EXPONENTIAL CHAIN DUAL TO RATIO AND REGRESSION TYPE ESTIMATORS OF POPULATION MEAN IN TWO-PHASE SAMPLING

Transcription

1 STATISTICA, anno LXXV, n. 4, 015 EXPONENTIAL CHAIN DUAL TO RATIO AND REGRESSION TYPE ESTIMATORS OF POPULATION MEAN IN TWO-PHASE SAMPLING Garib Nath Singh Department of Applied mathematics, Indian School of Mines, Dhanbad Mohd Khalid 1 Department of Applied mathematics, Indian School of Mines, Dhanbad 1. Introduction It is customary to utilize the information on auxiliary variables to improve the precision of estimates in sample surveys, if such is readily available or may be made available by incurring some additional cost. Ratio, product regression methods of estimation are the popular estimation procedures which utilize the information on auxiliary variable in terms of known population parameters at estimation stage. Sometimes, the population parameters of auxiliary variable may not be readily known at the time of generating estimates of population parameters of study variable. Two-phase (double) sampling happens to be a cost effective technique for generating the most reliable estimates of unknown population parameters of auxiliary variable in first phase sample. Ch (1975) introduced a technique of chaining the information of the second auxiliary variable to get a more precise estimate of the unknown population mean of first auxiliary variable in the first-phase sample in form of ratio estimator which is termed as chain-type ratio estimator. Further, his work was extended by Kiregyera (1980, 1984); Mukerjee et al. (1987), Srivnstava et al. (1989), Upadhyaya et al. (1990), Singh Singh (1991), Singh et al. (1994), Singh Upadhyaya (1995), Upadhyaya Singh (001), Singh (001), Kadılar Cıngı (003); Kadilar Cingi (004), Pradhan (005), Kadılar et al. (007), Gupta Shabbir (008), Singh et al. (008), Choudhury Singh (01), Ozgul Cingi (014), Singh Majhi (014), among others. Bahl Tuteja (1991) suggested exponential methods of estimation of population mean of study variable, which was further followed by Singh Vishwakarma (007), who studied the behavior of the exponential method of estimation in two-phase sampling. In follow up of the above works motivated with the generic nature of exponential method of estimation, we have proposed some new exponential chain-type estimators which are more relevant in producing 1 Corresponding Author. mohdkhalidkhan4870@gmail.com

2 380 G. N. Singh M. Khalid the precise estimates of the population mean of the study variable. Theoretical empirical properties of the proposed estimation procedures have been studied suitable recommendations are put forward to the survey practitioners.. Two-phase sampling setup Let U = (U 1, U,..., U N ) be a finite population of size N indexed by triplet characters (y, x, z). Our interest is to estimate the population mean Ȳ of study variable y in the presence of two auxiliary variables x z. Let x z be the first second auxiliary variables respectively, such that y is highly correlated with x while in compare to x, it is remotely correlated with z (ρ yx > ρ yz ). When population mean X of the first auxiliary variable x is unknown but information on second auxiliary variable z is available for all the units of the population, we use the two-phase sampling design which is described as follows. Consider a two-phase sampling design where in the first-phase a large (preliminary) sample s (s U)of size n is drawn following simple rom sampling without replacement (SRSWOR) scheme observed for two auxiliary variables x z to estimate X, while in the second phase a sub sample s s of size n is drawn from first-phase sample s by SRSWOR to observe the characteristic y under study. 3. Some estimators in two-phase sampling In this section, we consider some estimators of population mean in two phase sampling which are frequently used in survey literature s. The estimators along with their respective mean square errors are presented in the following sub- sections Estimators based on one auxiliary variable in two-phase sampling setup Ratio regression methods of estimation require the known population mean of auxiliary variable in the absence of such information, we go for two-phase sampling setup, where, we estimate the unknown population mean of auxiliary variable in the first- phase sample. For ready-made assessment the ratio regression estimators respectively in two-phase sampling setup along with their respective mean square errors up to the first order of approximations are reproduced below as: ȳ rd = ȳ x x (1) M(ȳ rd ) = Ȳ [ f 1 C y + ( C x ρ yx C y C x )] () ȳ lrd = ȳ + b yx (n)( x x) (3) M(ȳ lrd ) = Y [ f1 (1 ρ yx) + f ρ yx] C y (4)

3 Exponential chain dual to ratio regression type estimators 381 where b yx (n) is the sample regression coefficient of y on x based on sample s ȳ = 1 ( 1 y i, n n 1 ), N f = i s ( 1 1 n N x = 1 x i, x = 1 x n n i, f 1 = i s i s ) ( 1, = n 1 ) n k = n N n, S x = 1 N 1 N i=1 = (f 1 f ), C x = S x X, C y = S y Ȳ, (x i X), S y = 1 N 1 ρ yx be the correlation coefficient between the variables y x. N (y i Ȳ ), i=1 3.. Estimators based on two auxiliary variables in two-phase sampling setup Sometimes, information on more than one auxiliary variable is available at the estimation stage in sample surveys. Ch (1975) introduced a chain- type ratio estimator under two-phase sampling setup where he utilized the information on two auxiliary variables x z under the assumption that the population mean X of the auxiliary variable x is unknown information on auxiliary variable z is available for all the units of the population. The suggested estimator with their respective mean square error up to the first order approximations are reproduced as: ȳ rc = ȳ x Z (5) x z M(ȳ rc ) = Ȳ [ f 1 C y + (C x ρ yx C x C y ) + f (C z ρ yz C y C z ) ] (6) where Z is the population mean of the auxiliary variable z, z = 1 n i s z i, C z = S z Z, S z = 1 N 1 N (z i Z) ρ yz be the correlation coefficient between the variables y z. Further, Kiregyera (1980, 1984) extended the work of Ch (1975) suggested chain-type ratio to regression, regression to ratio regression to regression estimators of the population mean Ȳ of study variable y in two- phase sampling setup which utilized the information on two auxiliary variables x z under similar assumptions as considered by Ch (1975). The suggested estimators are given below with their respective mean square errors up to the first order of approximations as: ȳ k1 = ȳ [ x + b xz (n )( x Z ] z ) (7) M(ȳ k1 ) = Ȳ [ (C x + C y ρ yx C x C y ) + f C y + f ρ xz C x (ρ xz C x ρ yz C y ) ] (8) i=1 ȳ k = ȳ + b yx (n)( x rd x) (9)

4 38 G. N. Singh M. Khalid where x rd = x Z z M(ȳ k ) = Ȳ Cy [f 1 (1 ρ yx) ( + f ρ yx + ρ yx Cz Cx )] C z ρ yx ρ yz C x (10) ȳ k3 = ȳ + b yx (n)(x ld x) (11) where x ld [ x = + b xz (n )( Z ] z ) M(ȳ k3 ) = Ȳ Cy [ f3 (1 ρ yx) + f (1 + ρ xzρ yx ρ yx ρ yz ρ xz ) ] (1) where b xz (n ) is the sample regression coefficient of the variable x on z based on the sample s ρ xz be correlation coefficient between variables x z. 4. The proposed exponential chain-type dual to ratio regression estimators The intelligible use of auxiliary information for making estimates more precise has been a fascinating area of research in sample surveys. In the presence of auxiliary information Bhal Tuteja (1991) suggested an exponential type estimator to estimate the population mean of study variable their work was further extended by Singh Vishwakarma (007) in two-phase sampling setup. Motivated by the work related to the proposition of chain-type estimators in two-phase sampling setup, looking at the nice behavior of the exponential type estimators, we suggest a new exponential chain-type dual to ratio regression estimators to estimate the population mean Ȳ under two phase sampling setup with an assumption that the population mean X of the auxiliary variable x is unknown. The proposed estimators are formulated as: T 1 = ȳ exp x Z z x (13) x Z z + x [ x + b xz (n )( T = ȳ exp Z ] z ) x x + b xz (n )( Z, (14) z ) + x where z = N Z n z N n. 5. The Bias Mean square errors (MSE) of the proposed estimators T i (i = 1, ) Since the proposed estimators T i (i = 1, ) are exponential chain-type dual to ratio regression estimators, they are biased estimators of the population mean Ȳ, therefore, bias B (.) mean square errors M (.) of the estimators T i are derived up to the first order of approximations presented in the following theorems:

5 Exponential chain dual to ratio regression type estimators 383 Theorem 5.1. The bias of the estimators T i (i=1, ) to the first order of approximations are obtained as B(T 1 ) = Ȳ B(T ) = Ȳ [ ( 18 Cx 1 ρ yxc y C x ) + f ( k 8 C z k 4 C z k ] ρ yzc y C z ) [ f3 ( 3 8 C x 1 ρ yxc y C x ) +f ( 5 8 ρ xzcx 1 ρ xz Cx α 10 C z ZS xz + 1 ρ xz Cx α 003 C z ZS 1 z ρ xzρ yz C x C y ) ] (15) (16) Theorem 5.. The mean square errors of the estimators T i (i=1, ) to the first order of approximations are obtained as [ M(T 1 ) = Ȳ f 1 Cy + f ( k 4 C z kρ yz C y C z ) + 1 ] 4 (Cx 4ρ yx C y C x ) M(T ) = [f Ȳ 1 Cy f (ρ xzcx 4ρ yz ρ xz C y C x ) + 1 ] 4 (Cx 4ρ yx C y C x ) (17) (18) 6. Comparison of the estimators In this section, we compare the proposed estimators T i (i = 1, ) with respect to the estimators ȳ rd, ȳ lrd, ȳ rc, ȳ k1, ȳ k, ȳ k3 in terms of mean square criterions. The conditions for which the proposed estimators T i will dominate other estimators under mean square criterions are defined as: (i) The estimators T i (i = 1, ) are better than ȳ rd if M(T i ) M(ȳ rd ), which gives the conditions (3Cx 4ρ yx C y C x ) (k Cz 4kρ yz C z C y ) f (19) for the estimators T 1 T respectively. (3C x 4ρ yx C y C x ) (ρ xzc x 4ρ yz ρ xz C x C y ) f (0) (ii) The estimators T i (i = 1, ) are more efficient than ȳ lrd if M(T i ) M(ȳ lrd ), such that following conditions are satisfied for the estimators T 1 T respectively. (C x ρ yx C y ) (4kρ yz C y C z k Cz ) f (1) (4ρ yz C y C x 4ρ yxc y C x) (ρ xzc x 4ρ yz ρ xz C y C x ) f ()

6 384 G. N. Singh M. Khalid (iii) The estimators T i (i = 1, ) will dominate ȳ rc if M(T i ) M(ȳ rc ), which gives the results (Cx 4ρ yx C y C x ) ( k 4 1)C z (k )ρ yz C y C z ) } f (3) ( 3 4 C x ρ yx C y C x ) 1 4 ρ xzc x (ρ xz C x 4ρ yz C y ) + C z (ρ yz C y C z ) } f (4) for the estimators T 1 T respectively. (iv) The estimators T i (i = 1, ) are better than ȳ k1 if M(T i ) M(ȳ k1 ), subsequently, we get the following conditions 3(C x 4ρ yx C y C x ) 4 (ρ yz C y k C z) (ρ xz C x ρ yz C y ) } f (5) for the estimators T 1 T respectively. (C x 4 3 ρ yxc y C x ) ( 4 3 ρ xzρ yz C y C x ρ xzc x) f (6) (v) The estimators T i (i = 1, ) are preferable over ȳ k if M(T i ) M(ȳ k ), which leads to the conditions C y(ρ yx ρ yz C y ( 1 C x ρ yx C y ) C ) + (ρ z yx C x ρ yz ) Cz ( k 4 kρ yz Cy C z ) } f (7) (C x ρ yx C y ) } f (8) C 4ρ z yx C x Cy(ρ C z yx C x ρ yz ) + ρ xz C x (ρ xz C x 4ρ yz C y ) for the estimators T 1 T respectively. (vi) The estimator T i (i = 1, ) will influence over ȳ k3 if M(T i ) M(ȳ k3 ), which gives the conditions ( 1 C x ρ yx C y ) (ρ yx ρ xz ρ yx ρ yz ρ xz ) ( k 4 C z kρ yz C y C z ) } f (9) (C x ρ yx C y ) 4ρyx ρ xz C y(ρ yx ρ xz ρ yz ) ρ xz c x (ρ xz C x 4ρ yz C y ) } f (30) for the estimators T 1 T respectively.

7 Exponential chain dual to ratio regression type estimators Cost aspect The principle of optimization is an important criterion in sample surveys to make estimators more precise for given cost is an integral part of it. Since, twophase sampling setup is more prone to cost, therefore, it is essential to formulate a strategy which reduces the survey cost. Considering this fact, we derive the expressions for optimum sample sizes in primarily (first-phase) final (secondphase) samples respectively by considering the suitable cost function so that the mean square errors of the estimators are minimized. In most of the practical situations total cost is the linear function of samples selected at first second phases. Keeping this point in views, in this section, we define the following cost function which is given below: C = nc 1 + n (C + C 3 ), (31) where C is the total cost of the survey, C 1 is the cost per unit associated with the second phase sample of size n for collecting the information on study variable y C, C 3 be the cost per unit associated with the first phase sample of size n for collecting the information on auxiliary variable x z respectively. For large population, ignoring finite population correction (f.p.c), the mean square errors of the proposed estimators T i ( i = 1, ) can be expressed as where MSE(T i ) = 1 n V 1i + 1 n V i ; (i = 1, ), (3) V 11 = Ȳ (C y + C x 4 ρ yxc x C y ), V 1 = Ȳ ( k 4 C z C x 4 kρ yzc z C y + ρ yx C y C x ), V 1 = Ȳ (Cy + C x 4 ρ yxc x C y ), V = Ȳ ( ρ xzcx ρ yz ρ xz C x C y C x 4 4 +ρ yxc y C x ). Therefore, fixed for cost C 0 the objective function is defined as: ϕ i = M(T i ) + λ i (nc 1 + n (C + C 3 ) C 0 ); ( i = 1, ), (33) where λ i is the Lagrange s multipliers. It is assumed that C 1 > C > C 3. Utilizing the function of Lagrange s multipliers, the optimum values of n n for fixed cost C 0 which minimize the mean square errors of the estimators T i ( i = 1, ) in equation (3) are derived as: n opt = C 0 V1i C 1 V1i + V i C 1 (C + C 3 ) } ; (i = 1, ), (34) } n C o Vi opt = (C + C 3 )C 1 C1 V 1i + ; (i = 1, ). (35) V i (C + C 3 )}

8 386 G. N. Singh M. Khalid Substituting, the optimum values of n n derived in equation (34) (35) in equation (3), we get the optimum mean square errors of the estimators T i ( i = 1, ), as MSE(T i ) opt = 1 C 0 (C 1 V1i + V i (C + C 3 )) ; (i = 1, ). (36) 8. Empirical Study To analyze the performances of the proposed estimators T i (i = 1, ) with respect to the other estimators, we have computed percent relative efficiency of the estimators T i with respect to the estimators ȳ rd, ȳ lrd, ȳ rc, ȳ k1, ȳ k, ȳ k3 using three natural populations. The PRE of the estimators T i (i = 1, ) is calculated by using the following formula. [ ] MSE(δ) P RE = 100 ; (i = 1, ), MSE(T i ) where δ is some estimator of population mean Ȳ. Population source-i: Cochran (1977) Y =Number of placebo children. X=Number of paralytic polio cases in the placebo group. Z=Number of paralytic polio cases in the not inoculated group. N = 34, n = 10, n = 15, Ȳ = 4.9, X =.59, Z =.91, C y = 1.048, C z = 1.149, C x = , ρ yx = 0.736, ρ yz = , ρ xz = Population-II: Fisher (1936) Consisting of measurements on three variables, namely sepal width(y), sepal length(x), petal length (z) for 50 Iris flowers (versicolor) such that: N = 50, n = 9, n = 0, Ȳ =.770, C y = , C z = , C x = , ρ yx = , ρ yz = 0.559, ρ xz = Population-III: Shukla (1966) Y = Fiber. X= Height. Z= Base Diameter. N = 50, n = 8, n = 15, Ȳ = 4.9, X =.5840, C y = , C z = , C x = , ρ yx = , ρ yz = , ρ xz =

9 Exponential chain dual to ratio regression type estimators 387 TABLE 1 The percent relative efficiency (PRE) based on population I, II III of the proposed estimators T i (i = 1, ) with respect to the estimators ȳ rd, ȳ lrd, ȳ rc, ȳ k1, ȳ k ȳ k3. Proposed Known Estimators Estimators ȳ rd ȳ lrd ȳ rc ȳ k1 ȳ k ȳ k3 PRE for population I T T PRE for population II T T PRE for population III T T show the proposed estimators are not preferable. 9. Conclusions From Table-1, it is visible that the proposed estimators T i (i = 1, ) are dominating over the estimators ȳ rd, ȳ lrd, ȳ rc ȳ k1 for all population. The proposed estimators will also be preferable over the estimators ȳ k ȳ k3 for the populations, which satisfy the conditions derived in equations (7)-(8) (9)-(30) respectively. Looking on the nice behavior of the proposed estimator, this may be recommended to the survey practitioners for their practical applications. Acknowledgements Authors are thankful to the reviewer for his valuable suggestion. Authors are also thankful to the Indian School of Mines, Dhanbad, for providing financial assistance necessary infrastructure to carry out the present research work. References S. Bahl, R. Tuteja (1991). Ratio product type exponential estimators. Journal of Information Optimization Sciences, 1, no. 1, pp L. Ch (1975). Some ratio-type estimators based on two or more auxiliary variables. S. Choudhury, B. Singh (01). A class of chain ratio-cum-dual to ratio type estimator with two auxiliary characters under double sampling in sample surveys. Statistics in Transition-New Series, 13, no. 3, pp W. G. Cochran (1977). Sampling Techniques. New York: JohnWiley.

10 388 G. N. Singh M. Khalid R. A. Fisher (1936). The use of multiple measurements in taxonomic problems. Annals of eugenics, 7, no., pp S. Gupta, J. Shabbir (008). On improvement in estimating the population mean in simple rom sampling. Journal of Applied Statistics, 35, no. 5, pp C. Kadılar, H. Cıngı (003). A study on the chain ratio-type estimator. Hacettepe Journal of Mathematics Statistics, 3, pp C. Kadilar, H. Cingi (004). Ratio estimators in simple rom sampling. Applied Mathematics Computation, 151, no. 3, pp C. Kadılar, H. Çıngı, et al. (007). Ratio estimators using robust regression. Hacettepe Journal of Mathematics Statistics, 36, no.. B. Kiregyera (1980). A chain ratio-type estimator in finite population double sampling using two auxiliary variables. Metrika, 7, no. 1, pp B. Kiregyera (1984). Regression-type estimators using two auxiliary variables the model of double sampling from finite populations. Metrika, 31, no. 1, pp R. Mukerjee, T. Rao, K. Vijayan (1987). Regression type estimators using multiple auxiliary information. Australian Journal of Statistics, 9, no. 3, pp N. Ozgul, H. Cingi (014). A new class of exponential regression cum ratio estimator in two phase sampling. Hacettepe Journal of Mathematics Statistics, 43, no. 1. B. Pradhan (005). A chain regression estimator in two phase sampling using multi-auxiliary information. Bull. Malays. Math. Sci. Soc,, no. 8, p. 1. G. Shukla (1966). An alternative multivariate ratio estimate for finite population. Calcutta Statistical Association Bulletin, 15, no. 58-5, p. 15. G. Singh (001). On the use of transformed auxiliary variable in the estimation of population mean in two-phase sampling. Statistics in Transition, 5, no. 3, pp G. Singh, D. Majhi (014). Some chain-type exponential estimators of population mean in two-phase sampling. STATISTICS, 1. G. Singh, L. Upadhyaya (1995). A class of modified chain type estimators using two auxiliary variables in two-phase sampling. Metron, 53, no. 3-4, pp H. P. Singh, G. K. Vishwakarma (007). Modified exponential ratio product estimators for finite population mean in double sampling. Austrian Journal of Statistics, 36, no. 3, pp

11 Exponential chain dual to ratio regression type estimators 389 R. Singh, P. Chauhan, N. Sawan, F. Smarache (008). Improvement in estimating population mean using two auxiliary variables in two-phase sampling. arxiv preprint arxiv: V. Singh, G. Singh (1991). Chain type regression estimators with two auxiliary variables under double sampling scheme. Metron, 49, no. 1-4, pp V. K. Singh, H. P. Singh, H. P. Singh, D. Shukla (1994). A general class of chain estimators for ratio product of two means of a finite population. Communications in Statistics-Theory Methods, 3, no. 5, pp R. S. Srivnstava, S. Srivastava, B. Khare (1989). Chain ratio type estimator for ratio of two population m3ans using auxiliary characters. Communications in Statistics-Theory Methods, 18, no. 10, pp L. Upadhyaya, K. Kushwaha, H. Singh (1990). A modified chain ratio-type estimator in two-phase sampling using multi auxiliary information. Metron, 48, no. 1-4, pp L. Upadhyaya, G. Singh (001). Chain-type estimators using transformed auxiliary variable in two-phase sampling. Advances in Modelling Analysis-A-, 38, no. 1/, pp. 1a 10a. Summary The present work suggests some estimation procedures of population mean in two phase sampling. Chain-type exponential dual to ratio regression estimators have been proposed their properties are studied. Cost aspects of the suggested estimation procedures have been examined a recommendations are made to the survey practitioners for their practical applications. Keywords: Two- phase chain- type; auxiliary information; bias; mean square error; cost.