Efficient Generalized Ratio-Product Type Estimators for Finite Population Mean with Ranked Set Sampling

Abstract: In this paper we suggest two modified estimators of the population mean using the power transformation based on ranked set sampling (RSS). The first order approximation of the bias and of the mean squared error of the proposed estimators are obtained. A generalized version of the suggested estimators by applying the power transformation is also presented. Theoretically, it is shown that these suggested estimators are more efficient than the estimators in simple random sampling (SRS). A numerical illustration is also carried out to demonstrate the merits of the proposed estimators using RSS over the usual estimators in SRS.


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, Unyazici, and Cingi (2009) used this technique to improve ratio estimator given by Prasad (1989).Here we shall propose two modified estimators of population mean using power transformation using RSS based on auxiliary variable.
The classical ratio estimators given by Cochran (1940) for estimating the population mean Y is defined as ) , where y is the sample mean for study variable y and x , X are the sample mean and population mean, respectively, for the auxiliary variable x.
Austrian Journal of Statistics, Vol. 42 (2013), No. 3, 137-148 When the population coefficient of variation C x of the auxiliary variable x is known, Sisodia and Dwivedi (1981) give a modified ratio estimator for Y as Motivated by Sisodia and Dwivedi (1981), H. P. Singh and Kakran (1993) developed a ratio-type estimator for Y as where β 2 (x) is the known value of the coefficient of kurtosis of an auxiliary variable x.
Utilizing the information on the coefficient of variation C x and the coefficient of kurtosis β 2 (x) of the auxiliary variable x, Upadhyaya and Singh (1999) suggested the following ratio type estimators ) . (4) By applying the power transformation on the Upadhyaya and Singh (1999) estimators, H. P. Singh, Tailor, Singh, and Kim (2008) suggested modified estimators as where α and δ are suitably chosen scalars such that the mean squared errors of y U P 1(α) and y U P 2(δ) are minimum.
To the first degree of approximation, the mean squared error (MSE) of these estimators are with where ρ yx denotes the correlation coefficient between y and x.
2 Ratio Estimator in Ranked Set Sampling In ranked set sampling (RSS) m independent random sets are chosen (each of size m) and the units in each set are selected with equal probability and without replacement from a finite population of size N .The members of each random set are ranked with respect to the characteristic of the study variable or auxiliary variable.Then, the smallest unit is selected from the first ordered set and the second smallest unit is selected from the second ordered set.By this way, this procedure is continued until the unit with the largest rank is chosen from the m-th set.This cycle may be repeated r times, so mr = n units have been measured during this process.Thus, RSS and SRS have equivalent sample sizes n for comparison of their biases and efficiencies.
When we rank on the auxiliary variable, let (y [i] , x (i) ) denote the i-th judgment ordering of the study variable and the i-th perfect ordering for the auxiliary variable in the i-th set, where i = 1, . . ., m. Samawi and Muttlak (1996) define the ratio estimator for the population mean as where are the ranked set sample means for the variables y and x, respectively.
To the first degree of approximation, the MSE of the estimator y R,RSS is given by (after ignoring the finite population correction factor) ) , where 3 Suggested Estimators Based on Ranked Set Sampling Motivated by Sisodia and Dwivedi (1981) we suggest a ratio-type estimator for Y using RSS, when the population coefficient of variation of the auxiliary variable C x is known as To obtain the bias and the MSE of this estimator we put y with θ = 1 mr and similarly var(ε where Further to validate the first degree of approximation, we assume that the sample size is large enough to get |ε 0 | and |ε 1 | as small as possible such that the terms involving ε 0 or ε 1 in a degree greater than two are negligible.
The bias and the MSE of Motivated by H. P. Singh and Kakran (1993) we suggest another new ratio estimator in ranked set sampling as Similarly, the bias and the mean squared error are obtained, respectively, as bias(y Motivated by Upadhyaya and Singh (1999) we also proposed two more ratio type estimators considering both coefficients of variation and kurtosis using ranked set sampling as The bias and the MSE of y RSS,M M 3 can be found as follows.Since bias(y Similarly, the bias and mean squared error of the estimator y RSS,M M 4 can be obtained, respectively, by changing the place of coefficient of kurtosis and coefficient of variation, as bias(y By applying the power transformation on y RSS,M M 3 and y RSS,M M 4 given in ( 18) and ( 19), we now propose generalized estimators as The bias and MSE of the estimator y RSS,M M 3(α) to the first degree of approximation, are obtained next.We have bias and Similarly, the bias and mean squared error of the estimator y RSS,M M 4(δ) can be obtained, respectively, by changing the place of coefficient of kurtosis and coefficient of variation, as bias(y and 4 Optimality of α and δ The optimum value of α to minimize the MSE of y RSS,M M 3(α) can easily be found as zero of its derivative, i.e. ) Similarly, After substituting ( 26) and ( 27), respectively, in ( 24) and ( 25), we obtain the minimum mean squared error of the proposed estimators as where K = ρ Cy Cx .

Numerical Illustration
To compare the efficiencies of the various estimators of our study, we take a population of size N = 50 (see page 1111 in the appendix of S. Singh, 2003).The example considers the data of agricultural loans outstanding of all operating banks in different states of the USA in 1997, where y is the real estate farm loans (study variable) in 1000 $ and x is the non-real estate loans (auxiliary variable) in 1000 $.
From this population we have taken 100 ranked set samples with size m = 4 and number of cycles r = 3, so that n = mr = 12.For these 100 ranked set samples chosen, we have computed estimated MSE's of the proposed estimators y M M 1,RSS , y M M 2,RSS , y M M 3,RSS , y M M 4,RSS , y RSS,M M 3(α) , and y RSS,M M 4(δ) which are given in Table 2. shows the MSE's of the estimators y SD , y SK , y U P 1 , y U P 2 , y U P 1(α) , and y U P 2(δ) , given in H. P. Singh et al. (2008).

Conclusion
On comparing Table 1 with Table 2 and Table 3 for the 100 ranked set samples, we see that the MSE's of the proposed estimators are smaller than those of the estimators given by previous authors.As a result, all the proposed new ratio type estimators y M M 1,RSS , y M M 2,RSS , y M M 3,RSS , y M M 4,RSS , y RSS,M M 3(α) and y RSS,M M 4(δ) for the population mean using RSS are more efficient than the respective estimators y SD , y SK , y U P 1 , y U P 2 , y U P 1(α) and y U P 2(δ) under SRS.Thus, if the coefficient of variation and the coefficient of kurtosis are known for the auxiliary variable, then these proposed estimators are recommended for use in practice.Sisodia, B. V. S., and Dwivedi, V. K. (1981).A modified ratio estimator using coefficient of variation of auxiliary variable.Journal of the Indian Society of Agricultural Statistics, 33, 13-18. Upadhyaya, L. N., and Singh, H. P. (1999).Use of transformed auxiliary variable in estimating the finite population mean.Biometrical Journal, 41, 627-636.

Table 2 :
Estimated MSE's of different new estimators using RSS.Note that y RSS,M M 3(α) = y RSS,M M 4(δ)

Table 3 :
Estimated MSE's of different new estimators using RSS.Note that y RSS,M M 3(α) = y RSS,M M 4(δ)