International Research Journal of Innovations in Engineering and Technology - IRJIET International Open Access, Monthly, Peer Reviewed, Reputed Journal ISSN (online): 2581-3048

Impact Factor (2025): 6.9

DOI Prefix: 10.47001/IRJIET

Efficient Class of Ratio–Cum–Product Estimator of Population Mean in Two Phase Sampling in Presence of Two Auxiliary Variables with Application to Agricultural Data Sets

Abstract

In order to minimize errors in survey estimation situation, it is imperative that we adopt a sampling method that is capable of giving an improved results and which will be adjudged to be a better representation of the population. In this study, efficient class of ratio–cum-product estimators of population mean Y-  in Two Phase Sampling in presence of two auxiliary variables x and z was proposed. Members of the proposed class of estimator were obtained by varying the scalars associated with the proposed estimator, from where it was observed that the estimator produces the traditional sample mean ratio estimator Y- , the dual to Singh and Tailor (2005) estimator due to Tailor et al (2012), Singh and Tailor (2011) generalized version of the dual to ratio-cum-product estimator of the population mean Y-. Various attribute of the proposed estimator such as biases, relative biases, Mean Square Errors (MSEs), and optimal MSEs were derived for cases I and II to the quadratic form of Tailor’s series approximation. Theoretical proposition for evaluating efficiency was established and empirical study was conducted using Four (4) Agricultural data sets to ascertain the veracity of the theoretical proposition, from where it was found out from the results that the estimators TDS1TDS3, TDS4, TDS5 and TDS12, were more efficient in case I and estimators’ T*DS1T*DS3, T*DS4, T*DS4T*DS9, T*DS12 and T*DS13 were more efficient in case II, having been ascertained to have produced smaller MSEs. The gain in efficiency increases as the sample size decreases and was more significant in case II. Therefore, sub-sampling the second sample independent of the first sample is advantageous and recommended for appreciable gain in efficiency and superiority over the first phase in double sampling scheme.

Country : Nigeria

1 Etorti Imoke John2 Agada Anthony Nwachuya

  1. Department of Mathematics, School of Secondary Education Science programmes, Cross River State College of Education, Akamkpa-Nigeria
  2. Department of Mathematics, School of Secondary Education Science programmes, Cross River State College of Education, Akamkpa-Nigeria

IRJIET, Volume 8, Issue 1, January 2024 pp. 128-150

doi.org/10.47001/IRJIET/2024.801017

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References

  1. Al-jaraha, J. & Ahmed, M. S. (2002): The class of chain estimator for finite population variance using double sampling. Information and Management Sciences 13(21)13-18.
  2. Al-Omari, A.I., Jemain, A.A., Ibrahim, K., (2020). New ratio estimators of the mean using simple random sampling and ranked set sampling methods. Rev. Investig. Oper. 30 (2), 97-108.
  3. Aneesh, S. Hariharan, V., Gallucci, F., & Graig, H., (2013). Estimation of relative efficiency of Adaptive cluster versus  traditional Sampling designs applied to arrivals of sharks. Annals of Statistics Vol. 41, No.:1, 221-249. http://www.researchgate.net
  4. Al-Saleh, M.F., Darabseh, M., (2017). Inference on the skew normal distribution using ranked set sampling. International Journal of Computational. Theoretical Statistics. 4, 65-76.
  5. Bahl, S. & Tuteja, R. K (1991):  Ratio and product type exponential estimator. Journal of Information in Optimization Sciences 12(1): 159 -163.
  6. Chamu, S. & Singh, H. P (2014): An improved class of chain ratio-product type estimators in two-phase sampling using two auxiliary variables. Journal of Probability and Statistics vol. 2014, article ID 939701.
  7. Cingi, H. (2007):  Improvement in variable estimation in double sampling. Communications in Statistics-Theory and Methods 36 (11): 2075-2081.
  8. Cochran, W. G. (1940): The estimation of yield of cereal experiments by sampling for the yield of grains to total produce. Journal of Agricultural Science, 2075-2081.
  9. Freese, F. (1962): Elementary forest sampling. USDA Forester Service and Agricultural Handbook. 232.
  10. Gangele, R. K. & Choudhury, S.K (2013):  A note in estimation of population mean. In sample survey using auxiliary information. Mathematical Theory and Modeling 3(9), 9-14.
  11. Goswami, J.N., & Sukhatme, B. V. (1965): Ratio method of estimation in multi-phase sampling with several auxiliary variables. Journal of the Indian Society of Agricultural Statistics. 17(1): 83-103.
  12. Handique, B. K (2012): Timber volume estimation by double sampling using High resolution satellite data. The Indian Forester, vol. 140, No 3, pp 154-161.
  13. Hidiroglou, M. A & Sarndal, C. E. (1998): Use of auxiliary information foe two-phase sampling. Survey Methodology 24, 873-878.
  14. Kadilar, C. & Cingi, H (2006):  An improvement in estimating the population by using the correlation coefficient.
  15. Hacettepe Journal of Mathematics and Statistics, 359(1), 105-109.
  16. Khare, B. B. & Rehman H. U. (2015): Improved Ratio in regression type estimator for population mean using coefficient of variation of the study character in the presence of non-response. International Journal of Technology and Innovations, 14, 1-7.
  17. Kalita, D. & Singi, B. K (2013): Improved exponential chain Ratio and product –type estimators for finite population mean in double sampling. International Journals of Statistical Sciences 13, 21-37.
  18. Khare, B. B. & Srivastava, S. (1993): Estimation of population mean using auxiliary character in presence of non-response. National Academic Science Letters, India, 16(3), 111-114.
  19. Khare, B. B. & Srivastava, S. (1995):  Study of conventional and alternative two-phase sampling ratio, product and regression estimators in presence of non- response. National Academic Science Letters, India, 65(A), II 195-203.
  20. Khare, B. B.& Srivastava, S., (1997): Transformed ratio type estimators for the population mean in the presence of non-response. Communication in Statistics Theory and Methods, 26(7), 1779-1783.
  21. Kovalevsky, E. V., (1924). A sketch of history of survey sampling in Russia.  Journal of the Royal Statistical Society 148, 119-127. https://www.jstor.org
  22. Kumar, M. & Vishwakarma G. K. (2014): An improved class of chain Ratio-product type Estimators in two phase sampling using two auxiliary variables. Journal of Probability and Statistics Volume 2014, article ID 939701.
  23. Malik, K. A & Tailor, R (2013): Ratio type estimator of population mean in double sampling. International Journal of advanced Mathematics and Statistics i(1), 34-39.
  24. Mohanty, S. (1967): Combination of regression and ratio estimates. Journals of Indian Statistical Association. 31, 107-114.
  25. Murthy, M. N. (1967): Sampling Theory and Methods. Statistical publishing society Calcutta India.
  26. Neyman, J. (1938): Contribution to the theory of sampling human populations. Journal of American Statistical Association, 33(201): 101-116.
  27. Olkins I. (1958): Multivariate ratio estimation for finite populations. Biometrika 45: 154-165.
  28. Olusengun A. O. (2013): On the use of an Almost Unbiased Ratio Estimator in two – phase sampling scheme. The International Journal of Engineering and Science (IJES), 2(11), 118-123.
  29. Parrot D. L, Lhotka, J. M, Feridshouse, B.S (2012): Improving Woody Biomass Estimation Efficiency using double sampling. Forests 3, 179-189.
  30. Raj, D. (1965):  On a method of using multi-auxiliary information in sampling Surveys. Journal of the American Statistical Association 68: 270-277.
  31. Rao, P. S. R. S., & Mudholkar, G. S. (1967):  Generalized multivariate estimator for the mean of finite populations. Journal of the American Statistical Association 62: 1009-1012.
  32. Rao, P. S. R. S., (1987):  Ratio and regression estimates with sub sampling the non-respondents. Paper presented at a special contributed section of the non-respondents. Survey Methodology, 12(2), 217-230.
  33. Rathour, A.(2012): An alternative estimator for estimating the finite population mean using auxiliary information in sample surveys. ISRN  Probability and Statistics DOI 10.5402/2012/657682.
  34. Singh A. K., Sing, H. P. and Upadhyaya, L. N. (2011): Generalized Chain ratio estimator for finite population mean in two phase sampling. Journal of Indian Society of Agricultural Statistics, 54(3), 370-375.
  35. Singh, H.P & Sahoo, L. N (2014). An alternative class of estimators in double Sampling. http./maths.usm.my/bull Malaysian Mathematical science Society. 2(29), 89-94.
  36. Samuiddin, M.& Hanif, M. (2007). Estimation of population mean using single and two phase sampling with or without additional information. Pakistan Journals of Statistics 23(2), 99-118.
  37. Singh, H. P. & Espejo, M. R (2007): Double sampling ratio – product estimator of a finite population mean in sample surveys. Journal of Applied Statistics 34(1), 71-85.
  38. Singh H. P. & Joarder, A. H., (1998): Estimation of finite population variance using random non-response in survey sampling.  Metrika, 47, 241-249.
  39. Singh, H.P & Tailor, R. (2005): Estimation of finite population mean using known correlation coefficient between auxiliary characters. Statistica, 65(4), 401-418.
  40. Singh, H.P & Tailor, R. (2003): Use of known correlation coefficient in estimating the finite Population mean. Statistics in Transition 6(4): 555-560.
  41. Singh, H. P., Joarder A. H., & Tracy, D.S. (2000): Regression type estimator for random non-response in survey sampling. Statistica, anno LX, N0.:1 39-43.
  42. Singh, H. P.& Tracy, D. S., (2001): Regression type estimator for random non-response in survey sampling. Statistica, Anno LX, N0:1 39-43.
  43. Singh, H.P. & Vishwakarma, G.K (2007):  Modified Exponential Ratio and Product Estimators for finite population mean in double sampling. Austrian Journal of Statistics 36(3), 217 – 225.
  44. Singh, H. P., Chandra S.& Singh, S. (1993): Variance estimation using multi-auxiliary information for non-response in survey sampling. Statistica, anno LXIII, N0.: 1, 23-25.
  45. Singh, R. K. (1982): Generalized estimators of estimation of ratio and product of population parameters. Journals of Statistical Research 16, 1-2.
  46. Singh, B. K & Choudhury, S. (2012): Double sampling product –cum- Dual to ratio estimators of finite population mean in sample surveys. International Journals of Scientific and Engineering Research 3(4), 1-7.
  47. Singh, R. Kumar, M.& Smarandache, F.(2008): Almost Unbiased estimator for estimating population mean using known values of some population parameters. Pakistan Journal of Statistics and Operations Research.                 4(2):63- 76.
  48. Sing H.P., and Yadav A.(2018): An efficient family of ratio-cum-product estimators for finite population mean in sample surveys. Journal of Turkish Statistical Association, volume 11, No.: 1-2, pp 29-45.
  49. Solanki, R.S. & Singh, H.P (2013): An efficient class of ratio estimators for the population mean in double sampling using single auxiliary information. Communications in Statistics – Theory and Methods 42(1), 145 – 163.
  50. Spur, S. H. (1952): Forest inventory. The royal press company.
  51. Srinath, K. P. (1971): Multi-phase sampling in non-response or problems. Journals of American Statistical Association 66, 583-586.
  52. Srivastava, S.K (1965): An estimator of the mean of a finite population using several auxiliary variables. Journals of the Indian Statistical Society129: 531-533.
  53. Srivastava, S.K (1970): Two Phase sampling estimator in sample surveys. American Journals of Statistics 12, 23-27.
  54. Srivastava, S.K (1971): Generalized estimator for the mean of a finite population using multi-auxiliary information. Journal of the American Statistical Association 66: 404-407.
  55. Srivastava, S.K.& Tracy, D.S. (1980): A dual to ratio estimator in sampling surveys. Biometrica 67(1), 795-204.
  56. Steel R.G. D. & Torrie (1960): Principle and procedures of Statistics. McGraw Hill book Co.
  57. Sukhatme, B. V., (1962): Some ratio-type estimators in two-phase sampling. Journal of the American Statistical Association, 57:628-632.
  58. Toutenberg, H. & Srivastava. V. K.(1998). Estimation of ratio estimator for the population mean in survey sampling when some observations are missing. Metrika, 48, 177-187.
  59. Unnikrishan, N. K.& Kuntes, S. (1995). Optimality of an analogue of Basu’s estimator under a double sampling design. Sankhya, 57, 103-111.
  60. Vishwakarma, G. K., Zeeshan, S. M.(2021). Generalized ratio-cum-product estimator for finite population mean under two-phase sampling scheme. Journals of Modern Applied Statistical Methods: Vol. 19: issues1 Article 7 DOI:10.22237/jmasm/1608553320.
  61. Yedav, S. K. & Kadilar, C. (2013). Improved Exponential Type Ratio Estimator of population variance. Revista Colombiana de Estadistica vol. 36(1), 145-152.

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