Improved Ratio Estimators of Variance Based on Robust Measures

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1 Improved Ratio Estimators of Variance Based on Robust Measures Muhammad Abid 1, Shabbir Ahmed, Muhammad Tahir 3, Hafiz Zafar Nazir 4, Muhammad Riaz 5 1,3 Department of Statistics, Government College University, Faisalabad, 38000, Pakistan. Department of Mathematics, COMSATS Institute of Information Technology, Wah Cantt 47040, Pakistan 4 Department of Statistics, University of Sargodha, Sargodha, 40100, Pakistan. 5 Department of Mathematics and Statistics, King Fahad University of Petroleum and Minerals, Dhahran, 3161, Saudi, Arabia Abstract In this study, we develop some new estimators for estimating the population variance by utilizing the information on midrange and inter decile range of an auxiliary variable. A general class of estimators is also suggested. The derivations of the bias and the mean squared error are presented. Conditions are determined to verify the efficiency of the proposed estimators over existing estimators considered in this study. An Empirical study is also provided for illustration and verification. Moreover, a robust study is also carried out to evaluate the performance of proposed estimators as compared to existing estimators in case of extreme values. From the theoretical and empirical study, it is found that the suggested estimators perform more efficiently as compared to the existing estimators considered in this study. Keywords: Auxiliary Variable; Bias; Correlation Coefficient; Inter Decile Range; Mean Squared Error, Midrange. 1. PhD, Corresponding Author, mabid@zju.edu.cn, Cell # PhD, shabbiarahmad786@yahoo.com, Cell # PhD, tahirqaustat@yahoo.com, Cell # PhD, hafizzafarnazir@yahoo.com, Cell # PhD, riaz76qau@yahoo.com, Cell #

2 1. Introduction In survey research, if information is available on every unit of population and this information is correlated with the study variable, then such information is called auxiliary information. By utilizing this auxiliary information, one can propose numerous types of estimators for estimating the population variance by incorporating the product, the ratio and the regression methods of estimation. To the best of our knowledge, Neyman [1] was initially considering the use of auxiliary information in his study. After that, Cochran [] initiated the use of auxiliary information at estimation stage and proposed ratio estimator for estimating the population mean. The ratio estimator is most effective for estimating the population mean or variance when there exists a high positive correlation between the variables of interest and an auxiliary variable Suppose a finite population U = {U 1, U, U 3,, U N } consists of N different and identifiable units. Let Y be a measurable variable of interest with values Y i being ascertained on U i ; i = 1,,, N resulting in a set of observations Y = {Y 1, Y,, Y N }. The purpose of the measurement process is to estimate the population variance S Y = N 1 N i=1 (Y i Y ) by drawing a random sample from the population. The traditional ratio estimator for evaluating the population variance, S y, of the variable of interest Y is described as S r = s y s x S x (1.1) The bias and mean squared error (MSE) of the estimator given in Eq. (1) are respectively where,β (x) = μ 04 μ 0 B(S r ) = γs y [(β (x) 1) (λ 1)] MSE(S r ) = γs y 4 [(β (y) 1) + (β (x) 1) (λ 1)], β (y) = μ 40 μ 0, λ = μ (μ 0 μ 0 ) and μ rs = N 1 N i=1 (Y i Y ) r (X i X ) s.

3 Isaki [3] was the first to suggest variance base ratio and regression estimators by using one auxiliary variable for estimating the population variance. After that, Prasad and Singh [4] suggested new estimators and showed that their proposed estimators are more capable as compared to the estimators suggested by Isaki [3]. Arcos et al. [5] have also proposed some new estimators which performs better than the usual estimator and the other variance ratio estimators considered in this study. Comprehensive detail on the problem of creating competent estimators for estimating population variance can be seen in Agarwal and Sithapit [6], Ahmed et al. [7], Acros and Rueda [8], Al-Jararha and Al-Haj Ebrahem [9], Bhushan [10], Das and Tripathi [11], Cebrián and García [1], Gupta and Shabbir [13], Reddy [14], Singh et al. [15, 16], Upadhyaya and Singh [17] and Wolter [18]. The remainder of the article is planned as follows: In Section, we provide the detail explanation of the existing variance estimators. The formation of recommended estimators and the efficiency assessment of the proposed estimators with the usual estimator and the existing estimators are given in Section 3. Section 4 includes a practical study of suggested estimators. Robustness study of the proposed estimators is investigated in Section 5. Final comments and conclusion are presented in Section 6.. The Existing Estimators Upadhyaya and Singh [19] suggested the following estimator for S y by employing the information on kurtosis of an auxiliary variable. Upadhyaya and Singh [19] estimator is given as S 1 = s y ( S x + β (x) s x + β (x) ) 3

4 Kadilar and Cingi [0] proposed some new ratio estimators of variance and proved that these estimators perform efficiently as compared to the usual estimator of variance. Kadilar and Cingi [0] estimators are specified by S = s y ( S x +C x s x +C x S 3 = s y ( S x +β (x) s x +β (x) S 4 = s y ( S x β (x) +C x ) and s x β (x) +C x S 5 = s y ( S x C x +β (x) ). s x C x +β (x) Subramani and Kumarapandiyan [1] suggested an estimator of variance by adopting the information on the median of an auxiliary variable. Subramani and Kumarapandiyan [1] estimator is written as S 6 = s y ( S x +M d s x +M d ). Motivated by Upadhyaya and Singh [19] and Kadilar and Cingi [0], Subramani and Kumarapandiyan [] developed the following estimators S 7 = s y ( S x +Q 1 s x +Q S 8 = s y ( S x +Q 3 1 s x +Q S 9 = s y ( S x 3 +Q r s x +Q S 10 r = s y ( S x +Q d ) and s x +Q S 11 = s y ( S x +Q a d s x +Q a ). Making use of the information related to the deciles of an auxiliary variable, Subramani and Kumarapandiyan [3] introduced the following estimators for estimating the population variance S 1 = s y ( S x +D 1 s x +D S 13 = s y ( S x +D 1 s x +D S 14 = s y ( S x +D 3 s x +D S 15 = s y ( S x +D 4 ) and 3 s x +D S 16 = s y ( S x +D 5 ) 4 s x +D 5 S 17 = s y ( S x +D 6 s x +D S 18 = s y ( S x +D 7 6 s x +D S 19 = s y ( S x +D 8 7 s x +D S 0 = s y ( S x +D 9 ) and 8 s x +D S 1 = s y ( S x +D 10 ) 9 s x +D 10 Subramani and Kumarapandiyan [4] also suggested an estimator based on the median and coefficient of variation of an auxiliary variable. Subramani and Kumarapandiyan [4] estimator is given by, S = s y ( S x C x +M d ). s x C x +M d Khan and Shabbir [5] suggested an estimator as follows, 4

5 S 3 = s y ( S x ρ+q 3 ). s x ρ+q 3 3. The Suggested Estimator Encouraged by the work stated in section, we recommend some new ratio estimators of variance through the information on the midrange and inter decile range of an auxiliary variable. The midrange ((MR) is found as MR = X (1)+X (N), where X (1) and X (N) are the smallest and largest order statistics in a population of size N. Ferrell [6] verified that as the midrange is based on only extreme values of data, so this measure is sensitive to outliers. The next measure include in this study is the inter decile range (IDR) which is a difference between the 9 th and 1 st decile of an auxiliary variable and defined as: IDR = D 9 D 1. The major edge of IDR is that it is also robust against outliers (cf. Sohel et al. [7], Abid et al. [8, 9, 30, 31]). The proposed estimators can be specified as follows: S p1 = s y ( S x +MR s x +MR S p = s y ( S x +IDR s x +IDR S p3 = s y ( S x ρ+mr s x ρ+mr S p4 = s y ( S x ρ+idr s x ρ+idr S p5 = s y ( S x C x +MR ) and s x C x +MR S p6 = s y ( S x C x +IDR ). s x C x +IDR The proposed and the existing estimators also belong to the following general class of estimators for S y defined as S i = s y ( k 1S x + k k 1 s x + k ) where (k 1, k ) are either constant or function of known parameters of the population. The bias and the mean squared error (MSE) of S i can be obtained as follows: e 0 = s y S y S, e 1 = s x S x y S. Further, we can write, s y = S y (1 + e 0 ) and s x = S x (1 + e 1 ) and from x the definition of e 0 and e 1, we obtain E(e 0 ) = E(e 1 ) = 0, 5

6 E(e 0 ) = γ(β (y) 1 E(e 1 ) = γ(β (x) 1) and E(e 0 e 1 ) = γ(λ 1). The bias of the proposed estimators, S i, i = p1, p,, p6 is derived as: S i = s y ( k 1S x +k k 1 s x +k ); i = p1, p,, p6. S i = S k 1 S x + k y (1 + e 0 ) ( k 1 S ) x (1 + e 1 ) + k S i = S y (1 + e 0 ) ( (1+R i e 1 ) where, R i = ( S i = S y (1 + e 0 )(1 + R i e 1 ) 1 1 k 1S x ) k 1 S x+k Assuming R i e 1 < 1 so that (1 + R i e 1 ) 1 is expandable, expanding the right hand side of above equation and neglecting the terms of e s having power greater than two, we have; S i S y = S y e 0 S y R i e 1 + S y R i e 1 S y R i e 0 e 1 (3.1) Taking expectation on both sides of above equation, we will obtain E(S i S y ) = S y E(e 0 ) S y R i E(e 1 ) + S y R i E(e 1 ) S y R i E(e 0 e 1 ) So, B(S i ) = γs y R i [R i (β (x) 1) (λ 1)] The MSE of the proposed set of estimators, S i, i = p1, p,, p6 is derived as: Squaring both sides of Eq. (3.1 (S i S y ) = (S y e 0 S y R i e 1 + S y R i e 1 S y R i e 0 e 1 ) Expanding the right hand side of above equation and neglecting the terms of e s having power greater than two, we have; (S i S y ) = S 4 y e 0 + S 4 y R i e 1 S 4 y R i e 0 e 1 Taking expectation on both sides, we will obtain E(S i S y ) = (S 4 y E(e 0 ) + S 4 y R i E(e 1 ) S 4 y R i E(e 0 e 1 )) As we know the definition of MSE is: 6

7 MSE(S i ) = E(S i S y ) Hence, MSE (S i ) = 4 γsy [(β (y) 1) + R i (β (x) 1) R i (λ 1)] where i = 1,,,3 for existing estimators and i = p1, p,, p6 for proposed estimators. where, R i = ( k 1S x β k 1 S x +k (x) = μ 04 μ 0, β (y) = μ 40 μ 0 and λ = μ (μ 0 μ 0 ). The proper selections of k 1, k and R i are given in Table 1. [Insert Table 1 here] 3.1. Efficiency comparison This section presents the conditions in which the proposed estimators perform more capably as compared to the usual and the existing ratio estimators of variance. We have used the following notations of MSE(S pj ) and MSE(S i ) for the proposed and the existing estimators, respectively, for comparison purpose Comparison with usual ratio estimator of variance If the suggested estimators have lesser MSE values in comparison to the usual estimator, then they are more efficient. Mathematically, it is expressed as MSE(S pj ) < MSE(S r R pj (β (x) 1) R pj (λ 1) (β (x) 1) (λ 1) ( ) After solving Eq. (3 we get, R pj < (λ 1) (β (x) 1) 1. hence, MSE(S pj ) < MSE(S r if R pj < (λ 1) 1. (3.1.1.) (β (x) 1) where, j = 1,,, 6. 7

8 3... Comparisons with existing ratio estimators of variance The proposed estimators are considered to perform better if the MSE values of the suggested estimators are lesser than the MSE values of existing estimators and in algebraic form, it is written as MSE(S pj ) < MSE(S i R pj (β (x) 1) R pj (λ 1) R i (β (x) 1) R i (λ 1) (3...1) After solving Eq. (5 we get, R pj < (λ 1) (β (x) 1) R i. hence, MSE(S pj ) < MSE(S i if R pj < (λ 1) (β (x) 1) R i. (3...) where, i = 1,,, 3 and j = 1,,, Practical study The performances of the recommended estimators with respect to the usual and the existing ratio estimators of variance are judged by the use of 3 real populations. Populations 1 and are obtained from Murthy [3] page 8. Population 3 is taken from Cochran [33] page 15. In Table, we present the characteristics of 3 real populations. The values of the constants and biases of the existing and suggested estimators are reported in Tables 3 and 4, respectively, whereas, the MSE values are reported in Tables 5 and 6, correspondingly. [Insert Tables here] 8

9 From Tables 3 and 4, it is found that the recommended estimators have smallest values of constant by comparing with the usual and existing ratio estimators of variance. Therefore, they also also satisfy the conditions given in Eq. (3.1.1.) and Eq. (3...) which means that the proposed estimators performs better in comparison with the usual and existing estimators. The values of the biases of the recommended estimators are also lesser than usual estimator and the existing estimators (cf. Table 3 vs Table 4). Comparing the estimators suggested by Isaki [3], Upadhyaya and Singh [19], Kadilar and Cingi [0], Subramani and Kumarapandiyan [1,, 3, 4] and Khan and Shabbir [5] with the proposed estimators, it is revealed that recommended estimators have smallest values of MSE as compared to the existing estimators which confirm that the proposed estimators are more efficient (cf. Table 5 vs Table 6). [Insert Tables 3-6 here] 5. Robustness study of the proposed estimators As in the earlier sections, it is mentioned that measures used in this study such as midrange and inter decile range are robust against outliers. Thus when there exist outliers in the data these measures are performing efficiently as compared to other measures of locations. So, in this section, we evaluate the efficiency of our recommended estimators in case of outliers. For this purpose, we considered two real data set obtained from the Italian bureau of the environment protection- IBEP [34]. This data set consists of three variables: the entire quantity (tons) of reusable waste collection in Italy in 003 (Y the entire quantity of reusable waste collection in Italy in 00 (X 1 ) and the number of residents in 003 (X ). From Figures 1 and, we clearly see that there exist outliers in the data, so we expect the recommended estimators to outperform the usual estimator and the existing estimators. 9

10 [Insert Figure 1- here] The characteristics of two outliers data sets are reported in Table and the MSE values of the existing and the proposed estimators are given in Table 8. From Table 8, we reveal that the recommended estimators have least values of MSE in comparison to the usual and existing estimators, which proves that the proposed estimators are also more superior in the presence of outliers than the estimators used in this study. [Insert Tables 7-8 here] So, at the end, we may conclude that the performance of the proposed estimators is relatively better in comparison to the usual and existing ratio estimators of the variance in case of the data sets with and without outliers. 6. Summary and Conclusions To enhance the precision of estimators, the use of auxiliary information is very important in both the designing and estimation stages. This study developed some new ratio estimator of variance based on the information obtained from the midrange and inters decile range of an auxiliary variable. The proposed estimators are compared with the estimators introduced by Isaki [3], Upadhyaya and Singh [19], Kadilar and Cingi [0], Subramani and Kumarapandiyan [1,, 3, 4] and Khan and Shabbir [5]. From the results of this study, we have observed that the proposed estimators show better performance in terms of bias and mean squared error as compared with its competitors. The suggested estimators also perform more efficiently in the presence of extreme values as compared to the usual and existing estimators consider in this study. Hence, we strongly recommend the use of the proposed estimators over the estimators considered in this study when usual and unusual observations are present in the auxiliary variables. 10

11 Acknowledgement The authors are thankful to the anonymous reviewers and Editor for the constructive comments that helped in improving the last version of the paper. The author Muhammad Riaz is indebted to King Fahd University of Petroleum and Minerals (KFUPM Dhahran, Saudi Arabia for providing excellent research facilities. NOMENCLATURE Roman N n Y X X, Y x, y S y, S x s y, s x S xy C x, C y M d Q 1 Q 3 γ = 1 n Q r = Q 3 Q 1 Q d = Q 3 Q 1 Q a = Q 3+Q 1 MR = X (1)+X (N) IDR = D 9 D 1 B(. ) MSE(. ) S i S pj Subscript i j Greek ρ Population size Sample size Study variable Auxiliary variable Population means Sample means Population variance of Y and X Sample variance of Y and X Population covariance between X and Y Coefficient of variation Median of X First (lower) quartile Third (upper) quartile Inter quartile range Semi quartile range Semi quartile average Mid-range Inter decile range Bias of the Estimator Mean square error of the estimator Existing modified ratio type variance estimator of S y Proposed modified ratio type variance estimator of S y For existing estimators For proposed estimators Coefficient of correlation between X and Y 11

12 References 1. Neyman, J. "Contribution to the theory of sampling human populations", Journal of American Statistical Association, 33, pp (1938).. Cochran, W. G. "The estimation of the yields of the cereal experiments by sampling for the ratio of grain to total produce", Journal of Agricultural Sciences, 30, pp (1940). 3. Isaki, C. T. "Variance Estimation Using Auxiliary Information", Journal of the American Statistical Association, 78, pp (1983). 4. Prasad, B. and Singh, H. P. "Some Improved Ratio Type Estimators of Finite Population Variance in Sample Surveys", Communications in Statistics-Theory and Methods, 19, pp (1990). 5. Arcos, A., Rueda, M., Martinez, M. D., Gonzalez, S. and Roman, Y. Incorporating the Auxiliary Information Available in Variance Estimation", Applied Mathematics and Computation, 160, pp (005). 6. Agarwal, M. C. and Sithapit, A. B. "Unbiased Ratio Type Estimation", Statistics and Probability Letters, 5, pp (1995). 7. Ahmed, M. S., Raman, M. S. and Hossain, M. I. "Some Competitive Estimators of Finite Population Variance Using Multivariate Auxiliary Information", Information and Management Sciences, 11(1 pp (000). 8. Arcos, A. and Rueda, M. Variance Estimation Using Auxiliary Information: An Almost Unbiased Multivariate Ratio Estimator, Metrika, 45, pp (1997). 9. Al-Jararha, J. and Ebrahem, M. A-H. "A Ratio Estimator under General Sampling Design", Australian Journal of Statistics, 41( pp (01). 1

13 10. Bhushan, S. "Some Efficient Sampling Strategies Based on Ratio Type Estimator", Electronic Journal of Applied Statistical Analysis, 5(1 pp (01). 11. Das, A. K. and Tripathi, T. P. Use of Auxiliary Information in Estimating the Finite Population Variance, Sankhyã, 40, pp (1978). 1. Cebrián, A. and García, M. "Variance Estimation Using Auxiliary Information: An Almost Unbiased Multivariate Ratio Estimator", Metrika, 45(1 pp (1997). 13. Gupta, S. and Shabbir, J. "Variance Estimation in Simple Random Sampling Using Auxiliary Information", Hacettepe University Bulletin of Natural Sciences and Engineering Series B: Mathematics and Statistics, 37(1 pp (008). 14. Reddy, V. N. "On Transformed Ratio Method of Estimation", Sankhyã, 36, pp (1974). 15. Singh, H. P., Upadhyaya, U. D. and Namjoshi, U. D. "Estimation of Finite Population Variance", Current Science, 57(4 pp (1988). 16. Singh, H. P., Chandra, P. and Singh, S. "Variance Estimation Using Multi-Auxiliary Information for Random Non-Response in Survey Sampling", STATISTICA, 63, pp (003). 17. Upadhyaya, L. N. and Singh, H. P. "Almost Unbiased Ratio and Product-Type Estimators of Finite Population Variance in Sample Surveys", Statistics in Transition, 7(5 pp (006). 18. Wolter, K. M., Introduction to Variance Estimation, Springer-Verlag, Berlin (1985). 19. Upadhyaya, L. N. and Singh, H. P. "An Estimator of Population Variance that Utilizes the Kurtosis of an Auxiliary Variable in Sample Surveys", Vikram Mathematical Journal, 19, pp (1999). 13

14 0. Kadilar, C. and Cingi, H. "Improvement in Variance Estimation Using Auxiliary Information", Hacettepe University Bulletin of Natural Sciences and Engineering Series B: Mathematics and Statistics, 35(1 pp (006). 1. Subramani, J. and Kumarapandiyan, G. "Variance Estimation Using Median of the Auxiliary Variable", International Journal of Probability and Statistics, 1(3 pp (01).. Subramani, J. and Kumarapandiyan, G. "Variance Estimation Using Quartiles and their Functions of an Auxiliary Variable", International Journal of Statistics and Applications, (5 pp (01). 3. Subramani, J. and Kumarapandiyan, G. "Estimation of Variance Using Deciles of an Auxiliary Variable", Proceedings of International Conference on Frontiers of Statistics and Its Applications, 33, pp (01). 4. Subramani, J. and Kumarapandiyan, G. "Estimation of Variance Using Known Coefficient of Variation and Median of an Auxiliary Variable", Journal of Modern Applied Statistical Methods, 1(1 pp (013). 5. Khan, M. and Shabbir, J. "A Ratio Type Estimators for the Estimation of Population Variance Using Quartiles of an Auxiliary Variable", Journal of Statistics Applications and Probability, (3 pp (013). 6. Ferrell, E. B. "Control Charts Using Midranges and Medians", Industrial Quality Control, 9, pp (1953). 7. Sohel, R., Md, S. U. D., Habshah, M. and Imon, A. H. M. R. "Decile Mean: A New Robust Measure of Central Tendency", Chiang Mai Journal of Science, 39(3 pp (01). 14

15 8. Abid, M., Abbas, N., Nazir, Z. A. and Lin, Z. Enhancing the Mean Ratio Estimators for Estimating Population Mean Using Non-Conventional Location Parameters, Revista Colombiana de Estadistica, 39(1 pp (016). 9. Abid, M., Abbas, N. and Riaz, M. Improved Modified Ratio Estimators of Population Mean Based on Deciles, Chiang Mai Journal of Science, 43(1 pp (016). 30. Abid, M., Abbas, N., Sherwani, R. A. K. and Nazir, Z. A. Improved Ratio Estimators for the Population Mean Using Non-Conventional Measures of Dispersion, Pakistan Journal of Statistics and Operation Research, 1( pp (016). 31. Abid, M., Sherwani, R. A. K., Abbas, N. and Nawaz, T. Some Improved Modified Ratio Estimators Based on Decile Mean of an Auxiliary Variable, Pakistan Journal of Statistics and Operation Research, 1(4 pp (016). 3. Murthy, M. N., Sampling Theory and Methods, Statistical Publishing Society, Calcutta, India (1967). 33. Cochran, W. G., Sampling Techniques, John Wiley and Sons, New York (1977). 34. IBEP., (004) 15

16 Figure Captions Figure 1. Scatter graph of first auxiliary and study variables. Figure. Scatter graph of second auxiliary and study variables. Table Captions Table 1. The suitable choices of k 1 and k for existing and proposed estimators. Table. Characteristics of the populations. Table 3. The constants and biases of the existing ratio estimators of variance. Table 4. The related constants and biases of the proposed ratio estimators of variance. Table 5. The mean squared error values of the existing ratio estimators of variance. Table 6. The mean squared error values of the proposed ratio estimators of variance. Table 7. Characteristics of the populations of extreme values. Table 8. The mean squared error values of the existing and proposed ratio estimators of variance in case of extreme values populations. 16

17 Figure 1 17

18 Figure 18

19 Table 1 Estimators k 1 k S 1 1 β (x) S 1 C x S 3 1 β (x) S 4 β (x) C x S 5 C x β (x) S 6 1 M d S 7 1 Q 1 S 8 1 Q 3 S 9 1 Q r 1 Q d S 10 1 Q a S 11 1 D 1 S 1 1 D S 13 1 D 3 S 14 1 D 4 S 15 1 D 5 S 16 1 D 6 S 17 1 D 7 S 18 1 D 8 S 19 1 D 9 S 0 1 D 10 S 1 C x M d S ρ Q s 1 MR 1 IDR ρ MR ρ IDR S 3 S p1 S p S p3 S p4 S p5 S p6 C x C x MR IDR 19

20 Table Parameters Population 1 Population Population 3 N n Y X ρ S y C y S x C x β (x) β (y) λ M d Q Q Q r Q d Q a MR IDR

21 Table 3 Estimators Constant Population 1 Population Population 3 Population 1 Population Population 3 S r S S S S S S S S S S S S S S S S S S S S S S S 3 Bias 1

22 Table 4 Estimators Constant Population 1 Population Population 3 Population 1 Population Population 3 S p S p S p S p S p S p6 Bias Table 5 Estimators Population 1 Population Population 3 S r S S S S S S S S S S S S S S S S S S S S S S S 3

23 Table 6 Estimators Population 1 Population Population 3 S p S p S p S p S p S p Table 7 Parameters Population 4 Population 5 N n Y X ρ S y C y S x C x β (x) β (y) λ M d Q Q Q r Q d Q a MR IDR

24 Table 8 Estimators Population 4 Population 5 S r S S S S S S S S S S S S S S S S S S S S S S S S p S p S p S p S p S p6 4

25 Authors' Biographies Muhammad Abid obtained his MSc and MPhil degrees in Statistics from Quaid-i-Azam University, Islamabad, Pakistan, in 008 and 010, respectively. He did his PhD in Statistics from the Institute of Statistics, Zhejiang University, Hangzhou, China, in 017. He served as a Statistical Officer in National Accounts Wing, Pakistan Bureau of Statistics (PBS) during He is now serving as an Assistant Professor in the Department of Statistics, Government College University, Faisalabad, Pakistan, from 017 to present. He has published more than 15 research papers in research journals. His research interests include Statistical Quality Control, Bayesian Statistics, Non-parametric Techniques, and Survey Sampling. Shabbir Ahmad obtained his MSc degree in Statistics from the PMAS Arid Agriculture University Rawalpindi, Pakistan, in 00 and his MPhil degree in Statistics from Quaid i Azam University, Islamabad, Pakistan, in 005. He served as a Statistical Officer in the National Accounts Wing, Pakistan Bureau of Statistics, during 006 to 007. He obtained his PhD degree in Statistics at the Department of Mathematics, Institute of Statistics, Zhejiang University, Hangzhou, China, in 013. He is serving as an Assistant Professor in the Department of Mathematics, COMSATS Institute of Information Technology, Wah Cantonment, Pakistan. His current research interests include statistical process control and application of sampling techniques. His is shabbirahmad786@yahoo.com. Muhammad Tahir obtained his PhD in Statistics from Quaid-i-Azam University, Islamabad, Pakistan, in 017. Currently, he is an Assistant Professor of Statistics in Government College University, Faisalabad, Pakistan. He has published more than 5 research papers in national and international reputed journals. His research interests include Bayesian Inference, Reliability Analysis and Mixture Distributions. Hafiz Zafar Nazir obtained his MSc and M.Phil degrees in Statistics from the Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan, in 006 and 008, respectively. He did his PhD in Statistics from the Institute of Business and Industrial Statistics University of Amsterdam, The Netherlands, in 014. He served as a Lecturer in the Department of Statistics, University of Sargodha, Pakistan, during He is now serving as an Assistant 5

26 Professor in the Department of Statistics, University of Sargodha, Pakistan, from 015 until present. His current research interests include Statistical Process Control, Non-parametric techniques, and Robust methods. His address is Muhammad Riaz obtained his PhD in Statistics from the Institute for Business and Industrial Statistics, University of Amsterdam, The Netherlands, in 008. He holds the position of Professor in the Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. His current research interests include Statistical Process Control, Non-parametric Techniques, and Experimental Design. 6