Revista Colombiaa de Estadística Juio 2014, volume 37, o. 1, pp. 211 a 222 Geeralized Expoetial Type Estimator for Populatio Variace i Survey Samplig Estimadores tipo expoecial geeralizado para la variaza poblacioal e muestreo de ecuestas Amber Asghar 1,a, Aamir Saaullah 2,b, Muhammad Haif 2,c 1 Departmet of Mathematics & Statistics, Virtual Uiversity of Pakista, Lahore, Pakista 2 Departmet of Statistics, NCBA & E, Lahore, Pakista Abstract I this paper, geeralized expoetial-type estimator has bee proposed for estimatig the populatio variace usig mea auxiliary variable i siglephase samplig. Some special cases of the proposed geeralized estimator have also bee discussed. The expressios for the mea square error ad bias of the proposed geeralized estimator have bee derived. The proposed geeralized estimator has bee compared theoretically with the usual ubiased estimator, usual ratio ad product, expoetial-type ratio ad product, ad geeralized expoetial-type ratio estimators ad the coditios uder which the proposed estimators are better tha some existig estimators have also bee give. A empirical study has also bee carried out to demostrate the efficiecies of the proposed estimators. Key words: Auxiliary variable, Sigle-phase samplig, Mea square error, Bias. Resume E este artículo, de tipo expoecial geeralizado ha sido propuesto co el fi de estimar la variaza poblacioal a través de ua variables auxiliar e muestreo e dos fases. Alguos casos especiales del estimador medio y el sesgo del estimador geeralizado propuesto so derivados. El estimador es comprado teóricamete co otros dispoibles e la literatura y las codicioes bajos los cuales éste es mejor. U estudio empírico es llevado a cabo para comprar la eficiecia de los estimadores propuestos. Palabras clave: Iformació auxiliar, muestras e dos fases, error cuadrático medio, sesgo. a Lecturer. E-mail: zukhruf10@gmail.com b Lecturer. E-mail: chaamirsaaullah@yahoo.com c Associate professor. E-mail: drmiahaif@gmail 211
212 Amber Asghar, Aamir Saaullah & Muhammad Haif 1. Itroductio I survey samplig, the utilizatio of auxiliary iformatio is frequetly ackowledged to higher the accuracy of the estimatio of populatio characteristics. Laplace (1820) utilized the auxiliary iformatio to estimate the total umber of ihabitats i Frace. Cochra (1940) prescribed the utilizatio of auxiliary iformatio as a classical ratio estimator. Recetly, Dash & Mishra (2011) prescribed the few estimators with the utilizatio of auxiliary variables. Bahl & Tuteja (1991) proposed the expoetial estimator uder simple radom samplig without replacemet for the populatio mea. Sigh & Vishwakarma (2007), Sigh, Chauha, Sawa & Smaradache (2011), Noor-ul Ami & Haif (2012),Sigh & Choudhary (2012), Saaullah, Kha, Ali & Sigh (2012), Solaki & Sigh (2013b) ad Sharma, Verma, Saaullah & Sigh (2013) suggested expoetial estimators i sigle ad two-phase samplig for populatio mea. Estimatig the fiite populatio variace has great sigificace i various fields such as i matters of health, variatios i body temperature, pulse beat ad blood pressure are the basic guides to diagosis where prescribed treatmet is desiged to cotrol their variatio. Therefore, the problem of estimatig populatio variace has bee earlier take up by various authors. Gupta & Shabbir (2008) suggested the variace estimatio i simple radom samplig by usig auxiliary variables. Sigh & Solaki (2009, 2010) proposed the estimator for populatio variace by usig auxiliary iformatio i the preseces of radom o-respose. Subramai & Kumarapadiya (2012) proposed the variace estimatio usig quartiles ad their fuctios of a auxiliary variable. Solaki & Sigh (2013b) suggested the improved estimatio of populatio mea usig populatio proportio of a auxiliary character. Sigh & Solaki (2013) itroduced the ew procedure for populatio variace by usig auxiliary variable i simple radom samplig. Solaki & Sigh (2013a) ad Sigh & Solaki (2013) also developed the improved classes of estimators for populatio variace. Sigh et al. (2011), ad Yadav & Kadilar (2013) proposed the expoetial estimators for the populatio variace i sigle ad twophase samplig usig auxiliary variables. I this paper the motivatio is to look up some expoetial-type estimators for estimatig the populatio variace usig the populatio mea of a auxiliary variable. Further, it is proposed a geeralized form of expoetial-type estimators. The remaiig part of the study is orgaized as follows: The Sectio 2 itroduced the otatios ad some existig estimators of populatio variace i brief. I Sectio 3, the proposed estimator has bee itroduced, Sectio 4 is about the efficiecy compariso of the proposed estimators with some available estimators, sectio 5 ad 6 is about umerical compariso ad coclusios respectively. 2. Notatios ad some Existig Estimators Let (x i, y i ), i = 1, 2,..., be the pairs of sample observatios for the auxiliary ad study variables respectively from a fiite populatio of size N uder simple radom samplig without replacemet (SRSWOR). Let S 2 y ad s 2 y are variaces
Geeralized Expoetial Type Estimator for Populatio Variace 213 respectively for populatio ad sample of the study variable say y. Let X ad x are meas respectively for the populatio ad sample mea of the auxiliary variable say x. To obtai the bias ad mea square error uder simple radom samplig without replacemet, let us defie e 0 = s2 y Sy 2 Sy 2, e 1 = x X X s 2 y = Sy(1 2 + e 0 ), x = X(1 + e 1 ) where, e i is the samplig error, Further, we may assume that (1) E(e 0 ) = E(e 1 ) = 0 (2) Whe sigle auxiliary mea iformatio is kow, after solvig the expectatios, the followig expressio is obtaied as E(e 2 0) = δ 40, E(e2 1) = C2 x, E(e 0e 1 ) = δ 21C x where (3) µ pq (yi Ȳ )p (x i X) q δ pq = µ p/2 20 µq/2 02, ad µ pq = 1 N (p, q) be the o-egative iteger ad µ 02, µ 20 are the secod order momets ad δ pq is the momet s ratio ad C x = S x X is the coefficiet of variatio for auxiliary variable X. The ubiased estimator for populatio variace S 2 y = 1 N 1 N (Y i Ȳ )2 i is defied as ad its variace is t 0 = s 2 y (4) var(t 0 ) = s4 y δ 40 1 (5) Isaki (1983) proposed a ratio estimator for populatio variace i sigle-phase samplig as t 1 = s 2 Sx 2 y s 2 (6) x The bias ad the mea square error (MSE) of the estimator i (6), up to first order-approximatio respectively are Bias(t 1 ) = S2 y δ 04 δ 22 (7) MSE(t 1 ) S4 y δ 40 + δ 04 2δ 22 (8)
214 Amber Asghar, Aamir Saaullah & Muhammad Haif Sigh et al. (2011) suggested ratio-type expoetial estimator for populatio variace i sigle-phase samplig as S t 2 = s 2 2 y exp x s 2 x Sx 2 + s 2 (9) x The bias ad MSE, up to first order-approximatio is Bias(t 2 ) = S2 y δ04 8 δ 22 2 + 3 8 MSE(t 2 ) S4 y δ 40 + δ 04 4 δ 22 1 4 Sigh et al. (2011) proposed expoetial product type estimator for populatio variace i sigle-phase samplig as s t 3 = s 2 2 y exp x Sx 2 s 2 x + Sx 2 (12) The bias ad MSE, up to first order-approximatio is Bias(t 3 ) = S2 y δ04 8 + δ 22 2 5 8 MSE(t 3 ) S4 y δ 40 + δ 04 4 + δ 22 9 4 Yadav & Kadilar (2013) proposed the expoetial estimators for the populatio variace i sigle-phase samplig as t 4 = s 2 Sx 2 s 2 x y exp Sx 2 + (α 1)s 2 (15) x The bias ad MSE, up to first order-approximatio is Bias(t 4 ) = S2 y δ04 1 2α 2 (2α(1 λ) 1) MSE(t 4 ) S4 y (δ 40 1) + (δ 04 1) α 2 (1 2αλ) where, λ = δ22 1 δ 04 1 ad α = 1 λ. 3. Proposed Geeralized Expoetial Estimator Followig Bahl & Tuteja (1991), ew expoetial ratio-type ad product-type estimators for populatio variace are as X x t 5 = s 2 y exp (18) X + x (10) (11) (13) (14) (16) (17)
Geeralized Expoetial Type Estimator for Populatio Variace 215 t 6 = s 2 y exp x X x + X Equatios (18) ad (19) lead to the geeralized form as ( ) ( t EG = λ s 2 a x y exp α 1 = λ s X 2 y exp α + (a 1) x ) X x X + (a 1) x where the three differet real costats are 0 < λ 1, ad < α < ad a > 0. It is observed that for differet values of λ, α ad a i (20), we may get various expoetial ratio-type ad product-type estimators as ew family of t EG i.e. G = 0, 1, 2, 3, 4, 5. From this family, some examples of expoetial ratio-type estimators may be give as follows: It is oted that, for λ = 1, α = 0 ad a = a 0, t EG i (20) is reduced to (19) (20) t E0 = s 2 y exp(0) = s 2 y (21) which is a ubiased employig o auxiliary iformatio. For λ = 1, α = 0 ad a = 0, t EG i (20) is reduced to t E1 = s 2 y exp(1) (22) For λ = 1, α = 1 ad a = 2, t EG i (20) is reduced to X x t E2 = s 2 y exp = t 5 (23) X + x For λ = 1, α = 1 ad a = 1, t EG i (20) is reduced to X x t E3 = s 2 y exp X (24) Some example for expoetial product-type estimators may be give as follows: For λ = 1, α = 1 ad a = 2, t EG i (20) is reduced to { } X x t E4 = s 2 y exp = t 6 (25) X + x For λ = 1, α = 1 ad a = 1, t EG i (20) is reduced to { } X x t E5 = s 2 y exp X (26) 3.1. The Bias ad Mea Square Error of Proposed Estimator I order to obtai the bias ad MSE, (20) may be expressed i the form of e s by usig (1), (2) ad (3) as t EG = λ Sy(1 2 e 1 + e 0 ) exp α (27) 1 + (a 1)(1 + e 1 )
216 Amber Asghar, Aamir Saaullah & Muhammad Haif Further, it is assumed that the cotributio of terms ivolvig powers i e 0 ad e 1 higher tha two is egligible t EG λ Sy 2 1 + e 0 αe 1 a + α2 e 2 1 2a 2 αe 0e 1 (28) a I order to obtai the bias, subtract Sy 2 both sides ad takig expectatio of (28), after some simplificatio, we may get the bias as Bias(t EG ) S2 y λ {1 + α2 2a 2 C2 x α } a δ 21C x Sy 2 (29) Expadig the expoetials ad igorig higher order terms i e 0 ad e 1, we may have o simplificatio { t EG Sy 2 λ s 2 y 1 + e 0 αe } 1 1 (30) a Squarig both sides ad takig the expectatio we may get the MSE of (t EG ) from as (30) or MSE(t EG ) S4 y MSE(t EG ) S4 y λ {1 2 + (δ 40 1) 2 αa } δ 21C x + α2 a 2 C2 x + (1 2λ) λ 2 { 1 + (δ 40 1) 2ωδ 21 C x + ω 2 C 2 x} + (1 2λ) where, ω = α a, The MSE (t EG) is miimized for the optimal values of λ ad ω as, ω = δ 21 (C x ) 1 ad λ = (δ 40 δ 2 21) 1. The miimum MSE (t EG ) is obtaied as MSE mi (t EG ) S4 y 1 1 δ 40 δ 2 21 O substitutig the optimal values of λ = (δ 40 δ21) 2 1, α ad a ito (20), we may get the asymptotically optimal estimator as s 2 y δ21 ( t asym = δ 40 δ21 2 exp X x) (34) X + (C x 1) x The values of λ, α ad a ca be obtaied i prior from the previous surveys, for case i poit, see Murthy (1967), Ahmed, Rama & Hossai (2000), Sigh & Vishwakarma (2008), Sigh & Karpe (2010) ad Yadav & Kadilar (2013). I some situatios, for the practitioer it is ot possible to presume the values of λ, α ad a by employ all the resources, it is worth sesible to replace λ, α ad a i (20) by their cosistet estimates as (31) (32) (33) ˆω = ˆ δ 21 ( ˆ C x ) 1 ad ˆλ = ( ˆ δ 40 ˆ δ 2 21 ) 1 (35) ˆ δ 21, ad Ĉ respectively are the cosistet estimates of δ 21, ad C x.
Geeralized Expoetial Type Estimator for Populatio Variace 217 As a result, the estimator i (34) may be obtaied as ˆt asym = s 2 y δˆ 40 δ ˆ 21 2 exp δ21 ˆ ( X x) X + ( C ˆ x 1) x Similarly the MSE (t EG ) i (33) may be give as, MSE mi (ˆt asym ) s4 y 1 1 δˆ 40 δ ˆ 21 2 (36) (37) Thus, the estimator ˆt asym, give i (36), is to be used i practice. The bias ad MSE expressio for the ew family of t EG, ca be obtaied by puttig differet values of λ, α ad a i (29) ad (31) as Bias(t E2 ) S2 y 1 8 C2 x 1 2 δ 21C x Bias(t E3 ) S2 y 1 2 C2 x δ 21 C x Bias(t E4 ) S2 y 1 8 C2 x + 1 2 δ 21C x Bias(t E5 ) S2 y 1 2 C2 x + δ 21 C x MSE(t E2 ) S4 y MSE(t E3 ) S4 y MSE(t E4 ) S4 y MSE(t E5 ) S4 y (δ 40 1) δ 21 C x + 1 4 C2 x (δ40 1) 2δ 21 C x + C 2 x (δ 40 1) + δ 21 C x + 1 4 C2 x (δ40 1) + 2δ 21 C x + C 2 x 4. Efficiecy Comparisio of Proposed Estimators with some Available Estimators The efficiecy comparisos have bee made with the sample variace (t 0 ), Isaki (1983) ratio estimator (t 1 ), Sigh et al. (2011) ratio (t 2 ), ad product (t 3 ), estimators ad Yadav & Kadilar (2013) ratio (t 4 ), estimator usig (5),(8),(11),(14) ad (17) respectively with the proposed geeralized estimator ad class of proposed estimators. (38) (39) (40) (41) (42) (43) (44) (45)
218 Amber Asghar, Aamir Saaullah & Muhammad Haif MSE (t EG ) < Var (t 0 ) if δ 40 + 1 f 2 > 1 (46) MSE (t EG ) < MSE (t 1 ) if δ 40 + δ 04 2δ 22 + 1f > 1 (47) MSE (t EG ) < MSE (t 2 ) if δ 40 + δ 04 4 δ 22 1 4 + 1 f > 1 (48) MSE (t EG ) < MSE (t 3 ) if δ 40 + δ 04 4 + δ 22 9 4 + 1 f > 1 (49) MSE (t EG ) < MSE (t 4 ) if f(d δ 40) (δ 22 1) 2 (d f δ 40 δ21 2 ) > 1 (50) MSE (t E2 ) < Var (t 0 ) if 4 δ 21 > 1 C x MSE (t E2 ) < MSE (t 1 ) if 4(δ 40 2δ 22 + δ 21 C x + 1) Cx 2 > 1 (51) (52) MSE (t E2 ) < MSE (t 2 ) if (δ 40 4δ 22 + 4δ 21 C x + 3) Cx 2 > 1 (53) MSE (t E3 ) < Var (t 0 ) if 2 δ 21 > 1 C x MSE (t E3 ) < MSE (t 1 ) if (δ 04 2δ 22 + 2δ 21 C x + 1) Cx 2 > 1 (54) (55)
Geeralized Expoetial Type Estimator for Populatio Variace 219 MSE (t E3 ) < MSE (t 2 ) if ( δ04 4 δ 22 + 2δ 21 C x + 3 4 ) Cx 2 > 1 (56) MSE (t E4 ) < Var (t 0 ) if 4 δ 21 > 1 C x MSE (t E4 ) < MSE (t 3 ) if (δ 04 4δ 22 4δ 21 C x 1) Cx 2 > 1 (57) (58) MSE (t E5 ) < Var (t 0 ) if 2 δ 21 > 1 C x MSE (t E5 ) < MSE (t 3 ) if ( δ04 4 δ 22 2δ 21 C x 1 4 ) Cx 2 > 1 (59) (60) where f = δ 40 δ 2 21 ad d = δ 40 δ 04 δ 04 + 1. Whe the above coditios are satisfied the proposed estimators are more efficiet tha t 0, t 1, t 2, t 3 ad t 4. 5. Numerical Compariso I order to examie the performace of the proposed estimator, we have take two real populatios. The Source, descriptio ad parameters for two populatios are give i Table 1 ad Table 2 Table 1: Source ad Descriptio of Populatio 1 & 2. Populatio Source Y X 1 Murthy (1967, pg. 226) output umber of workers 2 Gujarati (2004, pg. 433) average (miles per gallo) top speed(miles per hour) The compariso of the proposed estimator has bee made with the ubiased estimator of populatio variace, the usual ratio estimator due to Isaki (1983), Sigh et al. (2011) expoetial ratio ad product estimators ad Yadav & Kadilar (2013) geeralized expoetial-type estimator. Table 3 shows the results of Percetage Relative Efficiecy (PRE) for Ratio ad Product type estimators. These estimators are compared with respect to sample variace.
220 Amber Asghar, Aamir Saaullah & Muhammad Haif Table 2: Parameters of Populatios. Parameter 1 2 N 25 81 25 21 Ȳ 33.8465 2137.086 X 283.875 112.4568 C y 0.3520 0.1248 C x 0.7460 0.4831 ρ yx 0.9136-0.691135 δ 40 2.2667 3.59 δ 21 0.5475 0.05137 δ 04 3.65 6.820 δ 22 2.3377 2.110 where ρ yx is the correlatio betwee the study ad auxiliary variable. Table 3: Percet Relative Efficiecies (PREs) for Ratio ad Product type estimators with respect to sample variace (t 0). Estimator Populatio 1 Populatio 2 t 0 = s 2 y 100 100 t 1 102.05 * t 2 214.15 * t 3 * 86.349 t 4 214.440 108.915 t E2 127.04 * t E3 125.898 * t E4 * 96.895 t E5 * 90.145 t EG 257.371 359.123 * shows the data is ot applicable 6. Coclusios Table 3 shows that the proposed geeralized expoetial-type estimator (t EG ) is more efficiet tha the usual ubiased estimator (t 0 ), Isaki (1983) ratio estimator, Sigh et al. (2011) expoetial ratio ad product estimators ad Yadav & Kadilar (2013) geeralized expoetial-type estimator. Further, it is observed that the class of expoetial-type ratio estimators t E2, ad t E3, are more efficiet tha the usual ubiased estimator ad Isaki (1983) ratio estimator. Furthermore, it is observed that the class of expoetial-type product estimators t E4 ad t E5, are more efficiet tha Sigh et al. (2011) expoetial product estimator.
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