STATISTICS IN TRANSITION new series, Jne 06 63 STATISTICS IN TRANSITION new series, Jne 06 Vol. 7, No., pp. 63 8 SOME EFFECTIVE ESTIMATION PROCEDURES UNDER NONRESPONSE IN TWOPHASE SUCCESSIVE SAMPING G. N. Singh, M. Khetan, S. Marya 3 ABSTRACT This work is designed to assess the effect of nonresponse in estiation of the crrent poplation ean in twophase sccessive sapling on two occasions. Sbsapling techniqe of nonrespondents has been sed and exponential ethods of estiation nder twophase sccessive sapling arrangeent have been proposed. Properties of the proposed estiation procedres have been exained. Epirical stdies are carried ot to jstify the sggested estiation procedres and sitable recoendations have been ade to the srvey practitioners. Key words: nonresponse, sccessive sapling, twophase sapling, ean sqare error, i replaceent strategy.. Introdction In collecting inforation throgh saple srveys, there ay arise neros probles; one of the is nonresponse. It freqently occrs in ail srveys, soe of the selected nits ay refse to retrn back the filled in qestionnaires. An estiate obtained fro sch an incoplete srvey ay be isleading, especially when the respondents differ significantly fro the nonrespondents, becase the estiate ay be a biased one. Hansen and Hrwitz (946) sggested a techniqe of sbsapling of nonrespondents to handle the proble of nonresponse. Cochran (977) and Fabian and Hynshik (000) extended the Hansen and Hrwitz (946) techniqe for the sitation when besides the inforation on the character nder stdy, inforation on axiliary character is also available. Recently, Chodhary et al. (004) Singh and Priyanka (007), Singh and Kar Departent of Applied Matheatics, Indian School of Mines, Dhanbad 86004. Eail: gnsingh_is@yahoo.co. Departent of Applied Matheatics, Indian School of Mines, Dhanbad 86004. Eail: kti.khetan@gail.co. 3 Departent of Applied Matheatics, Indian School of Mines, Dhanbad 86004.
64 G. N. Singh, M. Khetan, S. Marya: Soe effective estiation... (009, 00), Singh et al. (0) and Garcia engo (03) sed the Hansen and Hrwitz (946) techniqe for the estiation of poplation ean on the crrent occasion in twooccasion sccessive sapling. If the stdy character of a finite poplation is sbject to change over tie, a single occasion srvey is insfficient. For sch a sitation sccessive sapling provides a strong tool for generating reliable estiates over different occasions. Sapling on sccessive occasions was first considered by Jessen (94) in the analysis of far data. The theory of sccessive (rotation) sapling was frther extended by Patterson (950), Eckler (955), Rao and Graha (964), Sen (97, 97, 973), Gpta (979), Das (98) and Singh and Singh (00) aong others. In saple srveys, the se of axiliary inforation has shown its significance in iproving the precision of estiates of nknown poplation paraeters. When the poplation paraeters of axiliary variable are nknown before start of the srvey we go for twophase (doble) sapling strctre to provide the reliable estiates of the nknown poplation paraeters. Singh and Singh (965) sed twophase (doble) sapling for stratification on sccessive occasions. Recently, Singh and Prasad (0) and Singh and Hoa (04) applied twophase sapling schee with sccess in the estiation of the crrent poplation ean in twooccasion sccessive sapling. The ai of the present work is to stdy the effect of nonresponse when it occrs on varios occasions in twooccasion sccessive (rotation) sapling. Recently, Bahl and Tteja (99), Singh and Vishwakara (007) and Singh and Hoa (03) sggested exponential type estiators of poplation ean nder different realistic sitations. Motivated with the doinating natre of these estiators and tilizing the inforation on a stable axiliary variable with nknown poplation ean over both occasions, soe new exponential ethods of estiation have been proposed to estiate the crrent poplation ean in twophase sccessive (rotation) sapling arrangeent.. The Hansen and Hrwitz (946) techniqe of sbsapling of nonrespondents has been sed to redce the negative effects of nonresponse. Properties of the proposed estiators are exained and their epirical coparisons are ade with the siilar estiator and with the natral sccessive sapling estiator when coplete response is observed on both occasions. Reslts are interpreted and followed by sitable recoendations.. Saple strctres and sybols et U = (U, U,,,, U N) be the finite poplation of N nits, which has been sapled over two occasions. The character nder stdy is denoted by x(y) on the first (second) occasion respectively. It is assed that the nonresponse occrs only in stdy variable x(y) and inforation on an axiliary variable z (stable over occasion), whose poplation ean is nknown on both occasions, is available and positively correlated with stdy variable. Since we have assed that non
STATISTICS IN TRANSITION new series, Jne 06 65 response occrs on both occasions, the poplation can be divided into two classes those who will respond at the first attept and those who will not on both occasions. et the sizes of these two classes be N and N respectively on the first occasion and the corresponding sizes on the crrent (second) occasion be N and N, respectively. To frnish a good estiate of the poplation ean of the axiliary variable z on the first occasion, a preliinary saple of size n' is drawn fro the poplation by the siple rando sapling withot replaceent (SRSWOR) ethod, and inforation on z is collected. Frther, a secondphase saple of size n ( n' > n) is drawn fro the firstphase (preliinary) saple by the SRSWOR ethod and henceforth the inforation on the stdy character x is gathered. We asse that ot of selected n nits, n nits respond and n nit do not respond. et n h denote the size of sbsaple drawn fro the nonresponding nits in the saple on first occasion. A rando sbsaple s of = n nits is retained (atched) fro the responding nits on the first occasion for its se on the second occasion nder the assption that these nits will give coplete response on the second occasion as well. Once again, to frnish a fresh estiate of the poplation ean of the axiliary variable z on the second occasion, a preliinary (firstphase) saple of size ' is drawn fro the nonsapled nits of the poplation by the SRSWOR ethod and inforation on z is collected. A secondphase saple of size = (n) = n ( ' > ) is drawn fro the firstphase (preliinary) saple by the SRSWOR ethod and the inforation on stdy variable y is gathered. It is obvios that the saple size on the second occasion is also n. Here λ and (λ+ =) are the fractions of the atched and fresh saples, respectively, on the second (crrent) occasion. We asse that in the natched portion of the saple on the crrent (second) occasion nits respond and nits do not respond. et h denote the size of the sbsaple drawn fro the nonresponding nits in the fresh saple (s ) on the crrent (second) occasion. Hence, onwards, we se the following notations: X, Y, Z: The poplation eans of the variables x, y and z respectively. y, y, y, y, x h n, x n, x n, x h, z, z : The saple eans of the respective variables based on the saple sizes shown in sffices. ' ' z n, z : The saple eans of the axiliary variable z and based on the firstphase saples of sizes ' and n' respectively. ρ, ρ, ρ : The poplation correlation coefficients between the variables yx xz yz shown in sffices. S, S, S : The poplation variances of the variables x, y and z respectively. x y z S,S x y : The poplation variances of the variables x and y respectively in the nonresponding nits of the poplation. C, C, C : The coefficients of variation of the variables x, y and x y z z respectively.
66 G. N. Singh, M. Khetan, S. Marya: Soe effective estiation... C, C x y : The coefficients of variation of the variables x and y in the nonresponding nits of the poplation. N W= : The proportion of nonresponding nits in the poplation at first N N occasion. W= : The proportion of nonresponding nits in the poplation on N the crrent (second) occasion. n f = and f = n h h. 3. Forlation of estiation strategy To estiate the poplation ean Y on the crrent (second) occasion, two different estiators are considered one estiator T based on saple s of size drawn afresh on the second occasion and the second estiator T based on the saple s of size, which is coon to both occasions. Since the nonresponse occrs in the saples s n and s, we have sed the Hansen and Hrwitz (946) techniqe to propose the estiators T and T. Hence, the estiators T and T for estiating the crrent poplation ean Y are forlated as ' ' zz xn x z n T = yexp ' and T = yexp z +z x n +x z nx n +n xn h y + yh x n = and y =. n Cobining the estiators T and T, finally we have the following estiator of poplation ean Y on the crrent (second) occasion T = φ T + φ T (3.) φ (0 φ ) is the nknown constant (scalar) to be deterined nder certain criterions. 4. Properties of the estiator T Since the estiators T and T are exponential type estiators, the poplation ean Y are biased, therefore the reslting estiator T defined in
STATISTICS IN TRANSITION new series, Jne 06 67 eqation (3.) is also a biased estiator of Y. The bias B (.) and the ean sqare error M (.) of the estiator T are derived p to the first order of approxiations sing the following transforations: y ( e )Y, y ( e )Y, y ( e )Y, x ( e )X, x ( e )X, 3 4 n 5 x ( e )X, x ( e )X, z ( e )Z, z ( e )Z, z ( e )Z, ' ' n 6 n 7 8 9 0 ' z n ( e )Z, sch that E(e i) = 0, e i < i =,, 3,,. Under the above transforations, the estiators T and T take the following fors: T =Y(+e 3)exp (e0 e 9) + (e 0+e 9) and T =Y(+e )(+e )(+e 8) exp (e7e 4) + (e 7+e 4) Ths, we have the following theores: (4.) (4.) Theore 4.. and Proof Bias of the estiator T to the first order of approxiations is obtained as B(T) = φb(t )+(φ)b(t ) (4.3) 3 B(T ) = Y C ' z ρyzcycz 8 3 C x + ρxzcxcz ρxycycx n 8 BT =Y (f) W C x + ' Cz ρyzc ycz 8 n n The bias of the estiator T is given by B(T) = E TY = φe(ty) + (φ) E(TY) = φb(t )+ φ B(T ) (4.4) B(T ) = E T Y and B(T ) = E TY.
68 G. N. Singh, M. Khetan, S. Marya: Soe effective estiation... Sbstitting the expressions of T, and T fro eqations (4.) and (4.) in eqation (4.4), expanding the ters binoially and exponentially, taking expectations and retaining the ters p to the first order of saple sizes, we have the expressions for the bias of the estiator T as described in eqation (4.3). Theore 4.. The ean sqare error of the estiator T to the first order of approxiations is obtained as M( T) = φ M T + φ M T +φ φ C (4.5) W(f ) M(T ) = E T Y = ρ + + S 4 N ' yz y +ρ ρ + M(T ) = E T Y = S and Proof xz yx n 4 N y (f) + ' ρ yz + W n 4 n (4.6) (4.7) Sy C = E TY TY =. (4.8) N It is obvios that the ean sqare error of the estiator T is given by M(T) = E TY = E φ T Y + φ T Y = φ E T Y + φ E T Y +φ φ E T Y T Y = φ M(T )+ φ M T +φ φ C (4.9) Sbstitting the expressions of T, and T fro eqations (4.)(4.) in eqation (4.9), expanding the ters binoially and exponentially, taking expectations and retaining the ters p to the first order of saple sizes, we have the expression of the ean sqare error of the estiator T as it is given in eqation (4.5).
STATISTICS IN TRANSITION new series, Jne 06 69 Reark 4.. The expression of the ean sqare error in the eqation (4.5) is derived nder the assptions (i) that the coefficients of variation of nonresponse class are siilar to that of the poplation, i.e. C x = C x and C y = C, and (ii) since x and y y are the sae stdy variable over two occasions and z is the axiliary variable correlated to x and y, looking at the stability natre of the coefficients of variation, viz. Reddy (978), the coefficients of variation of the variables x, y and z in the poplation are considered eqal, i.e. C x = C y = C z. 5. Mini ean sqare error of the estiator T Since the ean sqare error of the estiator T in eqation (4.5) is the fnction of nknown constant φ, it is iniized with respect to φ, and sbseqently the i vale of φ is obtained as Now, sbstitting the vale of ean sqare error of T as M(T )C φ =. (5.) M(T )+M(T )C φ in eqation (4.5), we get the i M(T ).M(T )C M(T) =. M(T )+M(T )C (5.) Frther, sbstitting the vales fro eqations (4.6)(4.8) in eqation (5.), we get the siplified vale of M (T) which is given below: a+a + a Sy M(T) =. (5.3) a+a + a n 3 6 5 4 a = ac+k f, a = ad+cbk f, a = bd, a = ca+kf, a = ab+dkf, a = b, 3 4 5 6 a = (f+ta 0), b= a 0 ++(f )W, c = td +c +f (f)w, 4 d = f+(t )d + (f)w, k =, a 0 = ρ yz, c = +ρxz ρ xy, d = ρ yz, 4 4 4 n n n n f =,f =, f =, t = and t ' =. ' N n n h h
70 G. N. Singh, M. Khetan, S. Marya: Soe effective estiation... 6. Opti replaceent strategy Since the ean sqare error of the estiator T given in eqation (5.3) is the fnction of (fractions of the saple to be drawn afresh at the second occasion), the i vale of is deterined to estiate the poplation ean Y with axi precision and lowest cost. To deterine the i vale of, we iniized the ean sqare error of the estiator T given in eqation (5.3) with respect to, which reslts in qadratic eqation in and the respective soltions of, say ˆ, are given below: p +p+p 3=0 (6.) ˆ = p ± p p p 3 p = aa 5a a 4, p = aa 6a 3a 4 and p 3 = aa 6a 3a 5. p (6.) Fro eqation (6.) it is obvios that real vales of ˆ exist iff the qantities nder sqare root are greater than or eqal to zero. For any cobinations of correlations ρ yx, ρ xz and ρ yz, which satisfy the conditions of real soltions, two real vales of ˆ are possible. Hence, while choosing the vales of ˆ, it shold be reebered that 0 ˆ. If both the vales of ˆ satisfy the stated condition, we chose the saller vale of ˆ as it will help in redcing the cost of the srvey. All other vales of are inadissible. Sbstitting the adissible vale of ˆ, say (0), fro eqation (6.) into eqation (5.3), we have the i vale of the ean sqare error of T, which is shown below: a+ a + a Sy M(T ) =. (6.3) a+ a + a n (0) (0) 0 3 (0) (0) 6 5 4 7. Soe special cases Case : When nonresponse occrs only at first occasion When nonresponse occrs only at first occasion, the estiator for the ean Y on the crrent occasion ay be obtained as T = φ ξ + φ T (7.)
STATISTICS IN TRANSITION new series, Jne 06 7 ' zz ξ = y exp z ' and T is defined in section 3, φ (0 φ ) is +z the nknown constant (scalar) to be deterined nder certain criterions. 7.. properties of the estiator T Since the estiator T is exponential type estiator, it is biased for the poplation ean Y. The bias B (.) and the ean sqare error M (.) of the estiator T are derived p to the first order of approxiations siilar to that of the estiator T. Theore 7.. The bias of the estiator T to the first order of approxiations is obtained as B(T ) = φ B(ξ )+(φ )B(T ) (7.) 3 B(ξ ) = Y C ' z ρyzcycz 8 and B(T ) is defined in section 4. Theore 7.. The ean sqare error of the estiator T to the first order of approxiations is obtained as M( T ) = φ M ξ + φ M T +φ φ C (7.3) M(ξ ) = E ξ Y = ρ + S 4 N S y C = E ξ YT Y = N and M(T ) is defined in section 4. ' yz y (7.4) (7.5) Since the ean sqare error of the estiator T in eqation (7.3) is the fnction of nknown constant φ, it is iniized with respect to φ, and sbseqently the i vale of φ is obtained as M(T )C φ =. (7.6) M(ξ )+M(T )C
7 G. N. Singh, M. Khetan, S. Marya: Soe effective estiation... Now sbstitting the vale of φ in eqation (7.6), we get the i ean sqare error of the estiator T as M(ξ ).M(T )C M(T ) =, (7.7) M(ξ )+M(T )C Frther, sbstitting the vales fro eqations (4.7), (7.4) and (7.5) in eqation (7.7), we get the siplified vale of M (T ), which is given below: a+a + a Sy M(T ) = (7.8) a+a + a n 3 6 5 4 a = ad+cb k f, a 3 = b d, a 5 = ab +dkf, a 6 = b, b = a0, a and a 4 are defined in section 5. To deterine the i vales of, we iniized the ean sqare error of the estiator T given in eqation (7.8) with respect to, which reslts in qadratic eqation in and the respective soltions of below: p +p +p 3=0 ˆ = p ± p p p 3 p, say p = aa5a a 4, p = aa6a 3a 4 and p 3 = aa6a 3a 5. ˆ, are given (7.9) (7.0) (0) Sbstitting the adissible vale of ˆ, say, fro eqation (7.0) into eqation (7.8), we have the i vale of the ean sqare error of the estiator T, which is shown below: (0) (0) 0 a+ 3 a S + a y M(T ) =. (7.) a+ a + a n (0) (0) 6 5 4 Case : When nonresponse occrs only at second occasion When nonresponse occrs only at crrent (second) occasion, the estiator for the ean Y at crrent occasion ay be obtained as ' n n ξ = yexp x n +x z T = φ T + φ ξ (7.) x x z and T is defined in section 3,
STATISTICS IN TRANSITION new series, Jne 06 73 φ (0 φ ) is the nknown constant (scalar) to be deterined nder certain criterions. 7.. Properties of the estiator T Since the estiator T is exponential type estiator, it is biased for the poplation ean Y. The bias B (.) and the ean sqare error M (.) of the estiator T are derived p to the first order of approxiations siilar to that of the estiator T. Theore 7.3. The bias of the estiator T to the first order of approxiations is obtained as B(T ) = φ B(T )+(φ )B(ξ ) (7.3) 3 Bξ =Y C x + ρxzcxcz ρxycyc x + ' Cz ρ yzcycz n 8 n and B(T ) is defined in section 4. Theore 7.4. The ean sqare error of the estiator T to the first order of approxiations is obtained as (7.4) M( T ) = φ M T + φ M ξ +φ φ C M(ξ ) = Eξ Y = +ρ ρ + + ρ S n 4 N n (7.5) xz yx ' yz y and M(T ) is defined in section 4. y C = E T YξY = N S (7.6) Since the ean sqare error of the estiator T in eqation (7.4) is the fnction of nknown constant φ, it is iniized with respect to φ and sbseqently the i vale of φ is obtained as M(ξ )C φ = M(T )+M(ξ )C. (7.7)
74 G. N. Singh, M. Khetan, S. Marya: Soe effective estiation... Now sbstitting the vale of φ in eqation (7.7), we get the i ean sqare error of T as M(T ).M(ξ )C M(T ) = (7.8) M(T )+M(ξ )C Frther, sbstitting the vales fro eqations (4.6), (7.5) and (7.6) in eqation (7.8), we get the siplified vale of M (T ) which is given below: a + a + a Sy M(T ) = (7.9) a+ a + a n 3 6 5 4 a = ac +k f, a = ad +c bk f, a = bd, a = c a+kf, a = ab+d kf, 3 4 5 a = b, c = f+c +t d, d = (f)+d (t ). 6 To deterine the i vales of, we iniized the ean sqare error of the estiator T given in eqation (7.9) with respect to, which reslts in qadratic eqation in, and the respective soltions of, say ˆ, are given below: p +p +p 3 =0 (7.0) ˆ = p ± p p p 3 p (7.) p = a a5 a a 4, p = a a6a 3 a 4 and p 3 = a a6a 3 a 5. (0) Sbstitting the adissible vales of ˆ, say, fro eqation (7.) into eqation (7.9), we have the i vale of the ean sqare error of T, which is shown below: (0) (0) a + a + a S 0 3 y M(T ) =. (7.) a+ (0) (0) 6 a 5 + a 4 n 8. Coparison of efficiencies The percentage relative loss in efficiencies of the estiator T, T and T is obtained with respect to the siilar estiator and natral sccessive sapling estiator when the nonresponse is not observed on any occasion. The estiator ξ is defined nder siilar circstances as the estiator T bt nder coplete response, as the estiator ξ is the natral sccessive sapling estiator, and they are given as ξ = ψ ξ + ψ ξ ; (j=, ) (8.) j j j j j
STATISTICS IN TRANSITION new series, Jne 06 75 ' ' zz xn x z n ξ = yexp ', ξ = y, ξ = yexp, ξ = y +β yx (x n x ) z +z x n +x z Proceeding on a siilar line as discssed for the estiator T the i ean sqare errors of the estiators ξ j (j=,) are derived as and ' = q ± q q q 3 q ' ' 0 b+ 3 b + b S y M(ξ ) = ' ' b+ 6 b 5+ b4 n 0 Sy M(ξ ) = + (ρ xy) f. n (8.) (8.3) (fraction of the fresh saple for the estiator ξ ), b = ac +k f, b = ad +c b k f, b = b d, b = c a+kf, b = ab +d kf, 3 4 5 b = b, q = b b b b, q = b b b b and q = b b b b. 6 5 4 6 3 4 3 6 3 5 Reark 8.. To copare the perforances of the estiators T, T and T with respect to the estiators ξ j (j=, ), we introdce the following assptions: (i) ρ xz = ρ yz, which is an intitive assption, also considered by Cochran (977) and Feng and Zo (997), (ii) W=W (iii) f =f. The percentage relative losses in the precision of the estiators T, T and T with respect to ξ (j=, ) nder their respective iality conditions are given by j (0) M ξ j M T (0) MT M ξ j j MT (0) M ξ j j = 00, = 00 (0) (0) M T M T and j = 00; (j=, ) (0) M T For N = 5000, n' = 000, ' = 000, n = 500, t =0.50,t =0.50, f=0. and different choices of f, yx and yz, Tables 6 give the i vales of
76 G. N. Singh, M. Khetan, S. Marya: Soe effective estiation... (0) (0) (0),, and percentage relative losses j, j and (j=, ) in the precision j of the estiators T,T and T with respect to estiators ξ (j=, ). j Table. Percentage relative loss in the precision of the estiator T with respect ξ to W 0.05 0.0 0.5 0.0 yx f (0) (0) (0) (0) yz 0.6.5 0.6.0 0.6.5 0.6.0 0.6 640 0.467 0.36 0.404 8 0.3950 40 0.06 0.5890 0.4775 0.3985 0.338 0.557 0.45 0.3787 0.334 3.8.5435.5098.79 5.933 4.7480 4.709 5.3.45.4663.6439 3.090 4.6664 4.708 5.0344 5.743 8 0.3950 40 0.06 0.4580 0.666 03 0.495 0.557 0.45 0.3787 0.334 0.4844 0.4004 0.3398 36 5.933 4.7480 4.709 5.3 0.0748 8.850 8.870 9.988 4.6664 4.708 5.0344 5.743 8.4959 8.5863 9.796 0.3684 0.589 0.398 0.30 03 0.385 0.47 0.087 0.0880 0.579 0.454 0.359 0.3085 0.44 0.355 0.304 0.640 8.004 6.650 6.673 7.3438.8584 85.037.538 6.6733 6.7339 7.037 8.66.67.8338.6435 4.0 0.4580 0.666 03 0.495 0.0558 0.0385 0.03 0.077 0.4844 0.4004 0.3398 36 0.3636 0.3077 0.667 0.353 0.0748 8.850 8.870 9.988 4.6753.7754 3.0986 4.633 8.4959 8.5863 9.796 0.3684 4.3348 4.5833 5.58 7.459 Table. Percentage relative loss in the precision of the estiator T with respect to ξ W 0.05 0.0 0.5 0.0 yx f yz (0) (0) (0) (0) 0.6.5 0.6 640 0.467 0.36 0.404 6.7964 9.97 38.43 65.7067 8 0.3950 40 0.06 3.8003 6.59 35.070 6.564 0.589 0.398 0.30 03.986 4.959 3.3048 57.9460 0.4580 0.666 03 0.495 698.96 9.939 54.7839.0 0.6 8 0.3950 40 0.06 3.8003 6.59 35.070 6.564 0.4580 0.666 03 0.495 698.96 9.939 54.7839 0.385 0.47 0.087 0.0880 3.844 9.097 6.0759 49.576 0.0558 0.0385 0.03 0.077 5.8463 6.709 3.0 45.504.5 0.6 0.5890 0.4775 0.3985 0.338.036.38 9.5093 56.099 0.557 0.45 0.3787 0.334 4.565 8.5855 6.393 5.854 0.579 0.454 0.359 0.3085 6.70 6.70 3.4436 47.966 0.4844 0.4004 0.3398 36 8.05 4.593 5 44.370.0 0.6 0.557 0.45 0.3787 0.334 4.565 8.5855 6.393 5.854 0.4844 0.4004 0.3398 36 8.05 4.593 5 44.370 0.44 0.355 0.304 0.640.94 0.459 6.073 38.8 0.3636 0.3077 0.667 0.353 3.9665.6738.995 3.9609
STATISTICS IN TRANSITION new series, Jne 06 77 Table 3. Percentage relative loss respect to ξ Note: indicates (0) does not exist. in the precision of the estiator T with W 0.05 0.0 0.5 0.0 0.5489 0.3748 8 0.5667 0.39 6 ρyx f yz (0) 0.6.5 0.6.0 0.6.5 0.6.0 0.6 0.6356 0.54 0.477 0.363 0.6443 0.534 0.4368 0.3698 (0) 0.48 0.346 0.5457 7 0.667.0599 0.05 0.075 0.3345 0.4977 97 0.4057 0.655 754 0.5667 0.39 6 0.5984 0.4 0.3 0.6443 0.534 0.4368 0.3698 0.6604 0.54 0.454 0.386 (0) 7 0.667.0599 0.585.385.004 97 0.4057 0.655 754 0.3798 764.58.8759 0.583 0.4065 0.3089 0.658 0.4484 0.3445 0.655 0.534 0.4456 0.3780 0.675 0.5574 0.4705 0.406 (0) 0.4057 596.545 80.7458.8508 0 0.595 68.4345 0.5430.63.850.7098 0.5984 0.4 0.3 0.6497 0.473 0.3663 0.6604 0.54 0.454 0.386 0.6887 0.577 0.4859 0.463 0.585.385.004 890.954 3.64 0.3798 764.58.8759 0.696.49.3308 3.484 Table 4. Percentage relative loss respect to ξ in the precision of the estiator T with W 0.05 0.0 0.5 0.0 yx f yz (0) 0.6.5 0.6 0.5489 0.3748 8 (0).500 4.948 69.5345 0.5667 0.39 6 (0).9853 447 68.6579 0.583 0.4065 0.3089 (0).8349 40.305 67.8305 0.5984 0.4 0.3.6969 39.955 67.048.0 0.6 0.5667 0.39 6.9853 447 68.6579 0.5984 0.4 0.3.6969 39.955 67.048 0.658 0.4484 0.3445.459 39.068 65.6050 0.6497 0.473 0.3663.436 38.5697 64.3036.5 0.6 0.6356 0.54 0.477 0.363 0.37 3.7064 3.584 6 0.6443 0.534 0.4368 0.3698 4 3.4806 3.549 59.505 0.655 0.534 0.4456 0.3780 0.367 3.648 3.7456 58.76 0.6604 0.54 0.454 0.386 0.0486 3.0583 3.357 58.05.0 0.6 0.6443 0.534 0.4368 0.3698 4 3.4806 3.549 59.505 0.6604 0.54 0.454 0.386 0.0486 3.0583 3.357 58.05 0.675 0.5574 0.4705 0.406 0.53.6709 30.69 56.7079 0.6887 0.577 0.4859 0.463 0.646.344 9.958 55.4606 Note: indicates (0) does not exist.
78 G. N. Singh, M. Khetan, S. Marya: Soe effective estiation... Table 5. Percentage relative loss respect to ξ in the precision of the estiator T with W 0.05 0.0 0.5 0.0 ρyx f yz (0) 0.6.5 0.6.0 0.6.5 0.6.0 0.6 Note: indicates 55 0.4376 64 0.50 0.6634 0.338 0.305 7 0.5785 0.4668 0.3883 0.388 0.597 0.484 0.3573 0.3038 (0) 3.070.37.09.709 5.847 4.649 3.74 3.8495.3.89.786.496 4.3575 4.749 4.545 4.680 0.6634 0.338 0.305 7 0.06 0.63 0.0847 0.0707 0.597 0.484 0.3573 0.3038 0.498 0.3490 5 0.504 (0) does not exist. (0) 5.847 4.649 3.74 3.8495 9.08 6.36 5.673 5.880 4.3575 4.749 4.545 4.680 7.655 7.3 7.45 7.9947 0.4538 0.3 0.600 0.57 0.480 0.389 0.354 76 0.37 0.666 0.45 34 (0) 7.79 5.507 4.907 5.060 6.86 5.863 5.9574 6.4507 0.0446 9.57 9.6656 0.3593 0.06 0.63 0.0847 0.0707 0.498 0.3490 5 0.504 0. 8 0.538 0.333 9.08 6.36 5.673 5.880 7.655 7.3 7.45 7.9947.6475.089.59.908 Table 6. Percentage relative loss in the precision of the estiator T with respect to ξ W 0.05 0.0 0.5 0.0 yx f yz (0) 0.6.5 0.6.0 0.6.5 0.6.0 0.6 Note: indicates 55 0.4376 64 0.50 0.6634 0.338 0.305 7 0.5785 0.4668 0.3883 0.388 0.597 0.484 0.3573 0.3038 (0) 6.8086 9.4844 38.759 66.764 3.90 7.36 36.4038 63.906.90.403 9.9953 57.0593 3.946 9.858 7.3668 53.683 0.6634 0.338 0.305 7 0.06 0.63 0.0847 0.0707 0.597 0.484 0.3573 0.3038 0.498 0.3490 5 0.504 (0) does not exist. (0) 3.90 7.36 36.4038 63.906 0.3963 4.6045 33.7074 60.5469 3.946 9.858 7.3668 53.683 7.58 5.63 3.658 48.954 0.4538 0.3 0.600 0.57 0.480 0.389 0.354 76 0.37 0.666 0.45 34 (0).7490 5.6008 34.734 6.8388 5.749 7.633 5.06 50.683 9.6578 3.0360 0.687 44.3867 0.06 0.63 0.0847 0.0707 0.498 0.3490 5 0.504 0. 8 0.538 0.333 0.3963 4.6045 33.7074 60.5469 7.58 5.63 3.658 48.954.676.358 8.88 4.899
STATISTICS IN TRANSITION new series, Jne 06 79 9. Interpretations of reslts The following conclsions ay be drawn fro Tables 6: () Fro Tables and 5 it is clear that (0) (0) (a) For the fixed vales of W, ρ yx and f, the vales of, decrease with the increasing vales of ρ yz. This iplies that the higher the vale of ρ yz, the lower the fraction of a fresh saple reqired on the crrent occasion. (0) (0) (b) For the fixed vales of W, ρ yx and ρ yz, the vales of, decrease and, increase with the increasing vales of f. (c) For the fixed vales of W, ρ yz and f, no pattern is observed with the increasing vales of ρ yx. (0) (0) (d) For the fixed vales of f, ρ yz and ρ yx, the vales of, decrease and, increase with the increasing vales of W. This behavior shows that with the higher nonresponse rate one ay reqire to draw the saller saple on the crrent occasion, which redces the cost of a srvey. () Fro Tables and 6 it ay be seen that (a) For the fixed vales of W, ρ yx and f, the vales of, and, (0) (0) decrease with the increasing vales of ρ yz.. This iplies that if one ses the inforation on highly correlated axiliary variable, there is a significant gain in the precision of estiates. (b) For the fixed vales of W, ρ yx and ρ yz, the vales of and, increase with the increasing vales of f. (c) For the fixed vales of W, ρ yz and f, the vales of, increasing vales of ρ yx. (d) For the fixed vales of f, ρ yz and ρ yx, the vales of (0) (0), decrease increase with the, decrease and (0) (0), increase with the increasing vales of W. This pattern shows that the higher the nonresponse rate, the greater the loss. This behavior is practically jstified. (3) Fro Table 3 it is clear that (a) For the fixed vales of W, ρ yx and f, the vales of (0) decrease and increase with the increasing vales of ρ yz. This behavior indicates that if the inforation on highly correlated axiliary variable is available, it plays an iportant role in iproving the precision of estiates. (b) For the fixed vales of W, ρ yx and ρ yz, the vales of with the increasing vales of f. (0) and increase
80 G. N. Singh, M. Khetan, S. Marya: Soe effective estiation... (c) For the fixed vales of W, ρ yz and f, no pattern is visible with the increasing vales of ρ yx. (d) For the fixed vales of f, ρ yz and ρ yx, the vales of with the increasing vales of W. (0) and increase (4) Fro Table 4 it ay be seen that (0) (a) For the fixed vales of W, ρ yx and f, the vales of and decrease with the increasing vales of ρ yz. This iplies that negative loss is observed de to the presence of high correlation between the axiliary variables. This behavior is highly desirable. (0) (b) For the fixed vales of W, ρ yx and ρ yz, the vales of and increase with the increasing vales of f. This indicates that if a saller size of sbsaple is chosen, the loss in precision increases, as it was expected. (c) For the fixed vales of W, ρ yz and f no pattern is seen with the increasing vales of ρ yx. (d) For the fixed vales of f, ρ yz and ρ yx, the vales of with the increasing vales of W. 0. Conclsions and recoendations (0) and increase It ay be seen fro the above tables that for all cases the percentage relative loss in precisions is observed ver the i vale of exists, when nonresponse occrs on both occasions. Fro Tables, 3 and 5, it is seen that the loss is present de to the presence of nonresponse on each occasion, bt the negative ipact of nonresponse is very low, which jstifies the se of Hansen and Hrwitz (946) techniqe in the proposed estiation procedres. Fro Tables, 4 and 6, when the proposed estiators are copared with the natral sccessive sapling estiator, sbstantial profit is visible, which jstifies the intelligible se of axiliary inforation in the for of exponential ethods of estiation. Finally, looking at good behaviors of the proposed estiators one ay recoend the to srvey statisticians and practitioners for their practical applications. Acknowledgeents Athors are thankfl to the reviewers for their valable sggestions, which enhanced the qality of this paper. Athors are also thankfl to the Indian School of Mines, Dhanbad for providing financial assistance and necessary infrastrctre to carry ot the present research work.
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