Exponential Ratio Type Estimators of Population Mean under Non-Response

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1 Oen Journal of Statistics, 0,, 97-0 Published Online Februar 0 (htt:// htt://d.doi.org/0.6/ojs.0.00 Eonential Ratio Te Estimators of Poulation Mean under Non-Resonse Lovleen Kumar Grover, Parmdee Kaur Deartment of Mathematics, Guru Nanak Dev Universit, mritsar, India lovleen_@ahoo.co.in Received November, 0; revised December, 0; acceted December 0, 0 Coright 0 Lovleen Kumar Grover, Parmdee Kaur. This is an oen access article distributed under the Creative Commons ttribution License, which ermits unrestricted use, distribution, and reroduction in an medium, rovided the original work is roerl cited. In accordance of the Creative Commons ttribution License all Corights 0 are reserved for SCIRP and the owner of the intellectual roert Lovleen Kumar Grover, Parmdee Kaur. ll Coright 0 are guarded b law and b SCIRP as a guardian. BSTRCT This aer rooses some eonential ratio te estimators of oulation mean under the situations when certain observations for some samling units are missing. These missing observations ma be for either auiliar variable or stud variable. The biases and mean suare errors of the roosed estimators have been derived, u to the first order of aroimation. The roosed estimators are comared theoreticall with that of the eisting ratio te estimators defined b []. It has been found that the roosed eonential ratio te estimators erform better than the mean er unit estimator even for the low ositive correlation between stud variable and auiliar variable. Moreover, we obtained the conditions for which our roosed estimators are better than the corresonding ratio te estimators of []. To verif the theoretical results obtained, a simulation stud is carried out finall. KEYWORDS uiliar Variable; Bias; Mean Suare Error; Non-Resonse; Simle Random Samling without Relacement; Stud Variable Mathematical Subject Classification: 6 D 05. Introduction In surve samling situations, auiliar information is generall used to imrove the recision or accurac of the estimator of unknown oulation arameter of interest under the assumtion that all the observations in the samle are available. But in man surve samling situations, this assumtion is not true. This is the case of incomlete information which ma arise due to some non-resonse in the given samle. There are various ractical reasons for this incomlete information due to non-resonse like, ) unwillingness of the resondent to answer some articular uestions, ) accidental loss of information caused b unknown factors, ) failure on the art of investigator to collect correct information, etc. Such te of incomlete information is ver common in the studies related to medical research, market research surves, oinion olls, socio economic investigations, etc. In surve samling, when information about all the samling units is available then it is conventional to estimate unknown oulation mean of stud variable using ratio estimator rovided that there is a ositive correlation between stud variable and auiliar variable (see []). But, when information about all the units is not available then the traditional comlete data estimating rocedures could not be used straight forwardl to analze the data. [] discussed the situation that certain incomlete information ma occur on either stud variable Corresonding author. OPEN CCESS

2 98 L. K. GROVER, P. KUR or auiliar variable or both of these variables. In the earl investigation with non-resonse situations of surve samling, [,-6] considered the roblem of estimation of oulation mean. Further investigation is carried out from various angles recentl b man authors viz. [7-], etc. We have seen that [] considered various ratio te estimators of oulation mean, under different situations, when some of the observations on either the stud variable or auiliar variable are missing. In this aer, we have assumed the same situations of non-resonse as considered b them. In Section, we have roosed the corresonding eonential ratio te estimators of oulation mean for these situations. In Section, we have obtained the eressions for the biases and mean suare errors (MSE) of the roosed estimators. In Section, the comarisons (with resect to biases and mean suare errors) have been made for the roosed estimators with the corresonding eisting estimators of []. Finall, in Section 5, a simulation stud has been erformed to suort the theoretical results obtained earlier in this aer.. Notations and the Proosed Estimators Let and denote the ositive valued stud variable and ositive valued auiliar variable resectivel. ssume that there is a ositive correlation between and. Let ( Yi, X i) ; i =,,,, N denote the values of bivariate (, ) on the i th unit of oulation of size N. Consider a samle, sa s, of size n is drawn with simle random samling without relacement (SRSWOR) from this oulation. Now, the roblem of interest is to estimate the unknown oulation mean Y of stud variable when the oulation mean X of auiliar variable is assumed to be known. It is assumed that ) observations of (, ), namel (, ),(, ),,, n ) measured on selected units in the samle are comletel available. In addition to these available observations, let,,, denote the available observations of variable on other units in the samle but the corresonding observations of variable are missing on these samle units. Similarl, we have a set of other available observations of variable, namel,,, in the samle but the associated values of variable are missing on these samle units. Thus we have the following sub samles of the samle s : For ) samling units, the observations ( i, i) of (, ) are known which constitutes a sub samle s s Samle s of size n For the other samling units, the observations s i of are known but the corresonding observations of are not known which constitutes a sub samle s s For the remaining samling units the observations i of are known but the corresonding observations of are not known which constitutes a sub samle s We note that s = s s s and s s = φ an emt set for i, j =,, and i j. i j Here, the uantities lete observations on bivariate Let and denote the numbers of distinct samling units in the samle s with incom, and these must be random in nature. OPEN CCESS

3 L. K. GROVER, P. KUR 99 = i s n i i s i s = i s n i s = samle mean of based on subsamle s ( ) = s i i s ) ) n + = s s = n + [] defined the following four ratio te estimators of Y : ( samle mean of based on subsamle ) ( samle mean of based on subsamle ) ( samle mean of based on subsamle ) ( samle mean of based on subsamle ) ( samle mean of based on subsamle s s) r = X ( based on subsamle s ) () ) = X = X ( based on subsamle s s ) () r n + n + r = X = X s s n ( based on subsamle ) () () n n + r = X = X ( based on whole samle s) (5) n n + In the ordinar circumstances, when there is no non-resonse then [5] introduced an eonential ratio te estimator of Y which is better than mean er unit estimator of Y even for the low ositive correlation between and. On the other hand, the ordinar ratio estimator of Y (due to []) is better than mean er unit estimator for high ositive correlation between and and under certain conditions. On taking this advantage of eonential ratio te estimators and then considering the concet of ratio te estimators defined b [], we have got a motivation to roose the following eonential ratio te estimators of Y : X = e, based on subsamle s X + Re ) ) n + X X Re = e e, ( based on subsamle s ) = s X (7) + n + X + X n + X Re = e e, ( based on subsamle ) = s s (8) X + n X + ) ) n + X X ( n ) + Re = e e, ( based on whole samle ) = s X ( n ) + n + X + (9) (6) OPEN CCESS

4 00 L. K. GROVER, P. KUR These four roosed estimators will now be comared with the corresonding four estimators of [] with resect to their biases and mean suare errors.. Biases and Mean Suare Errors of Proosed Estimators To obtain the biases and mean suare errors of the roosed estimators, we roceed as follows. Let U =, V =, U =, V = (0) X Y X Y Now we state the following lemma. Lemma.: Under SRSWOR, we have the following eectations: where = = = = 0, EU EV EU EV EU = f+ C, EV = f+ C, EUV = f ρcc, + EU = C, EV = C, N N EUU = EUV = EVV = EUV = 0 N f E C ( X X) C Y Y ( N ) N + =, =, i = i n N N X i= N Y i= N ρ = corr (, ) = ( Yi Y) ( Xi X), and XYC C i= E denotes the unconditional eectation based on the whole samle s. Proof: See endi. Remark.: We note that f f + and f f + alwas hold good, where as f < f, hold good ac- > cording as <. > From (0), we can rewrite the following: ( ), ( ), ( ), ( ) = + U X = + V Y = + U X = + V Y () () Now we state the following lemma. Lemma.: On using () and retaining the terms u to second degree of can write the following: U VU = Re Y Y U V s s s and U, V, U s V onl, we () ) ) ) Y Y U U V UU U n 8 n n n n n n Re = U UV U V 8 ) U n n Re Y = Y U V UV V UV n n n n () (5) OPEN CCESS

5 L. K. GROVER, P. KUR 0 ) Y Y U U UU U n 8 n n n n n n Re = + + ) V UV U V n n n + U + V UV U V 8 n n n n n + n n n n n Proof: See endi. Theorem.: The eressions of biases and mean suare errors of the four roosed eonential ratio te estimators of Y, u to first order of aroimation, are Bias ( Re ) = Yf + C ρcc 8 C MSE ( Re ) = Y f + + C ρcc C Bias ( Re ) = Yf ρcc 8 MSE( Re ) = Y C f + C f ρc C f + Bias ( Re ) = Y C f ρc C f 8 + MSE( Re ) = Y C f + C f ρc C f + Bias ( Re ) = Y C f ρc C f 8 C MSE ( Re ) = Y f + f C ρcc f () where f+ = E, f= E, f= E n N n N n N Proof: See endi. Remarks.: The eressions for biases and mean suare errors of the roosed estimators (as obtained in Theorem.) involve some unknown oulation arameters like Y, C, C and ρ. To find the estimates of these biases and mean suare errors, the general ractice in surve samling is to relace the unknown oulation arameters with their resective consistent estimators based on the same samle. To test the sueriorit of our roosed estimators over the eisting estimators of [], we comare the biases and the mean suare errors of these estimators.. Comarison of Proosed Estimators with Eisting Estimators To comare the biases and mean suare errors of the roosed estimators with the estimators defined b [], we reuire the eressions of their biases and mean suare errors, u to first order of aroimation and these are given in Table. Remark.: Note that MSE ( ) = f Y C and Bias ( ) = 0. Remark.: While comaring the biases and mean suare errors, we have taken and as fied uantities in the given samle, so we must have (6) (7) (8) (9) (0) () () () OPEN CCESS

6 0 L. K. GROVER, P. KUR E =, E =, E =. n n n n n n Remark.: [6] has shown that the values of arameter K C = ρ remain stable in an reetitive surve. So while comaring biases and mean suare errors of various estimators, we shall tr to find the conditions on the values of K under which one estimator is suerior to the other estimator. In the resent situations, we also note that the value of K alwas lies in the interval 0 < K <. Theorem.: U to the terms of order n, we have 88 5 Bias ( Re ) < Bias ( r ) if K 0,, 96 (5) Proof: See endi. Theorem.: U to the term of order C 88 5 Bias ( Re ) < Bias ( r ) if K 0,, 96 (6) 88 f 5 f + + Bias ( Re ) < Bias ( r ) if K 0,, 96 f f (7) 88 f 5 f Bias ( Re ) < Bias ( r ) if K 0,, 96 f f (8) n, we have MSE ( Re ) < MSE ( r ) if K < (9) MSE ( Re ) < MSE ( r ) if K < (0) f + MSE ( Re ) < MSE ( r ) if K < () f f MSE ( Re ) < MSE ( r ) if K < () f Proof: See endi. From the above two theorems, we see that our roosed estimators are suerior to the eisting estimators un- der some ver simle conditions. Theorem.: U to the term of order n, we have C MSE ( r ) < MSE ( ), if K > + f C f + () Table. Biases and mean suare errors of the eisting estimators, u to first order of aroimation. Estimator Bias (.) MSE (.) r ( Yf C ρc C + ) Y f ( C + C ρcc + ) r ( Y fc ρcc f ) Y ( f C + fc ρcc f + ) r ( Y f C ρcc f + ) Y ( C f + C f ρc C f + ) r ( Y fc ρcc f ) Y ( fc + fc ρcc f ) OPEN CCESS

7 L. K. GROVER, P. KUR 0 C f f MSE ( r ) < MSE ( ), if K > + C + f () f + MSE ( r ) < MSE ( ), if K > (5) f f MSE ( r ) < MSE ( ), if K > (6) f Re <, if > + C f + ( ) MSE MSE K C f+ f MSE ( Re ) < MSE ( ), if K > + C f C f (7) (8) f + MSE ( Re ) < MSE ( ), if K > (9) f f MSE ( Re ) < MSE ( ), if K > (0) f Proof: This theorem can be roved in the similar wa as Theorem.. Corollar.: On combining Theorems. and., we have the following results: ) The mean suare error of the roosed estimator is less than that of both and C f C f K +,, rovided that C f <. + C f + ) The mean suare error of the roosed estimator Re is less than that of both C f+ f C f+ f K +,, rovided that C f <. C f ) The mean suare error of the roosed estimator Re is less than that of both f f + + K,. f f ) The mean suare error of the roosed estimator Re Re is less than that of both r if r and if r and if r and if f f K,. f f Remark.: From the results of Theorem., we can conclude that four roosed eonential ratio te estimators are reall better than mean er unit estimator for even the lower ositive values of K (or euivalentl even for lower ositive correlation between and ). 5. Simulation Stud To suort the facts roved in earlier sections of this aer, we erform a simulation stud here. For this urose, we have taken an emirical oulation of size from [7] [age No. 77]. In this oulation, variable is area under wheat in 97 and variable is area under wheat in 97. For this oulation, we have following reuisite arameters: X = 856., Y = 08.88, C = 0.86, C = 0.7, ρ = 0.5, K = 0.8 OPEN CCESS

8 0 L. K. GROVER, P. KUR We have simulated samle (with SRSWOR) 50 times from the above fied oulation using R software (version..0). While erforming a simulation stud, we use the following stes in seuence: ) samle, sa s, of size n is drawn from the fied oulation using simle random samling without relacement (SRSWOR). ) Take the fied values of missingness rates, that is, and. ) Randoml, we deleted observations from the set of observations of stud variable and observations from the set of observations of auiliar variable. ) Identif the subsamles, namel s, s and s. 5) The values of the estimators r, r, r, r, Re, Re, Re, Re and are calculated for each trilet ( n,, ). 6) Calculate the variances (or aroimate mean suare errors) of these estimators b using their 50 values that are obtained from 50 different simulated samles drawn from the given fied oulation. We have taken the different values of trilet (,, ) n as shown in Table. In Tables and, we have mentioned the variances of values of various estimators, considered in this aer, obtained for the simulated 50 different samles drawn from the given oulation on taking various values of samle sizes and values of missingness rates. Table. Considered samle sizes and missingness rates. n = with =, = =, = =, = 5 =, = = 5, = n = 7 with =, = =, = =, = 5 =, = = 5, = Table. Variances of various estimators for samle size n =. Missingness rate Variances of various estimators for some fied values of and r Re r Re r Re r Re =, = =, = =, = =, = = 5, = Table. Variances of various estimators for samle size n = 7. Missingness rate Variances of various estimators for some fied values of and r Re r Re r Re r Re =, = =, = =, = =, = = 5, = OPEN CCESS

9 L. K. GROVER, P. KUR 05 Discussion, Conclusions and Interretations For the given fied oulation, K = 0.8, which satisfies all the conditions, obtained in the results (9) to (), for all values of trilet ( n,, ), considered in Tables and. Therefore, the variances of all the roosed eonential ratio te estimators are less than that of all the corresonding eisting ratio te estimators of []. It can be verified through Tables and. gain for the given fied oulation, K = 0.8, which does not satisf all the conditions, obtained in the results () to (0), for all values of trilet ( n,, ), considered in Tables and. Therefore, the variances of all the eight ratio te estimators of Y [four roosed estimators and four [] s estimators] considered in this aer are not less than that of the mean er unit estimator. It can be verified again through Tables and. REFERENCES [] H. Toutenberg and V. K. Srivastava, Estimation of Ratio of Poulation Means in Surve Samling When Some Observations re Missing, Metrika, Vol. 8, No., 998, htt://d.doi.org/0.007/pl [] W. G. Cochran, The Estimation of Yields of Cereal Eeriments b Samling for the Ratio Gain to Total Produce, Journal of griculture Societ, Vol. 0, No., 90, htt://d.doi.org/0.07/s [] D. S. Trac and S. S. Osahan, Random Non-Resonse on Stud Variable versus on Stud as Well as uiliar Variables, Statistica, Vol. 5, No., 99, [] B. B. Khare and S. Srivastava, Transformed Ratio Te Estimators for the Poulation Mean in the Presence of Non Resonse, Communications in Statistics Theor and Methods, Vol. 6, No. 7, 997, htt://d.doi.org/0.080/ [5] H. Toutenberg and V. K. Srivastava, Efficient Estimation of Poulation Mean Using Incomlete Surve Data on Stud and uiliar Characteristics, Statistica, Vol. 6, No., 00,. -6. [6] H. J. Chang and K. C. Huang, Ratio Estimation in Surve Samling When Some Observations re Missing, Information and Management Sciences, Vol., No., 00,. -9. htt://d.doi.org/0.06/s (0) [7] C. N. Bouza, Estimation of the Poulation Mean with Missing Observations Using Product Te Estimators, Revista Investigación Oeracional, Vol. 9, No., 008, [8] M. K. Chaudhar, R. Singh, R. K. Shukla, M. Kumar and F. Smarandache, Famil of Estimators for Estimating Poulation Mean in Stratified Samling under Non-Resonse, Pakistan Journal of Statistics and Oerational Research, Vol. 5, No., 009, [9] M. Ismail, M. Q. Shahbaz and M. Hanif, General Class of Estimators of Poulation Mean in Presence of Non-Resonse, Pakistan Journal of Statistics, Vol. 7, No., 0, [0] C. Kadilar and H. Cingi, Estimators for the Poulation Mean in the Case of Missing Data, Communications in Statistics Theor and Methods, Vol. 7, No., 008, htt://d.doi.org/0.080/ [] S. Kumar and H. P. Singh, Estimation of Mean Using Multi uiliar Information in Presence of Non-Resonse, Communications of the Korean Statistical Societ, Vol. 7, No., 00,. 9-. htt://d.doi.org/0.55/ckss [] N. Nangsue, djusted Ratio and Regression Te Estimators for Estimation of Poulation Mean When Some Observations re Missing, World cadem of Science, Engineering and Technolog, Vol. 5, 009, [] M. M. Rueda, S. Gonza lez and. rcos, Indirect Methods of Imutation of Missing Data Based on vailable Units, lied Mathematics and Comutation, Vol. 6, No., 005, htt://d.doi.org/0.06/j.amc [] M. M. Rueda, S. Gonza lez and. rcos, Estimating the Difference between Two Means with Missing Data in Samle Surves, Model ssisted Statistics and lication, Vol., No., 005, [5] M. M. Rueda, S. Gonza lez and. rcos, General Class of Estimators with uiliar Information Based on vailable Units, lied Mathematics and Comutation, Vol. 75, No., 006,. -8. htt://d.doi.org/0.06/j.amc [6] M. Rueda, S. Martínez, H. Martínez and. rcos, Mean Estimation with Calibration Techniues in Presence of Missing Data, Comutational Statistics and Data nalsis, Vol. 50, No., 006, htt://d.doi.org/0.06/j.csda [7] D. Shukla, N. S. Thakur and D. S. Thakur, Utilization of Non-Resonse uiliar Poulation Mean in Imutation for Missing Observations, Journal of Reliabilit and Statistical Studies, Vol., No., 009, [8] H. P. Singh and S. Kumar, General Famil of Estimators of Finite Poulation Ratio, Product and Mean Using Two Phase Samling Scheme in the Presence of Non-Resonse, Journal of Statistical Theor and Practice, Vol., No., 008, htt://d.doi.org/0.080/ [9] H. P. Singh and S. Kumar, General Procedure of Estimating the Poulation Mean in the Presence of Non-Resonse under OPEN CCESS

10 06 L. K. GROVER, P. KUR Double Samling Using uiliar Information, Statistics and Oerations Research Transactions, Vol., No., 009, [0] H. P. Singh and S. Kumar, Imroved Estimation of Poulation Mean under Two-Phase Samling with Sub Samling the Non- Resondents, Journal of Statistical Planning and Inference, Vol. 0, No. 9, 00, htt://d.doi.org/0.06/j.jsi [] H. P. Singh and S. Kumar, Combination of Regression and Ratio Estimate in Presence of Nonresonse, Brazilian Journal of Probabilit and Statistics, Vol. 5, No., 0, htt://d.doi.org/0./0-bjps7 [] S. Singh, H. P. Singh, R. Tailor, J. llen and M. Kozak, Estimation of Ratio of Two Finite-Poulation Means in the Presence of Non-Resonse, Communications in Statistics Theor and Methods, Vol. 8, No. 9, 009, htt://d.doi.org/0.080/ [] H. P. Singh and R. S. Solanki, Estimation of Finite Poulation Means Using Random Non-Resonse in Surve Samling, Pakistan Journal of Statistics and Oeration Research, Vol. 7, No., 0,. -. [] H. P. Singh and R. S. Solanki, General Procedure for Estimating the Poulation Parameter in the Presence of Random Non- Resonse, Pakistan Journal of Statistics, Vol. 7, No., 0, [5] S. Bahl and R. K. Tuteja, Ratio and Product Te Eonential Estimators, Journal of Information and Otimization Sciences, Vol., No., 99, htt://d.doi.org/0.080/ [6] V. N. Redd, Stud on the Use of Prior Knowledge on Certain Poulation Parameters in Estimation, Sankha, Sr C, Vol. 0, 978, [7] D. Singh and F. S. Chaudhar, Theor and nalsis of Samle Surve Designs, New ge International (P) Limited Publishers, New Delhi, 00. OPEN CCESS

11 L. K. GROVER, P. KUR 07 endi Proof of Lemma.: Taking EU = E = E( ) = E( E( /, )) X X X where E denotes the conditional eectation based on the sub samle (either s or s or s ) under the condition that both and are fied for the given samle s. where E ( ) s = 0 X = = X Now EU E ( s ) ( X) = /, is the samle mean of based on whole comlete samle s. Hence the reuired result. gain taking ( ) EU = E E U /, = E C = Cf+, n N E ( U /, ) = C n N Hence the reuired result again. Similarl, we can rove the other results of Lemma.. Proof of Lemma.: Due to comle form of estimator Re, we are recalling estimator as defined in (9). ) ) n + X ( n ) + = e ( n ) n + X + Re )( UX + X) + ( U X + X) X )( VY + Y) + ( V Y + Y) ) = e n )( UX + X) + ( U X + X) X + ) On retaining the terms onl u to second degrees of s s s s U, V, U and V, we have ) UU ( n ) ) ( n ) n + n V + V n n Re = Y U + U n n 8 n U U ( using ( ) ) n n n U U n Re = Y U + U + UU + + V n 8 n n n 8 n n n n n UV U V + V UV U V n ) n ) n n n n n OPEN CCESS

12 08 L. K. GROVER, P. KUR endi ) ) Y = Y U + U + UU U n 8 n n n n n n Re + 8 n n U V UV U V n n n n n n + + V UV U V n n n n n which is the reuired result. Similarl, we can rove the other results of Lemma.. Proof of Theorem.: gain due to comle nature of estimator So, from (6), we have The bias of ) Re ), we rove the results () and () onl. Y = Y U + U + UU U n 8 n n n n n n Re Re + 8 n n U V UV U V n n n n n n + + V UV U V n n n n n, u to the terms of order ( Re ) = ( Re ) = ( Re ) Bias E Y E Y ) ) n, is given b ) n n n = YE U + U + UU U n 8 n n n + V UV U V n n n + U + V UV U V 8 n n n n n n n n n n ( n )( n ) ) n n n = Y EU + EU + EUU 8) ) ) n EU + EU + EV EUV n 8 n n n n n EUV + EV EUV n n n n n EUV On using the values of eectations, as obtained in Lemma., and then simlifing, we have Hence the result () is roved. Bias ( Re ) = Y C f ρc C f 8 OPEN CCESS

13 L. K. GROVER, P. KUR 09 endi gain b definition, the mean suare error of The mean suare error of Re ) Re is given b ( Re ) = ( Re ) MSE E Y, u to the terms of order n, is ) MSE ( ) Y E U U UU U n 8 n n n n n n Re = V UV U V n n n + U + V UV U V 8 n n n n n n n n n n On retaining the terms onl u to second degrees of ) U, V, U and V s s s s, we have MSE ( ) Y E U U V V n n n n n n Re = n n n + UU UV UV n n n n n n UV UV + VV n n n n n n n = Y EU + EU + E n n n ) ( V) ) n n n + E ( UU ) E ( UV ) E UV n n n n + EV n n n n )) n EUV EUV + EVV On using the values of eectations, as obtained in Lemma. and then simlifing, we have C MSE ( Re ) = Y f + f C C C f ρ Hence the result () is roved. In the similar wa, we can rove the other results (7) to (). Proof of Theorem.: Here we give the roof of the result (8) onl. The roofs of other results of this Theorem are similar. Taking (on using () and Table ) ( Re ) < ( r ) Bias Bias if Y C f ρcc f < Y fc CC f 8 ( ρ ) OPEN CCESS

14 0 L. K. GROVER, P. KUR 55 ρ C Cρ or if C f f 0 + f f < 6 C 8 C (on suaring both sides of the ineualit and then simlifing) where C + > where K = ρ C or if 8 fk 0 f fk 55 f or if K K + > 0 8 f 8 f ( K K )( K K ) f f or if > 0, (.) 5 f 88 f K = and =. 96 f K f Noting that K > K. For ( K K)( K K) > 0, we have the following two cases: Either case : When ( K K ) > 0 and ( K K ) > 0 then it imlies that K > K and K > K. Thus we must have K > K. Or case : When ( K K ) < 0 and ( K K ) < 0 then it imlies that K < K and K < K. Thus we must have K < K. On combining Case and Case, we conclude that for ( K K)( K K) > 0 we must have either K > K or K < K, that is, K ( 0, K) ( K, ). (.) Thus from the results (.) and (.), we have got that 88 f 5 f Bias ( Re ) < Bias ( r ) if K 0,, 96 f f Hence the result (8) is roved. Proof of Theorem.: We give here the roof of the result () onl. The roofs of other results of this Theorem are similar. Taking (on using () and Table ) Hence the roof of result () is comlete. MSE ( Re ) < MSE ( r ) C f + fc ρ CC f < fc + fc CC f ρ if C f ρ CC f < fc CC f ρ or if or if f K < f. OPEN CCESS