Sampling Techniques: Methods and Applications

Muhammad Hanif
Vice Rector, Research NCBA&E, Lahore, Pakistan

Muhammad Qaiser Shahbaz
Department of Statistics, King Abdul Aziz University, Jeddah, Saudi Arabia

Munir Ahmad
National College of Business Administration and Economics, Lahore, Pakistan

Series: Mathematics Research Developments
BISAC: MAT029000

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Special issue: Resilience in breaking the cycle of children’s environmental health disparities
Edited by I Leslie Rubin, Robert J Geller, Abby Mutic, Benjamin A Gitterman, Nathan Mutic, Wayne Garfinkel, Claire D Coles, Kurt Martinuzzi, and Joav Merrick

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Sampling techniques have been widely used in almost all areas of life. The method of drawing a sample is very important for estimation of population characteristics. In this book we have made an attempt to discuss popular sampling designs and estimation methods that can be used for estimation of population characteristics.

This book presents two popular sampling design categories, namely the sampling of units and the sampling of groups of units. We have discussed designs that can be used for the sampling of units (for example, simple random sampling) and designs that are used for the sampling of groups (for example, cluster and multistage sampling).

The availability of supplementary information provides a basis to improve the efficiency of estimates. This book discusses estimation methods with and without the use of supplementary information. Two popular methods which use supplementary information – namely, ratio and regression estimators – have been discussed in detail in this book alongside their design and model based study.

The probabilities of population unit selection plays an important role in estimation. In this regard, the sampling designs are classified into two broader categories, namely equal probability sampling and unequal probability sampling. This book discusses in detail both of these sampling designs. The unequal probability sampling design has been discussed in the context of the Hansen–Hurwitz (1943) estimator, Horvitz–Thompson (1952) estimator and some special estimators.

The model based study of various estimators provides insight about their behavior under a linear stochastic model. This book provides a detailed discussion about properties of various estimators under a linear stochastic model both in equal and unequal probability sampling. Finally, the book presents useful material on multiphase sampling.

This book can be effectively used at undergraduate and graduate levels. The book is helpful for research students who want to pursue their career in sampling. The book is also helpful for practitioners to know the application of various sampling designs and estimators. (Imprint: Nova)

Dedication

Preface

Chapter 1. Basic Sampling Theory

Chapter 2. Simple Random Sampling

Chapter 3. Stratified and Systematic Random Sampling

Chapter 4. Single Stage Cluster Sampling

Chapter 5. Multistage Sampling (with Equal Probabilities)

Chapter 6. Ratio and Regression Estimators

Chapter 7. Sampling with Probability Proportional to Size (with replacement)

Chapter 8. Sampling with Unequal Probabilities without Replacement: Horvitz – Thompson Estimator

Chapter 9. Sampling with Unequal Probabilities without replacement: Special Estimators

Chapter 10. Design and Model Based Sampling Inference – I

Chapter 11. Design and Model Based Inference – II

Chapter 12. Multistage Cluster Sampling (Using Unequal Probabilities)

Chapter 13. Multiphase Sampling

Index

[1] Agarwal, R., Singh, D. and Singh, P. (1984) “Systematic sampling using varying probabilities”, J. Ind. Soc. Agri. Stat. 31(1), 99 – 109.
[2] Agarwal, R., Singh, P. and Singh, D. (1984) “IPPs sampling scheme through grouping”, Biometrical J. 26, 527 – 533.
[3] Ahmad, Z. Shahbaz, M. Q. and Hanif, M. (2013). Two Phase Sampling, Cambridge Scholars Publishing, UK.
[4] Arnab, R. (1978a). On strategies of sampling finite populations on successive occasions with varying probabilities. Sankhya, C, 41, 141 – 155.
[5] Arnab, R. (1979). An addendum to Singh and Singh's paper on random non-response in unequal probability sampling. Sankhya, C, 41, 138–140.
[6] Basu, D. (1971) “An essay on the logical foundation of survey sampling – part one” Foundation of Statistical Inference. Holt, Rinehard and Winston, Edited by Godambe and Sprott.
[7] Bayless, D. L. and Rao, J. N. K. (1970) “An empirical study of stabilities of estimators and variance estimators in unequal probability sampling for n = 3 and 4”, J. Amer. Stat. Assoc. 65, 1645 – 1667.
[8] Beale, E. M. L. (1962). Some Uses of Computers in Operational Research, Industrielle organization, 31, 27–28.
[9] Biyani, S. H. (1980). On inadmissibility of the Yates-Grundy variance estimator in unequal probability sampling. JASA, 75, 709–712.
[10] Blackwell, D. (1947) Conditional expectation and unbiased sequential estimation, Ann. Math. Stat., 18, 105–110.
[11] Bowley, A. L. (1913). Working class households in Reading. J. Roy. Statist. Soc. 76, 672–701.
[12] Bowley, A. L. (1926). Measurements of precision attained in sampling. Bull. Inst. Inte. Statist. 22, 1–62.
[13] Brewer, K. R. W. (1963a) “A model of systematic sampling with unequal probabilities”, Aust. J. Stat. 5, 5 – 13.
[14] Brewer, K. R. W. (1963b). “Ratio estimation and finite population: some results deducible from assumption of an underlying stochastic process”, Aust. J. Stat. 5, 93 – 105.
[15] Hanif, M. and Brewer, K. R. W. (1977). Comparison of Poisson sampling with some selection procedures for unequal probabilities. Punjab Univ. J. Math. (Lahore) 10/11, 33 – 39.
[16] Brewer, K. R. W. (1979). “A class of robust sampling design for large scale surveys”, J. Amer. Stat. Assoc. 74(4), 911 – 915.
[17] Brewer, K. R. W. and Hanif, M. (1969a). Sampling without replacement with probability of inclusion proportional to size. I: Methods using Horvitz-Thompson estimator. Unpublished Manuscript.
[18] Brewer, K. R. W. and Hanif, M. (1969b). Sampling without replacement with probability of inclusion proportional to size. II: Methods using Special estimators. Unpublished Manuscript.
[19] Brewer, K. R. W. and Hanif, M. (1969a). “Sampling without replacement with probability of inclusion proportional to size – I: Methods using Horvitz – Thompson estimator”, Unpublished manuscript.
[20] Brewer, K. R. W. and Hanif, M. (1983). “Sampling with Unequal Probabilities”, Lecture notes to Statistics, No. 15, Springer – Verlag.
[21] Brewer, K. R. W., Early, L. J. and Hanif, M. (1980). Poisson, modified Poisson and allocated sampling. J. Statist. Planning and Inference, 10, 15 – 30.
[22] Brewer, K. R. W., Early, L. J. and Joyce, S. F. (1972). “Selecting several samples from a single population”, Aust. J. Stat., 14, 231–239.
[23] Butt, N. S. and Shahbaz, M. Q. (2004). Modification of Murthy’s estimator using general selection procedure for unequal probability sampling without replacement. Pak. J. Statist. 20(2), 297–302.
[24] Carroll, J. L. and Hartley, H. O. (1964). “The systematic method of unequal probability sampling without replacement”, Abstract in Biometrics, 20, 908 – 909.
[25] Cassel, C. M., Sarndal, C. E. and Wretman, J. H. (1976). “Some results on generalized difference estimation and generalized regression estimation for finite populations”, Biometrika, 63, 615–620.
[26] Chand, L. (1975). Some ratio type estimators based on two or more auxiliary variables. Unpublished Ph.D. thesis, Iowa State University, Ames, Iowa (USA).
[27] Cassel, C. M., Sarndal, C. E. and Wretman, J. H. (1976). “Foundation of inference in survey sampling”, John Wiley and Sons, Inc. New York.
[28] Chaudhuri, A. and Mukhopadhyay, P. (1978). A note on how to choose the sample size for Horvitz-Thompson estimation. Bull. Cal. Stat. Assoct., 27, 149–154.
[29] Chaudhuri, A. and Vos, J. W. E. (1986). “Unified Theory and Strategies of Survey Sampling”, North Holland series in Statistics and Probability.
[30] Cochran, W. G. (1942). Sampling theory when the sampling units are of unequal sizes. J. Amer. Statist. Assoc., 37, 199–212.
[31] Cochran, W. G. (1946). “Relative accuracy of systematic and stratified random samples for a certain class of populations.” Annals of Mathematical Statistics 17: 164–177.
[32] Cochran, W. G. (1977). “Sampling Techniques”, John Wiley and Sons, Inc., New York.
[33] Connor, W. S. (1966). “An exact formula for the probability that two specified sampling units will occur in a sample drawn with unequal probabilities”, J. Amer. Stat. Assoc., 61, 384 – 390.
[34] Cumberland, W. G. and Royall, R. M. (1981). “Prediction model and unequal probability sampling”, J. Roy. Stat. Soc. B, 43, 353 – 367.
[35] Das, A. C. (1951). “On two phase sampling and sampling with varying probabilities”, Bull. Inter. Stat. Inst. 33(2), 105 – 112.
[36] Das, A. K. and Tripathi, T. P., Admissible estimators for quadratic forms in finite populations., Bull. Inter. Stat. Inst., 1977, Second Edition, 47,4, 132–135.
[37] Deming(1960). Sampling Design in Business Research. Wiley Classics Library.
[38] Durbin, J. (1953a) “Some results in sampling when the units are selected with unequal probabilities”, J. Roy. Stat. Soc, B, 15, 262–269.
[39] Durbin, J. (1953b) Sampling with unequal probabilities of selection within strata. Unpublished to date. Seen by the author.
[40] Durbin, J. (1967). “Design of multistage surveys for estimation of sampling errors”, Appl. Stat., 16, 152 – 164.
[41] Fellegi, I. P. (1963). “Sampling with varying probabilities without replacement: rotating and non – rotating samples”, J. Amer. Stat. Assoc. 58, 183 – 201.
[42] Finney, D. J. [1948]. Random and Systematic Sampling in Timber Surveys. Forestry 22, 1–36.
[43] Fisher, R. A. (1934). Statistical Methods for Research Workers, 5th Ed. Oliver and Boyd, London.
[44] Ronald A. Fisher and Frank Yates (1948). Statistical Tables for Biological, Agricultural and Medical Research. 3rd Edition. Edinburgh and London, 13(3), 26–27.
[45] Foreman, E. K. and Brewer, K. R. W. (1971). “The efficient use of supplementary information in standard sampling procedures”, J. Roy. Stat. Soc. B, 33, 391 – 400.
[46] Gauss, C. F. (1809). Theoria motus corporum coellestium, Werke, 7, K. Gesellschaft Wissenschaft. Göttingen. English translation: C. H. Davis (ed.), Dover, 1963).
[47] Gini, C. and Galvani, L. (1929). Di una applicazione del metodo rappresentativo all’ultimo censimento italiano della popolazione (10 dicembri, 1921). Annali di Statistica, Series 6, 4, 1–107.
[48] Godambe, V. P. and Joshi, V. M. (1965). “Admissibility and Bayes estimation in sampling finite population, I, II and III”, Ann. Math. Stat. 36, 1707–1742.
[49] Godambe, V. P. (1960). An Optimum Property of Regular Maximum Likelihood Estimation. The Annals of Mathematical Statistics. 31(4), 1208–1211.
[50] Goodman, L. A. and Hartley, H. O. (1958). The precision of unbiased ratio-type estimators. Journal of the American Statistical Association, 53, 491–508.
[51] Goodman, R., Kish, L. (1950). Controlled Selection–A Technique in Probability Sampling. Journal of the American Statistical Association, 45, 350–372.
[52] Kish, L. (1949). “A Procedure for Objective Respondent Selection within the Household”, Journal of the American Statistical Association, 44, pp. 380–387.
[53] Gram, J. P. (1883). Om beregning af en bevoxnings masse ved hjelp af prvetr½r (In Danish). Tidsskrift Skovbruk, 6, 137–198.
[54] Hajek, M. (1942). Marcus Hajek 1861–1941. Practica oto-rhino-laryngologica, 4, 175–176.
[55] Hajek, J. (1964). “Asymptotic theory of rejective sampling with varying probabilities from a finite population”, Ann. Math. Stat. 35, 1491–1523.
[56] Hajek, J. (1971). “Comments on An essay on the logical foundation of survey sampling, part one, by Basu” in Foundation of Statistical Inference, Godambe, V. P. and Sprott, D. A. (editors) Holt, Rinheart and Winston, Toranto, 326.
[57] Hajek, J. (1981). “Sampling from a finite population” Marcel Dekker Inc. New York.
[58] Hanif, M. (1969). Multistage cluster sampling. Unpublished MA Thesis, University of New South Wales.
[59] Hanif, M. (1994). “Design and model based sampling inference”, Unpublished Ph. D. Thesis.
[60] Hanif, M. (1974). “An asymptotic variance formula for sampling without replacement with unequal probabilities”, Libyan J. Sci. A, 47 – 51.
[61] Hanif, M. and Ahmad, M. (1977). Optimum level of clusterings for multipurpose sample surveys in Libya. Bull. Internat. Statist. Inst. XLVII, Book 4, 224–227.
[62] Hanif, M., Shahbaz, M. Q. and Ahmad, Z. (2010). Some Improved Estimators in Multiphase sampling, Pak. J. Stat. Vol. 26(1), 195–202.
[63] Hanif, M., Mukhopadhyay, P. and Bhattacharyya, S. (1993). On estimating the variance of Horvitz and Thompson estimator. Pak. J. Statist., A. 9, 123–136.
[64] Hanif, M., Samiuddin, M. and Shahbaz, M. Q. (2004). A simple general procedure for unequal probability sampling without replacement. Pak. J. Statist. Vol. 20(1), 73–80.
[65] Hanif, M. and Brewer, K. R. W. (1979 – 1980). “Generalization of the Horvitz and Thompson estimator”, Punj. Uni. J. Math. 12–13, 123–34.[66] Hanif, M., Shahbaz, M. Q. and Ahmad, Z. (2010). Some Improved Estimators in Multiphase sampling, Pak. J. Statist., 26(1), 195–202.
[67] Hansen, M. H. and Hurwitz, W. N. (1942). Relative efficiencies of various sampling units in population inquires. JASA, 37, 89–94.
[68] Hansen, M. H. and Hurwitz, W. N. (1943). “On the theory of sampling from a finite population”, Ann. Math. Stat. 14, 333–362.
[69] Hansen, M. H., Hurwitz, W. N., & Madow, W. G. (1993). Sample survey methods and theory. New York: Wiley.
[70] Hansen, M. H. and Hurwitz, W. N. and Madow, W. G. (1953). Sample Survey Methods and Theory. Vol. 2, John Wiley and Sons, New York.
[71] Hansen, M. H., Madow, W. G. and Tepping, B. J. (1983). “An evaluation of model dependent and probability sampling inference in sample surveys, with discussion”, J. Amer. Stat. Assoc. 78, 776–807.
[72] Hanurav, T. V. (1962). “Some Sampling schemes in probability sampling”, Sankhya, A. 24, 428.
[73] Hartley, H. O. and Rao, J. N. K. (1962). “Sampling with unequal probabilities and without replacement”, Ann. Math. Stat. 33, 350–374.
[74] Hartley, H. O. and Ross, A. (1954). Unbiased ratio estimators. Nature, 174, 270–271.
[75] Hendricks, W. A. (1944). The relative efficiencies of groups of farms as sampling units. Journal of the American Statistical Association, 39(227), 366–376.
[76] Herzel, A. (1986). “Sampling without replacement with unequal probabilities with pre-assigned joint inclusion probabilities of any order” Metron, 44, 1–4, 49–68.
[77] 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.
[78] Ikeda, T. (1950). A consideration concerning the selection of samples (in Japanese). Research Memoir of The Institute of Statistical Mathematics 5, 12.
[79] Isaki, C. T. and Fuller, W. A. (1982). “Survey design under the regression super – population model”, J. Amer. Stat. Assoc., 77, 209–218.
[80] Isaki, C. & Fuller, W. A. (1982). Survey design under the regression superpopulation model. J. Amer. Statist. Assoc., 77, 89–96.
[81] Rao, J. N. K. (1962). On the estimation of relative efficiency of sampling procedures. AISM 14, 143–150.
[82] Rao, J. N. K. (1965) “On two simple schemes of unequal probability sampling without replacement”, J. Ind. Stat. Assoc. 3, 173–180.
[83] Jessen, R. J. (1942) Statistical Investigation of a Sample Survey for Obtaining Farm Facts. Iowa State College of Agriculture and Mechanic Arts, Agricultural Experiment Station, Research Bulletin 304.
[84] Jessen, R. J. (1969). “Some methods of probability non – replacement sampling”, J. Amer. Stat. Assoc. 64, 175–193.
[85] Joshi, V. M. 1970. Note on the admissibility of Sen-Yates-Grundy estimator. Sankhya, Ser. A, 32: 431–438.
[86] Kendall and Buckland (1960). A Dictionary of Statistical Terms, Longman Publishers.
[87] Kiar, A. N. (1897). Sur les methodes representatives ou typopogiques appliquees a la statistique. Bulletin de I’Institut International de Statistique, 180–185.
[88] Kiar, A. N. (1901). Sur les methodes representatives ou typopogiques. Bulletin of the International Statistical Institute, 13, 66–70.
[89] Kiaer, A. (1903). Sur les methodes representatives ou typopogiques. Bulletin de I’Institut International de Statistique, 13: 66–78.
[90] Kiregyera, B. (1980). A chain ratio type estimator in finite population double sampling using two auxiliary variables. Metrika. 27, 217–223.
[91] Kiregyera, B. (1984). A Regression-Type Estimator using two Auxiliary Variables and Model of Double sampling from Finite Populations. Metrika. 31, 215–226.
[92] Kumar, P., Gupta, V. K. and Agarwal, S. K. (1985) “On inclusion probability proportional to size sampling scheme”, J. Stat. Plan. Infer. 12, 127–131.
[93] Lahiri, D. B. (1951). A method for sample selection providing unbiased ratio estimators. Bull. Inst. Inter. Statist., 33–140.
[94] Lanke, J. (1974). “On non-negative variance estimators in survey sampling”, Sankhya, C, 36(1), 33–42.
[95] Lanke, J. (1972). Sampling distinguishable elements with replacement. The Annals of Mathematical Statistics, 43(4), 1329–1332.
[96] Laplace, P. S. (1778). Mémoire sur les probabilités. Mémoires de l’Académie Royale des sciences de Paris, 9, 221–332.
[97] Madow, W. G. (1949). “On the theory of systematic sampling II”, Ann. Math. Stat. 20, 333–354.
[98] Madow, W. G., & Madow, L. H. (1944). On the theory of systematic sampling, I. The Annals of Mathematical Statistics, 15(1), 1–24.
[99] Mahalanobis, P. C. (1940). A sample survey of the acreage under jute in Bengal. Sankhyā: The Indian Journal of Statistics, 511–530.
[100] Mahalanobis, P. C. (1942a). Statistical definition of standard yield of crops. Sankhyā: The Indian Journal of Statistics, 97–98.
[101] Mahalanobis, P. C. (1942b). Sample surveys Presidential Address (Mathematics & Statistics Section). In Proceedings of the Indian Science Congress. 25–46.
[102] Mahalanobis, P. C. (1944). On large-scale sample surveys. Philosophical Transactions of the Royal Society of London B: Biological Sciences, 231(584), 329–451.
[103] Mahalanobis, P. C. (1952). Some aspects of the design of sample surveys. Sankhyā: The Indian Journal of Statistics, 1–7.
[104] Mickey, M. R. (1959). Some finite population unbiased ratio and regression estimators. Journal of the American Statistical Association, 54(287), 594–612.
[105] Midzuno, H. (1950). An outline of the Theory of Sampling Systems, Annals of the Institute of Statistical Mathematics, Vol. 3, No. 2.
[106] Midzuno, H. (1951). On the sampling system with probability proportionate to sum of sizes. Annals of the Institute of Statistical Mathematics, 3(1), 99–107.
[107] Midzuno, H. (1952). On the sampling system with probability proportional to sum of sizes. Ann. Inst. Stat. Math. 3, 99–107.
[108] Sen, A. R. (1952). Present status of probability sampling and its use in the estimation of characteristics. Econometrica, 20 103.
[109] Mohanty, S. (1967). Combination of Regression and Ratio Estimate. Jour. Ind. Statist. Asso., 5,16–19.
[110] Murthy, M. N. (1963). Generalised unbiased estimation in sampling from finite populations. Sankhyā B 25, 245–262.
[111] Murthy, M. N. (1968). On designing and conducting multi-subject household enquiries with reference to a permanent survey organization. Sankhyā: The Indian Journal of Statistics, Series B, 367–382.
[112] Narain, R. D. (1951). “On sampling without replacement with varying probabilities”, J. Ind. Soc, Agri. Stat. 3, 169–175.
[113] Neyman, J. (1934). On the two different aspects of representative method: The method of stratified sampling and the method of purposive selection. J. Roy. Statist. Soc., 97, 558–606.
[114] Ogus, J. L. and Clark, D. F. (1971). “The annual survey of manufacturers: a report on Methodology”, Technical Paper # 24, U. S. Bureau of Census, Washington.
[115] Pascual, J. N. (1961). Unbiased ratio estimators in stratified sampling, J. Amer. Statis. Assoc., 56, 70 -87.
[116] Patel, H. C. and Dharmadhikari, S. W. (1977). Admissibility of Murthy's and Midzuno's estimators within the class of linear unbiased estimators of finite population totals. Sankhyā: Series C, 40, 21–28.
[117] Patel, H. C. 1974. A note on necessary best estimators in the class of linear unbiased estimators. Sankhyā, C, 36, 195–196.
[118] Pathak, P. K. (1961). “Use of order statistics in sampling without replacement”, Sankhya, A, 23, 409–414.
[119] Pathak, P. K. (1962a). On simple random sampling with replacement. Sankhya, 24(A), 287–302.
[120] Pathak, P. K. (1962b). On sampling units with unequal probabilities. Sankhya, 24(A), 315–326.
[121] Pathak, P. K. (1967a) “Asymptotic efficiency of Des Raj’s strategy – I”, Sankhya A, 29, 283–298.
[122] Pearson, K. (1900). Mathematical contributions to the theory of evolution. VII. On the correlation of characters not quantitatively measurable. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 195, 1–405.
[123] Prabhu-Ajgaonkar, S. G. (1984). A note on necessary best estimator of order two (Corr: V32 p64). Metrika, 31, 1–4.
[124] Quenouille, M. H. (1956). Note on bias in estimation. Biometrika, 43, 353–360.
[125] Raj, D. (1954). “Ratio estimation in sampling with equal and unequal probabilities”, Jour. Ind. Soc. Agri. Stat., 6 (2), 127–138.
[126] Raj, D. (1956a) “Some estimators in sampling with varying probabilities without replacement”, J. Amer. Stat. Assoc. 60, 278–284.
[127] Raj, D. (1956b) “A note in determination of optimum probabilities in sampling without replacement”, Sankhya, 17, 197–200.
[128] Raj, D. (1966). Some remarks on a simple procedure of sampling without replacement. J. Amer. Statist. Assoc., 61, 391–396.
[129] Raj, D. (1972). “The Design of Sample Surveys”, McGraw – Hill, New York.
[130] Rao, C. R. (1949). Sufficient statistics and minimum variance unbiased estimation, Proc. Camb. Philo. Soc., 45, 213–218.
[131] Rao, J. N. K. (1961). “On the estimate of variance in unequal probability sampling”, Ann. Inst. Stat. Math., 13(1), 57–60.
[132] Rao, J. N. K. (1963a). “On two systems of unequal probability sampling without replacement”, Ann. Inst. Stat. Math., 15, 67–72.
[133] Rao, J. N. K. (1963b). “On three procedures of unequal probability sampling without replacement”, J. Amer. Stat. Assoc., 58, 192–215.
[134] Rao, J. N. K. (1965). “On two simple schemes of unequal probability sampling without replacement”, J. Ind. Stat. Assoc. 3, 173–180.
[135] Rao, J. N. K. (1966). “On the relative efficiency of some estimators in pps sampling for multiple characteristics”, Sankhya, A, 28, 61–70.
[136] Rao, P. S. R. S. (1969). Comparison of four ratio type estimators under a model. J. Amer. Statist. Assoc., 64, 574–580.
[137] Rao, P. S. R. S. (1979a). On applying the jack-knife procedure to the ratio estimator. Sankhya C, 41, 115–126.
[138] Rao, P. S. R. S. (1979b). Theory of the MINQUE - a review. Sankhya-B 39, 201–210.
[139] Rao, P.S.R.S. (1979c). Variance estimation for the ratio estimator of the mean of a finite population. Research report, U.S. Bureau of the Census and the University of Rochester.
[140] Rao, J. N. K. and Bayless, D. L. (1969). “An empirical study of the stabilities of estimators and variance estimators in unequal probability sampling of two units per stratum”, J. Amer. Stat. Assoc., 64, 540–549.
[141] Rao, P. S. R. S. and Rao, J. N. K. (1971). Small sample results for ratio estimators. Biometrika, 58, 625–630.
[142] Rao, J. N. K. and Singh, M. P. (1973). “On the choice of estimators in survey sampling”, Aust. Jour. Stat., 15, 95–104.
[143] Rao, J. N. K. and Vijayan, K. (1977). On estimating the variance in sampling with probability proportional to aggregate size. JASA., 80, 620–630.
[144] Rao, J. N. K., Hartley, H. O. and Cochran, W. G. (1962). “On a simple procedure of unequal probability sampling without replacement”, J. Roy. Stat. Soc., B, 24, 482–491.
[145] Robinson, P. M. (1982). On the convergence of the Horvitz-Thompson estimator. Australian Journal of Statistics, 24(2):234–238.
[146] Robinson, P. M. and Tsui, K. W. (1979). On Brewer's asymptotic analysis in robust sampling designs for large scale surveys. Tech. Bep. No. 79–43, University of British Columbia.
[147] Roy, D. C. (2003). A regression type estimates in two-phase sampling using two auxiliary variables. Pak. J. Statist. 19(3), 281–290.
[148] Roy, J., and I. M. Chakravarti (1960), “Estimating the mean of a finite population.” Annals of Mathematical Statistics 31, 392–398.
[149] Royall, R. M. (1970). “On finite population sampling theory under certain regression models”, Biometrika, 57, 377–387.
[150] Royall, R. M. (1976). Likelihhod functions in finite population sampling theory, Biometrika, 63, 605–614.
[151] Royall, R. M. (1976b). The linear least-squares prediction approach to two-stage sampling. Journal of the American Statistical Association, 71, 657–664.
[152] Royall, R. M. and Herson, J. (1973a) “Robust estimation in finite populations – I”, J. Amer. Stat. Assoc. 68, 880–889.
[153] Royall, R. M. and Herson, J. (1973b) “Robust estimation in finite populations – II”, J. Amer. Stat. Assoc. 68, 890–893.
[154] Samiuddin, M. and Asad, H. (1981). “A simple procedure of unequal probability sampling”, Biometrika, 68(3), 728–731.
[155] Samiuddin, M. and Hanif, M. (2006). Estimation in two phase sampling with complete and incomplete information. Proc. 8th Islamic Countries Conference on Statistical Sciences. 13, 479–495.
[156] Samiuddin, M., Hanif, M. and Asad, H. (1978). “Some remarks on PPS sampling without replacement”, Presented at Fourth Australian Statistical Conference, Canberra.
[157] Samiuddin, M., Kattan, A. K. A., Hanif, M. and Asad, H. (1992). “Some remarks on models, sampling schemes and estimators in unequal probability sampling”, Pak. J. Stat., 8(1), A, 1–18.
[158] Sampford, M. R. (1967). “On Sampling without replacement with unequal probabilities of selection”, Biometrika, 54, 499–513.
[159] Sarndal, C. E. (1980). On -inverse weighting versus best linear weighting in probability sampling. Bk, 67, 639–650.
[160] Sarndal, C. E. (1980). How survey methodologists communicates. JOS, 1, 49–63.
[161] Sarndal, C. E. (1984). “Design – consistent and model – dependent estimation for small domains”, J. Amer. Stat. Assoc. 79(3), 624–631.
[162] Seng, Y. P. (1951). Historical survey of the development of sampling theories and practice,” Jour. Roy. Stat. Soc., Ser. A, 114, 440–457.
[163] Sen, A. R. (1952). Present status of probability sampling and its use in the estimation of a characteristic. Econometrika, 20–103.
[164] Sen, A. R. (1953). “On the estimate of the variance in sampling with varying probabilities”, J. Ind. Soc. Agri. Stat., 5, 119–127.
[165] Sethumadhavi, R. and Rajagopalan, M. (1974). Use of PPS with 3-P sampling procedure. Sankhya-C, 36(3), 167–172.
[166] Shahbaz (2003). Sampling with unequal probabilities and without replacement, Unpublished PhD Thesis, NCBA&E, Lahore, Pakistan.
[167] 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.
[168] Shahbaz, M. Q., Hanif, M. and Samiuddin, M. (2002). On some estimators in unequal probability sampling. J. Statist. Theory and Appl. Vol. 1(4), 239–252, (USA)
[169] Singh, S. and Singh, R. (1979). On random non response in unequal probability sampling. Sankhya Ser., C(41), 127–137.
[170] Smith, H. F. (1938a). “An experimental law describing heterogeneity in the yields of agricultural crops”, J. Agri. Science, 28, 1–23.
[171] Smith, H. F. (1938b). Report of a preliminary statistical investigation of flowering dates of plants recorded in the phenological reports of the Royal Meteorological Society. Quarterly Journal of the Royal Meteorological Society, 64(273), 23–46.
[172] Som, R. K. (1973). “A Manual of Sampling Techniques”, Heinemann Educational Book Ltd. U.K.
[173] Stuart, A. (1954). Multistage sampling with prelminary random stratification of first stage units. Rev. Internat. Statist. Inst. 32, 193–201.
[174] Stuart, A. (1963). Calculation of Spearman's Rho for Ordered Two-Way Classifications. The American Statistician, 17(4), 23–24.
[175] Sukhatme, P. V. (1947a). The problems of plot size in large-scale surveys. JASA, 42, 297–310.
[176] Sukhatme, P. V. (1947b). Use of small size plots in yield surveys. Nature, 160–542.
[177] Sunter, A. B. (1977). “List sequential sampling with equal or unequal probabilities without replacement”, Inter. Stat. Rev. 54(1), 33–50.
[178] Tchuprow, A. A. (1923). On the mathematical expectation of the moments of frequency distributions in the case of correlated observations. Metron, 2, 461–493, 646–680.
[179] Tin, M. (1965). Comparison of some ratio estimators. J. Amer. Statist. Assoc., 60, 294–307.
[180] Tsui, K. W. (1983). “A class of admissible estimators of a finite population total.” Annals of the Institute of Statistical Mathematics, Part A 35, 25–30.
[181] Vijayan, K. (1975) “On estimating the variance in unequal probability sampling”, Jour. Amer. Stat. Asoc., 70, 713–716.
[182] Vijayan, K. (1966). On Horvitz-Thompson and Des Raj Estimators. Sankhya Ser., A(28), 87–92.
[183] Vithala R. Rao (1984). Pricing Research in Marketing: the state of the art. The journal of business, 57(1), 39–60.
[184] Yates, F. (1948). Systematic sampling. Transactions, Royal Society, London, A241, 345–377.
[185] Yates, F. (1949). Sampling Methods for Censuses and Surveys (1st ed.). Charles Griffin, London.
[186] 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.
[187] Yates, F., and P. M. Grundy (1953), “Selection without replacement from within strata with probability proportional to size.” Journal of the Royal Statistical Society, Series B 15, 253–261.
[188] Yates, F. (1953). Sampling methods for Censuses and Surveys. Charles Griffin & Company, London.
[189] Yogi, A. K. and Gupta, P. C. (1975). On the non-existence of second and higher order necessary best estimators. Journal of the Indian Society of Agricurtural Statistics. 27, 55–66.

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