Erciyes Üiversitesi Fe Bilimleri Estitüsü Dergisi Cilt 36, Sayı 3, 2020 Erciyes Uiversity Joural of Istitue Of Sciece ad Techology Volume 36, Issue 3, 2020 Improvemet i Expoetial Estimators of Populatio Mea usig Iformatio o Auxiliary Attribute Tolga ZAMAN 1 * * 1 Çakırı Karateki Uiversity, Faculty of Sciece, Departmet of Statistics, ÇANKIRI (Alıış / Received: 20.08.2019, Kabul / Accepted: 03.03.2020, Olie Yayılama / Published Olie: 25.12.2020) Keywords Expoetial Estimators, Simple Radom Samplig, Mea Square Error, Auxiliary Attribute, Efficiecy Abstract: This paper proposes ew expoetial estimators combiig ratio estimators for estimate populatio mea of study variable usig iformatio about populatio proportio possessig certai attributes. It is obtaied mea square error (MSE) equatios for all proposed ratio expoetial estimators ad is show that all proposed expoetial estimators are always more efficiet tha the ratio estimators. I additio, these results are supported by a applicatio with origial data sets. Yardımcı Niteliğe İlişki Bilgi Kullaılarak Kitle Ortalamasıı Üstel Tahmi Edicilerideki Gelişme Aahtar Kelimeler Üstel Tami Ediciler, Basit Ragele Örekleme, Hata Kareler Ortalama Yardımcı Özellik Etkilik Öz: Bu makale, belli özelliğe sahip ola kitle oraı hakkıdaki bilgiyi kullaarak çalışma değişkeii ortalama tahmii içi öerile tahmi edicileri birleştire yei üstel tahmiciler öermektedir. Öerile tüm üstel tahmi ediciler içi hata kareler ortalaması (HKO) deklemleri elde edilmiştir ve öerile bütü üstel tahmi edicileri, ora tahmicileride her zama daha verimli olduğu gösterilmiştir. Ek olarak, bu souçlar orijial veri setleri içere bir uygulama tarafıda desteklemektedir. *İlgili Yazar, email: tolgazama@karateki.edu.tr 1. Itroductio There are may situatio whe auxiliary iformatio is available i the form of attributes. For example sex is a good auxiliary attribute while dealig with height, ad the breed of a cow is a good auxiliary attribute while estimatig milk productio [1], crop variety is used as a auxiliary attribute i estimatig the yield of wheat [2], etc. There are some recet studies o the estimators usig the iformatio of the auxiliary attribute i Literature, such as, Shabbir ad Gupta [3], Koyucu [4], Malik ad Sigh [5], Zama [6, 7] ad Zama ad Kadilar [8] The Naik ad Gupta [1] estimator for the populatio mea Y of the variate of study, which make use of iformatio regardig the populatio proporito posseessig certai attribute, is defied by y NG = P (1.1) y p Let y i be ith characteristic of the populatio ad φ i is the case of possessig certai attributes. If ith uit has the desired characteristic, it takes the value 1, if ot the the value 0. That is; 1, if ith uit of the populatio possesses attribute φ i = { 0, otherwise where y the sample mea of the study variable ad a = i=1 φ i be the the total cout of the uits that possess certai attribute sample. p = a shows the ratio of these uits ad it is assumed that the populatio proportio P of the form of attribute φ is kow. 325
Improvemet i Expoetial Estimators of Populatio Mea usig Iformatio o Auxiliary Attribute The MSE of the Naik ad Gupta [1] estimator is MSE(y NG Y 2 ( 2 2ρ pb + 2 ) (1.2) where, f = N ; N is the umber of uits i the populatio; is the populatio coefficiet of variatio the form of attribute ad is the populatio coefficiet of variatio of the study variable. Followig Bahl ad Tuteja [9], Zama ad Kadilar [10] proposed ratio expoetial estimators i order to estimate populatio mea of study variable y, usig iformatio about populatio proportio possessig certai attributes i simple radom samplig; (kp + l) (kp + l) t ZK = y exp [ (kp + l) + (kp + l) ] (1.3) where k( 0), l are either real umber or the fuctios of the kow parameters of the attribute such as, β 2 (φ) ve ρ pb. The followig table presets estimators of the populatio mea which ca be obtaied by suitable choice of costats k ad l Estimators Table 1. The Proposed Estimators by Zama ad Kadilar [10] P p t ZK1 = y exp ( P + p ) Sigh et al. (2007) estimator P p t ZK2 = y exp ( Values of k 1 0 P + p + 2β 2 (φ) ) 1 β 2 (φ) P p t ZK3 = y exp ( ) P + p + 2 1 P p t ZK4 = y exp ( ) P + p + 2ρ pb 1 ρ pb β 2 (φ)(p p) t ZK5 = y exp [ ] β 2 (φ)(p + p) + 2 β 2 (φ) (P p) t ZK6 = y exp [ (P + p) + 2β 2 (φ) ] β 2 (φ) (P p) t ZK7 = y exp [ ] (P + p) + 2ρ pb ρ pb ρ pb (P p) t ZK8 = y exp [ ] ρ pb (P + p) + 2 ρ pb β 2 (φ)(p p) t ZK9 = y exp [ ] β 2 (φ)(p + p) + 2ρ pb β 2 (φ) ρ pb ρ pb (P p) t ZK10 = y exp [ ρ pb (P + p) + 2β 2 (φ) ] ρ pb β 2 (φ) I Table 1,, β 2 (φ) ad ρ pb are, respectively, coefficiet of variatio belogig to ratio of uits possessig certai attributes, coefficiet of populatio kurtosis ad populatio correlatio coefficiet betwee ratio of uits possessig certai attributes ad study variable. y ad p are sample mea belogig to study variable ad sample proportio possessig certai attributes, respectively. The MSE ad bias of this ratio estimator is as follows; B(t ZKi Y (λ 2 i C 2 p λ i ρ pb ) (1.4) MSE(t ZKi Y 2 [λ 2 i C 2 p 2λ i ρ pb + C 2 y ], i = 2,,10 (1.5) where, λ 1 = 1 2 ; λ 2 = P 2(P+β 2 (φ)) ; λ 3 = P 2(P+ ) ; λ 4 = P 2(P+ρ pb ) ; λ 5 = β 2 (φ)p 2(β 2 (φ)p+ ) ; l 326
Improvemet i Expoetial Estimators of Populatio Mea usig Iformatio o Auxiliary Attribute λ 6 = P 2( P+β 2 (φ)) ; λ 7 = 2. Suggested Estimators P 2( P+ρ pb ) ; λ 8 = ρ pbp 2(ρ pb P+ ) ; λ 9 = β 2 (φ)p 2(β 2 (φ)p+ρ pb ) ; λ 10 = ρ pb P 2(ρ pb P+β 2 (φ)) Followig Kadilar ad Cigi [11], it is proposed the expoetial estimators combiig ratio expoetial estimators tzk 1 ad t ZKi (i = 2,3,,10) as follows; t ZKi = ωt 1 + (1 ω)zt i ; (i = 2,3,,10) (2.1) where ω is a real costat to be determied such that the MSE of t ZKi is miimum. It is obtaied the MSE ad bias equatios for these proposed estimators usig Taylor series as; (for details, please see the Appedix A) where, λ 2 = MSE(t ZKi Y 2 [C 2 p ( ω 2 2 + λ i ωλ i ) 2ρ pb ( ω 2 + λ i ωλ i ) + C 2 y ] (2.2) P ; λ 2(P+β 2 (φ)) 3 = P ; λ P 2(P+ ) 4 = ; λ 2(P+ρ pb ) 5 = β 2 (φ)p ; λ 2(β 2 (φ)p+ ) 6 = P 2( P+β 2 (φ)) λ 7 = P 2( P+ρ pb ) ; λ 8 = ρ pbp 2(ρ pb P+ ) ; λ 9 = β 2 (φ)p 2(β 2 (φ)p+ρ pb ) ; λ 10 = ρ pb P 2(ρ pb P+β 2 (φ)) We ca have the optimal values of ω (2.2) by followig equatios: (for details, please see the Appedix B). ω opt = 2 (ρ pb λ i ) (1 2λ i ) (2.3) It is obtaied miimum MSE of the proposed estimators usig the optimal equatios of ω i (2.3). All proposed estimators have the same miimum MSE as follows: MSE mi (t ZKi Y 2 [C 2 y (1 ρ 2 pb )] ; i = 2,3,,10 (2.4) 3. Efficiecy Comparisos I this sectio, it is compared the MSE of the proposed expoetial estimators i (2.4) with the MSE of the Naik- Gupta [1] estimator, the ratio expoetial estimator suggested by Sigh et al. [12] ad ratio estimators listed i Table1. Comparig the MSE of the proposed estimators, give i (2.1), with the ratio estimator suggested by Naik-Gupta [1], give i (1.1), we have the followig coditios; MSE(t ZKi ) < MSE(y NG ) i = 2,3,,10 1 f Y 2 [C 2 y (1 ρ 2 pb )] < 1 f Y 2 (C 2 y 2ρ pb + C 2 p ) (ρ pb ) 2 > 0 (3.1) Whe the coditios (3.1) is satisfied, the proposed expoetial estimators are more efficiet tha the ratio estimator suggested by Naik-Gupta [1]. Comparig the MSE of the proposed expoetial estimators, give i (2.1), with the MSE of the ratio expoetial estimator suggested by Sigh et al. [12], give i (1.3), we have the followig coditios; MSE mi (t ZKi ) < MSE(t ZK1 ) 327
Improvemet i Expoetial Estimators of Populatio Mea usig Iformatio o Auxiliary Attribute 1 f Y 2 [ 2 (1 ρ pb 2 )] < 1 f Y 2 [ C 2 p 4 ρ pb + C 2 y ] (ρ pb 2 ) 2 > 0 (3.2) Whe the coditios (3.2) is satisfied, the proposed expoetial estimators are more efficiet tha the ratio estimator suggested by Sigh et al. [12]. Comparig the MSE of the proposed expoetial estimators, give i (2.1), with the MSE of the ratio expoetial estimator suggested by Zama ad Kadilar [10], give i Table 1, we have the followig coditios; MSE mi (t ZKi ) < MSE(t ZKi ) 1 f Y 2 [C 2 y (1 ρ 2 pb )] < 1 f Y 2 [λ 2 i C 2 p 2λ i ρ pb + C 2 y ] (λ i ρ pb ) 2 > 0 (3.3) It is iferred that all proposed expoetial estimators are more efficiet that all ratio estimators i give i Table 1 i all coditios, because the coditio give i (3.3) is always satisfied. 4. Numerical Illustratio It is used the teacher ad wdbc data sets to calculate efficiecy of estimators which are give i Table 2 ad Table 3. I this sectio, we use the data set i Zama et al. [13] i order to compare the efficiecies betwee the proposed estimators, give i (2.1), with the ratio estimators, give i Sectio 1, based o MSE equatios. The MSE of these estimators are computed as give i (1.2), (1.5) ad (2.3) ad these estimators are compared to each other with respect to their MSE values. The data is defied as followig; y = the umber of teachers 1, if the umber of teachers is more tha 60 φ i = { 0, otherwise Table 2. Populatio 1 Data Statistics N:111 Y: 29.279 λ 2 :0.0146 λ 6 :0.0382 λ 10 :0.0117 : 30 P: 0.117 λ 3 :0.0203 λ 7 :0.1441 β 2 (φ): 3.898 : 0.872 λ 4 :0.0640 λ 8 :0.0164 ρ pb : 0.797 : 2.758 λ 5 :0.0709 λ 9 :0.1819 As secod example, the data for the empirical study is take from populatio data set cosidered by Sukhatme [14] The data is defied as followig; y = Number of villages i the circles 1, if A circle cosistig more tha five vilages φ i = { 0, otherwise Table 3. Populatio 2 Data Statistics N:89 Y: 3.3596 λ 2 :0.0171 λ 6 :0.0433 λ 10 :0.0132 328
Improvemet i Expoetial Estimators of Populatio Mea usig Iformatio o Auxiliary Attribute : 20 P: 0.1236 λ 3 :0.0221 λ 7 :0.1508 β 2 (φ): 3.4917 : 0.6008 λ 4 :0.0695 λ 8 :0.0171 ρ pb : 0.766 : 2.6779 λ 5 :0.0694 λ 9 :0.1802 I Tables 2 ad 3, it is observed the statistics about the populatios. Note that the sample sizes as = 30, = 20 ad use simple radom samplig [15]. We would like to recall that sample size has o effect o efficiecy comparisos of estimators, as show i Sectio 3. Tablo4. MSE values of the Ratio Estimators MSE Estimator Populatio 1 Populatio 2 t NG 94.532 2.2168 t ZK1 t ZK2 t ZK3 t ZK4 t ZK5 t ZK6 t ZK7 t ZK8 t ZK9 t ZK10 Proposed 15.5403 14.7247 14.2948 11.3891 10.9827 13.0318 7.6304 14.5909 6.5614 14.9436 5.7840 0.4030 0.1404 0.1356 0.0981 0.0981 0.1171 0.0666 0.1404 0.0654 0.1442 0.0652 I Table 4, values of MSE, which are computed usig equatios preseted i Sectios 1 ad 2, are give. Whe we examie Table 4, it is observed that the proposed expoetial estimators have the smallest MSE value amog all ratio estimators give Sectio 1. This is a expected results, as metioed i Sectio 3. From the result of this umerical illustratio, it is deduced that all proposed expoetial estimators are more efficiet tha all ratio estimators for this data set. 5. Coclusios It is developed ew expoetial estimators combiig ratio estimators cosidered is Sectio 1 usig iformatio about populatio proportio possessig certai attributes i simple radom samplig ad obtaied miimum MSE equatio for proposed estimators. Theoretically, It is demostrated that all proposed expoetial estimators are always more efficiet tha all ratio estimators give Sectio 1. These theoretical results are supported by a applicatio with origial data sets. 329
Improvemet i Expoetial Estimators of Populatio Mea usig Iformatio o Auxiliary Attribute Refereces [1] Naik, V.D., Gupta, P.C. 1996. A ote o estimatio of mea with kow populatio proportio of a auxiliary character. Jour. Id. Soc. Agr. Stat., 48 (2), 151-158. [2] Jhajj, H. S., Sharma, M. K., Grover, L. K. 2006. A family of estimators of populatio mea usig iformatio o auxiliary attribute. Pakista Joural of Statistics-All Series-, 22(1), 43. [3] Shabbir, J., Gupta, S. 2010. Estimatio of the fiite populatio mea i two phase samplig whe auxiliary variables are attributes. Hacettepe Joural of Mathematics ad Statistics, 39(1), 121-129. [4] Koyucu, N. 2012. Efficiet estimators of populatio mea usig auxiliary attributes. Applied Mathematics ad Computatio, 218(22), 10900-10905. [5] Malik, S., Sigh, R. 2013. A improved estimator usig two auxiliary attributes. Applied mathematics ad computatio, 219(23), 10983-10986. [6] Zama, T. 2018. New family of estimators usig two auxiliary attributes. Iteratioal Joural of Advaced Research I Egieerig & Maagemet (IJAREM), 4(11), 11-16. [7] Zama, T. 2018. Modified Ratio Estimators Usig Coefficiet of Skewess of Auxiliary Attribute. Iteratioal Joural of Moder Mathematical Scieces, 16(2), 87-95. [8] Zama, T., Kadilar, C. 2019. New class of expoetial estimators for fiite populatio mea i two-phase samplig. Commuicatios i Statistics-Theory ad Methods, 1-16. [9] Bahl, S., Tuteja, R.K. 1991. Ratio ad product type expoetial estimator, Iformatio ad Optimizatio Scieces XII (I), 159-163, 1991. [10] Zama, T., Kadilar, C. 2019. Novel family of expoetial estimators usig iformatio of auxiliary attribute. Joural of Statistics ad Maagemet Systems, 1-11. [11] Kadilar C., Cigi, H. 2006. Improvemet i estimatig the populatio mea i simple radom samplig. Applied Mathematics Letters. 19. 75-79 [12] Sigh, R., Chauha, P., Sawa, N., Smaradache, F. 2007. Ratio-product type expoetial estimator for estimatig fiite populatio mea usig iformatio o auxiliary attribute. i Auxiliary Iformatio ad a Priori Values i Costructio of Improved Estimators edited by Sigh, R., Chauha, P., Sawa, N. ad Smaradache, F., Reaissace High Press, 18-32. [13] Zama, T., Saglam, V., Sagir, M., Yucesoy, E., Zobu, M. 2014. Ivestigatio of some estimators via Taylor series approach ad a applicatio. America Joural of Theoretical ad Applied Statistics, 3(5), 141-147. [14] Sukhatme, P.V.1957. Samplig theory of surveys with applicatios. The Idia Society of Agrcultural Statistics, New Delhi, pp. 279-280 [15] Çıgı, H. 1994. Samplig Theory, Akara: Hacettepe Uiversity Press. [16] Wolter, K.M. 1985. Itroductio to Variace Estimatio, (Spriger-Verlag). Appedices Appedix A. I geeral, Taylor series method for k variables ca be give as; k where ad h(x 1, x 2,, x k ) = h(x 1, X 2,, X k) + d j (x j X j) + R k (X k, α) + O k k k d j = h(x 1, x 2,, x k ) α j R k (X k, α) = 1 2 h(x 1, X 2,, X k) (x j X j)(x i 2! X i) + O k X ix j j=1 j=1 where O k represets the terms i the expasio of the Taylor series of more tha the secod degree [9]. Whe it is omitted the term R k (X k, α), we obtai Taylor series method for two variables as follows; h(p, y ) h(p, Y ) j=1 h(c, d) (p P) + c P,Y h(c, d) d (y Y ) Y, P 330
Improvemet i Expoetial Estimators of Populatio Mea usig Iformatio o Auxiliary Attribute where, h(p, y ) = t ZKi ad h(p, Y ) = Y MSE equatios of the proposed estimators give i (2.1) compute as follows: t ZKi Y (ωy exp [ P p ] + (1 ω)y exp [(kp+l) (kp+l) P+p p + ( ωy 2P (kp+l)+(kp+l) ]) P,Y (ωy exp [ P p ] + (1 ω)y exp [(kp+l) (kp+l) P+p y (p P) (kp+l)+(kp+l) ]) Y k + (1 ω) ( )) (p P) + (y Y ) 2kP + 2l P,Y (y Y ) E(t ZKi Y ) 2 [( ω2 2 Y 4P 2 + (1 Y 2 k 2 ω)2 ( (2kP + 2l) 2) + 2 ( ωy ) (1 ω) ( Y k )) V(p) 2P 2kP + 2l 2 ( ωy 2P + Y k 2kP + 2l ωy k ) Cov(p, y ) + V(y )] 2kP + 2l Y 2 [( ω2 4P 2 + (1 k 2 2ω(1 ω)k ω)2 ( (2kP + 2l) 2) + 2P(2kP + 2l) ) V(p) 2 Y ( ω 2P + k 2kP + 2l ωk ) Cov(p, y ) 2kP + 2l + V(y ) Y ] 2 Appedix B 1 f Y 2 [( ω2 4 + (1 ω)2 λ 2 i + ω(1 ω)λ i ) C 2 p 2ρ pb ( ω 2 + λ i ωλ i ) + C 2 y ] MSE(t ZKi Y 2 [( ω 2 2 + λ i ωλ i ) C 2 p 2ρ pb ( ω 2 + λ i ωλ i ) + C 2 y ] ; i = 2,3,,10 (A. 1) It has the optimal values of α by followig equatios: MSE(t ZKi ) = 1 f ω Y 2 [2 ( ω 2 + λ i ωλ i ) ( 1 2 λ i) C 2 p 2ρ pb ( 1 2 λ i)] = 0 ( ω 2 + λ i ωλ i ) ( 1 2 λ i) 2 = ρ pb ( 1 2 λ i) ω 2 + λ i ωλ i = ρ pb ω opt = 2 (ρ pb λ i ) (1 2λ i ) (B. 1) It is obtaied miimum MSE of the proposed estimators usig the optimal equatios of ω opt i (B. 1). MSE mi (t ZKi Y 2 [( ω 2 opt 2 + λ i ω opt λ i ) C 2 p 2ρ pb ( ω opt 2 + λ i ω opt λ i ) + C 2 y ] (B. 2) ω opt 2 + λ i ω opt λ i = ω opt ( 1 2 λ i) + λ i = = ρ pb λ i 2λ i ρ pb + 2λ i 2 ρ pb λ i 2λ i ρ pb + 2λ i 2 + λ i 2λ i 2 1 2λ i 1 2λ i + λ i 331
Improvemet i Expoetial Estimators of Populatio Mea usig Iformatio o Auxiliary Attribute Usig (B. 3) i (B. 1), we have = ρ pb 2λ C i ρ pb ρ p b (1 2λ p C i ) p = = ρ 1 2λ i 1 2λ pb (B. 3) i MSE mi (t ZKi 2 Y C 2 y C [(ρ pb ) C 2 y 2ρ pb (ρ pb ) + C 2 p ] p MSE mi (t ZKi Y 2 [C 2 y (1 ρ 2 pb )] ; i = 2,3,,10 (B. 3) I this paper, It is used the fuctio give i (B.3) Equatio to calculate MSE values of the proposed expoetial estimators. 332