The number of sample points in S1, S2, and S3 is given by (N-2n-1), (N-2n-1), and (Nn)-2(N-2n-1), respectively.By definition of covariance, we ?2nc1c2].(17)Theorem?��[c1(xmax??xmin?)+c2(ymax??ymin?)??=��Syx?n��(N?1)??2nc1c2]?��[c1(xmax??xmin?)+c2(ymax??ymin?)??=Cov?(y?,x?)?N?nN(N?1)?��[c1(xmax??xmin?n)(N?2n?1)?c2(ymax??ymin?n)(N?2n?1)+2c1c2(N?2n?1)]??=(Nn)?1[��s��S(y??Y?)(x??X?)]?(Nn)?1?+c1c2��s��S1+��s��S2]??c2��s��S2(y??Y?)?��s��S1(y??Y?)??c1��s��S2(x??X?)?��s��S1(x??X?)?+��s��S3(y??Y?)(x??X?)?+��s��S2(y??Y?)(x??X?)?=(Nn)?1[��s��S1(y??Y?)(x??X?)?+��s��S3(y??Y?)(x??X?)]?+��s��S2(y??c1?Y?)(x??c2?X?)?=(Nn)?1[��s��S1(y?+c1?Y?)(x?+c2?X?)?haveCov?(y?c11,x?c21) necessary 2 ��If a sample of size n units is drawn from a population of size N units, then the covariance between y-c12 and x-c22, when they are negatively ��[c1(xmax??xmin?)+c2(ymax??ymin?)?2nc1c2].

(18)The?correlated, is given byCov?(y?c12,x?c22)=��Syx+n��N?1 above Theorem 2 can be proved similarly as Theorem 1.We define the following relative error terms.Let e0=(y-c1-Y-)/Y- and e1=(x-c2-X-)/X-, such thatE(e0)=E(e1)=0,(19)E(e02)=��Y?2[Sy2?2nc1N?1(ymax??ymin??nc1)],(20)E(e12)=��X?2[Sx2?2nc2N?1(xmax??xmin??nc2)],(21)E(e0e1)=��Y??X?[Syx?nN?1��c2(ymax??ymin?)+c1(xmax??xmin?)?2nc1c2].(22)Expressing Y-^RC in terms of e’s, we haveY?^RC=Y?(1+e0)(1+e1)?1.(23)Expanding and rearranging right-hand side of (23), to first degree of approximation, we have(Y?^RC?Y?)?Y?(e0+e1?e0e1+e12).(24)Using (24), the bias of Y-^RC is given byB(Y?^RC)?��X?[R(Sx2?2nc2N?1(xmax??xmin??nc2))?Syx?nN?1��(c2(ymax??ymin?)+c1(xmax??xmin?)?2nc1c2)],(25)where R=Y-/X-.

Using (24), the mean square error of Y-^RC, to the first degree of approximation, is given ��[(c1?Rc2)(ymax??ymin?)?R(xmax??xmin?)?n(c1?Rc2)].(27)To?byM(Y?^RC)?��[Sy2?2nc1N?1(ymax??ymin??nc1)+R2Sx2?2nc2N?1(xmax??xmin??nc2)?2RSyx?nN?1��(c2(ymax??ymin?)+c1(xmax??xmin?)?2nc1c2)](26)orM(Y?^RC)?M(y?R)?2��nN?1 find optimum values of c1 and c2, we differentiate (27) with respect to c1 and c2 ?2n(c1?Rc2)=0(28)Here??2n(c1?Rc2)=0,?M(Y?^RC)?c2=0?(ymax??ymin?)?R(xmax??xmin?)?as?M(Y?^RC)?c1=0?(ymax??ymin?)?R(xmax??xmin?) we have one equation with two unknowns so unique, solution is not possible, so we let c2 = (xmax ? xmin )/2n, and then c1 = (ymax ? ymin )/2n.For optimum values of c1 and c2, the optimum mean square error of Y-^RC is given ��[(ymax??ymin?)?R(xmax??xmin?)]2.

(29)Similarly?byM(Y?^RC)opt?M(y?R)?��2(N?1) the bias and mean square error or optimum mean square error of Y-^PC are, respectively, given ��[(c1+Rc2)(ymax??ymin?)+R(xmax??xmin?)?n(c1+Rc2)].(31)For??2nc1c2}],(30)M(Y?^PC)?M(y?P)?2��nN?1?��{c2(ymax??ymin?)+c1(xmax??xmin?)?byB(Y?^PC)?��X?[Syx?nN?1 optimum values of Drug_discovery c1 and c2, the optimum mean square error of Y-^PC is given ��[(ymax??ymin?)?R(xmax??xmin?)]2.