ARobustRegressionTypeEstimatorforEstimatingPopulationMeanunderNonNormalityinthePresenceofNonResponse

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Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 15 Issue 7 Version 1.

(USA Online ISSN: 49-466 & Print ISSN: 0975-5896 A Robust Regression Type Estimator for Estimating Population Mean under Non-Normality in the Presence By Sanjay Kumar Central University of Rajasthan,

In this study, we incorporate robust modified maximum likelihood estimators (MMLEs into regression type estimator in the presence of non-response and their properties have been obtained theoretically.

We show that utilization of MMLEs in estimating finite populations mean leads to robust estimates, which is very advantageous when we have non-normality or other common data anomalies such as

1 Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 15 Issue 7 Version 1.0 Year 015 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA Online ISSN: & Print ISSN: A Robust Regression Type Estimator for Estimating Population Mean under Non-Normality in the Presence By Sanjay Kumar Central University of Rajasthan, India Abstract- In sampling theory, regression type estimators are extensively used to estimate the population mean when the correlation between study and auxiliary variables is high. In this study, we incorporate robust modified maximum likelihood estimators (MMLEs into regression type estimator in the presence of non-response and their properties have been obtained theoretically. For the support of the theoretical outcomes, simulations under several super-population models have been made. We study the robustness properties of these modified estimators. We show that utilization of MMLEs in estimating finite populations mean leads to robust estimates, which is very advantageous when we have non-normality or other common data anomalies such as outliers. Keywords: regression type estimator, modified maximum likelihood, robust linear regression, super-population, simulation study, non-response. GJSFR-F Classification : MSC 010: 93D1 ARobustRegressionTypeEstimatorforEstimatingPopulationMeanunderNonNormalityinthePresenceofNonResponse Strictly as per the compliance and regulations of : 015. Sanjay Kumar. This is a research/review paper, distributed under the terms of the Creative Commons Attribution- Noncommercial 3.0 Unported License permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

A Robust Regression Type Estimator for Estimating Population Mean under Non- Normality in the Presence Sanjay Kumar Abstract- In sampling theory, regression type estimators are extensively used to

We show that utilization of MMLEs in estimating finite populations mean leads to robust estimates, which is very advantageous when we have noormality or other common data anomalies such as outliers.

Introduction The use of auxiliary information in sample survey have been considered mainly in the field of agricultural, biological, medical and social sciences at the stage of plaing, designing,

In sampling theory, the ratio method of estimation uses the auxiliary information which is correlated with the study variable to improve the precision which results in improved estimators when the

2 A Robust Regression Type Estimator for Estimating Population Mean under Non- Normality in the Presence Sanjay Kumar Abstract- In sampling theory, regression type estimators are extensively used to estimate the population mean when the correlation between study and auxiliary variables is high. In this study, we incorporate robust modified maximum likelihood estimators (MMLEs into regression type estimator in the presence of non-response and their properties have been obtained theoretically. For the support of the theoretical outcomes, simulations under several super-population models have been made. We study the robustness properties of these modified estimators. We show that utilization of MMLEs in estimating finite populations mean leads to robust estimates, which is very advantageous when we have noormality or other common data anomalies such as outliers. Keywords: regression type estimator, modified maximum likelihood, robust linear regression, super - population, simulation study, non-response. I. Introduction The use of auxiliary information in sample survey have been considered mainly in the field of agricultural, biological, medical and social sciences at the stage of plaing, designing, selection of units and devising the estimation procedure. In sampling theory, the ratio method of estimation uses the auxiliary information which is correlated with the study variable to improve the precision which results in improved estimators when the regression of yyon xx is linear and passes through origin. When the regression of yy on xx is linear, it is not necessary that the line should always passes through origin. Under such conditions, it is more appropriate to use the regression type estimators and the correlation between study and auxiliary variables is high. Sometimes, it may not be possible to collect complete information for all the units selected in the sample due to non-response. Estimation of the population mean in sample in the presence of non-response has been considered by Hansen and Hurwitz (1946, Rao (1986, 1987 and several other authors. Let YY and XX be the population mean of the main study variableyy and the auxiliary variablexx for the population UU: (UU 1, UU,.... UU NN. The population UU is supposed to be composed of NN 1 responding and NN non-responding units. From the population of size NN, a sample of size is selected by using SRSWOR method of sampling and it was observed that 1 units respond and units don t respond. Further, by making extra 43 Author: Department of Statistics, Central University of Rajasthan Bandarsindri, Kishangarh , Ajmer, Rajasthan, India. sanjay.kumar@curaj.ac.in

3 effort, a sub-sample of size rr = (KK > 1 is drawn from KK non-responding units by using SRSWOR method of sampling. Hence, we have 1 units from respondent group and rr units from non-respondent group of the population in the sample for which the value of the yy character is obtained. Hansen and Hurwitz (1946 proposed the unibiased estimator for YY, which is given as follows: 44 where yy 1 and yy yy = 1 yy 1 + yy, (1.1 are the sample means based on 1 and rr units respectively. The estimator yy is unbiased and the VV(yy is given by VV(yy = ff SS yy + WW (KK 1 SS yy( (1. where ff = NN ; WW NN ii = NN ii (ii = 1,, SS NN yy and SS yy( are the population mean squares of the character yy for the whole population and for the non-responding part of the population. The regression type estimator in the presence of non-response (Rao1990when the population mean XX is known, is given by = + θθ LL (XX xx, (1.3 where, θθ LL = SS yyyy ss xx is the regression coefficient obtained by least square estimation.ss yyyy and ss xx (sample mean square denote the unibiased estimates of SS yyyy and SS xx based on 1 + rr observations and observations respectively. The bias and mean square error (MSE of the traditional regression estimator is given by where, θθ LL = BB( = CCCCCC(xx, θθ LL (1.4 MMMMMM( = 1 1 SS NN yy + θθ LL SS xx θθ LL SS yyyy + WW (KK 1 SS yy(, (1.5 SS yyyy SS xx, SS xx is the population variance of the auxiliary variable, SS yyyy is the population covariance between the study variable and the auxiliary variable. We know that the regression estimator is useful in estimating the finite population mean when the information on the auxiliary variable is available, however this is known to be quiet sensitive to outliersas studied by Farrell and Barrera(006 and Gwet and Rivest (199. In sample survey studies, non-normal distributions are very common in practice as found in Cochran (1977, Jenkinset. al.(1977, Chambers (1986and Farrell and Barrera(007. In this paper, we study robust modified maximum likelihood estimator (MMLE into regression type estimator (Rao 1990in the presence of non-response and provide their properties theoretically. We specially focus on the situation where the error term is not normally distributed. We obtain the mean square error of the proposed regression estimator theoretically and found the conditions under which the proposed regression type estimator in the presence of non-response has less mean square error than the

4 corresponding regression type estimator. We support the theoretical result with simulations under several super population models and study the robustness property of the modified regression estimator. We show that utilization of MMLE for estimating finite populations mean results to robust estimate, which is very fruitful when we have non-normality or other common data anomalies such as outliers. II. Non-Normal Errors and Proposed Regression Estimator. For the linear regression model,yy ii = θθxx ii + ee ii ; ii = 1,,,,let the distribution of the error term follows the long tailed symmetric family. ff (ee = LLLLLL(pp, σσ = ΓΓΓΓ pp σσ KK ΓΓ 1 ΓΓ pp 1 KK ee σσ ; < ee <, (.1 where, KK = pp 3, pp is the shape parameter (ppis known with EE(ee ii = 0and VV(ee ii = σσ. Here it can be obtained that the kurtosis of (.1 is μμ 4 μμ = 3KK/(KK. The coefficients of kurtosis of the LTS family that we consider in this family are, 6, 4.5, 4.0 for p=.5, 3.5, 4.5, 5.5 respectively. We realize that when pp =, (.1 reduces to a normal distribution. The likelihood equations obtained from the likelihood function of (.1 are expressions in terms of the intractable functions. gg (zz ii = zz ii KK (zz ii, where, zz ii = ee ii (ii = 1,,, and do not have explicit solutions. σσ The robust MMLE which is known to be asymptotically equivalent to the MLE are obtained in following three steps: 1. The likelihood equations are expressed in terms of the ordered variate zz (ii = ee (ii. σσ. The function gg(zz ii are replaced by their linear approximations and 3. The resulting equations are solved for the parameters. The solutions which are explicit functions of the concomitant observations(yy [ii], xx [ii], ii = 1,,, are where, KK = ii=1 ββ ii yy [ii] xx [ii] / ii=1 ββ ii xx [ii] LL = θθ TT = KK + LLσσ TT, andσσ TT = GG + GG + 4 (, (. ii=1 αα iixx [ii] ββ ii xx ii=1 [ii] αα ii = KK, GG = (pp/kk ii=1 αα ii (yy [ii] KKxx [ii], (.3 CC = (pp/kk ββ ii (yy [ii] KKxx [ii] 3 tt [ii] {1+(1/KKtt [ii] ii=1 } andββ ii = 1 (1/KKtt [ii] {1+(1/KKtt [ii] }, (.4 45 where, the approximate tt (ii values are obtained from the equation

5 46 tt (ii h(zzdddd = ii + 1 ; 1 ii, where h(zz is the distribution of zz = ee/σσ In the same linear model, yy ii = θθxx ii + ee ii ; ii = 1,,,, now we suppose that the error term has one of the distributions in the skewed family namely, generalised logistic distribution which is given by ff(ee = rr σσ exp ( ee/σσ {1+exp ( ee/σσ} ; rr+1 < ee <, (.5 where, rr is the shape parameter with EE(ee ii = σσ {ΨΨ(rr ΨΨ(1}andVV(ee ii = σσ {ΨΨ (rr + ΨΨ (1}. Here ΨΨ(xx = Г (xx/г(xx is the psi function and ΨΨ (xx is its derivative. For,rr < 1, rr = 1, andrr > 1,(.5 represents negatively skewed, symmetric and positively skewed distribution respectively. The coefficient of skewness and kurtosis of the generalised logistic distribution which we consider in this study are computed from the moment generating function rr ΓΓ(rr+tttt ΓΓ(1 tttt MM ee (tt = and rris given below: ΓΓ(rr+1 rr-values Skewness Kurtosis The likelihood equations obtained from (.5 can be expressed in terms of the ordered variates zz (ii, (ii = 1,,,, and in whole the intractable function gg zz (ii = 1. These functions are linearised as we have done in the LTS family case. The 1+EE{zz (ii } solutions of the MMLE equations are the MMLEs which are given as follows: θθ TT = KK MMσσ TT and σσ TT = DD+ DD +4 ( 1, (.6 where, KK can be calculated from the formula (.3 by replacing αα ii, ββ ii aaaaaatt ii with 1 and tt (ii = log (qq rr ii 1,qq ii = 1 respectively. +1 In equation (.6 for calculating θθ TT and the following equations and αα ii = 1+ee tt (ii +tt (ii ee tt (ii, ββ (1+ee tt (ii ii = ee tt (ii, (.7 (1+ee tt (ii MM = ii=1 iixx [ii] pp ii xx ii=1 [ii] where ii = αα ii (rr ffffff 1 ii σσ TT, MM, DD and EEvalues are calculated from, DD = (rr + 1 ii=1 ΔΔ ii (yy [ii] KKxx [ii] (.8 EE = (rr + 1 ii=1 ββ ii yy [ii] KKxx [ii], (.9

6 Islam et. al. (001 showed that the MMLEs given in (.6 are more efficient and robust then their corresponding least square estimators (LSEs when the error term is from the skewed family (.5. In this study we calculate the MMLE θθ TT from (. if the error term is from LLLLLL(pp, σσor from (.6 if the error is from GGGG(rr, σσand modify the traditional regression estimator in the presence of non-response as given in (1.1 to achieve efficient estimator under non-normality, which is given by = + θθ TT (XX xx, (.10 The bias and mean square error of the proposed estimator (.10 are given by BB( = CCCCCC(xx, θθ TT (.11 MMMMMM( = 1 1 NN SS yy + θθ TT SS xx θθ TT SS xxxx + WW (KK 1 SS yy( (.1 In order to compare the MSE of the proposed estimator in (.10 with the MSE of the regression type estimator = + θθ LL (XX xx, we get the following conditions under which the proposed estimator is more efficient than the regression type estimator. MMMMMM( MMMMMM( ifθθ TT θθ LL or θθ TT θθ LL (.13 NOTE:- In general the shape parameter pp in (.1 and rr in (.5 may not be known. Using the least square estimator and constructing q-q plots (with the observed values as in Hamilton(199, one can easily determine the closest distribution for the error term. Since the families (.1 and (.5 include a very large variety of location scale distribution, one can easily determine an approximate distribution for the error by using one of the two families given in study. III. Simulation Study In this study for the simulation, we have used R-programming software. In the super population models generated, we use the model yy ii = θθxx ii + ee ii, ii = 1,,, NN, (3.1 where, we generate ee ii and xx ii independently and calculate yy ii for ii = 1,,, NN. Let the errors ee 1, ee,, ee NN be the random observations from a super population either from (.1 or (.5. Let UU NN denotes the corresponding finite population consists of NNpairs (xx 1, yy 1, (xx, yy,..., (xx NN, yy NN. To calculate the MSE of the proposed estimator in (.10, we calculate for all possible samples NN simple random samples of size from UU NN. Since NN is extremely large, so we conduct all Monte-Carlo studies as follows. We take NN = 500 in each simulation and generate UU 500 pairs from an assumed super population. From the generated finite population UU 500, we have selected a sample of size ( = 14, 19, 6, 40, 70 by simple random sampling without replacement. From each selected sample, the last 43% (3, 4, 6, 8, 11 respectively of units have been considered as non-responding units. Now, we choose at random S= samples for all the possible 500 samples of size ( = 14, 19, 6, 40, 70, which gives values 47

7 48 of. To compare the efficiency of the proposed estimator under different models for a given, we calculate the values of mean square errors as follows: MMMMMM( = 1 SS (TT SS jj =1 llllll YY,MMMMMM( = 1 (tt SS jj SS =1 llllll YY and where, YY = 1 NN NN ii=1 YY ii MMMMMM( = 1 SS ( YY, For setting the population correlation ρρ yyyy is sufficiently high, which choose the value of parameter θθ in the model = θθθθ + ee, such that the correlation coefficient between study variable (yy and auxiliary variable (xx is ρρ yyyy to determine the value of θθ that satisfied this condition, we follow a similar way given by Rao and Beegle (1967 and write the population correlation between the study variable (yy and the auxiliary variable (xx.for example if XX~UU(0,1, the value of θθ for which the population correlation between y and x becomes θθ = 1σσ ρρ yyyy 1 ρρ for the LTS family and θθ = yyyy 1σσ φφ (rr+φφ(1 ρρ yyyy 1 ρρ yyyy value of θθ family and θθ = SS jj =1 for the skewed family. Similarly, if xxis generated from EEEEEE (1, the for which the population correlation becomes θθ = σσ ρρ yyyy 1 ρρ yyyy σσ φφ (rr+φφ(1 ρρ yyyy 1 ρρ yyyy for the skewed family. for the symmetric Here we take σσ = 1, in all situations without loss of generality and calculate the require parameter θθ for which ρρ yyyy = IV. Comparison of Efficiencies of the Proposed Estimator We consider four different super-population models given below to see how much efficiency we gain with the proposed modified estimator, when the condition (.13 is satisfied under non-normality: I. xx~uu(0,1andee~llllll(pp, 1 and independent of xx. II. xx~eeeeee(1andee~llllll(pp, 1 and independent of xx. III. xx~uu(0,1andee~gggg(rr, 1 and independent of xx. IV. xx~eeeeee(0.5andee~gggg(rr, 1 and independent of xx. For the models (1 to (4, the values of θθwhich makes the population correlation ρρ yyyy = 0.75 are given in table 1. Table 1 : Parameter values of θθused in models (1 (4 that give ρρ yyyy = 0.75 Population pp Model ( Model ( Population rr Model ( Model (

8 Here, we note that for the LTS family (.1, the value of θθ does not depend on the shape parameterpp. To verify that the super-population are generated appropriately, we provide a scatter graph and error distribution of model to pp = 4.5 for model ( in the figure 1 and in the figure. Similarly for the model (3 with rr = 0.5 in the figure 3 and in the figure 4, a scatter graph and error distribution is provided. Figure 1 : A scatter graph of the study variable and auxiliary variable obtained from model ( for pp = Figure : Generated error distribution obtained from model ( for pp = 4.5

9 Figure 3 : A scatter graph of the study variable and auxiliary variable obtained from model (3 for rr = Figure 4 : Generated error distribution obtained from model (3 for rr = 0.5. Relative efficiencies are calculated as RRRR = MMMMMM( MMMMMM(. 100, where, MSE(.and relative efficiency (RE are given in the table for the model (1 and ( and in the table 3 for the model (3 to and (4.

10 From the table, we see that the proposed estimator is more efficient than the regression estimator in the presence of non-response because the theoretical condition is satisfied. We also observe that when sample size increases, mean square error decreases. From the table 3, we also observe that the proposed estimator in the presence of non-response is more efficient than the regression estimator because the theoretical condition is satisfied. It is also clear that when sample size increases, mean square error decreases. pp =.5 pp = 4.5 pp = 5.5 pp =.5 pp = 4.5 pp = 5.5 Table : Mean square error and efficiencies of the estimators under super-populations (1-. Est. Est. MMMMMMMMMM(11: xx~uu(00, 11aaaaaa ee~llllll(pp, * ( ( ( ( ( ( (15.11 ( ( ( ( ( ( ( ( ( ( ( ( ( (146.8 (156.9 (169.8 ( ( ( ( ( (151.9 (143.8 MMMMMMMMMM(: xx~eeeeee(1aaaaaa ee~llllll(pp, * ( ( ( ( ( ( (15.96 ( (15.14 ( ( ( ( ( ( ( (15.1 ( (165.6 ( ( ( ( ( ( (157.1 ( ( (*Efficiencies are in the parenthesis Table 3 : Mean square error and efficiencies of the estimators under super-populations (3-4. ( ( rr = 0.5 MMMMMMMMMM(33: xx~uu(00, 11aaaaaa ee~gggg(rr, 11 Est *

11 5 rr = 1.5 rr =.0 rr = 0.5 rr = 1.5 rr = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( MMMMMMMMMM(4: xx~eeeeee(0.5aaaaaa ee~gggg(rr, 1 Est (190.0 (18.89 (174.5 ( ( (19.8 ( ( ( ( ( (18.07 (199.9 ( ( ( ( (01.71 ( ( ( (04.8 (17.59 ( ( (00.6 (04.60 (17.69 ( ( (*Efficiencies are in the parenthesis V. Robustness of the Proposed Estimator The outliers in sample data are normally a in centered problem for survey statistician [1]. In practice, the shape parameters pp in LLLLLL(pp, σσ, and rr in GGGG(rr, σσ might be mis-specified. Therefore, it is very important for estimators to have efficiencies of robustness estimates such as an estimator is full efficient or nearly so for an assumed model and maintains high efficiencies for plausible to the assumed model. Here, we take NN = 500 and σσ = 1 without loss of generality and we study the robustness property of proposed estimator under different outlier models as follows. We assume xx~uu(0,1 and the error termee~llllll(pp = 3.5, σσ = 1. We determine our super-population model as follow: (5. True model: LLLLLL(pp = 3.5, σσ = 1 (6. Dixon s outliers model: NN NN oo observations from LLLLLL(3.5, 1and NN oo (we don t know which form LLLLLL(3.5,.0 (7. Mis-specified model: LLLLLL(4.0, 1 Here, we realize that the model (5, the assumed super population model is given for the purpose of comparison and the models (6 and (7 are taken as its plausible alternatives. Here we have assumed the super population model LLLLLL(3.5, 1for estimating θθ TT. The coefficients (αα ii, ββ ii from (.4 are calculated with pp = 3.5 and are used in models (5 and (6. NN oo in model (6 is calculated from the formula( Ref 1. Chambers, R. L. (1986, Outlier robust finite population estimation. Journal of the American Statistical Association, 81,

12 NN = 50 for NN = 500.We standardised the generated ee ii ss, (ii = 1,,, NN in all the models to have the same variance as that of LLLLLL(3.5, 1 i.e. it should be equal to 1. The simulated values of MSE and the relative efficiency are given in table 4. Here the estimators θθ LL, θθ TT are both location invariant estimators so that both of them are the estimators of θθ under all the models described above. Here theoretical condition (.13 is satisfied for model (5. From the table 4, we see that in the presence of non-response, the proposed estimator is more efficient than the regression estimator because the theoretical condition is satisfied. We also observe that when sample size increases, mean square error decreases. Now we assumed that the error term ee is from the skewed family and xxis from UU (0,1. We assumed the model to be GGGG(3,1 and determine our super population as (8 True model: GGGG(3,1 (9 Dixon outlier model: NN NN oo observations from GGGG(3,1 and NN oo (we don t know which from GGGG(3,, where (NN oo = NN = 50. (10 Mis-specified model: GGGG(5,1. The model (8 is assumed as a super population model and all other models (9 and(10 are taken as its plausible alternatives. The generated ee ii ss, (ii = 1,,, NN were standardized in the models (9 and (10 to have the same variance as that of GGGG(3,1 i.e. VV(ee ii = {ΨΨ (4 + ΨΨ (1},where ΨΨ (xx is the derivative of the psi function. The simulated values of the MSEs and relative efficiencies of the estimators and relative efficiency under models (8 to(10 are given in the table 4. Also, from the table 4, we see that the proposed estimator is more efficient than the regression estimator since the theoretical condition is satisfied. Also, it is clear that when sample size increases, mean square error decreases. Table 4 : Mean square errors and efficiencies under super-populations (5 (7 for LTS family and under super-populations (8 (10 for skewed family Est TTTTTTTT MMMMMMMMMM(5: xx~uuuuuu(0, 1aaaaaa ee~llllll(3.5, * ( ( ( (16.0 DDDDDDDDDD oooooooooooooo MMMMMMMMMM(6: xx~uuuuuu(0, 1aaaaaa (NN NN 0 ee~llllll(3.5,1 + NN 0 ee~llllll(3.5, * ( ( ( ( ( ( ( (09.0 MMMMMM ssssssssssssssssss MMMMMMMMMM(7: xx~uuuuuu(0, 1aaaaaa ee~llllll(4.0,1 TTTTTTTT MMMMMMMMMM(8: xx~uuuuuu(0, 1aaaaaa ee~gggg(3.0, (18.00 ( ( ( ( ( (185.6 (199. ( ( ( ( DDDDDDDDDD oooooooooooooo MMMMMMMMMM(9: xx~uuuuuu(0, 1aaaaaa (NN NN 0 ee~gggg(3.0,1 + NN 0 ee~gggg(3.0, MMMMMM ssssssssssssssssss MMMMMMMMMM(10: xx~uuuuuu(0,1aaaaaa ee~gggg(5.0,

13 (15.14 (10.97 ( ( (151.1 ( ( (1.86 ( ( (15.75 ( (*Efficiencies are in the parenthesis 54 VI. Determination of Shape Parameter In order to determine whether when a particular density is appropriate for the error term, a Q-Q plot of the ordered estimated residuals which are calculated using the LSEs (Least Square Estimation, ee ii = yy ii + θθ LL xx ii, (ii = 1,,, are plotted against population quantiles for that density. The population quantiles tt ii are determined from the equation tt (ii tt(uudddd = ii +1 ; 1 ii, where is the sample size. The Q-Q plot that closely approximates a straight line would be assumed to be the most appropriate. VII. Conclusion In this study, we show that when the error term is not normal which is applicable in most areas, MML integrated regression estimator( in the presence of non-response can improve the efficiency of regression estimator. We also show that the MML integrated regression estimator ( (modified regression estimators is robust to outliers as well as other data anomalies. References Références Referencias 1. Chambers, R. L. (1986, Outlier robust finite population estimation. Journal of the American Statistical Association, 81, Cochran, W.G. (1977. Sampling Techniques. John Wiley & Sons, New York. 3. Farrell, P. J., & Barrera, M. S. (006. A comparison of several robust estimators for a finite population mean. Journal of Statistical Studies, 6, Farrell, P.J., Barrera, S.M. (007. A comparison of several robust estimators for a finite population mean. J. Stat. Stud. 6, Gwet, J. P., & Rivest, L. P. (199. Outlier resistant alternatives to the ratio estimator. Journal of the American Statistical Association, 87, Hamilton, L. C. (199. Regression with graphics. California: Brooks, Cole Publishing Company. 7. Hansen, M. H. and Hurwitz, W. N. (1946. "The problem of nonresponse in sample surveys", J. Amer. Stat. Assoc.,41, Islam, M. Q., Tiku, M.L. and Yildirim, F. (001. Non-normal regression I skew distributions. Commun. in Statist.: Theo. and Meth., 30, Jenkins, O. C., Ringer, L.J., Hartley, H.O. (1977. Root estimators. Journal of the American Statistical Association, 68, Rao, J. N. K., & Beegle, L. D. (1967. A Monte Carlo study of some ratio estimators. Sankhy a Ser. B, Rao, P. S. R. S. (1986. "Ratio estimation with subsampling the Non-respondents", Survey Methodology, 1(, Rao, P. S. R. S. (1987. "Ratio and regression estimates with subsampling the nonrespondents", Paper presented at a special contributed session of the International Statistical Association Meetings, Sept,, -16, 1987,Tokyo, Japan.

14 13. Rao, P. S. R. S. (1990. Regression estimators with subsampling of non-respondents. Data Quality Control. Theory and Pragmatics (Gunar e. Liepins and V.R.R. Uppuluri, Eds. Marcel Dekker, New York,

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ANewApproximationtoStandardNormalDistributionFunction. A New Approximation to Standard Normal Distribution Function

ANewApproximationtoStandardNormalDistributionFunction. A New Approximation to Standard Normal Distribution Function Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 7 Issue 6 Version.0 Year 207 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals