Review Article

Some Efficient Methods to Remove Bias in Ratio and Product Types Estimators in Ranked Set Sampling

Nitu Mehta and V. L. Mandowara

  • Page No:  276 - 282
  • Published online: 31 Mar 2022
  • DOI : HTTPS://DOI.ORG/10.23910/1.2022.2771a

  • Abstract
  •  nitumehta82@gmail.com

Ranked set sampling is one method to potentially increase precision and reduce costs by using quantitative or qualitative information to obtain a more representative sample. Use of auxiliary information has shown its significance in improvement of efficiency of estimators of unknown population parameters. Ratio estimator is used when auxiliary information in the form of population mean of auxiliary variable at estimation stage for the estimation of population parameters when study and auxiliary variable are positively correlated. In case of negative correlation between study variable and auxiliary variable, Product estimator is defined for the estimation of population mean. This paper proposed the problem of reducing the bias of the ratio and product estimators of the population mean in ranked set sampling (RSS). This paper suggested several type unbiased estimators of the finite population mean using information on known population parameters of the auxiliary variable in ranked set sampling. An important objective in any statistical estimation procedure is to obtain the estimators of parameters of interest with more precision. The Variance of the proposed unbiased ratio and product estimators are obtained up to first degree of approximation. Theoretically, it is shown that these suggested estimators are more efficient than the unbiased estimators in Simple random sampling. A numerical illustration is also carried out to demonstrate the merits of the proposed estimators using RSS over the usual estimators in SRS.

Keywords :   RSS, unbiased estimators, ratio product estimators, auxiliary variables

  • INTRODUCTION

    The literature on Ranked set sampling describes a great variety of techniques for using auxiliary information to obtain more efficient estimators. Ranked set sampling was first suggested by McIntyre (1952) to increase the efficiency of estimator of population mean. Kadilar et al. (2009) used this technique to improve ratio estimator given by Prasad (1989) and Bouza (2008) used this to improve the product estimator. MandowaraandMehta (2013) suggestedefficient generalized Ratio-Product type estimators using RSS. Jeelani and Bouza (2015) suggested new ratio method of estimation under ranked set sampling. Mehtaand Mandowara (2012, 2016) proposed a better estimator of population mean with power transformation and modified ratio-cum-product estimator in RSS. Here we shall propose two modified Methods to construct unbiased ratio and product type estimators of population mean using Ranked set sampling based on auxiliary variable. The Variance of the proposed unbiased ratio and product tpye estimators are obtained up to first degree of approximation...


  • QUENOUILLE’S METHOD IN RANKED SET SAMPLING

    In this section, we apply the technique of Quenouille (1956) to Ranked set sampling to reduce bias from ratio estimator and constructing the unbiased ratio estimator of the population mean. Here we draw a sample of size 2n units from a population of N units. We divide the sample of 2n units into two equal halves each of size n. The sample based on 2n units is called the pooled sample...


  • PROPOSED EXACTLY UNBIASED RATIO TYPE ESTIMATOR IN RSS

    Motivated by Hartley and Ross (1954), we suggest a method of proposing an exactly unbiased ratio-type estimator for the population mean...


  • SUGGESTED EXACTLY UNBIASED PRODUCT TYPE ESTIMATOR IN RSS

    Adapting the estimator given by Singh Sarjinder (2003) and utilizing the product estimator for the population mean in ranked set sampling given by Bouza (2008), we suggest a method of proposing an exactly unbiased Product type estimator for the population...


  • EFFICIENCY COMPARISON

    On comparing (1.5) and (1.7) with (3.4) and (4.3) respectively, we obtain...


  • NUMERICAL ILLUSTRATION

    To compare efficiencies of various estimators of our study, here, we take a population of size N=50 on page 1111(Appendix) from the book entitled “Advanced Sampling Theory with Applications”, Vol.2, by Sarjinder Singh, published from Kluwer Academic Publishers. The example considers the data of Agricultural loans outstanding of all operating banks in different states of USA in 1997, where y is real estate farm loans (study variable) in $000 and x is the non-real estate loans (auxiliary variable) in $000...


  • CONCLUSION

    We have proposed two modified Methods to construct unbiased ratio and product type estimators using RSS and obtained variance of the proposed unbiased ratio and product type estimators. The variance of proposed estimators have been compared with the variance of   SRS estimators and found these proposed estimators have smaller variance than corresponding estimators. This theoretical result has been supported by the above example.


  • Reference
  • Bouza, Carlos N., 2008. Ranked set sampling for the product estimator. Revista Investigation Operational 29(3), 201–206.

    Cochran, W.G., 1940. Some properties of estimators based on sampling scheme with varying probabilities. Australian Journal of Statistics 17, 22–28.

    Hartley, H.O., Ross, A., 1954. Unbiased ratio estimators. Nature 174, 270–271.

    Jeelani, M.I., Bouza, C.N., 2015. New ratio method of estimation under ranked set sampling. Revista Investigación Operational 36, 151–155.

    Kadilar, C., Unyazici, Y., Cingi, H., 2009. Ratio estimator for the population mean using ranked set sampling, Statistical Papers 50, 301–309.

    McIntyre, G.A., 1952. A method of unbiased selective sampling using ranked sets, Australian Journal of Agricultural Research 3, 385–390.

    Mandowara, V.L., Mehta, N., 2013. Efficient generalized ratio-product type estimators for finite population mean with ranked set sampling. Austrian Journal of Statistics 42(2), 137–148.

    Mehta, N., Mandowara, V.L., 2012. A better estimator of population mean with power transformation based on ranked set sampling. Statistics in Transition-new Series 13(3), 551–558.

    Mehta, N., Mandowara, V.L., 2016. A Modified ratio-cum-product estimator of finite population mean using ranked set sampling. Communication and Statistics-Theory and Methods 45(2), 267–276.

    Mehta, N., Mandowara, V.L., 2020. Conflicts of interests - a general procedure for estimating finite population mean using ranked set sampling. Revista Investigacion Operacional 41(6), 902–903.

    Murthy, M.N., 1964. Product method of estimation. Sankhya-A 26, 69-74.

    Prasad, B., 1989. Some improved ratio type estimators of population mean and ratio in finite population sample surveys. Communication and Statistics-Theory and Methods 18, 379–392.

    Quenouille, M.H., 1956. Notes on bias in estimation. Biometrika 43, 353–360.

    Samawi, H.M., Muttlak, H.A., 1996. Estimation of ratio using rank set sampling. The Biometrical Journal 38, 753–764.

    Singh, S., 2003. Advanced sampling theory with application. Vol. I, Kluwer Academic Publishers, Netherlands.


Cite

1.
Mehta N, M VL, owara . Some Efficient Methods to Remove Bias in Ratio and Product Types Estimators in Ranked Set Sampling IJBSM [Internet]. 31Mar.2022[cited 8Feb.2022];13(1):276-282. Available from: http://www.pphouse.org/ijbsm-article-details.php?article=1588

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