Ratio Estimators in Simple Random Sampling Using Information on Auxiliary Attribute

Transcription

1 ajesh Singh, ankaj Chauhan, Nirmala Sawan School of Statistics, DAVV, Indore (M.., India Florentin Smarandache Universit of New Mexico, USA atio Estimators in Simle andom Samling Using Information on Auxiliar Attribute ublished in: ajesh Singh, ankaj Chauhan, Nirmala Sawan, Florentin Smarandache (Editors AUXILIAY INFOMATION AND A IOI VALUES IN CONSTUCTION OF IMOVED ESTIMATOS enaissance High ress, Ann Arbor, USA, 007 (ISBN-0: (ISBN-3:

2 Abstract Some ratio estimators for estimating the oulation mean of the variable under stud, which make use of information regarding the oulation roortion ossessing certain attribute, are roosed. Under simle random samling without relacement (SSWO scheme, the exressions of bias and mean-squared error (MSE u to the first order of aroximation are derived. The results obtained have been illustrated numericall b taking some emirical oulation considered in the literature. AMS Classification: 6D05. Ke words: roortion, bias, MSE, ratio estimator. 7

3 . Introduction The use of auxiliar information can increase the recision of an estimator when stud variable is highl correlated with auxiliar variable x. There exist situations when information is available in the form of attribute, which is highl correlated with. For examle a Sex and height of the ersons, b Amount of milk roduced and a articular breed of the cow, c Amount of ield of wheat cro and a articular variet of wheat etc. (see Jhajj et al., []. Consider a samle of size n drawn b SSWO from a oulation of size N. Let i and i denote the observations on variable and resectivel for i th unit ( i,,...n. Suose there is a comlete dichotom in the oulation with resect to the resence or absence of an attribute, sa, and it is assumed that attribute takes onl the two values 0 and according as i, if ith unit of the oulation ossesses attribute 0, otherwise. Let N A i and i n a denote the total number of units in the oulation and i i samle resectivel ossessing attribute. Let A and N of units in the oulation and samle resectivel ossessing attribute. a denote the roortion n Taking into consideration the oint biserial correlation between a variable and an attribute, Naik and Guta (996 defined ratio estimator of oulation mean when the 8

4 rior information of oulation roortion of units, ossessing the same attribute is available, as follows: t (. here is the samle mean of variable of interest. The MSE of t u to the first order of aroximation is [ + S S S ] f MSE (t (. n where n f, N Y N N, ( i Y S, N S, ( i i N i N S ( i ( i Y N. i In the resent aer, some ratio estimators for estimating the oulation mean of the variable under stud, which make use of information regarding the oulation roortion ossessing certain attribute, are roosed. The exressions of bias and MSE have been obtained. The numerical illustrations have also been done b taking some emirical oulations considered in the literature.. The suggested estimator Following a and Singh (98, we roose the following estimator t + b ( * (. s where b, s * + b (, s ( n n i i and s. n i n ( i ( i Y 9

5 emark : When we ut b 0 in (., the roosed estimator turns to the Naik and Guta (996 ratio estimator t given in (.. MSE of this estimator can be found b using Talor series exansion given b f (, f (c,d f (c,d f (, + ( + ( Y (.,Y c c,y where * f (, and (,Y f. Exression (. can be alied to the roosed estimator in order to obtain MSE equation as follows: * (( + b ( /,Y ( + b + ( + ( Y,Y ( Y + B,Y (( + b ( * (Y + B E( V( Cov(, + V( 4 3 /,Y ( Y (Y + B + (Y B V( Cov(, + V( (.3 where B S S ρs S. ρ S S S, is the oint biserial correlation coefficient. Now, MSE(t E( ( Y + B V ( ( Y + B Cov (, + V( (.4 0

6 Simlifing (.4, we get MSE of t as MSE f ( t [ S + S ( ρ ] (.5 n Several authors have used rior value of certain oulation arameters (s to find more recise estimates. Searls (964 used Coefficient of Variation (CV of stud character at estimation stage. In ractice this CV is seldom known. Motivated b Searls (964 work, Sen (978, Sisodia and Dwivedi (98, and Uadhaa and Singh (984 used the known CV of the auxiliar character for estimating oulation mean of a stud character in ratio method of estimation. The use of rior value of Coefficient of Kurtosis in estimating the oulation variance of stud character was first made b Singh et al. (973. Later, used b and Searls and Intaraanich (990, Uadhaa and Singh (999, Singh (003 and Singh et al. (004 in the estimation of oulation mean of stud character. ecentl Singh and Tailor (003 roosed a modified ratio estimator b using the known value of correlation coefficient. In next section, we roose some ratio estimators for estimating the oulation mean of the variable under stud using known arameters of the attribute such as coefficient of variation C, Kurtosis ( ( 3. Suggested Estimators We suggest following estimator ρ. β and oint biserial correlation coefficient + b( (m + m (m + m t (3. where m ( 0, m are either real number or the functions of the known arameters of the attribute such as C, ( ( ρ. β and

7 The following scheme resents some of the imortant estimators of the oulation mean, which can be obtained b suitable choice of constants m and m : Estimator Values of m m t + b( + b( ( β ( [ + β ( ] + β ( t [ ] + b( C t + t 3 C ( + C + b ( [ + ρ ] 4 ( + ρ + b ( ( [ β ] ( β ( + C ( 5 C t + + b ( ( [ C ( ] + β C + β ( t 6 β C C β ( t7 C + b ( ( [ ] + ρ C + ρ C ρ ρ t + + b( ( [ ρ ] ρ + C C 8 ρ C t + b ( ( [ β ] ( + ρ β ( + ρ 9 β ( ρ + b( ( [ ρ ( ] + β ρ + β ( t0 ρ β (

8 Following the aroach of section, we obtain the MSE exression for these roosed estimators as [ S + S ( ρ ] f MSE(ti i, ( i,,3,..., 0 (3. n where Y, Y + β (, 3 Y, + C 4 Y, + ρ Yβ ( 5, β( + C 6 YC C + β (, 7, C YC + ρ Yρ 8, ρ + C 9 Yβ ( β ( + ρ, 0 Yρ. ρ + β ( 4. Efficienc comarisons It is well known that under simle random samling without relacement (SSWO the variance of the samle mean is V( (4. f n S From (4. and (3., we have V( MSE(ti 0, i,,..., 0 ρ > S S i (4. When this condition is satisfied, roosed estimators are more efficient than the samle mean. 3

9 Now, we comare the MSE of the roosed estimators with the MSE of Naik and Guta [] estimator t. From (3. and (. we have MSE(t MSE(t 0, ( i,,..., 0 [ + K ] i S ρ i (4.3 S where K C ρ. C 5. Emirical Stud The data for the emirical stud is taken from natural oulation data set considered b Sukhatme and Sukhatme []: Number of villages in the circles and A circle consisting more than five villages N 89, Y 3.36, 0.36, ρ 0.766, C 0.604, C.9, β ( In the below table 5. ercent relative efficiencies (E of various estimators are comuted with resect to. 4

10 Table 5.: E of different estimators of Y with resect to. Estimator E (., 00 t.6 t 7.36 t t t t t t t t t From table 5., we observe that the roosed estimators t i ( i,..., 0 which uses some known values of oulation roortion erforms better than the usual samle mean and Naik and Guta [] estimator t. Conclusion: 5

11 We have suggested some ratio estimators for estimating Y which uses some known value of oulation roortion. For ractical uroses the choice of the estimator deends uon the availabilit of the oulation arameters. eferences Jhajj, H. S., Sharma, M. K. and Grover, L. K., A famil of estimators of oulation mean using information on auxiliar attribute. akistan Journal of Statistics, (, (006. Naik, V. D. and Guta,. C., A note on estimation of mean with known oulation roortion of an auxiliar character. Journal of the Indian Societ of Agricultural Statistics, 48 (, 5-58 (996. a, S. K. and Singh,. K., Difference-cum-ratio te estimators. Journal of the Indian Statistical Association, 9, 47-5 (98. Searls, D. T., The utilization of known coefficient of variation in the estimation rocedure. Journal of the American Statistical Association, 59, 5-6 (964. Searls, D. T. and Intaraanich,., A note on an estimator for the variance that utilizes the kurtosis. The American Statistician, 44, (990. Sen, A.., Estimation of the oulation mean when the coefficient of variation is known. Communications in Statistics Theor and Methods A, 7, (978. Singh, G. N., On the imrovement of roduct method of estimation in samle surves. Journal of the Indian Societ of Agricultural Statistics, 56 (3, (003. Singh H.. and Tailor,., Use of known correlation coefficient in estimating the finite oulation mean. Statistics in Transition, 6, (003. Singh H.., Tailor,., Tailor,. and Kakran, M. S., An imroved estimator of oulation mean using ower transformation. Journal of the Indian Societ of Agricultural Statistics, 58 (, 3-30 (004. 6

12 Singh, J., ande, B. N. and Hirano, K., On the utilization of a known coefficient of kurtosis in the estimation rocedure of variance. Annals of the Institute of Statistical Mathematics, 5, 5-55 (973. Sisodia, B. V. S. and Dwivedi, V. K., A modified ratio estimator using coefficient of variation of auxiliar variable. Journal of the Indian Societ of Agricultural Statistics, 33 (, 3-8 (98. Sukhatme,. V. and Sukhatme, B. V., Samling Theor of Surves with Alications. Iowa State Universit ress, Ames IOWA,970. Uadhaa, L. N. and Singh, H.., On the estimation of the oulation mean with known coefficient of variation. Biometrical Journal, 6, 95-9 (984. Uadhaa, L. N. and Singh, H.., Use of transformed auxiliar variable in estimating the finite oulation mean. Biometrical Journal, 4, (999. 7