ISSN 684-8403 Journal of Statistics Volume 3, 06. pp. 4-57 Comparison of Estimators in Case of Lo Correlation in Muhammad Shahad Chaudhr and Muhammad Hanif Abstract In this paper, to Regression-Cum-Eponential Estimators have been proposed to estimate the population mean using population mean of single auiliar variable in. The epressions for the Mean Square Error and Bias of the proposed Estimators have been derived. A simulation stud has been carried out to demonstrate and compare the efficiencies and precisions of the Estimators. The proposed Estimators have been compared ith Ratio, Regression, Eponential Ratio Estimators in usual sampling, the Hansen-Hurit and the Ratio Estimators in hen there is lo positive correlation beteen stud variable and auiliar variable. Keords Transformed population, Epected final sample sie, Within netork variances, Estimated relative bias, Estimated percentage relative efficienc. Introduction In Surve Sampling, to assess the thickness of entities that are clustered has been a main problem. Eamples of such clustered population comprise: plants and animals of rare and ding out species, fisheries, drug users, HIV and AIDS patients, etc. The Conventional Random Sampling Designs ma be useless and often fail to give samples ith meaningful information for such population. Government College Universit, Lahore, Pakistan Email: almoeed@hotmail.com National College of Business Administration and Economics, Lahore, Pakistan Email: drmianhanif@gmail.com
4 Muhammad Shahad Chaudhr and Muhammad Hanif The (ACS) method is suitable for the rare and clustered population. In ACS an initial sample is selected b a Conventional Sampling Design then the surrounding area of each unit selected in the first sample is measured. A condition C is defined to take in a unit in the sample. All adjacent units is added and investigated if the predefined condition C (usuall C > 0, here Y is the stud variable) is fulfilled and this procedure keep on. This process stops hen a ne unit does not meet the condition. All the units studied (including the initial sample) composed the final sample. The set consisting of those units that met the condition is knon as a netork. The units that do not satisf the condition are knon as edge units. A cluster is a combination of netork units ith connected edge units. Auiliar information s are used to increase the precision of the Estimators of the population mean of Y. Surve statisticians frequentl eercise auiliar information to estimate population parameters; mean and variance. To reduce the error in estimates surveors are alas probing for efficient estimation methods. The development goes on in various forms of Estimators. The available Estimators, in the case of ACS are relativel simple and most of these are based upon information of single auiliar variable. Thompson (990) first proposed the idea of the ACS scheme and introduced modified Hansen-Hurit (943) and Horvit-Thompson (95) tpe Estimators and Rao-Blackell versions of Estimators. As an indispensable issue related to sampling technique, the sampling efficienc as eamined ith eamples. This article launched a variet of subsequent research efforts on ACS. Hoever, ho those adaptive design factors including the predefined condition (or magnitude of the critical value), the definition of the neighborhood affects the efficienc of ACS in comparison ith the non-adaptive design (simple random sampling), and possible challenges arising from case to case are not concretel touched. Drver (003) found that ACS performs ell in a uni-variate setting. The efficienc of ACS depends on the relationship of the variables ith one another in a multivariate setting. The simulation on real data of blue-inged and red-inged results shos that Horvit-Thompson Tpe Estimator as the most efficient Estimator using the condition of one tpe of duck to estimate that tpe of duck. The Simple Random Sample can be more efficient than an adaptive one hen the prediction as on one tpe of duck and the condition as on another tpe of duck. The ACS is more efficient than SRS hen the condition as a function of
Comparison of Estimators in Case of Lo Correlation in to tpes of ducks. For highl correlated variables the ACS performs ell for the parameters of interest. Chao (004) proposed the Ratio Estimator in ACS and shoed that it produces better estimation results than the original Estimator of, and a Ratio Estimator under a comparable Conventional sampling design. This article is meant to be an initial investigation of the utiliation of the auiliar information in ACS. Drver and Chao (007) discussed the Classical Ratio Estimator in ACS and proposed to ne Ratio Estimators under ACS, one of hich is unbiased for ACS designs. The result shos that the proposed Estimators can be considered as a robust alternative of the conventional ratio Estimator, especiall hen the correlation beteen the variable of interest and the auiliar variable is not high enough for conventional ratio Estimator to have satisfactor performance.. Process: Consider a finite population of N units is labelled,,3,,n and denoted as u = {u,u,,u N }. Consider a small initial sample s o of sie n ith n < N hich is selected b simple random sample ithout replacement (SRSWOR). The first sample is chosen b traditional sampling process in an ACS procedure and then the predefined neighboring units for all the units of the first sample is considered for a particular condition C, sa > 0. If an of the units in the initial sample satisf condition C ( i c or i c here c is a constant), there neighboring units are added to the sample and observed. In general, if the characteristic of interest is found at a particular area then e continue to locate around that area for more information. Further, if an neighboring unit satisfies the condition then its neighborhoods are also sampled and the process goes on. This iterative process stops hen the ne unit does not satisf condition C. The neighborhood can be defined b social and institutional relationships beteen units and can be decided in to as. The first-order neighborhood consists of the sampling unit itself and four adjacent units denoted as east, est, north, and south. The second-order neighborhood (Figure ) consists of first-order neighboring units and the units including northeast, northest, southeast, and southest units. There are in total eight neighborhood quadrates including the first-order neighborhood and the second order neighborhood. All the units including the initial sample composed the final sample. 43
44 Muhammad Shahad Chaudhr and Muhammad Hanif A netork is a set consisting of units that satisf the specified condition (usuall =). A unit that does not satisf the predefined condition in the first sample is the netork of sie one. The edge units are those hich do not satisf the specified criteria. A cluster is a miture of netork units ith associated edge units. Clusters ma have overlapping edge units. The netorks do not have common elements such that the union of the netorks becomes the population. Thus, it is possible to partition the population of all units into a set of eclusive and complete netorks. The netorks related to clusters can be denoted b A, A, A 3,, A n or the can be shon ith darker lines around the quadrates. These ma be shaded as ell. The edge units can be denoted ith open circles ( ). The entire region is partitioned into N rectangular or square units of equal sie that can be set in a lattice sstem. The rectangular or square units are called quadrates. The units (u,u,,u N ) form a disjoint and ehaustive partition of the entire area so that units labels (,,,N) categories the position of N quadrates. In ACS the population is measured in terms of quadrates onl. The unit i hich is related to the stud variable i has a population vector of -values, = (,, 3,, N ). No consider an eample to understand the ACS procedure using first-order neighborhoods and the condition C that i. Figure shos that the region is partitioned into N =50 quadrates of equal sie and that the population is divided into three clusters of sie greater than one. Figure 3 shos that the three netorks are denoted ith A, A, and A 3 and the edge units are denoted ith open circles ( ). According to the netork smmetr assumption an unit in A i, a netork ill lead to the selection of all units in that netork A i. There are 35 netorks of sie one as an unit that does not meet the condition is a netork of sie one. There are three clusters. Each cluster can be decomposed into a netork (that satisfies C) and individual netorks of sie (that do not satisf C) i.e. edge units. Thus, netork A has 6 units ith 8 edge units. Netork A has 4 units ith 9 edge units. Netork A 3 has 5 units ith 7 edge units. Thus, the cluster containing A has 6 + 8 =4 units. The cluster containing A has 4 + 9 =3 units and cluster containing A 3 has 5 + 7 = units. Clusters are not necessaril disjoint and ma have common edge units. The clusters containing A and A 3 have one overlapping edge unit. Figure 4 shos that the cluster containing A and the edge units are shaded, the overlapping edge unit of A and A 3 is also shaded.
Comparison of Estimators in Case of Lo Correlation in. Some Estimators in Simple Random Sampling Let N be the total number of units in the population. A random sample of sie n is selected b using Simple Random Sampling WithOut Replacement. The stud variable and auiliar variable are denoted b and ith their population means and, population standard deviation S and S, coefficient of variation C and C respectivel. Also ρ represent population correlation coefficient beteen and Y, and n. N Cochran (940) and Cochran (94) proposed the Classical Ratio and Regression Estimators t t (.) (.) The Mean Square Error (MSE) of the Estimators of eq. (.) and eq. (.) are MSE( t) Y C C CC (.3) MSE t Y C (.4) respectivel. Bahl and Tuteja (99) proposed the Eponential Ratio and Eponential Product Estimators to estimate the population mean. t 3 ep The Mean Square Error and Bias of the Eponential Ratio Estimator t 3 are MSE( t 3 ) C Y C CC 4 3 CC Bias( t3) Y C 8 3. Some Estimators in (.5) (.6) (.7) Let an initial sample of n units is selected ith a Simple Random Sample WithOut Replacement (SRSWOR). The neighboring units are added to the 45
46 Muhammad Shahad Chaudhr and Muhammad Hanif sample if the -value of a sampled unit meets a certain condition C, for eample i c here c is a constant. If A i is the netork including unit i, then unit i ill be included in the final sample either, an unit of netork A i is selected as part of the initial Simple Random Sample or an unit of a netork for hich unit i is an edge unit is selected. Suppose a finite population of sie N is labeled as,,3,,n. Let the auiliar variable i be correlated ith the variable of interest i, such that,,. N and,,. N. Let and denotes the average -value and average -value in the netork i i hich includes unit i such that, i j m and i j i j A m i i j Ai respectivel, here m i is the sie of the cluster A i. ACS can be considered as Simple Random Sampling WithOut Replacement hen the averages of netorks are considered (Drver and Chao, 007 and Thompson, 00). So, consider the notations and are the sample means of the stud and auiliar variables in the transformed population respectivel, such that, n n i i n n i and. Consider C and C represents population coefficient of variations of the stud and auiliar variables respectivel and ρ represent population correlation coefficient beteen and in the ACS i.e. hen average of the netork assumed. Let us define, Y e and e. (3.) Y here, e and e are the Relative Sampling Errors of the stud and auiliar variables respectivel, such that, Ee 0 C E e E e and and Ee C E e e C C (3.) (3.3) Thompson (990) developed an Unbiased Estimator for population mean Y in ACS based on a modification of the Hansen-Hurit Estimator hich can be used hen sampling is ith replacement or ithout replacement. Units that do not satisf C are ignored if the are not in the initial sample. In terms of the n netorks (hich ma not be unique), intersected b the initial sample. i
Comparison of Estimators in Case of Lo Correlation in 47 t n 4 i, (3.4) n i here, i m j t 4 is, i j A i N Var( t 4) i Y N (3.5) i Drver and Chao (007) proposed a Modified Ratio Estimator for the population mean keeping in vie. i is0 t ˆ (3.6) 5 R i is0 The Mean Square Error of t 5 is N MSE( t 5) i Ri N (3.7) i here, R is the population ratio beteen i and in the transformed population. i Chutiman (03) proposed a Modified Regression Estimator for the population mean of the stud variable in. t6 ( ) (3.8) here, S SS Y C (3.9) S S C The approimate Mean Square Error of t 6 is 6 MSE( t ) S ( ) YC ( ) (3.0) Shahad and Hanif (05) proposed a Generalied Eponential Estimator for the population mean in ACS using to auiliar variables. t Z GE ep (3.) ( a ) Z ( b )
48 Muhammad Shahad Chaudhr and Muhammad Hanif For, 0, and a, the Estimator t GE ma be obtained as t 7 ep The Bias and Mean Square Error of the Estimator t 7 are 3C CC Bias( t 7 ) Y 8 MSE( t 7 ) = C Et7 Y Y C CC 4 4. Proposed Estimators in (3.) (3.3) (3.4) Folloing the Bahl and Tuteja (99) and Shahad and Hanif (05) the Proposed Modified Regression-Cum-Eponential Ratio Estimators in ACS ith one auiliar variable are t 8 ep (4.) t 9 t4 t 7 ep (4.) 4. Bias and Mean Square Error of Estimator t 8 : The Estimator eq. (4.) ma be ritten as t ( e ) 8 Y e ep (4..) ( e) e e t8 Y Ye ep (4..) e e t8 Y Ye ep (4..3) 4 e e e t8 Y Ye 4 8 e 3e t8 Y Ye 8 (4..4) (4..5)
Comparison of Estimators in Case of Lo Correlation in Appling epectations on both sides of eq. (4..5), and using notations eq. (3.), e get, Bias( t 8 )= 3C Et8 Y (4..6) 8 In order to derive Mean Square Error of eq. (4.), e have eq. (4..3) b ignoring the term degree or greater as e t8 Y Ye (4..7) Taking square and epectations on the both sides of eq. (4..7), and using notations eq. (3. and 3.3), MSE( t 8 ) = C E( t8 Y ) Y C Y CC. 4 (4..8) Differentiate.r.t to β and equate to ero, e get, Y CC (4..9). C 4 Substitute eq. (4..9) into eq. (4..8), e get minimum Mean Square Error, C MSE( t8) min Y C (4..0) C 4 4. Bias and Mean Square Error of Estimator t 9 : The Estimator eq. (4.) ma be ritten as t ( e ) 9 Y e Y e ep (4..) ( e) t e 9 e Y Ye Y ( e )ep (4..) 4 e e e t9 Y Ye Y ( e ) (4..3) 4 8 e e e ee t9 Y Ye Y e (4..4) 4 8 49
50 Muhammad Shahad Chaudhr and Muhammad Hanif Appling epectations on both sides of eq. (4..4), e get, Bias ( t 9 ) = 3 C CC Et9 Y Y (4..3) 8 In order to derive mean square error of eq. (4.), e have eq. (4..) b ignoring the term degree or greater as t9 Y e Ye Y e ep (4..4) Ye t9 Y Ye Y Ye (4..5) Taking square and epectations on the both sides of eq. (4..5) Y C MSE( t9) ( t9 Y ) Y C Y Y C 4 Y C Y C C Y C C. (4..6) Differentiate.r.t to β and equate to ero, e get, CC C. (4..7) C C CC 4 Substitute eq. (4..7) into eq. (4..6), e get minimum Mean Square Error, CC C MSE( t 9) min Y C. C C C CC 4 (4..8) 5. Results and Discussion To compare the efficienc of proposed Estimators ith the other Estimators, a population is used and performed simulations for the thorough stud. The condition C for added units in the sample is > 0. The -values are obtained and averaged for keeping the sample netork according to the condition and for each
Comparison of Estimators in Case of Lo Correlation in sample netork -values are obtained and averaged. For the simulation stud ten thousands iteration as run for each Estimator to get accurac estimates ith the simple random sampling ithout replacement and the initial sample sies of 5, 0, 5, 0 and 5. The epected final sample sie is sum of the probabilities of inclusion of all quadrates and it varies from sample to sample in ACS. Let, E(v) denotes the epected final sample sie in ACS. For the comparison, the sample mean from a SRSWOR based on E(v) has variance using the usual formula. ( N E ( ) ( v Var )) (5.) NE() v The estimated Mean Square Error of the estimated mean is ^ r MSE t ( t Y ) (5.) * * r i here, t * is the value for the relevant Estimator for sample i, and r is the number of iterations. The Estimated Relative Bias is defined as ^ RBias t * r ( t* ) Y r i Y The percentage relative efficienc is, Var( ) PRE ^ MSE t * 00 (5.3) (5.4) 5. Population: This real population contains ring-necked ducks (RND) and blueinged teal (BWT) data collected b Smith et al., (995) are counts of to species of aterfol in 50 00-km quadrates in central Florida. The ring-necked ducks (Table ) is taken as the stud variable the blue-inged teal (Table ) is taken as auiliar variable. There found a ver lo correlation 0.0978 beteen the both tpes of fol and this correlation increases to 0.65 in the transformed population (Table 3 and 4). Thus, there is a lo correlation beteen the sampling unit level and high correlation at the netork (region) level. Drver and Chao (007) shoed that usual Estimators in SRSWOR perform better than ACS Estimators for strong correlation at unit level but performs orse hen having the strong correlation at netork level. 5
5 Muhammad Shahad Chaudhr and Muhammad Hanif The overall variance of the stud variable is 39753 and for auiliar variable the variance is 37668 hile in the transformed population these variances reduce to 74763.9 and 37656, respectivel. The ithin netork variance of the stud variable for the netork (4000, 0, 3500, 00, 34, 75, 335, 4, 97) is 9755495 and for the corresponding values of the auiliar variable (0, 0, 0, 3, 5, 4, 4, ) the ithin netork variance is 69.5. The overall variances are found to lo as compare to the ithin netork variances in the stud variable population. ACS is preferable than the Conventional sampling if the ithin netork variances are large enough as compare to overall variance (Drver and Chao 007). 6. Conclusion The Estimated Relative Bias (Table 5) of the Regression Estimators in ACS goes quickl to ero hen sample sie increases as compare to the other Estimators in usual sampling methods. The Percentage Relative Efficienc (Table 6) of the ACS Estimators is much higher than the SRS Estimators. The Regression Estimator in ACS has maimum Percentage Relative Efficienc for the initial sample sie and starts increasing for comparable sample sies, hile proposed Eponential Regression Estimator t 8 also has higher PRE than the Ratio Estimator and Eponential Ratio Estimator in ACS. Thus, the Regression and Eponential Regression Estimators in ACS perform much better than the other Conventional Estimators and Ratio and Eponential Ratio Estimators in ACS even though there is a ver lo positive correlation among the stud and the auiliar variables, under the given conditions. Drver and Chao (007) treated 0/0 as ero for the Ratio Estimator. The usual Ratio Estimator and Ratio Estimator in ACS did not perform and return no value (*) and infinit (**) for the initial sample sies 5, 0, 5 and 0, respectivel. In this simulation stud 0/0 is not assumed as 0. The use of Regression and Eponential Regression Estimators is better in ACS than assuming an unlikel assumption for the Ratio Estimator in. Thus, Regression and Eponential Regression Tpe Estimators are much suitable and vigorous for patch, rare and clustered population. The results of relative efficiencies have shon the poor performance of usual Estimators under SRSWOR and the Eponential Ratio Estimators because of the eak correlation beteen stud variable and auiliar variables at the unit level. For the future research, the case of lo negative correlation beteen the stud and auiliar variables ma also be eplored in ACS. Moreover, the logarithmic, trigonometric,
Comparison of Estimators in Case of Lo Correlation in inverse trigonometric and hperbolic Estimators ma also be studied and eplored in Conventional and Adaptive Sampling. Some Estimators are given in Table 8 for the future research. Acknoledgments The authors are grateful to Yves G. Berger, Universit of Southampton, UK for the programming advice to simulation studies and valuable suggestions regarding the improvement of this paper. 53 Southeast East Northeast South Sampling Unit North Southest West Northest Figure: Second-order neighbourhoods (Diagonal cells) 0 0 0 0 0 0 0 0 0 0 0 4 3 0 0 3 0 0 0 3 4 0 0 6 0 3 5 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 Figure : Population ith three clusters A A A A 3 A A A A A 3 A 3 A 3 A A A A 3 Figure 3: Netorks ith edge units 0 0 0 0 0 0 0 0 0 0 0 4 3 0 0 3 0 0 0 3 4 0 0 6 0 3 5 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 Figure 4: Cluster containing A, Common edge unit of A and A 3
54 Muhammad Shahad Chaudhr and Muhammad Hanif Table: Ring-necked ducks data (Smith et al., 995) as stud variable() for population 0 00 00 75 0 0 0 0 675 0 4000 3500 34 335 4 0 0 35 0 55 0 0 0 0 97 0 4 0 85 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 83 Table: Blue-inged teal data (Smith et al., 995) as auiliar variable() for population 0 0 3 5 0 0 0 0 0 0 0 0 0 4 4 0 0 0 03 0 0 0 0 0 3 0 3639 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 77 Table3: Average of the netork () of stud variable() for population 0 6.78 6.78 6.78 0 0 0 0 675 0 6.78 6.78 6.78 6.78 6.78 0 0 35 0 55 0 0 0 0 6.78 0 4 0 85 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 83 Table4: Average of the netork () of auiliar variable for population 0 5.33 5.33 5.33 0 0 0 0 0 0 5.33 5.33 5.33 5.33 5.33 0 0 0 03 0 0 0 0 0 5.33 3 0 3639 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 77
Comparison of Estimators in Case of Lo Correlation in Table 5: Estimated Relative Bias of the population for different sample sies n E(v) t t t 3 t 4 t 5 t 6 t 7 t 8 t 9 5 6.4 * -0.06.45 0.003 * -0.003.33 0.05 0.30 0 4.3 * 0.0 0.999 0.006 * 0.006.040 0.003 0. 5 9.33 ** -0.0 0.856-0.00 ** -0.00 0.885 0.00 0.95 0 33.7 ** -0.0 0.70-0.003 9.805 0.004 0.73 0.00 0.74 5 36.54 0.667-0.009 0.594-0.00 0.077-0.003 0.586 0.005 0.58 Table 6: Percentage Relative Efficiencies for the population for the Estimators based on E(v) Simple Random Sampling E(v) t t t 3 t 4 t 5 t 6 t 7 t 8 t 9 6.4 00 * 0. 4.98 3. * 33.9 4.47 4.44 78.60 4.3 00 * 00.66 6.0 39.6 * 47.8.90 40.85 90.0 9.33 00 0 00.93 5.9 6. 0.00 7.9. 63.94 00.39 33.7 00 0 0.08 6.09 79.03 0.0 9.5.55 8.46 08.95 36.54 00 0 0.0 5.83 94.83 0.0 05.33 0.8 99.06 7.60 55 Table 7: Descriptive measures of the population 8.4 37668 Y 466.66 C 68.5779 0.0979 39753 C 47.035 0.65 8.4 37656 C 68.577 N 50 466.64 74763.9 C 85.36 C 0
Muhammad Shahad Chaudhr and Muhammad Hanif 56 Table 8: Some Proposed Estimators for future research SRS ACS SRS ACS SRS ACS ln ln sin sin sin sin ln ln sin sin sin sin ln ln sin sin sin sin sinh sinh sinh sinh cosh cosh In General SRS In General ACS ln ( ) ( ) Z a Z b ln ( ) ( ) Z a Z b ( ) ( ) Z TRG a Z b ( ) ( ) Z TRG a Z b ( ) ( ) Z INVTRG a Z b ( ) ( ) Z INVTRG a Z b ( ) ( ) Z TRGH a Z b ( ) ( ) Z TRGH a Z b ( ) ( ) Z INVHTRG a Z b ( ) ( ) Z INVHTRG a Z b here, TRG = Trigonometric functions INVTRG = Inverse Trigonometric functions TRG H = Hperbolic Trigonometric functions INVHTRG = Inverse Hperbolic Trigonometric functions
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