• No results found

Suggested Area of future research

131

Generally, the discussions above have shown that no single estimator is efficient for all populations and conditions and a rough idea of the magnitude of the correlation between the study and auxiliary variables and hence the size measures would provide insight into which estimator would be best for the target population.

Secondly, the idea of the ratio of coefficients of variations, skewness and kurtosis as related with the correlation coefficient would help in the specification of estimators.

Whereas the information of the target populations is not available to the survey statistician, this study have shown that among the estimators in the class defined by c=1,2,3 and 4 there is the one estimator that is best for estimating population total.

Thus, in this era of information technology, it would be easier to identify such estimator when the suggested estimators are run simultaneously.

132

132 REFERENCES

1 Adhikary, A.K.(2009). Improving the Hansen-Hurwitz Estimator in PPSWR sampling. Calcutta Stat. Assoc. Bull., 61,(6), 267-287.

2 Agarwal M.C and Jain, N.(1989). A new predictive product estimator.

Biometrika, 78, 822-823.

3 Agarwal, M.C and Panda, .B.(1993). Multivariate Product Estimators. Jour.

Ind. Soc. Agri. Statist., 45(3), 354-371.

4 Alodat, N.A(2009). On Unequal Probability sampling Without Replacement sample size 2. Int. J. Open Problems Comp. Math., 2, 1,108-112.

5 Amahia, G.N, Chaubey, Y.P and Rao, T.J(1989). Efficiency of a new PPS sampling for multiple characteristics. Journal of Statistical Planning and Inference, (21),75-84.

6 Arnab, R.(1990). On commuttivity of design and model expectations in randomized response surveys. Communications in Statistics, 19: 3751-3757.

7 Arnab, R.(2001). Estimation of a finite population total in varying proability sampling for multi-character surveys. Metrika, 54,159-177.

8 Basu, D(1958). On sampling with and without replacement. Sankhya, 28, 287-294.

9 Basu, D.(1971). An essay on the logical foundation of survey sampling I”

Foundation of statistical inference. Holt, Rinehard and Winston. Edited by Godambe and Sprott.

10 Bansal M.L and Singh,R.(1985). An alternative estimator for multiple

characteristics in PPS sampling. Journal of Statistical. Planning and Inference., 21, 75-84.

11 Bansal ML, Singh R (1989) An alternative estimator for multiple

characteristics corresponding to Horvitz and Thompson estimator in probability proportional to size and without replacement sampling. Statistica, Anno. XLIX, 3:447-452.

133

12 Bansal ML, Singh R (1990) An alternative estimator for multiple

characteristics in RHC sampling scheme. Commun. Statist. Theory

Meth. 19(5):1777-1784

13. Bedi P.K (1995) An alternative estimator in Midzuno scheme for multiple characteristics. Commun. Statist. Simula.: 17-30.

14 Bedi, P.K and Rao, T.J.(1997). PPS method of estimation under a transformation. Jour. Ind. Soc. Agri. Statist., 184-195.

15 Brewer K.R.W.(1963). Ratio estimation in finite populations. Some results deducible from the assumption of an underlying stochastic process.

Australian Journal of Statistics, 5, 93-105.

16 Brewer, K.R.W. (1979). A class of robust sampling designs for large-scale surveys. Jour. Amer. Stat. Assoc. 74, 911-915.

17 Brewer, M.L and Hanif, M.(1983). Sampling with unequal probabilities.

Edited by Brillinger, D; Fienberg, S; Gani, J; Hartigan, J; and Krickeberg, K. Springer-Verlag, New York.

18 Carroll, J.L and Hartley H.O(1964): The symmetric method of unequal probability sampling without replacement. Biometrika, 63, 614-620.

19 Chakravarti, I.M.(1954). On a problem of planning a multistage survey for multiple correlated character. Sankhya, 14, 211-216.

20 Chaudhuri A. (2001a). Using Randomized Response from a complex survey to estimate a sensitive proportion in a dichotomous finite population.

Jour. Statist. Planning Inference. 95, 37-42.

21 Chaudhuri A. (2001b). Estimating sensitive proportions from unequal probability samples using randomized responses. Pak. J. statist.

1793), 259-270.

22 Chaudhuri A.(2002). Estimating sensitive proportions from randomized responses in Unequal Probability sampling. Calcutta Statistical Bulletin, 52, 205-208.

23 Chaudhuri A.(2010). Essentials of Survey Sampling. PHI Learning Private Limited, New Delhi.

134

24 Chauhudri, A and Adhikary, A.K.(1990). Variance estimation with randomized response. Communications in Statistics: Theory and Methods, 19(3), 1119-1125.

25 Chaudhuri, A and Dihidar, K.(2009). Rao-Hartley sampling with competitive estimators. Calcutta Stat. Assoc. Bull., 61(6), 227-237.

26 Chaudhuri A and Pal S(2002). On certain alternative Mean square estimators in complex surveys. J. statist. Planning Inference, 104, 363-375.

27 Chaudhuri, A. and Vos, J.W.E.(1986). “Unified Theory and strategies of survey sampling”, North Holland Series in Statistics and Probability.

28 Chikkagoudas, M.S.(1967). On PPP- sampling with and without replacement.

Australian Jour. Statistics, 9(3),109-118

29 Cochran, W.G.(1946). Relative accuracy of systematic and stratified random samples for a certain class of population. Ann. Math. Statist. 17, 164-177.

30 Cochran, W.G.(1977): Sampling Techniques. 3rd edition, Wiley, New York.

31 Dalenius, T(1962). Recent advances in sample survey theory and methods.

Annals of Maths. Stat., 33(2), 323-349

32 Das, A.C.(1951). “On two phase sampling and sampling with varying probabilities”, Bull. Inter. Stat. Inst. 33(2), 105-112.

33 Das, A.K and Tripathi, T.P(1978): Unse of auxiliary information in estimating the finite population variance. Sankhya, C, 139-148.

34 Das, A.K. and Tripathi, T.P.(1980). Sampling strategies for population mean when the coefficient of variation of an auxiliary character is known.

Sankhya C, 42, 76-86.

35 Deshpande, M.N.(1978): A new sampling procedure with varying probabilities. Jour. of Ind. Soc. Of Agr. Stat., 30, 1110-114.

36 Dodge Y, Rousson V.(2000). Direction of Dependence in a regression line.

Commun. Stat. Theory Methods, 29(9-10), 1945-1955

135

37 Dodge, Y., and Rousson, V. (2001). On asymmetric properties of the correlation coefficient in the regression setting. The American Statistician, 55, 51 - 54.

38 Dodge Y and Yandegari(2009). On direction of dependence, metrika, online 39 Durbin, J. (1967). Some results in sampling when the units are selected with

unequal probabilities. J. Roy. Stat. Soc, B, 15, 262–269.

40 Ekaette, I.E(2008). A class of alternative estimators for Multi-characteristics in PPS sampling Scheme. Unpublished Ph.D thesis, Univeristy of Ibadan, Nigeria.

41 Fellegi, I.(1963): Sampling with varying probabilities without replacement:

rotating and non-rotating samples. Jour. Amer. Stat. Assoc, 58, 183-201.

42 Gajendra, K.V, Singh, H.P and singh, S.(2010). A family of estimators of population mean using multi-auxilliary variate and post-stratification.

Nonlinear Analysis: Modelling and control, 15(2),233-253.

43 Godambe, V.P.(1982). Estimation in survey sampling robustness and

optimality (with discussions). Jour. Amer. Stat. Assoc., 77, 393-406.

44 Godambe, V.P(1955). A unified theory of sampling from finite population.

Jour. Roy.Statist.Soc. B. 17, 269-278.

45 Godambe, V.P(1956). A new approach to sampling finite population, I, II.

Jour. Roy. Statist. Soc. B. 28, 310-328.

46 Godambe V.P and Thompson, M.E.(1977). Robust near optimal estimation in survey practice. Bull. Int. Stat. Inst. 47:3, 129-146.

47 Godambe, V.P and Joshi V.M.(1965). Admissibility and Bayes‟ estimation in sampling finite population I. Ann. Math. Statist. 36, 1707-1722.

48 Goodman, R. and Kish, L.(1950). Controlled selection – a technique in probability sampling. Jour. Amer. Statist. Assoc. 77, 393-405.

49 Grewal, I. S, Bansal M. L. and Singh. (1997): An alternative estimator for multiple characteristics using randomized response technique in PPS sampling. Aligarh Jour. Statist, 19, 51-65.

136

50 Gupta V.K., Nigam A.K and Kumar P.(1982): On a family of sampling

schemes with inclusion probability proportional to size. Biometrika, 69, 191-196.

51 Hajek, J.(1958). Some contributions to the theory of probability exactly proportional to size. Jour. Roy. Statist. Soc. B. 16, 236-238.

52 Hajek, J.(1964). Asymptotic Theory of rejective sampling with varying

probabilities from a finite population. Ann. Math. Stat. 35, 1491-1523.

53 Hanif M.and Ahmad, M.(2001). Approximate variance formula for variance of Horvitz Thompson Estimator using first order inclusion probabilities.

ISOS, VII, Lahore.

54 Hanif M. and Brewer, K.R.W.(1980). Sampling with unequal probabilities without replacement: A review. Inter. Stat. Rev., 48(3), 317-335.

55 Hansen, M. H. and Hurwitz, W. N. (1943). On the theory of sampling from a finite population. Ann. Math. Stat. 14, 333 – 362.

56 Hanurav, T.V.(1962). Some sampling schemes in Probability sampling.

Sankhya, A, 24, 421-428.

57 Hanurav, T.V.(1967). Optimum utilization of auxiliary information: πPS sampling of to units from a stratum.. Jour. Roy. Soc. Statist., B(23), 374-391.

58 Hartley H.O and Rao, J.N.K.(1962). Sampling with unequal probabilities and without replacement. Ann. Math. Stat. 33, 350-374.

59 Harzel, A.(1986). Sampling without replacement with unequal probabilities sample designs with pre-assigned joint inclusion- probabilities of any order. Metron, XLIV(1), 49-68.

60 Horvitz, D. G. and Thompson. D. J. (1952). A generalization of sampling without replacement from a finite universe. J. Amer. Stat. Assoc. 47, 663 – 685.

61 Jensen, R.J.(1942). Statistical Investigation of a sample survey for obtaining farm facts. Iowa Agric. Exp. Station Res. Bull. 304.

137

62 Kumar P, Agarwal SK (1997) Alternative estimators for the population totals in multiple characteristic survey. Commun. Statist.- Theory Meth.

26(10):2527- 2537

63 Lahiri, D.B.(1951). A method of sample selection providing unbiased ratio estimates. Bull. Internat. Statist. Inst. XXXIII, Book 2, 133-140.

64 Lui, Kung-Jong(1990). Modified Product estimators of finite population mean in finite sampling. Commun. Statist. – Theory Meth; 19(10), 3799- 3807.

65 Madow, W.G.(1948): On the limiting distributions of estimates based on sample from finite universes. Ann. Math. Stat., 19, 535-545.

66 Madow, W.G.(1949): On the theory of systematic sampling II. Ann. Math.

Stat., 20, 333- 354.

67 Mahalanobis, P.C.(1944). On large scale sample surveys. Philos. Trans. Roy.

Soc. B 231,329-451.

68 Mangat N.S and Singh R.(1993). Sampling with varying probabilities without replacement. A review. Aligarh Jour. Stat. 12 and 13: 75-105

69 Marriot, F.H.C(1990). Dictionary of Statistical Terms. Longman.

70 Menedez, E and Ferrales, J.(1989). A generalized Ratio estimator. Trab.

Estadist, 4, 3-11.

71 Midzuno, H.(1950): an outline of the theory of sampling systems. Ann. Inst. of Statist. Math. 1149-156.

72 Midzuno, H.(1952): on the sampling system with probability proportional to the sum of sizes. Ann. Inst. of Statist. Math. 3, 99-107.

73 Muddapur, M.V. (2003). On directional dependence in a regression line.

Communications in Statistics. Theory and Methods, 32, 2053 - 2057.

74 Mukhopadhyay P.(1972): A sampling scheme to realize a pre-assigned set of inclusion probabilities of first two order. Cal. Stat. Assoc, Bull., 21, 87-122.

75 Mukhopadhyay P.(1977): Robust estimators of finite population total under certain linear regression models. Sankhya, C, 39, 71-87.

138

76 Mukhopadhyay P.(1978): Estimating the variance of a finite population under super-population model. Metrika, 25, 115-122.

77 Mukhopadhyay P.(1982): Optimum strategies for estimating the variance of a finite population under super-population model. Metrika, 25, 115-122.

78 Mukhopadhyay P.(1984):Optimum estimation of finite population variance under generalized random permutation models. Cal. Stat. Assoc, Bull.,33, (129-130), 93-104.

79 Mukhopadhyay P.(1991): Varying probability without replacement sampling designs, A Review. ISI Tech. Rep. No. ASC/91/3.

80 Mukhopadhyay P.(1996): Inferential problems in survey sampling. New Age Int. Publishers, New Delhi.

81 Murthy, M.N.(1957): Ordered and Unordered estimators in sampling without replacement. Sankhya, 18, 379-390.

82 Murthy M.N(1964). Product Method of estimation, Sankhya, A, 26, 64-74 83 Narain, R.D. (1951). On sampling without replacement with varying

probabilities. J. Ind. Soc. Agric. Stat. 3, 169-175.

84 Neyman, J.(1934). On the two different aspects of the representative methods:

the method of stratified sampling and the method of purposive selection. Jour. Roy. Stat. Soc, 97, 558-625.

85 Olkin, I.(1958). Multivariate ratio estimators for finite populations. Biometrika, 45, 154-165

86 Pandey, B.N and Dubey, V.(1988). Modified product estimator using

coefficient of variation of auxiliary variate. Assam Statistical Review, 2, part 2, 64-66.

87 Pathak, P.K(1966). An estimator in PPS sampling for multiple characteristics.

Sankhya, A, 28(1), 35-40.

88 Raj, D. (1956). Some estimators in sampling with varying probabilities without replacement. J. Amer. Stat. Assoc. 51, 269 – 284.

89 Rao, J.N.K(1963). On the estimate of variance in unequal probability sampling. Ann. Inst. Statist. Math. 13(1), 57-60.

139

90 Rao, J.N.K(1963). On two systems of unequal probability sampling without replacement. Ann. Inst. Statist. Math. 15, 67-72.

91 Rao, J.N.K.(1965). On two simple schemes of unequal probability sampling without replacement. J. Ind. Stat. Assoc. 3, 173-180.

92 Rao JNK (1966a) Alternative estimators in the PPS sampling for multiple characteristics. Sankhya, 28(A):47-60.

93 Rao JNK (1966b). On the relative efficiency of some estimators in PPS sampling for multiple characteristics. Sankhya, 28(A):61-70.

94 Rao, J.N.K.(1978). Sampling designs involving unequal probabilities of selection and robust estimation of a finite population total.

Contributions to survey sampling and applied statistics. Papers in honour of H.O. Hartley, H.A. David (ed.), Academic Press, 69-87.

95 Rao T.J (1993a) On certain alternative estimators for multiple characteristics in varying probability sampling. Jour. Indian Soc. Agril. Statist.

45(3):307-318

96 Rao T.J (1993b) On certain problems of sampling design and estimation for multiple characteristics. Sankhya , 55(B):372-381.

97 Rao, J.N.K and Hartley, H.O.(1962). Sampling with unequal probabilities and without replacement. Annal Maths. Stat. 33, 350-375.

98 Rao J.N.K., Hartley, H.O., Cochran W.G. (1962) On a simple procedure of unequal probability sampling without replacement. Jour. Roy. Stat.

Soc. 24(B):482-491.

99 Rao, J.N.K and Singh, M.P.(1973). On the choice of estimators in survey sampling. Aust.Jour. Stat. 15, 95-104.

100 Robson D.S (1957). Applications of multivariate polykays to the theory of unbiased ratio-type estimation. Journ. Amer. Stat. assoc., 52, 409-414.

101 Rodgers, J.L. and Nicewander, W.A. (1988). Thirteen ways to look at the correlation coefficient. The American Statistician, 42, 59 - 66.

140

102 Rovine, M.J., and von Eye, A. (1997). A 14th way to look at a correlation coefficient: correlation as the proportion of matches. The American Statistician, 51, 42 - 46.

103 Royall, R.M.(1970): On finite population sampling theory under certain linear regression models. Biometrika, 57, 377-387.

104 Royall, R.M.(1971a). Linear regression in finite population sampling theory.

Foundation of statistical inference, V.P. Godambe and D.A.Sprott (eds.). Holt, Reinchart and Winston, Toronto, Canada, 259-279 105 Royall, R.M.(1971b). Finite population sampling – on label estimation. Ann.

Math. Stat., 41, 1774-1779.

106 Royall, R.M and Cumberland, W.G.(1981a): An empirical study of the ratio estimator and estimator of its variance. Jour. Amer. Stat.

Assoc., 76, 66-77.

107 Royall, R.M and Cumberland, W.G.(1981b):The finite population linear regression estimator and estimator of its variance- an empirical study. Jour. Amer. Stat. Assoc., 76, 924-930.

108 Royall, R.M and Herson, J.(1973b). Robust estimation in finite population II.

Statification on a size variable. Jour. Amer. Statisti. Assoc. 68, 890-893.

109 Sahoo J.(1995). An alternative unbiased product estimator. Jour. of statistical Research, 29, 1, 51-54

110 Sahoo, L.N., Das, B.C and Singh, G.N.(2006). A note on an IPPS sampling scheme. Allemeines Statistisches Archiv, 90, 385-393.

111 Sahoo, L.N., Mishra, G. And Senapati, S.C.(2005). A new sampling scheme with inclusion probability proportional to size. Jour. of Stat. Theory and applications. 4, 361-369.

112 Sahoo J, Sahoo L, Mohanty S (1994) Unequal probability sampling using a transformed auxiliary variable. Metron: 71-83

113 Sahoo L.N, Senapati, S.C and Mangaraj, A.K(2010). A class of IPPS sampling scheme. Revista Investigacion Operacional, 31, 3, 217-224.

141

114 Sampford, M.R.(1967): On sampling without replacement with unequal probability selection. Biometrika, 54, 499-513.

115 Sankarmanayana, K.(1969). A IPPS sampling scheme using Lahiri‟s method of selection. J. Ind. Soc. Agri. Stat., 21, 58-66.

116 Sarndal C.W., Swensson, B and Wretman, J.H.(1992). Model assisted survey sampling. Springer- Verlag

117 Saxena, P.R, Singh, P and Srivastava(1986): An unequal probability sampling scheme. Biometrika, 73, 761-763.

118 Sen, A.R.(1953). On the estimate of the variance in sampling with varying probabilities. Jour. Indian Soc. Agric. Statist. 6, 119-127.

119 Senapati, S.C., Sahoo, L.N and Mishra, G.(2006). N a scheme of two units with inclusion probability proportional to size. Australian Journal of Statistics, 35, 445-454.

120 Sengupta S.(1981). A Sampling scheme to realize inclusion probabilities exactly proportional to size. Jour. Ind. Soc. Agric. Statist. 33(2), 119-127.

121 Shahbaz, M.Q.(2004). A new estimator of population total using Durbin‟s Selection procedure for unequal probability sampling, Pak. J. Statist., 20(2), 287-294.

122 Shahbaz, M.Q. and Hanif, M(2003). A simple procedure for unequal

probability sampling without replacement and a sample of size 2. Pak.

J. Stat., vol 19(1), 151-160.

123 Shimizu, S., and Kano, Y. (2006). Use of non-normality in structural equation modeling: Application to direction of causation. Journal of Statistical Planning and Inference, 136.

124 Sidhu, S.S, Bansal, M.L and Singh S.(2007). Estimation of sensitive multi-characters using unknown values of unrelated questions. Applied Mathematical Sciences.vol 1 (37), 1803-1820.

142

125 Sinha B.K(1973): on sampling schemes to realized pre-assigned set of

inclusion probabilities of first two orders. Calcutta Stat. Assoc. Bull., 22,69-110.

126 Smith, H.F.(1938). An empirical law describing heterogeneity in the yields of agricultural crops. Jour. agri. Sci. 28, 1-23.

127 Srivastava, K.M and Srivastatva, N.(2009). Statistical Inference: Testing of Hypothesis. PHI Learning Private Ltd, New Delhi.

128 Srivenkataramana T(1980). A dual to ratio estimator in sample surveys.

Biometrika 67: 199-204.

129 Singh, G.N.(2003). An improvement of product method of estimation in sample surveys. Journ. Ind. Soc. Agri. Stat., 56(3),267-275.

130 Singh, H.P.(1986). A note on unbiased product-type estimators. Alig. J.

Statist., 6, 45-52

131 Singh, M.P.(1967). Ratio cum product method of estimation, Metrika, 12, 34-42.

132 Singh, H.R and Espejo, M.R.(2003). On linear and ratio-product estimation of finite population mean. The statistician, 52(1), 59-67.

133 Singh S, Grewal I.S and Joarder A.(2004). General class of estimators in multi-character surveys. Statistical papers, 45, 571-582.

134 Singh S and Horn S (1998) An alternative estimator for multi-character surveys. Metrika, 48, 99-107

135 Singh, H.P.and Kumar, S.(2009). A general class of DSS estimators of

population Ratio, product and mean in the presence of non-response based on the sub-sampling of non-respondents. Pak. J. Statist., 26(1), 203-208.

136 Singh, R and Singh, H.P.(1998). Almost Unbiased Ratio and Product –type estimators in systematic sampling. Questioniio, 22, 3, 403-416.

137 Singh S., Singh, H.P., Tailor, Allen, J. and Grewal I.(2002). Class of estimators for estimating ratio of two population means in the presence of random non-response. Working paper.

143

138 Singh, H.P and Tailor, R.(2003). Use of known correlation coefficient in estimating the finite population mean. Statistics in Transition, 6, 555-560.

139 Singh, H.P and Taylor, R.(2005). Estimation of population mean with known coefficient of variation of auxiliary characters. Statistica Anna LXV, 301-313

140 Sigh,H.P and UpadhayaL.N(1986). A dual to modified ratio estimator using coefficient of variation of auxiliary variable. Proceedings of national Academy of Sciences, India, 56, A, Part 4, 336-340.

141 Singh, V.K, Singh, G.N and Shukla, D(1993). A general class of estimators using auxiliary information for estimating population variance.

Sankhya, C,42 87-96

142 Sisodia,B.V.S and Dwivedi,V.K(1981). A modified ratio estimator using coefficient of variation of auxiliary variable. Journ. Ind. Soc. Agri.

Stat., 33(2), 13-18.

143 Sukhatme, P.V.(1935). Contributions to the theory of representative method.

Journ. Roy. Stat. Soc., 2,253-268.

144 Sungur, E.A. (2005). A note on directional dependence in regression setting.

Communications in Statistics. Theory and Methods, 34, 1957 - 1965.

145 Tripathi, T.P.(1969). A regression-type estimator in sampling with PPS and with replacement. Aust. Jour. Stat., 11, 140-148.

146 Tripathi, T.P.(1973). Double sampling for inclusion probabilities and regression method of estimation. Jour. Ind. Stat. Assoc., 10,33-46.

147 Tripathi, T.P.(1976). On double sampling for multivariate ratio and difference methods of estimation. Jour. Ind. Stat. Assoc., 28, 33-54.

148 Turkey, J.(1950). Some sampling simplified. Jour. Amer. Stat. Assoc., 45, 501-519.

149 Vijayan, K.(1966). On Horvitz-Thompson and Des Raj estimators. Sankhya, A(28), 87-92.

144

150 Vijayan, K.(1968). An exact πPS sampling scheme – Generalization of a method of Hanurav. Jour. Roy. Stat. Soc., B(30), 556-566.

151 Warner, S.L.(1965). Randomized Rsponse. A survey Technique for

eliminating evasive answers bias. Jour. Amer. Stat. Assoc., 60, 63-69.

152 Wishart, J.(1952). Moment coefficient of k-statistics in samples from a finite population. Biometrika, 39, 1-13.

153 Yates, F.(1948). Systematic sampling. Phil. Trans. Roy. Soc. London, A(241), 345-377.

154 Yates, F. and Grundy, P. M. (1953). Selection without replacement from within strata with probability proportional to size. J. Roy. Stat. Soc., B, 15,

153 – 161.

155 Yates, F and Zacopany, I.(1935). The estimation of the efficiency of sampling with special reference to sapling for yield in cereal experiment. J. Agri.

Sci., 25, 543-577.

145 APPENDIX A

STUDY POPULATIONS UTILIZED IN THIS STUDY

Table 46: Study Population I

S/No x y Ρ S/No x Y ρ

1 3 11 0.395 16 5 10 0.395

2 4 7 0.395 17 6 9 0.395

3 5 9 0.395 18 3 5 0.395

4 8 8 0.395 19 3 7 0.395

5 12 8 0.395 20 9 9 0.395

6 11 9 0.395 21 6 6 0.395

7 8 8 0.395 22 7 12 0.395

8 9 12 0.395 23 8 9 0.395

9 11 10 0.395 24 8 6 0.395

10 10 9 0.395 25 9 9 0.395

11 8 8 0.395 26 11 11 0.395

12 9 14 0.395 27 11 10 0.395

13 7 12 0.395 28 10 14 0.395

14 8 10 0.395 29 5 8 0.395

15 8 10 0.395 30 3 7 0.395

Table 47: Study Population II

S/No X y Ρ

1 41 36 0.162

2 43 47 0.162

3 54 41 0.162

4 39 47 0.162

5 49 47 0.162

6 45 45 0.162

7 41 32 0.162

8 33 37 0.162

9 37 40 0.162

10 41 41 0.162

11 47 37 0.162

12 39 48 0.162

146

Table 48:Study Population III

S/No X y Ρ

1 100 3 -0.32

2 88 8 -0.32

3 20 9 -0.32

4 17 11 -0.32

5 60 5 -0.32

6 77 9 -0.32

7 51 5 -0.32

8 69 4 -0.32

9 66 6 -0.32

10 77 9 -0.32

11 68 2 -0.32

12 36 4 -0.32

13 74 4 -0.32

14 33 5 -0.32

15 54 6 -0.32

16 55 6 -0.32

17 77 6 -0.32

Table 49: Population IV

S/No x y Ρ

1 6.8 20 -0.77

2 6.2 23 -0.77

3 5.5 38 -0.77

4 0.85 86 -0.77

5 0.71 92 -0.77

6 9 16 -0.77

7 1.4 81 -0.77

8 4.5 53 -0.77

9 3.8 42 -0.77

10 2.1 62 -0.77

11 4.85 39 -0.77

12 3.197 35 -0.77 13 0.443 87 -0.77 14 0.468 91 -0.77

15 0.59 84 -0.77

16 0.339 75 -0.77 17 0.161 54 -0.77 18 0.787 64 -0.77 19 0.069 26 -0.77 20 0.11 100 -0.77

147 APPENDIX B

MSE OF THE PROPOSED ALTERNATIVE ESTIMATORS FOR THE FOUR STUDY POPULATIONS Table50: Relative Efficiency based on Expected MSE (MSE) of alternative estimators as compared with HHE for population I

g Rho Estimator a β2 G a β2 G a β2

0 0.162 𝜏 𝑐 134.319 a + 0 β2 1 11 a + 0 β2 2 0.9153 a + 0 β2

𝜏 𝑔,𝑐=1 131.281 a + 0.01167 β2 10.890 a + 0.01167 β2 0.94453 a + 0.01167 β2 𝜏 𝑔,𝑐=2 131.706 a + 0.01608 β2 11.1282 a + 0.01608 β2 0.9557 a + 0.01608 β2 𝜏 𝑔,𝑐=3 131.787 a + 0.01687 β2 11.1432 a + 0.01687 β2 0.95771 a + 0.01687 β2 𝜏 𝑔,𝑐=4 131.801 a + 0.017 β2 11.1457 a + 0.017 β2 0.95804 a + 0.017 β2

Table51: Relative Efficiency based on Expected MSE (MSE) of alternative estimators as compared with HHE for population II

g Rho Estimator a β2 G a β2 G A β2

0 0.395 𝜏 𝑐 1042.98 a + 0 β2 1 29 a + 0 β2 2 0.96253 a + 0 β2

𝜏 𝑔,𝑐=1 827.65 a + 0.03733 β2 28.5959 a + 0.03733 β2 1.09179 a + 0.03733 β2 𝜏 𝑔,𝑐=2 826.513 a + 0.06859 β2 25.5262 a + 0.06859 β2 1.20495 a + 0.06859 β2 𝜏 𝑔,𝑐=3 854.467 a + 0.08467 β2 31.5779 a + 0.08467 β2 1.26148 a + 0.08467 β2 𝜏 𝑔,𝑐=4 861.352 a + 0.09176 β2 32.0436 a + 0.09176 β2 1.286 a + 0.09176 β2

148

Table 52: Relative Efficiency based on Expected MSE (MSE) of alternative estimators as compared with HHE for population III

g Rho Estimator A β2 G a β2 G a β2

0 0.5 𝜏 𝑐 329.723 a + 0 β2 1 16 a + 0 β2 2 0.92725 a + 0 β2

𝜏 𝑔,𝑐=1 260.038 a + 0.04894 β2 14.9824 a + 0.04894 β2 1.04657 a + 0.04894 β2 𝜏 𝑔,𝑐=2 254.2 a + 0.13684 β2 14.8622 a + 0.13684 β2 1.24678 a + 0.13684 β2 𝜏 𝑔,𝑐=3 257.917 a + 0.22283 β2 17.2335 a + 0.22283 β2 1.42167 a + 0.22283 β2 𝜏 𝑔,𝑐=4 261.977 a + 0.28608 β2 18.0611 a + 0.28608 β2 1.54175 a + 0.28608 β2

0 0.32 𝜏 𝑐 329.723 a + 0 β2 1 16 a + 0 β2 2 0.92725 a + 0 β2

𝜏 𝑔,𝑐=1 254.197 a + 0.10403 β2 15.6325 a + 0.10403 β2 1.17519 a + 0.10403 β2 𝜏 𝑔,𝑐=2 259.189 a + 0.24371 β2 14.5097 a + 0.24371 β2 1.46198 a + 0.24371 β2 𝜏 𝑔,𝑐=3 264.557 a + 0.32309 β2 18.5333 a + 0.32309 β2 1.60951 a + 0.32309 β2 𝜏 𝑔,𝑐=4 266.811 a + 0.35449 β2 18.927 a + 0.35449 β2 1.66577 a + 0.35449 β2

149

Table 53: Relative Efficiency based on Expected MSE (MSE) of alternative estimators as compared with HHE for population IV

g Rho Estimator a β2 G a β2 G a. β2

0 0.91 𝜏 𝑐 2292.36 a + 0 β2 1 16 a + 0 β2 2 0.28083 a + 0 β2

𝜏 𝑔,𝑐=1 711.025 a + 0.01273 β2 9.15609 a + 0.01273 β2 0.23119 a + 0.01273 β2 𝜏 𝑔,𝑐=2 430.8 a + 0.03008 β2 7.5652 a + 0.03008 β2 0.20608 a + 0.03008 β2 𝜏 𝑔,𝑐=3 322.159 a + 0.05026 β2 6.83628 a + 0.05026 β2 0.18485 a + 0.05026 β2 𝜏 𝑔,𝑐=4 266.236 a + 0.07415 β2 6.41053 a + 0.07415 β2 0.16365 a + 0.07415 β2

0 0.775 𝜏 𝑐 2292.36 a + 0 β2 1 16 a + 0 β2 2 0.28083 a + 0 β2

𝜏 𝑔,𝑐=1 346.544 a + 0.04391 β2 7.00906 a + 0.04391 β2 0.19106 a + 0.04391 β2 𝜏 𝑔,𝑐=2 222.85 a + 0.11534 β2 6.0349 a + 0.11534 β2 0.13087 a + 0.11534 β2 𝜏 𝑔,𝑐=3 183.251 a + 0.22511 β2 5.59361 a + 0.22511 β2 0.0497 a + 0.22511 β2 𝜏 𝑔,𝑐=4 164.902 a + 0.38348 β2 5.23992 a + 0.38348 β2 0.06507 a + 0.38348 β2

150

Table 54: Estimates of MSE using conventional and alternative estimators in PPSWOR sampling scheme for population I.

g Rho Estimator A β2 g A β2 G a. β2

0 0.162 𝜏 𝑐 757.7251 a+ 0.000 β2 1 61.04531 a+ 0.000 β2 2 4.995357 a+ 0.000 β2 𝜏 𝑔,𝑐=1 720.7753 a+ 0.038 β2 59.61553 a+ 0.038 β2 5.010464 a+ 0.038 β2 𝜏 𝑔,𝑐=2 719.4419 a+ 0.051 β2 59.7652 a+ 0.051 β2 5.045536 a+ 0.051 β2 𝜏 𝑔,𝑐=3 719.346 a+ 0.054 β2 59.79987 a+ 0.054 β2 5.052158 a+ 0.054 β2 𝜏 𝑔,𝑐=4 719.3336 a+ 0.054 β2 59.80576 a+ 0.054 β2 5.053256 a+ 0.054 β2

Table 55: Estimates of MSE using conventional and alternative estimators in PPSWOR sampling scheme for population II.

g Rho Estimator A β2 g A β2 G a. β2

0 0.39 𝜏 𝑐 22609.09 a+ 0.000 β2 1 504.0447 a+ 0.000 β2 2 13.96817 a+ 0.000 β2 𝜏 𝑔,𝑐=1 13450.04 a+ 0.429 β2 398.2594 a+ 0.429 β2 13.74707 a+ 0.429 β2 𝜏 𝑔,𝑐=2 12709.7 a+ 0.721 β2 404.3067 a+ 0.721 β2 14.66844 a+ 0.721 β2 𝜏 𝑔,𝑐=3 12598.15 a+ 0.859 β2 410.831 a+ 0.859 β2 15.16344 a+ 0.859 β2 𝜏 𝑔,𝑐=4 12577.8 a+ 0.917 β2 414.0036 a+ 0.917 β2 15.37947 a+ 0.917 β2

151

Table 56: Estimates of MSE using conventional and alternative estimators in PPSWOR sampling scheme for population III.

g Rho Estimator A β2 g A β2 G a. β2

0 0.32 𝜏 𝑐 3672.619 a+ 0.00 β2 1 153.996 a+ 0.00 β2 2 7.43251 a+ 0.00 β2 𝜏 𝑔,𝑐=1 2299.603 a+ 0.39 β2 117.8413 a+ 0.39 β2 7.1915 a+ 0.39 β2 𝜏 𝑔,𝑐=2 2171.411 a+ 0.79 β2 119.7481 a+ 0.79 β2 8.017956 a+ 0.79 β2 𝜏 𝑔,𝑐=3 2152.597 a+ 1.00 β2 122.0586 a+ 1.00 β2 8.471023 a+ 1.00 β2 𝜏 𝑔,𝑐=4 2149.15 a+ 1.09 β2 123.0378 a+ 1.09 β2 8.645418 a+ 1.09 β2

Table 57: Estimates of MSE using conventional and alternative estimators in PPSWOR sampling scheme for population IV.

G Rho Estimator A β2 g A β2 G a. β2

0 0.77 𝜏 𝑐 250723 a+ 0.0 β2 1 1039.224 a+ 0.0 β2 2 8.235249 a+ 0.0 β2

𝜏 𝑔,𝑐=1 20067.68 a+ 0.7 β2 165.6597 a+ 0.7 β2 4.465283 a+ 0.7 β2 𝜏 𝑔,𝑐=2 9544.162 a+ 1.1 β2 111.7766 a+ 1.1 β2 4.502901 a+ 1.1 β2 𝜏 𝑔,𝑐=3 6518.028 a+ 1.6 β2 96.10361 a+ 1.6 β2 4.980096 a+ 1.6 β2 𝜏 𝑔,𝑐=4 5191.764 a+ 2.2 β2 90.86967 a+ 2.2 β2 5.711622 a+ 2.2 β2

152 APPENDIX C

MSE OF THE PROPOSED ALTERNATIVE ESTIMATORS FOR THE THEORETICAL DISTRIBUTIONS OF THE MEASURE OF SIZE VARIABLES.

Table 58:Expected MSE of linear alternative estimators as compared with HHE for the theoretical Normal Distribution for population I

g Rho Estimator A β2 G A β2 g A β2

0 0 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=2 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=3 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=4 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2

0 0.1 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 132.3884 a + 0.018428 β2 11.19912 a + 0.018428 β2 0.962711 a + 0.018428 β2 𝜏 𝑔,𝑐=2 131.8173 a + 0.017061 β2 11.14775 a + 0.017061 β2 0.958255 a + 0.017061 β2 𝜏 𝑔,𝑐=3 131.8047 a + 0.017032 β2 11.14629 a + 0.017032 β2 0.958115 a + 0.017032 β2 𝜏 𝑔,𝑐=4 131.8039 a + 0.01703 β2 11.14618 a + 0.01703 β2 0.958104 a + 0.01703 β2

0 0.162 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 133.267 a + 0.020548 β2 11.27445 a + 0.020548 β2 0.969096 a + 0.020548 β2 𝜏 𝑔,𝑐=2 131.8603 a + 0.017162 β2 11.15207 a + 0.017162 β2 0.958647 a + 0.017162 β2 𝜏 𝑔,𝑐=3 131.8084 a + 0.01704 β2 11.14674 a + 0.01704 β2 0.95816 a + 0.01704 β2 𝜏 𝑔,𝑐=4 131.8045 a + 0.017031 β2 11.14625 a + 0.017031 β2 0.958112 a + 0.017031 β2

153

0 0.5 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 146.3695 a + 0.053071 β2 12.36458 a + 0.053071 β2 1.05984 a + 0.053071 β2 𝜏 𝑔,𝑐=2 135.2385 a + 0.025334 β2 11.44082 a + 0.025334 β2 0.983076 a + 0.025334 β2 𝜏 𝑔,𝑐=3 132.6938 a + 0.019164 β2 11.22548 a + 0.019164 β2 0.964953 a + 0.019164 β2 𝜏 𝑔,𝑐=4 132.0511 a + 0.017617 β2 11.16953 a + 0.017617 β2 0.960173 a + 0.017617 β2

0 0.9 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 193.7988 a + 0.184093 β2 16.23106 a + 0.184093 β2 1.37615 a + 0.184093 β2 𝜏 𝑔,𝑐=2 178.0379 a + 0.138435 β2 14.95324 a + 0.138435 β2 1.272178 a + 0.138435 β2 𝜏 𝑔,𝑐=3 166.9784 a + 0.107597 β2 14.05313 a + 0.107597 β2 1.198654 a + 0.107597 β2 𝜏 𝑔,𝑐=4 158.9573 a + 0.085902 β2 13.39801 a + 0.085902 β2 1.144957 a + 0.085902 β2

0 1 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 217.3183 a + 0.255567 β2 18.12977 a + 0.255567 β2 1.529953 a + 0.255567 β2 𝜏 𝑔,𝑐=2 217.3183 a + 0.255567 β2 18.12977 a + 0.255567 β2 1.529953 a + 0.255567 β2 𝜏 𝑔,𝑐=3 217.3183 a + 0.255567 β2 18.12977 a + 0.255567 β2 1.529953 a + 0.255567 β2 𝜏 𝑔,𝑐=4 217.3183 a + 0.255567 β2 18.12977 a + 0.255567 β2 1.529953 a + 0.255567 β2

154

Table59:Expected MSE of linear alternative estimators as compared with that of HHE for the theoretical chi square Distribution for population I

G Rho Estimator A β2 G A β2 g A β2

0 0 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=2 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=3 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=4 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2

0 0.1 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 135.6999 a + 0.034894 β2 11.57401 a + 0.034894 β2 1.003628 a + 0.034894 β2 𝜏 𝑔,𝑐=2 131.9305 a + 0.01807 β2 11.16635 a + 0.01807 β2 0.960684 a + 0.01807 β2 𝜏 𝑔,𝑐=3 131.8139 a + 0.017127 β2 11.14796 a + 0.017127 β2 0.958342 a + 0.017127 β2 𝜏 𝑔,𝑐=4 131.8048 a + 0.01704 β2 11.14634 a + 0.01704 β2 0.958127 a + 0.01704 β2

0 0.162 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 141.1793 a + 0.054889 β2 12.10459 a + 0.054889 β2 1.055195 a + 0.054889 β2 𝜏 𝑔,𝑐=2 132.2603 a + 0.020108 β2 11.20976 a + 0.020108 β2 0.965799 a + 0.020108 β2 𝜏 𝑔,𝑐=3 131.8506 a + 0.017452 β2 11.15414 a + 0.017452 β2 0.959148 a + 0.017452 β2 𝜏 𝑔,𝑐=4 131.8107 a + 0.017096 β2 11.1474 a + 0.017096 β2 0.958267 a + 0.017096 β2

155

0 0.5 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 242.7276 a + 0.410393 β2 21.32146 a + 0.410393 β2 1.903468 a + 0.410393 β2 𝜏 𝑔,𝑐=2 153.6932 a + 0.097924 β2 13.27229 a + 0.097924 β2 1.165272 a + 0.097924 β2 𝜏 𝑔,𝑐=3 137.6106 a + 0.042065 β2 11.76194 a + 0.042065 β2 1.022119 a + 0.042065 β2 𝜏 𝑔,𝑐=4 133.5498 a + 0.026226 β2 11.35419 a + 0.026226 β2 0.98136 a + 0.026226 β2

0 0.9 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 2980.232 a + 13.59931 β2 261.2076 a + 13.59931 β2 23.21023 a + 13.59931 β2 𝜏 𝑔,𝑐=2 1038.571 a + 3.896042 β2 91.90541 a + 3.896042 β2 8.257873 a + 3.896042 β2 𝜏 𝑔,𝑐=3 591.4351 a + 1.841051 β2 52.38213 a + 1.841051 β2 4.712762 a + 1.841051 β2 𝜏 𝑔,𝑐=4 411.4918 a + 1.072098 β2 36.39137 a + 1.072098 β2 3.269943 a + 1.072098 β2

2 1 𝜏 𝑐 0.915304 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 2078113 a + 2064369 β2 25797900 a + 2064369 β2 2078113 a + 2064369 β2 𝜏 𝑔,𝑐=2 2078113 a + 2064369 β2 25797900 a + 2064369 β2 2078113 a + 2064369 β2 𝜏 𝑔,𝑐=3 2078113 a + 2064369 β2 25797900 a + 2064369 β2 2078113 a + 2064369 β2 𝜏 𝑔,𝑐=4 2078113 a + 2064369 β2 25797900 a + 2064369 β2 2078113 a + 2064369 β2

156

Table60:Expected MSE of linear alternative estimators as compared with that of HHE for the theoretical Uniform Distribution for population I

G Rho Estimator A β2 G A β2 g A β2

0 0 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=2 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=3 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=4 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2

0 0.1 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=2 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=3 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=4 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2

0 0.162 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=2 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=3 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=4 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2

0 0.5 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=2 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=3 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=4 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2

157

0 0.9 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=2 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=3 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=4 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2

0 1 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=2 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=3 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=4 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2

158

Table61:Expected MSE of linear alternative estimators as compared with that of HHE for the theoretical Gamma Distribution for population I

G Rho Estimator A β2 G A β2 g A β2

0 0 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=2 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=3 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2 𝜏 𝑔,𝑐=4 131.8038 a + 0.01703 β2 11.14617 a + 0.01703 β2 0.958103 a + 0.01703 β2

0 0.1 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 137.1507 a + 0.024525 β2 11.52525 a + 0.024525 β2 0.98454 a + 0.024525 β2 𝜏 𝑔,𝑐=2 131.808 a + 0.016693 β2 11.14059 a + 0.016693 β2 0.957157 a + 0.016693 β2 𝜏 𝑔,𝑐=3 131.7977 a + 0.016981 β2 11.14507 a + 0.016981 β2 0.957964 a + 0.016981 β2 𝜏 𝑔,𝑐=4 131.8032 a + 0.017025 β2 11.14605 a + 0.017025 β2 0.958089 a + 0.017025 β2

0 0.162 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 145.307 a + 0.038028 β2 12.15674 a + 0.038028 β2 1.033699 a + 0.038028 β2 𝜏 𝑔,𝑐=2 132.1061 a + 0.016787 β2 11.15541 a + 0.016787 β2 0.957588 a + 0.016787 β2 𝜏 𝑔,𝑐=3 131.7878 a + 0.016847 β2 11.14234 a + 0.016847 β2 0.957581 a + 0.016847 β2 𝜏 𝑔,𝑐=4 131.7994 a + 0.016996 β2 11.1454 a + 0.016996 β2 0.958006 a + 0.016996 β2

0 0.5 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 274.4888 a + 0.279584 β2 22.45316 a + 0.279584 β2 1.8628 a + 0.279584 β2 𝜏 𝑔,𝑐=2 162.8866 a + 0.067723 β2 13.54195 a + 0.067723 β2 1.143754 a + 0.067723 β2 𝜏 𝑔,𝑐=3 140.0335 a + 0.029269 β2 11.7467 a + 0.029269 β2 1.001621 a + 0.029269 β2 𝜏 𝑔,𝑐=4 133.8967 a + 0.019296 β2 11.28071 a + 0.019296 β2 0.966165 a + 0.019296 β2

159

0 0.9 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 2642.948 a + 8.162984 β2 214.5193 a + 8.162984 β2 17.59732 a + 8.162984 β2 𝜏 𝑔,𝑐=2 1094.886 a + 2.556025 β2 88.56306 a + 2.556025 β2 7.247816 a + 2.556025 β2 𝜏 𝑔,𝑐=3 658.2176 a + 1.233234 β2 53.28872 a + 1.233234 β2 4.368034 a + 1.233234 β2 𝜏 𝑔,𝑐=4 466.8206 a + 0.723698 β2 37.88626 a + 0.723698 β2 3.114942 a + 0.723698 β2

0 1 𝜏 𝑐 134.3193 a + 0 β2 1 11 a + 0 β2 2 0.915304 a + 0 β2

𝜏 𝑔,𝑐=1 45003.38 a + 216.4849 β2 3642.2 a + 216.4849 β2 297.0335 a + 216.4849 β2 𝜏 𝑔,𝑐=2 45003.38 a + 216.4849 β2 3642.2 a + 216.4849 β2 297.0335 a + 216.4849 β2 𝜏 𝑔,𝑐=3 45003.38 a + 216.4849 β2 3642.2 a + 216.4849 β2 297.0335 a + 216.4849 β2 𝜏 𝑔,𝑐=4 45003.38 a + 216.4849 β2 3642.2 a + 216.4849 β2 297.0335 a + 216.4849 β2

160

Table62:Expected MSE of linear alternative estimators as compared with that of HHE for the theoretical Normal Distribution for population II

G Rho Estimator A β2 G A β2 g A β2

0 0 𝜏 𝑐 1051.816 a + 0 β2 1 29 a + 0 β2 2 0.96206 a + 0 β2

𝜏 𝑔,𝑐=1 865.8536 a + 0.108431 β2 32.74248 a + 0.108431 β2 1.343367 a + 0.108431 β2 𝜏 𝑔,𝑐=2 865.8536 a + 0.108431 β2 32.74248 a + 0.108431 β2 1.343367 a + 0.108431 β2 𝜏 𝑔,𝑐=3 865.8536 a + 0.108431 β2 32.74248 a + 0.108431 β2 1.343367 a + 0.108431 β2 𝜏 𝑔,𝑐=4 865.8536 a + 0.108431 β2 32.74248 a + 0.108431 β2 1.343367 a + 0.108431 β2

0 0.1 𝜏 𝑐 1051.816 a + 0 β2 1 29 a + 0 β2 2 0.96206 a + 0 β2

𝜏 𝑔,𝑐=1 845.8217 a + 0.087788 β2 31.35343 a + 0.087788 β2 1.269936 a + 0.087788 β2 𝜏 𝑔,𝑐=2 863.5582 a + 0.106219 β2 32.59258 a + 0.106219 β2 1.33555 a + 0.106219 β2 𝜏 𝑔,𝑐=3 865.621 a + 0.108208 β2 32.72737 a + 0.108208 β2 1.342581 a + 0.108208 β2 𝜏 𝑔,𝑐=4 865.8303 a + 0.108408 β2 32.74097 a + 0.108408 β2 1.343289 a + 0.108408 β2

0 0.395 𝜏 𝑐 1051.816 a + 0 β2 1 29 a + 0 β2 2 0.96206 a + 0 β2

𝜏 𝑔,𝑐=1 824.0128 a + 0.042208 β2 28.52622 a + 0.042208 β2 1.105386 a + 0.042208 β2 𝜏 𝑔,𝑐=2 837.3864 a + 0.07753 β2 30.67661 a + 0.07753 β2 1.233096 a + 0.07753 β2 𝜏 𝑔,𝑐=3 852.7463 a + 0.095335 β2 31.85808 a + 0.095335 β2 1.296896 a + 0.095335 β2 𝜏 𝑔,𝑐=4 860.3805 a + 0.103107 β2 32.38195 a + 0.103107 β2 1.324527 a + 0.103107 β2

0 0.5 𝜏 𝑐 1051.816 a + 0 β2 1 29 a + 0 β2 2 0.96206 a + 0 β2

𝜏 𝑔,𝑐=1 830.7452 a + 0.030244 β2 27.94087 a + 0.030244 β2 1.062463 a + 0.030244 β2 𝜏 𝑔,𝑐=2 827.6869 a + 0.06214 β2 29.69287 a + 0.06214 β2 1.17751 a + 0.06214 β2 𝜏 𝑔,𝑐=3 841.8129 a + 0.083104 β2 31.04279 a + 0.083104 β2 1.253137 a + 0.083104 β2 𝜏 𝑔,𝑐=4 852.5789 a + 0.095158 β2 31.84625 a + 0.095158 β2 1.296269 a + 0.095158 β2

161

0 0.9 𝜏 𝑐 1051.816 a + 0 β2 1 29 a + 0 β2 2 0.96206 a + 0 β2

𝜏 𝑔,𝑐=1 967.1424 a + 0.001732 β2 28.07782 a + 0.001732 β2 0.966477 a + 0.001732 β2 𝜏 𝑔,𝑐=2 914.6792 a + 0.005591 β2 27.63903 a + 0.005591 β2 0.978196 a + 0.005591 β2 𝜏 𝑔,𝑐=3 880.8265 a + 0.010472 β2 27.48049 a + 0.010472 β2 0.993993 a + 0.010472 β2 𝜏 𝑔,𝑐=4 858.5772 a + 0.015843 β2 27.49419 a + 0.015843 β2 1.01207 a + 0.015843 β2

0 1 𝜏 𝑐 1051.816 a + 0 β2 1 29 a + 0 β2 2 0.96206 a + 0 β2

𝜏 𝑔,𝑐=1 1051.816 a + 0 β2 29 a + 0 β2 0.96206 a + 0 β2

𝜏 𝑔,𝑐=2 1051.816 a + 0 β2 29 a + 0 β2 0.96206 a + 0 β2

𝜏 𝑔,𝑐=3 1051.816 a + 0 β2 29 a + 0 β2 0.96206 a + 0 β2

𝜏 𝑔,𝑐=4 1051.816 a + 0 β2 29 a + 0 β2 0.96206 a + 0 β2

162

Table63:Expected MSE of linear alternative estimators as compared with that of HHE for the theoretical chi square Distribution for population II

G Rho Estimator A β2 G A β2 g A β2

0 0 𝜏 𝑐 36308.52 a + 0 β2 1 29 a + 0 β2 2 0.921861 a + 0 β2

𝜏 𝑔,𝑐=1 829.6751 a + 0.908103 β2 63.92168 a + 0.908103 β2 5.783388 a + 0.908103 β2 𝜏 𝑔,𝑐=2 829.6751 a + 0.908103 β2 63.92168 a + 0.908103 β2 5.783388 a + 0.908103 β2 𝜏 𝑔,𝑐=3 829.6751 a + 0.908103 β2 63.92168 a + 0.908103 β2 5.783388 a + 0.908103 β2 𝜏 𝑔,𝑐=4 829.6751 a + 0.908103 β2 63.92168 a + 0.908103 β2 5.783388 a + 0.908103 β2

0 0.1 𝜏 𝑐 36308.52 a + 0 β2 1 29 a + 0 β2 2 0.921861 a + 0 β2

𝜏 𝑔,𝑐=1 664.2674 a + 0.526584 β2 47.1417 a + 0.526584 β2 4.117038 a + 0.526584 β2 𝜏 𝑔,𝑐=2 808.5584 a + 0.856564 β2 61.79399 a + 0.856564 β2 5.570925 a + 0.856564 β2 𝜏 𝑔,𝑐=3 827.5057 a + 0.902771 β2 63.70318 a + 0.902771 β2 5.761554 a + 0.902771 β2 𝜏 𝑔,𝑐=4 829.4576 a + 0.907568 β2 63.89977 a + 0.907568 β2 5.781199 a + 0.907568 β2

0 0.395 𝜏 𝑐 36308.52 a + 0 β2 1 29 a + 0 β2 2 0.921861 a + 0 β2

𝜏 𝑔,𝑐=1 481.2405 a + 0.141577 β2 25.77527 a + 0.141577 β2 2.022115 a + 0.141577 β2 𝜏 𝑔,𝑐=2 603.9834 a + 0.400059 β2 40.85706 a + 0.400059 β2 3.498724 a + 0.400059 β2 𝜏 𝑔,𝑐=3 717.119 a + 0.642909 β2 52.54465 a + 0.642909 β2 4.651189 a + 0.642909 β2 𝜏 𝑔,𝑐=4 780.3777 a + 0.789056 β2 58.95113 a + 0.789056 β2 5.287552 a + 0.789056 β2

0 0.5 𝜏 𝑐 36308.52 a + 0 β2 1 29 a + 0 β2 2 0.921861 a + 0 β2

𝜏 𝑔,𝑐=1 473.1934 a + 0.092116 β2 22.29896 a + 0.092116 β2 1.675077 a + 0.092116 β2 𝜏 𝑔,𝑐=2 534.9548 a + 0.260827 β2 33.237 a + 0.260827 β2 2.752832 a + 0.260827 β2 𝜏 𝑔,𝑐=3 635.1936 a + 0.464789 β2 44.13358 a + 0.464789 β2 3.820684 a + 0.464789 β2 𝜏 𝑔,𝑐=4 715.7946 a + 0.63993 β2 52.40998 a + 0.63993 β2 4.637846 a + 0.63993 β2

163

0 0.9 𝜏 𝑐 36308.52 a + 0 β2 1 29 a + 0 β2 2 0.921861 a + 0 β2

𝜏 𝑔,𝑐=1 1100.386 a + 0.009496 β2 17.73342 a + 0.009496 β2 0.991647 a + 0.009496 β2 𝜏 𝑔,𝑐=2 704.5309 a + 0.020492 β2 17.47576 a + 0.020492 β2 1.088707 a + 0.020492 β2 𝜏 𝑔,𝑐=3 574.6131 a + 0.032946 β2 18.0484 a + 0.032946 β2 1.199124 a + 0.032946 β2 𝜏 𝑔,𝑐=4 516.352 a + 0.047071 β2 18.98861 a + 0.047071 β2 1.320375 a + 0.047071 β2

0 1 𝜏 𝑐 36308.52 a + 0 β2 1 29 a + 0 β2 2 0.921861 a + 0 β2

𝜏 𝑔,𝑐=1 36308.52 a + 0 β2 29 a + 0 β2 0.921861 a + 0 β2

𝜏 𝑔,𝑐=2 36308.52 a + 0 β2 29 a + 0 β2 0.921861 a + 0 β2

𝜏 𝑔,𝑐=3 36308.52 a + 0 β2 29 a + 0 β2 0.921861 a + 0 β2

𝜏 𝑔,𝑐=4 36308.52 a + 0 β2 29 a + 0 β2 0.921861 a + 0 β2

164

Table64:Expected MSE of linear alternative estimators as compared with that of HHE for the theoretical Gamma Distribution for population II

G Rho Estimator A β2 G A β2 g A β2

0 0 𝜏 𝑐 3166.908 a + 0 β2 1 29 a + 0 β2 2 0.9393 a + 0 β2

𝜏 𝑔,𝑐=1 845.3701 a + 0.646209 β2 50.66768 a + 0.646209 β2 3.654303 a + 0.646209 β2 𝜏 𝑔,𝑐=2 845.3701 a + 0.646209 β2 50.66768 a + 0.646209 β2 3.654303 a + 0.646209 β2 𝜏 𝑔,𝑐=3 845.3701 a + 0.646209 β2 50.66768 a + 0.646209 β2 3.654303 a + 0.646209 β2 𝜏 𝑔,𝑐=4 845.3701 a + 0.646209 β2 50.66768 a + 0.646209 β2 3.654303 a + 0.646209 β2

0 0.1 𝜏 𝑐 3166.908 a + 0 β2 1 29 a + 0 β2 2 0.9393 a + 0 β2

𝜏 𝑔,𝑐=1 738.2882 a + 0.430939 β2 41.03276 a + 0.430939 β2 2.864421 a + 0.430939 β2 𝜏 𝑔,𝑐=2 832.1993 a + 0.619401 β2 49.51382 a + 0.619401 β2 3.559611 a + 0.619401 β2 𝜏 𝑔,𝑐=3 844.024 a + 0.643463 β2 50.55002 a + 0.643463 β2 3.644645 a + 0.643463 β2 𝜏 𝑔,𝑐=4 845.2352 a + 0.645933 β2 50.65589 a + 0.645933 β2 3.653336 a + 0.645933 β2

0 0.395 𝜏 𝑐 3166.908 a + 0 β2 1 29 a + 0 β2 2 0.9393 a + 0 β2

𝜏 𝑔,𝑐=1 624.5125 a + 0.148824 β2 26.6606 a + 0.148824 β2 1.670031 a + 0.148824 β2 𝜏 𝑔,𝑐=2 697.506 a + 0.34845 β2 37.07197 a + 0.34845 β2 2.539546 a + 0.34845 β2 𝜏 𝑔,𝑐=3 773.4345 a + 0.501154 β2 44.27541 a + 0.501154 β2 3.130129 a + 0.501154 β2 𝜏 𝑔,𝑐=4 814.3856 a + 0.583344 β2 47.94323 a + 0.583344 β2 3.43078 a + 0.583344 β2

0 0.5 𝜏 𝑐 3166.908 a + 0 β2 1 29 a + 0 β2 2 0.9393 a + 0 β2

𝜏 𝑔,𝑐=1 636.0752 a + 0.102242 β2 24.1271 a + 0.102242 β2 1.441977 a + 0.102242 β2 𝜏 𝑔,𝑐=2 651.5555 a + 0.247725 β2 31.9719 a + 0.247725 β2 2.118866 a + 0.247725 β2 𝜏 𝑔,𝑐=3 718.6735 a + 0.391549 β2 39.16339 a + 0.391549 β2 2.711176 a + 0.391549 β2 𝜏 𝑔,𝑐=4 772.5635 a + 0.499413 β2 44.19631 a + 0.499413 β2 3.123647 a + 0.499413 β2

165

0 0.9 𝜏 𝑐 3166.908 a + 0 β2 1 29 a + 0 β2 2 0.9393 a + 0 β2

𝜏 𝑔,𝑐=1 1356.109 a + 0.009721 β2 22.63467 a + 0.009721 β2 0.973363 a + 0.009721 β2 𝜏 𝑔,𝑐=2 964.2044 a + 0.023587 β2 21.43753 a + 0.023587 β2 1.037457 a + 0.023587 β2 𝜏 𝑔,𝑐=3 801.4107 a + 0.038699 β2 21.40077 a + 0.038699 β2 1.114033 a + 0.038699 β2 𝜏 𝑔,𝑐=4 718.0063 a + 0.054962 β2 21.86685 a + 0.054962 β2 1.198751 a + 0.054962 β2

0 1 𝜏 𝑐 3166.908 a + 0 β2 1 29 a + 0 β2 2 0.9393 a + 0 β2

𝜏 𝑔,𝑐=1 3166.908 a + 0 β2 29 a + 0 β2 0.9393 a + 0 β2

𝜏 𝑔,𝑐=2 3166.908 a + 0 β2 29 a + 0 β2 0.9393 a + 0 β2

𝜏 𝑔,𝑐=3 3166.908 a + 0 β2 29 a + 0 β2 0.9393 a + 0 β2

𝜏 𝑔,𝑐=4 3166.908 a + 0 β2 29 a + 0 β2 0.9393 a + 0 β2

166

Table65:Expected MSE of linear alternative estimators as compared with that of HHE for the theoretical Normal Distribution for population III

G Rho Estimator A β2 G A β2 g A β2

0 0 𝜏 𝑐 309.9455 a + 0 β2 1 16 a + 0 β2 2 0.935397 a + 0 β2

𝜏 𝑔,𝑐=1 270.3297 a + 0.080318 β2 17.38381 a + 0.080318 β2 1.193971 a + 0.080318 β2 𝜏 𝑔,𝑐=2 270.3297 a + 0.080318 β2 17.38381 a + 0.080318 β2 1.193971 a + 0.080318 β2 𝜏 𝑔,𝑐=3 270.3297 a + 0.080318 β2 17.38381 a + 0.080318 β2 1.193971 a + 0.080318 β2 𝜏 𝑔,𝑐=4 270.3297 a + 0.080318 β2 17.38381 a + 0.080318 β2 1.193971 a + 0.080318 β2

0 0.1 𝜏 𝑐 309.9455 a + 0 β2 1 16 a + 0 β2 2 0.935397 a + 0 β2

𝜏 𝑔,𝑐=1 266.0149 a + 0.065452 β2 16.84624 a + 0.065452 β2 1.144693 a + 0.065452 β2 𝜏 𝑔,𝑐=2 269.8331 a + 0.078741 β2 17.326 a + 0.078741 β2 1.188758 a + 0.078741 β2 𝜏 𝑔,𝑐=3 270.2794 a + 0.080159 β2 17.37799 a + 0.080159 β2 1.193447 a + 0.080159 β2 𝜏 𝑔,𝑐=4 270.3247 a + 0.080302 β2 17.38323 a + 0.080302 β2 1.193918 a + 0.080302 β2

0 0.5 𝜏 𝑐 309.9455 a + 0 β2 1 16 a + 0 β2 2 0.935397 a + 0 β2

𝜏 𝑔,𝑐=1 263.614 a + 0.022441 β2 15.52963 a + 0.022441 β2 1.002413 a + 0.022441 β2 𝜏 𝑔,𝑐=2 262.2456 a + 0.046533 β2 16.20042 a + 0.046533 β2 1.081737 a + 0.046533 β2 𝜏 𝑔,𝑐=3 265.1611 a + 0.062033 β2 16.72554 a + 0.062033 β2 1.133323 a + 0.062033 β2 𝜏 𝑔,𝑐=4 267.4639 a + 0.070797 β2 17.03741 a + 0.070797 β2 1.162444 a + 0.070797 β2

0 0.51 𝜏 𝑐 309.9455 a + 0 β2 1 16 a + 0 β2 2 0.935397 a + 0 β2

𝜏 𝑔,𝑐=1 263.8905 a + 0.021649 β2 15.51342 a + 0.021649 β2 0.999858 a + 0.021649 β2 𝜏 𝑔,𝑐=2 262.1098 a + 0.04539 β2 16.16357 a + 0.04539 β2 1.077935 a + 0.04539 β2 𝜏 𝑔,𝑐=3 264.9177 a + 0.061009 β2 16.68967 a + 0.061009 β2 1.129916 a + 0.061009 β2 𝜏 𝑔,𝑐=4 267.2529 a + 0.070047 β2 17.01041 a + 0.070047 β2 1.159955 a + 0.070047 β2

167

0 0.9 𝜏 𝑐 309.9455 a + 0 β2 1 16 a + 0 β2 2 0.935397 a + 0 β2

𝜏 𝑔,𝑐=1 293.5411 a + 0.00118 β2 15.64636 a + 0.00118 β2 0.937744 a + 0.00118 β2 𝜏 𝑔,𝑐=2 282.6542 a + 0.00393 β2 15.46246 a + 0.00393 β2 0.945173 a + 0.00393 β2 𝜏 𝑔,𝑐=3 275.2801 a + 0.007513 β2 15.38519 a + 0.007513 β2 0.955621 a + 0.007513 β2 𝜏 𝑔,𝑐=4 270.2481 a + 0.011528 β2 15.37709 a + 0.011528 β2 0.967828 a + 0.011528 β2

0 1 𝜏 𝑐 309.9455 a + 0 β2 1 16 a + 0 β2 2 0.935397 a + 0 β2

𝜏 𝑔,𝑐=1 309.9455 a + 0 β2 16 a + 0 β2 0.935397 a + 0 β2

𝜏 𝑔,𝑐=2 309.9455 a + 0 β2 16 a + 0 β2 0.935397 a + 0 β2

𝜏 𝑔,𝑐=3 309.9455 a + 0 β2 16 a + 0 β2 0.935397 a + 0 β2

𝜏 𝑔,𝑐=4 309.9455 a + 0 β2 16 a + 0 β2 0.935397 a + 0 β2

168

Table66:Expected MSE of linear alternative estimators as compared with that of HHE for the theoretical chi square Distribution for population III

G Rho Estimator A β2 G A β2 g A β2

0 0 𝜏 𝑐 403234.6 a + 0 β2 1 16 a + 0 β2 2 0.910672 a + 0 β2

𝜏 𝑔,𝑐=1 263.1843 a + 0.152949 β2 23.35668 a + 0.152949 β2 2.21809 a + 0.152949 β2 𝜏 𝑔,𝑐=2 263.1843 a + 0.152949 β2 23.35668 a + 0.152949 β2 2.21809 a + 0.152949 β2 𝜏 𝑔,𝑐=3 263.1843 a + 0.152949 β2 23.35668 a + 0.152949 β2 2.21809 a + 0.152949 β2 𝜏 𝑔,𝑐=4 263.1843 a + 0.152949 β2 23.35668 a + 0.152949 β2 2.21809 a + 0.152949 β2

0 0.1 𝜏 𝑐 403234.6 a + 0 β2 1 16 a + 0 β2 2 0.910672 a + 0 β2

𝜏 𝑔,𝑐=1 239.4568 a + 0.116728 β2 20.7772 a + 0.116728 β2 1.953667 a + 0.116728 β2 𝜏 𝑔,𝑐=2 260.5395 a + 0.148842 β2 23.07283 a + 0.148842 β2 2.189023 a + 0.148842 β2 𝜏 𝑔,𝑐=3 262.9169 a + 0.152533 β2 23.32801 a + 0.152533 β2 2.215155 a + 0.152533 β2 𝜏 𝑔,𝑐=4 263.1575 a + 0.152907 β2 23.35381 a + 0.152907 β2 2.217796 a + 0.152907 β2

0 0.32 𝜏 𝑐 403234.6 a + 0 β2 1 16 a + 0 β2 2 0.910672 a + 0 β2

𝜏 𝑔,𝑐=1 204.6528 a + 0.064235 β2 16.66747 a + 0.064235 β2 1.528718 a + 0.064235 β2 𝜏 𝑔,𝑐=2 238.9565 a + 0.115977 β2 20.72179 a + 0.115977 β2 1.947977 a + 0.115977 β2 𝜏 𝑔,𝑐=3 254.7554 a + 0.139925 β2 22.44937 a + 0.139925 β2 2.12516 a + 0.139925 β2 𝜏 𝑔,𝑐=4 260.4126 a + 0.148645 β2 23.0592 a + 0.148645 β2 2.187627 a + 0.148645 β2

0 0.51 𝜏 𝑐 403234.6 a + 0 β2 1 16 a + 0 β2 2 0.910672 a + 0 β2

𝜏 𝑔,𝑐=1 190.5905 a + 0.0366 β2 14.30043 a + 0.0366 β2 1.27604 a + 0.0366 β2

𝜏 𝑔,𝑐=2 212.0788 a + 0.075779 β2 17.6153 a + 0.075779 β2 1.627522 a + 0.075779 β2 𝜏 𝑔,𝑐=3 232.9032 a + 0.106926 β2 20.04655 a + 0.106926 β2 1.878588 a + 0.106926 β2 𝜏 𝑔,𝑐=4 246.5 a + 0.127344 β2 21.55191 a + 0.127344 β2 2.033165 a + 0.127344 β2

169

0 0.9 𝜏 𝑐 403234.6 a + 0 β2 1 16 a + 0 β2 2 0.910672 a + 0 β2

𝜏 𝑔,𝑐=1 279.6471 a + 0.004675 β2 12.08126 a + 0.004675 β2 0.954416 a + 0.004675 β2 𝜏 𝑔,𝑐=2 218.6081 a + 0.010113 β2 12.19525 a + 0.010113 β2 1.009845 a + 0.010113 β2 𝜏 𝑔,𝑐=3 199.2812 a + 0.015843 β2 12.5611 a + 0.015843 β2 1.068948 a + 0.015843 β2 𝜏 𝑔,𝑐=4 191.8762 a + 0.02179 β2 13.02836 a + 0.02179 β2 1.129732 a + 0.02179 β2

0 1 𝜏 𝑐 403234.6 a + 0 β2 1 16 a + 0 β2 2 0.910672 a + 0 β2

𝜏 𝑔,𝑐=1 403234.6 a + 0 β2 16 a + 0 β2 0.910672 a + 0 β2

𝜏 𝑔,𝑐=2 403234.6 a + 0 β2 16 a + 0 β2 0.910672 a + 0 β2

𝜏 𝑔,𝑐=3 403234.6 a + 0 β2 16 a + 0 β2 0.910672 a + 0 β2

𝜏 𝑔,𝑐=4 403234.6 a + 0 β2 16 a + 0 β2 0.910672 a + 0 β2

170

Table67:Expected MSE of linear alternative estimators as compared with that of HHE for the theoretical Gamma Distribution for population III

G Rho Estimator A β2 G A β2 g A β2

0 0 𝜏 𝑐 408.1397 a + 0 β2 1 16 a + 0 β2 2 0.924323 a + 0 β2

𝜏 𝑔,𝑐=1 267.1293 a + 0.225082 β2 19.99049 a + 0.225082 β2 1.707981 a + 0.225082 β2 𝜏 𝑔,𝑐=2 267.1293 a + 0.225082 β2 19.99049 a + 0.225082 β2 1.707981 a + 0.225082 β2 𝜏 𝑔,𝑐=3 267.1293 a + 0.225082 β2 19.99049 a + 0.225082 β2 1.707981 a + 0.225082 β2 𝜏 𝑔,𝑐=4 267.1293 a + 0.225082 β2 19.99049 a + 0.225082 β2 1.707981 a + 0.225082 β2

0 0.1 𝜏 𝑐 408.1397 a + 0 β2 1 16 a + 0 β2 2 0.924323 a + 0 β2

𝜏 𝑔,𝑐=1 254.8876 a + 0.174802 β2 18.37435 a + 0.174802 β2 1.538015 a + 0.174802 β2 𝜏 𝑔,𝑐=2 265.6993 a + 0.21948 β2 19.81166 a + 0.21948 β2 1.689342 a + 0.21948 β2 𝜏 𝑔,𝑐=3 266.9841 a + 0.224515 β2 19.97242 a + 0.224515 β2 1.7061 a + 0.224515 β2 𝜏 𝑔,𝑐=4 267.1148 a + 0.225025 β2 19.98869 a + 0.225025 β2 1.707793 a + 0.225025 β2

0 0.32 𝜏 𝑐 408.1397 a + 0 β2 1 16 a + 0 β2 2 0.924323 a + 0 β2

𝜏 𝑔,𝑐=1 242.293 a + 0.097498 β2 15.90531 a + 0.097498 β2 1.264434 a + 0.097498 β2 𝜏 𝑔,𝑐=2 254.6474 a + 0.173738 β2 18.33994 a + 0.173738 β2 1.53435 a + 0.173738 β2 𝜏 𝑔,𝑐=3 262.6203 a + 0.207236 β2 19.41955 a + 0.207236 β2 1.648347 a + 0.207236 β2 𝜏 𝑔,𝑐=4 265.6311 a + 0.219212 β2 19.80308 a + 0.219212 β2 1.688447 a + 0.219212 β2

0 0.51 𝜏 𝑐 408.1397 a + 0 β2 1 16 a + 0 β2 2 0.924323 a + 0 β2

𝜏 𝑔,𝑐=1 246.8547 a + 0.053877 β2 14.68803 a + 0.053877 β2 1.104994 a + 0.053877 β2 𝜏 𝑔,𝑐=2 243.883 a + 0.115055 β2 16.45269 a + 0.115055 β2 1.327845 a + 0.115055 β2 𝜏 𝑔,𝑐=3 251.8237 a + 0.160816 β2 17.92204 a + 0.160816 β2 1.48963 a + 0.160816 β2 𝜏 𝑔,𝑐=4 258.362 a + 0.189755 β2 18.85717 a + 0.189755 β2 1.589201 a + 0.189755 β2

171

0 0.9 𝜏 𝑐 408.1397 a + 0 β2 1 16 a + 0 β2 2 0.924323 a + 0 β2

𝜏 𝑔,𝑐=1 336.9437 a + 0.003625 β2 14.87091 a + 0.003625 β2 0.932355 a + 0.003625 β2 𝜏 𝑔,𝑐=2 299.0812 a + 0.011033 β2 14.39677 a + 0.011033 β2 0.954576 a + 0.011033 β2 𝜏 𝑔,𝑐=3 276.7375 a + 0.019979 β2 14.24224 a + 0.019979 β2 0.984073 a + 0.019979 β2 𝜏 𝑔,𝑐=4 262.8367 a + 0.029656 β2 14.26646 a + 0.029656 β2 1.017636 a + 0.029656 β2

0 1 𝜏 𝑐 408.1397 a + 0 β2 1 16 a + 0 β2 2 0.924323 a + 0 β2

𝜏 𝑔,𝑐=1 408.1397 a + 0 β2 16 a + 0 β2 0.924323 a + 0 β2

𝜏 𝑔,𝑐=2 408.1397 a + 0 β2 16 a + 0 β2 0.924323 a + 0 β2

𝜏 𝑔,𝑐=3 408.1397 a + 0 β2 16 a + 0 β2 0.924323 a + 0 β2

𝜏 𝑔,𝑐=4 408.1397 a + 0 β2 16 a + 0 β2 0.924323 a + 0 β2

172

Table68:Expected MSE of linear alternative estimators as compared with that of HHE for the theoretical Normal Distribution for population IV

G Rho Estimator A β2 G A β2 g A β2

0 0 𝜏 𝑐 340.2711 a + 0 β2 1 16 a + 0 β2 2 0.863202 a + 0 β2

𝜏 𝑔,𝑐=1 345.281 a + 0.105493 β2 20.39472 a + 0.105493 β2 1.275766 a + 0.105493 β2 𝜏 𝑔,𝑐=2 345.281 a + 0.105493 β2 20.39472 a + 0.105493 β2 1.275766 a + 0.105493 β2 𝜏 𝑔,𝑐=3 345.281 a + 0.105493 β2 20.39472 a + 0.105493 β2 1.275766 a + 0.105493 β2 𝜏 𝑔,𝑐=4 345.281 a + 0.105493 β2 20.39472 a + 0.105493 β2 1.275766 a + 0.105493 β2

0 0.1 𝜏 𝑐 340.2711 a + 0 β2 1 16 a + 0 β2 2 0.863202 a + 0 β2

𝜏 𝑔,𝑐=1 333.9158 a + 0.085473 β2 19.44206 a + 0.085473 β2 1.204443 a + 0.085473 β2 𝜏 𝑔,𝑐=2 344.0406 a + 0.103351 β2 20.29305 a + 0.103351 β2 1.268205 a + 0.103351 β2 𝜏 𝑔,𝑐=3 345.1559 a + 0.105278 β2 20.38448 a + 0.105278 β2 1.275005 a + 0.105278 β2 𝜏 𝑔,𝑐=4 345.2684 a + 0.105472 β2 20.39369 a + 0.105472 β2 1.27569 a + 0.105472 β2

0 0.775 𝜏 𝑐 340.2711 a + 0 β2 1 16 a + 0 β2 2 0.863202 a + 0 β2

𝜏 𝑔,𝑐=1 314.4131 a + 0.007303 β2 15.98776 a + 0.007303 β2 0.90676 a + 0.007303 β2 𝜏 𝑔,𝑐=2 309.0091 a + 0.019926 β2 16.42917 a + 0.019926 β2 0.958809 a + 0.019926 β2 𝜏 𝑔,𝑐=3 310.4453 a + 0.033057 β2 16.98626 a + 0.033057 β2 1.009936 a + 0.033057 β2 𝜏 𝑔,𝑐=4 314.3506 a + 0.045325 β2 17.54289 a + 0.045325 β2 1.056594 a + 0.045325 β2

173

0 0.9 𝜏 𝑐 340.2711 a + 0 β2 1 16 a + 0 β2 2 0.863202 a + 0 β2

𝜏 𝑔,𝑐=1 325.2462 a + 0.001664 β2 15.8957 a + 0.001664 β2 0.879025 a + 0.001664 β2 𝜏 𝑔,𝑐=2 316.7422 a + 0.005399 β2 15.94085 a + 0.005399 β2 0.898166 a + 0.005399 β2 𝜏 𝑔,𝑐=3 312.0518 a + 0.010142 β2 16.0718 a + 0.010142 β2 0.919003 a + 0.010142 β2 𝜏 𝑔,𝑐=4 309.738 a + 0.015374 β2 16.25346 a + 0.015374 β2 0.940586 a + 0.015374 β2

0 1 𝜏 𝑐 340.2711 a + 0 β2 1 16 a + 0 β2 2 0.863202 a + 0 β2

𝜏 𝑔,𝑐=1 340.2711 a + 0 β2 16 a + 0 β2 0.863202 a + 0 β2

𝜏 𝑔,𝑐=2 340.2711 a + 0 β2 16 a + 0 β2 0.863202 a + 0 β2

𝜏 𝑔,𝑐=3 340.2711 a + 0 β2 16 a + 0 β2 0.863202 a + 0 β2

𝜏 𝑔,𝑐=4 340.2711 a + 0 β2 16 a + 0 β2 0.863202 a + 0 β2

174

Table69:Expected MSE of linear alternative estimators as compared with that of HHE for the theoretical chi square Distribution for population IV

G Rho Estimator A β2 G A β2 g A β2

0 0 𝜏 𝑐 1614.477 a + 0 β2 1 16 a + 0 β2 2 0.890109 a + 0 β2

𝜏 𝑔,𝑐=1 356.0437 a + 0.771814 β2 37.5296 a + 0.771814 β2 4.656567 a + 0.771814 β2 𝜏 𝑔,𝑐=2 356.0437 a + 0.771814 β2 37.5296 a + 0.771814 β2 4.656567 a + 0.771814 β2 𝜏 𝑔,𝑐=3 356.0437 a + 0.771814 β2 37.5296 a + 0.771814 β2 4.656567 a + 0.771814 β2 𝜏 𝑔,𝑐=4 356.0437 a + 0.771814 β2 37.5296 a + 0.771814 β2 4.656567 a + 0.771814 β2

0 0.1 𝜏 𝑐 1614.477 a + 0 β2 1 16 a + 0 β2 2 0.890109 a + 0 β2

𝜏 𝑔,𝑐=1 294.6133 a + 0.470549 β2 28.77923 a + 0.470549 β2 3.450415 a + 0.470549 β2 𝜏 𝑔,𝑐=2 348.3631 a + 0.732278 β2 36.44641 a + 0.732278 β2 4.506554 a + 0.732278 β2 𝜏 𝑔,𝑐=3 355.2567 a + 0.767739 β2 37.41869 a + 0.767739 β2 4.641199 a + 0.767739 β2 𝜏 𝑔,𝑐=4 355.9648 a + 0.771405 β2 37.51848 a + 0.771405 β2 4.655027 a + 0.771405 β2

0 0.775 𝜏 𝑐 1614.477 a + 0 β2 1 16 a + 0 β2 2 0.890109 a + 0 β2

𝜏 𝑔,𝑐=1 292.1775 a + 0.027213 β2 12.0722 a + 0.027213 β2 1.069921 a + 0.027213 β2 𝜏 𝑔,𝑐=2 229.7782 a + 0.06192 β2 13.39109 a + 0.06192 β2 1.322983 a + 0.06192 β2 𝜏 𝑔,𝑐=3 220.6601 a + 0.105742 β2 15.37721 a + 0.105742 β2 1.614858 a + 0.105742 β2 𝜏 𝑔,𝑐=4 226.5605 a + 0.158389 β2 17.659 a + 0.158389 β2 1.931919 a + 0.158389 β2