the BVSRS sign test via Pitmans asymptotic efficiency and asymptotic power invest
12、igated see Ref: 9 In this paper, we present Ranked Set Sampling variance with its relative efficiency and Clusters Sampling variance with its relative efficiency. The paper is arranged as follows: In section 2, we give the mathematical expressions of both techniques. The comparison of RSS and Cluste
13、rs sampling variance through illustrative example is presented in section 3 using pollution data, the discussion of the results is in section 4 and conclusion is given in section 5. 2.1Ranked Set Sampling Design:Ranked set sampling is a methodology designed for the cases where sampling is complex an
14、d expensive so that small, but well spread-out samples are desired. According to the environmental sampling module 15 by Jorgen Lauridsen the ranked set sample mean is defined as: 1the ranked set sample mean is an unbiased estimate for . Using the theory for order statistics, the reduced order stati
15、stics 2where, and are the mean and standard deviation for Y. Each T(i) has mean and variance yi and qi and 3Then, 4Now, 5Also, the relative efficiency of and is given by 62.2Cluster Sampling DesignWe assume that the population consist of N clusters of M elements each and n clusters are selected from
16、 N clusters by simple random sampling without replacement:Denote by = value of the characteristic under study for the jth element, (j = 1,2N) and in the ith cluster (i = 1,2,N). 1 2 3 4It is clear that . This is so since the cluster are of equal size.Clearly is an unbiased estimate of and its varian
17、ce is given by 5where, It then follows that the efficiency of a cluster as the unit of sampling compared with that of an element is given by 6where, 3.0 Comparison of Ranked Set Sampling and Cluster Sampling using Pollution data3.1 Procedure on Ranked Set SamplingIn order to plan an RSS design, we m
18、ust therefore choose a set size that is typically small, around two, three or four, to minimize ranking error. Call the set size M, where M is the number of sample units allocated to each set. The steps are as follows:Step 1: Randomly select M2 sample units from the population.Step 2: Allocate the M
19、2 selected units as randomly as possible into M sets, each of size MStep 3: Without yet knowing any values for the variable of interest, rank the units with each set based on a perception of relative values for this variables. This may be based on personal judgement or done with measurements of a co
20、variate that is correlated with the variable of interest.Step 4: Choose a sample for actual analysis for by including the smallest ranked unit in the first set then the second smallest ranked unit in the second set, continuing in this fable ion until the largest ranked unit is selected in the least
21、set.Step 5: Repeat step 1 through step 4 for r cycles until the desired sample, size n = mr, is obtained for analysis for example, m = 2 and r = 5.Table 1: The computation analysis of 10 sample units of RSSCyclesM1M2173,42076,84075,130321466970135848200288,44099,15093,7951339157430573520503112,42012
22、0,660116,5403521176260339488004128,280128,290128,2855053006140505128,310188,200158,2551.021201197E101793406050Total5308076131405720052.044681877E101890555150The table above shows that the computation analysis of 10 sample units are actually analyzed to obtain ranked set sampling mean and its varianc
23、e. Therefore, also its sampling variance is given by= 2.034517817E16 3.2Cluster Sampling DesignCluster sampling is one in which groups (called clusters) of the numbers of the population are sample as units. Unlike stratified sampling the groups (cluster) may not necessary be homogenous; it could be
24、on the basis of geographical proximity. The clusters are then selected using a simple random sampling method. All elements in the selected groups are investigated.For example m = 2 and cluster = 5Table 2: The 5 clusters each consist 2 units and 10 elementsSample unitsClusterM1M2176,8407342075130288,
25、44099150937953120,6601124201165404128,2801282901282855128310188,200158255572005The table above, shows the 5 clusters each consist 2 units and 10 elements. This is actually analysed to obtained the cluster mean and its variance. We have,= 114401.Since is an unbiased estimator of and its variance is g
26、iven by= 10756138.873.4Efficiency of ranked set sampling R.E = 3.420773.5Efficiency of cluster samplingThe estimator of population mean of equation (4) is made up of element. The relative efficiency of with respect to base on a sample of nm elements drawn by SRS without replacement from cluster in t
27、he population and is given by = 232.7175758 232.724.0 DiscussionThe ranked set sampling (RSS) estimator of a population mean is an unbiased and at least as precise as the cluster sampling estimator. In section 3, where comparison were made the results shown in table 1 & 2 that the variance of ranked
28、 set sampling is less than the variance of cluster sampling with the same number of quantifications. Also the relative efficiency of ranked set sampling is smaller than the relative efficiency of cluster sampling. Therefore, the smaller the variance of a sampling technique the more efficient it is.
29、From the results obtained, this established the fact that RSS is more efficient and precise than the cluster sampling. 5.0ConclusionWith the results from the computations, this paper reveal that the variance of RSS is smaller than the variances of cluster sampling, this implies that RSS is more effi
30、cient than cluster sampling ( is more efficient than ). This encouraging outcome occurred because results obtained through RSS are likely to be regularly spaced than those obtained through cluster sampling, and therefore are more representative of the population, then regardless of ranking errors, t
31、he RSS estimator of a mean is unbiased and at least as precise as the cluster sampling estimator with some number of quantification. In addition, the computation of RSS is simple and easy because we have control over which individuals of the population enter the sample.References1Barnett, V. (2004).
32、 Environmental Statistics, Published by John Willey and Sons Ltd. Pp 199-1212Chen, Z.and Bai, Z.H. (1998). The Optimal ranked-set sampling Scheme for Parametric families Sanya A. Accepted.3Chen, Z. (2001). On Optimal ranked-set Sample Scheme for Inference on Population J. Statistical.4Dell, F.R and
33、Clutter, J.L (1972). Ranked set Sampling Theory with order Statistics Background. Biometrika 28, 545-5555Jorgen, Lauridsen (2005). Environmental Sampling http:/Statmaster. du.dk/courses/st 118/ module 15/index html.6McIntyre, G.A. (1952). A method of Unbiased Selective Sampling, using ranked-set,Aus
34、tral J. Afr. Res; 3, 385-390.7 Samawi, H.M, Al-Saheh M.F (2004) On bivariate rankedset sampling for distribution and qunatile estimation and qunatile interval estimation using ratio estimator. Common Statistic. Theory and methods, 33 (8), 1801-1819.8Samawi, H.M; Al-Salah, M.F. and Al-Saidy O. (2008)
35、. The Matched Pairs Sign Test using Bivariate Ranked-set Sampling. Georgia Sourthern University,Technical Report series, pp 1-13.9Sukhatme, P.V and Sukhatme, B.N (1954). Sampling Theory of Surveys with Applications, Published by P.S Jayasinghe pp 222-22710Takawasi, K. and Wakimoto, K. (1968). On Unbiased Estimates of Population mean based on the Sample Stratified by means of ordering. Ann.ihst. Statist. Math. 30, 814-824