Abstract
This paper considers some efficient combined and separate classes of estimators of the population mean in the presence of bivariate auxiliary information under stratified ranked set sampling. The mean square error expressions of the proffered combined and separate classes of estimators are derived to the first order of approximation. The theoretical conditions are obtained under which the proffered combined and separate classes of estimators perform better than the existing combined and separate class of estimators. Subsequently, numerical and simulation studies are performed using real and artificially generated populations. The numerical and simulation results are found to be rewarding, showing the superiority of the proffered estimators over the existing estimators.
1. Introduction
It is evident from sample surveys that the convenient utilization of auxiliary information may always boost the efficiency of the estimator. This information may be used either at the design phase (sampling design) or at the estimation phase or at both phases. It is a well-known fact that when this information is employed for the estimation phase, the ratio, exponential, product, and regression-type estimators are often used in distinct aspects. Various esteemed authors have considered this auxiliary information at the estimation stage and envisaged a wide range of modified estimators till date. However, the literature encompasses a variety of estimators in the presence of multiple auxiliary information. Koyuncu and Kadilar [1] introduced a family of estimators of the population mean using stratified simple random sampling . Tailor et al. [2] suggested the ratio-cum-product estimator in for estimating the population mean. Tailor and Chouhan [3] envisaged the ratio-cum-product-type exponential estimator of the finite population mean. Lone et al. [4] employed a generalized ratio-cum-product-type exponential estimator in , whereas Lone et al. [5] mooted enhanced separate classes of estimators of the population mean. Muneer et al. [6] investigated a class of combined estimators in . Recently, Muneer et al. [7] investigated the apparent family of chain exponential estimators in .
The concept of ranked set sampling was initiated by Mclntyre [8] but did not furnish any mathematical support. Takahasi and Wakimoto [9] improved the lacuna of the method by furnishing the necessary mathematical theory. Dell and Clutter [10] demonstrated that the mean under is an unbiased estimator of the population mean under the condition of perfect and imperfect ranking. Muttlak [11] suggested the estimation of parameters in simple linear regression under . Samawi and Muttlak [12] shown that ranking of the denominator variable of the ratio estimator improves the efficiency. The interested readers may refer to some recent works such as Bhushan and Kumar [13–16] and Bhushan et al. [17, 18] for a comprehensive study about . In extensive surveys, when the information on more than one auxiliary variable is available, then, Abu-Dayyeh et al. [19] introduced two estimators under for estimating the population mean. Following Olkin [20], Mehta and Mandowara [21] suggested an improved ratio estimator under . Khan and Shabbir [22] introduced a generalized exponential-type ratio-cum-ratio estimators using and for estimating the population mean. This study is aimed at proffering some efficient classes of combined and separate estimators in the presence of bivariate auxiliary information using .
Neutrosophic statistics is an extension of classical statistics and is applied when the data is coming from a complex process or from an uncertain environment. Smarandache [23] considered a neutrosophic set as a generalization of the intuitionistic fuzzy set. Smarandache [24] discussed the neutrosophic measure, neutrosophic integral, and neutrosophic probability. Alhabib et al. [25] discussed some neutrosophic probability distributions. Aslam et al. [26] proposed a new diagnosis test under the neutrosophic statistics with an application to diabetic patients, whereas Aslam et al. [27] devised a vague data analysis using neutrosophic the Jarque-Bera test. Aslam [27] suggested the neutrosophic statistical test for counts in climatology. Tahir et al. [28] investigated neutrosophic ratio-type estimators for estimating the population mean. Recently, Vishwakarma and Singh [29] introduced a generalized estimator for computation of the population mean under neutrosophic with an application to solar energy data. In future, we intend to extend the current study using neutrosophic statistics.
1.1. Sampling Methodology
The procedure of ranked set sampling is based on selecting independent random samples of size units with equal probability and with replacement from a population of size units. The units of each random sample are now ranked with respect to the variable of choice. Let the study variable be denoted by and the two auxiliary variables by and . Then, trivariate samples are selected randomly from the population and distributed into sets, each of size units. Let the ranking be performed over each unit of the auxiliary variable . Now, from the first sample, the unit with the smallest rank of , together with the variables and associated with the smallest rank of , is measured. From the second sample of the same size, the variables and associated with the second smallest rank of are measured. This procedure is continued until the variables and associated with the highest rank of are measured from the sample. This process completes a single cycle. The whole procedure is repeated times to get a desired sample of size units.
The procedure of stratified ranked set sampling is consisting of quantifying trivariate random samples from the stratum of the population. These quantified random samples are fixed up into sets, each of size units. The procedure of is now employed on each set to get sets of ranked set samples, each of size units. These ranked set samples are jointly formed sets, each of size units. Iterate this procedure times for each stratum to get the desired stratified ranked set sample of size units.
Consider a finite population based on identifiable units with a study variable and two auxiliary variables and associated with each unit of the population. Let the population be divided into disjoint strata with stratum based on , units. Let (, , ), (, , ),…,(, , ) be the stratified ranked set sample for cycle in the stratum. Here, and are the judgement order for the variables and and is the order statistics for variable in the sample at the cycle of the stratum. Let , , and be the stratified ranked set sample means corresponding to the population means , , and of variables , , and , where is the weight in stratum . Let , and be the stratified ranked set sample means corresponding to the population means , and of variables , , and in stratum . Let , , , , , and be the sample variances and covariances corresponding to the population variances and covariances , , , , , and in the stratum . Let , , and be the population coefficient of variation of variables , , and , respectively.
To derive the of the proffered combined estimators, the following notations will be utilized throughout the paper. such that and
On the basis of (2), it can be written as follows: where , , , , , , .
Again, to determine the properties of the separate estimators, the following notations will be used throughout the paper. such that , where , , ,
The paper is divided into some sections. Section 2 is devoted to the existing combined and separate estimators, whereas Section 3 deals with the proffered combined classes of estimators. The suggested combined and separate estimators are given in Section 3 along with their properties. The theoretical comparisons of the proffered and existing combined and separate estimators are given in Section 4. The theoretical results are illustrated through a simulation study in Section 5. Lastly, the study is concluded in Section 6.
2. Review of Literature
This section considers the review of existing combined and separate classes of estimators.
2.1. Combined Estimators
The classical combined ratio estimator of population mean using bivariate auxiliary information under is defined as follows:
The classical combined regression estimator of population mean based on bivariate auxiliary information under is defined as follows: where and are the regression coefficients of on and , respectively.
Motivated by Khoshnevisan et al. [30] and Koyuncu and Kadilar [1], we introduce a general family of combined estimators for population mean under as follows: where , , , and are suitably chosen scalars, whereas , and , are either real numbers or the function of known parameters of variables and such as standard deviation, coefficients of variation, coefficient of skewness, coefficient of kurtosis, and correlation coefficient.
On the lines of Tailor and Chouhan [3], one may suggest the following combined class of the estimator of population mean under as follows: where is a duly opted scalar.
Following Lone et al. [4], we define a generalized combined ratio-cum-product-type exponential estimator in as follows: where and are suitably chosen scalars.
Following Lone et al. [5], one may introduce a combined estimator for estimating population mean under as follows:
On the lines of Muneer et al. [6], one may investigate a combined class of estimators under as follows: where and are duly opted scalars.
Khan and Shabbir [22] suggested a generalized combined exponential-type ratio-cum-ratio estimators under as follows: where and are duly opted scalars.
Following Muneer et al. [7], one can introduce a combined chain ratio exponential family of estimators in as follows:
On the lines of Searls [31], the above-combined chain ratio exponential family of estimators becomes where , , , , , and are duly opted scalars; assumes values −1, 0, and +1 in order to form different new and existing estimators.
It is noticed that the minimum of the Koyuncu and Kadilar- [1] type estimator , [4, 5] estimator and and Khan and Shabbir- [22] estimator are equal to the minimum of classical regression estimator .
The of the estimators considered in this section is discussed in Appendix A.
2.2. Separate Estimators
The classical separate ratio estimator of population mean using bivariate auxiliary information under can be defined as follows:
The classical separate regression estimator of population mean under bivariate auxiliary information using is as follows: where and are the regression coefficients of on and in stratum , respectively.
Following Koyuncu and Kadilar [1], one may consider a general family of separate estimators for population mean under as follows: where , , , and are suitably chosen constants whereas , and , are either real numbers or the function of known parameters of variables and , respectively, in stratum .
Following Tailor and Chouhan [3], one may suggest the following separate class of the estimator of the population mean using bivariate auxiliary information under as follows: where is a duly opted scalar.
On the lines of Lone et al. [4], we consider a generalized separate ratio-cum-product-type exponential estimator in as follows: where and are suitably chosen scalars.
The separate version of the Lone et al. [5] estimator for estimating population mean under is defined as follows:
Following Muneer et al. [6], we introduce a separate class of estimators in as follows: where and are duly opted scalars.
Khan and Shabbir [22] suggested a generalized separate exponential-type ratio-cum-ratio estimator under as follows: where and are duly opted scalars.
Motivated by Muneer et al. [7], a separate chain ratio exponential family of estimators is defined in as follows:
On the lines of Searls [31], the above separate chain ratio exponential family of estimators becomes where , , , and and is a duly opted scalar; assumes values −1, 0, and +1 in order to form different new and existing estimators.
It is to be noted that the minimum of Koyuncu and Kadilar- [1] type estimator , Lone et al.- [4, 5] type estimator , , and Khan and Shabbir- [22] estimator are similar to the minimum of classical regression estimator .
The of the estimators discussed in this section is given in Appendix B for ready reference.
3. Proffered Estimators
Motivated by the works of Bhushan et al. [32, 33], we have extended the work of Bhushan et al. [17] using bivariate auxiliary information under .
3.1. Combined Estimators
We propose some improved combined classes of estimators based on bivariate auxiliary information under as follows: where , , , are suitably chosen scalars.
Theorem 1. The of the proffered combined class estimators to the first order of approximation is given by
Proof. The outline of the derivation is given in Appendix C for ready reference.
Corollary 2. The minimum at the optimum values of , , and is given by
Proof. The outline of the derivation is given in Appendix C for ready reference.
3.2. Separate Estimators
We propose some improved separate classes of estimators based on bivariate auxiliary information under as follows: where , , and , are suitably chosen scalars.
Theorem 3. The of the proffered separate classes of estimators to the first order of approximation is given by
Proof. The outline of the derivation is given in Appendix C for ready reference.
Corollary 4. The minimum at the optimum values of , , and , , is given by
Proof. The outline of the derivation is given in Appendix D for ready reference.
4. Theoretical Conditions
4.1. Combined Estimators
On comparing the minimum of the proffered combined classes of estimators and the existing combined estimators from (27) with (47), (48), (50), (52), (56), (58), (60), (62), (63), and (65), we obtain the theoretical conditions given as follows:
If condition (31) holds, then, the proffered combined classes of estimators perform better than the other existing combined estimators.
4.2. Separate Estimators
On comparing the minimum of the proffered separate classes of estimator with the existing separate estimators from (30) with (81), (82), (84), (86), (90), (92), (94), (96), (97), and (99), we get the following theoretical conditions.
If conditions (32)–(42) hold, then, the proffered separate classes of estimators , become superior than the other existing separate estimators.
4.3. Comparison of Proffered Combined and Separate Estimators
On comparing the minimum of proffered combined and separate classes of estimators and , , we get
If the ratio estimate is veritable and the relationship between auxiliary and study variables within each stratum is a straight line passing through the origin, then, the last term of (43) is broadly small and it vanishes.
Moreover, unless is invariant from stratum to stratum, separate estimators probably become more efficient in each stratum if the sample in each stratum is large enough so that the approximate formula for is valid and the cumulative bias that can affect the proffered estimators is negligible, whereas the proffered combined estimators are to be preferably recommended with only a small sample in each stratum (see [34]).
5. Numerical Study
To enhance the credibility of the theoretical development of the proposed combined and separate classes of estimators, we have conducted a numerical study using two real populations which are described as follows.
Population 5 (Sarndal et al. [35], p. 529). is the population in thousands during 1985, is the population in thousands during 1975, is the total number of seats in the municipal council, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and .
Population 6 (National Horticulture Baard [36]). is productivity (MT/hectare), is production (000 tons), is the area (000 hectares), , , , , , , .70, , , , , , , , , , , , , , , , , and .
We have calculated the and of different estimators by using the above populations. The is calculated with respect to the usual mean estimator using the following expression.
where is the of existing and proposed combined and separate estimators. The results of the numerical study for the above populations are reported in Table 1 by and . The numerical results show the dominance of the proposed combined and separate classes of estimators and , , respectively, in terms of lesser and greater over the combined and separate usual mean estimator, classical ratio, and regression estimators, Koyuncu and Kadilar- [1] type estimator, Tailor and Chouhan- [3] type estimator, [4, 5] type estimators, Khan and Shabbir- [22] type estimator, and Muneer et al.- [6, 7] type estimators. Also, the proposed combined and separate class of estimators and attain the lesser and greater among the proposed classes of estimators in both the populations.
6. Simulation Study
To generalize the results of the numerical study, we have conducted a simulation study over a hypothetically generated normal population. The simulation procedure is explained in the following points: (i)Trivariate random observations of size 600 units are drawn from a trivariate normal distribution with parameters , , , , , and and different amounts of correlation coefficients , and (ii)The population generated above is divided into 3 equal strata, and a stratified ranked set sample of size 12 units with a number of cycles 4 and set size 3 is drawn from each stratum(iii)Compute the required statistics(iv)Iterate the above steps 10000 times to calculate the and of various combined and separate classes of estimators using the following expression:
The and of the combined and separate classes of estimators are calculated using (45) and (46), respectively, and the results are reported for various values of correlation coefficients in Tables 2 and 3 which exhibit the ascendancy of the proposed combined and separate classes of estimators over the existing combined and separate classes of estimators.
7. Conclusion
In this paper, we proffer some improved classes of estimators along with their properties using bivariate auxiliary information in . The proffered estimators dominate the other existing estimators under the conditions stated in Section 4. The numerical and simulation studies are performed using real and artificially generated populations with various amounts of correlation coefficients. The results of numerical and simulation studies are reported in terms of and from Tables 1 to 3. From the perusal of the results of Tables 1–3, the following conclusions are drawn: (i)In Table 1, the proffered combined classes of estimators perform better than the existing combined estimators, namely, conventional mean estimator , classical ratio and regression estimators & , Koyuncu and Kadilar- [1] type estimator , Tailor and Chouhan- [3] type estimator , Lone et al.- [4, 5] type estimators & , Khan and Shabbir [22] estimator , and Muneer et al.- [6, 7] type estimators , , & in both the populations(ii)In Table 1, the proffered separate classes of estimators , dominate the existing separate estimators, namely, conventional mean estimator , classical ratio and regression estimators & , Koyuncu and Kadilar- [1] type estimator , Tailor and Chouhan- [3] type estimator , Lone et al.- [4, 5] type estimators & , Khan and Shabbir [22] estimator , and Muneer et al.- [6, 7] type estimators , & (iii)In Table 2, the proffered combined classes of estimators , perform better than the existing combined estimators. The similar inclination in the results of separate estimators can also be observed in Table 3(iv)In Tables 1, 2, and 3, the proffered combined and separate classes of estimators and , respectively, become superior among the proffered combined and separate classes of estimators(v)In Tables 1, 2, and 3, the proffered separate classes of estimators perform better than the proffered combined classes of estimators in terms of lesser and greater (vi)From the simulation results of Tables 2 and 3, it has been observed that the increases as the correlation coefficients decrease
Thus, the proffered classes of estimators are preferred for the computation of the population mean when bivariate auxiliary information is available.
Appendix
A. MSEs of Existing Combined Estimators
The of the existing combined classes of estimators is tabulated as follows: where
The optimum values of the scalars involved in the estimators are given as follows:
B. MSEs of Existing Separate Estimators
This section considers the of the existing separate estimators. where
The optimum values of the constants involved in the estimators are obtained by minimizing expression w.r.t. the constants as follows:
C. MSEs of Proposed Combined Estimators
This section addresses the outline of the proof of Theorem 1 and Corollary 2 of Section 3.1.
Consider the estimator
Using the notations defined in the earlier section, we get
Squaring and taking expectation both sides of (116), we will get the of the estimator up to first order of approximation as follows
The optimum values of , , and can be obtained by minimizing (117) w.r.t. , , and as follows: where and .
Putting , , and in (117), we get the minimum as follows:
In a similar way, we can calculate the of other estimators as follows where
The optimum values of the scalars involved are given hereunder:
D. MSEs of Proposed Separate Estimators
This section addresses the outline of the proof of Theorem 3 and Corollary 4 of Section 3.2.
Consider the estimator
Using the notations defined in the earlier section, we get
Squaring and taking expectation both sides of (130), we will get the of the estimator up to the first order of approximation as follows:
The optimum values of , , and can be obtained by minimizing (131) w.r.t. , , and as follows where and .
Putting , and in (131), we get the minimum as
Similarly, we can calculate the of other estimators as follows where
The optimum values of the scalars involved are given hereunder as follows:
Data Availability
The descriptive statistics of the artificially generated populations are given in the manuscript.
Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the present study.
Acknowledgments
This research was supported by the Princess Nourah Bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R259), Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia.