A ratio chain-type exponential estimator for finite population mean using double sampling.

In this article, we have proposed a ratio chain-type exponential estimator for finite population mean of the study variable under double sampling scheme using auxiliary variables. The large sample properties of the suggested strategy are derived up to first order, of approximation, and its competence conditions are carried out under which the suggested estimator is performed better than the other existing estimators discussed in the literature. An empirical study shows that the suggested strategy is more efficient than the other relevant competing estimators under two phase sampling scheme.

Introduction and literature review

To increase the precision of estimators for population mean of the study variable under double sampling design, a lot of works have been done in the field of sample survey and when the study variable is strongly connected with the auxiliary variables the precision of the estimators can be more and more. Using the knowledge of the auxiliary variables several authors have proposed different estimation technique for finite population mean of the study variable, Sukhatme (1962), have developed a general ratio-type estimator. Chand (1975), have suggested two chain ratio-type estimators to estimate the population mean using two auxiliary variables (Kiregyera 1980, 1984; Srivnstava et al. 1990; Bahl and Tuteja 1991; Srivastava 1970; Cochran 1977; Singh et al. 2006, 2007, 2011; Dash and Mishra 2011; Singh and Choudhury 2012; Khare et al. 2013; Khare and Rehman 2013) etc.

Let us consider a finite population of size N of different units . Let y and x be the study and the auxiliary variable with corresponding values and respectively for i-th unit is defined on a finite population U.

Let and be the corresponding population means of the study as well as auxiliary variable respectively. Also let and be the corresponding population variances of the study as well as auxiliary variable respectively and let and be the coefficient of variation of the study as well as auxiliary variable respectively, and be the correlation coefficient between x and y.

Let y and x be the study and the auxiliary variable with corresponding values and respectively for i-th unit in the sample. Let and be the corresponding unbiased sample means of the study as well as auxiliary variable respectively.

Also let and be the corresponding unbiased sample variances of the study as well as auxiliary variable respectively.

Let , and be the co-variances between their respective subscripts respectively. Similarly is the corresponding sample regression coefficient y on x based on a sample of size n. Also and are the coefficients of variation of the study and auxiliary variables respectively.

The usual unbiased estimator to estimate the population mean of the study variable is

The variance of the estimator up to first order of approximation is, given by

The usual ratio and regression estimators in two phase sampling and their mean square error are, given as follows

The mean squared error and variance are given belowwhere and

Chand (1975), proposed the following chain ratio-type estimator in double sampling by incorporating the knowledge of two auxiliary variables, the suggested estimator is, given by

The mean square error of the suggested estimator is, given as

Kiregyera (1984), suggested the following chain-type exponential estimators in two phase sampling, the suggested estimators are given as

The mean square errors of the suggested estimators, up to first order of approximation are, given as follows

Searls (1964), proposed an estimation procedure for population mean using known knowledge of the coefficient of variation of the auxiliary variablewhere and

Khare and Rehman (2013), have proposed improved chain type estimators for population mean using auxiliary information, the suggested estimators are given bywhere , and are constants.

The mean square errors of the suggested estimators, are, given byandwhere the optimum values of , and are and .

Also , and

Singh et al. (2013), recommended a class of exponential chain ratio-product type estimator for estimating population mean using two auxiliary variables, aswhere and are suitably chosen constants, such that

The minimum mean square error of the suggested estimator is given as followswhere the optimum value of is .

The proposed estimator

Let us consider a finite population of size N units. To estimate the population mean of the variable of interest y taking values , in the existence of two auxiliary variables say x and z taking values and respectively for the ith unit . We assume that there is high correlation between y and x as compared to the correlation between y and z (i.e. ). When the population of the auxiliary variable x is unknown, but information on the other cheaply auxiliary variable z closely related to x but compared to x remotely to y, is available for all the units in a population. In such a situation we use two phase sampling. In the two phase sampling scheme a large initial sample of size is drawn from the population by using simple random sample without replacement sampling (SRSWOR) scheme and measure x and z to estimate . In the second phase, we draw a sample (subsample) of size from first phase sample of size , i.e. () by using SRSWOR or directly from the population U and observed the study variable y.

Under the given probability sampling, we have proposed a chain-ratio-type exponential estimator for finite population mean of the study variable y, given bywhere and are the unknown constants, whose value is to be determined for optimality conditions.

To obtain the properties of the proposed estimator we define the following relative error terms and their expectations.

Let and

Such that For i = 0, 1, 2.

Equation (21) can be further simplified as given by

Further simplifying, up to order one

Expanding the right hand side of the Eq. (23), up to first order of approximation and subtracting from both sides we get

On squaring and taking expectation on both sides of (24), we get mean square error up to first order of approximation, given as

Differentiating Eq. (25) w.r.t to and we get the optimum values of and respectively as given by, where the optimum values are and respectively.

On substituting the optimum value of and in Eq. (25), we get the minimum mean square error of the proposed estimators, given as follows

Efficiency comparison

In this section, we have obtained some conditions by comparing the mean square errors of the estimators under which the proposed estimator performs better than the other existing estimators. The proposed estimator is more efficient if the given conditions are satisfied.

By (26) and (2), if,

By (26) and (5), if,

By (26) and (6), if,

By (26) and (8), if,

By (26) and (11), if,

By (26) and (12), if,

By (26) and (17), if,

By (26) and (18), if,

By (26) and (20), if,

Numerical study

To verify the theoretical conditions under the efficiency comparison numerically, we have taken a real data set from the literature. The description and the necessary data statistics of the data is, given below.

For percent relative efficiency we use the following formula.for j = 1, 2, 3, 4, 5, 6, 7, 8 and m.

Population

The data from the population of 100 records of resale of homes from February 15 to April 30, 1993 from the files maintained by the Albuquerque Board of realtors on selling price ($) as a study variable y square feet of living space as an auxiliary variable x and annual taxes ($) as an additional variable z have been taken. The numerical values of the parameters of the population are given as.

Conclusion

On the basis of mean square errors and the percent relative efficiencies of the estimators as shown in Tables 1 and 2, it has been observed that the performance of the proposed estimator is better than the other relevant existing estimators discussed in the literature of survey sampling, which reveals the usefulness of suggested method in practice and would work very well in practical surveys.

Table 1

Mean square errors (MSE’s) of the estimators

Estimators	Mean squared errors (MSE’s)
	n′ = 60, n = 30	n′ = 65, n = 35	n′ = 70, n = 40
y¯_0	3583.66	2852.30	2303.78
y¯_1	1691.43	1355.15	1087.35
y¯_2	1690.47	1354.39	1086.73
y¯_3	1490.62	1192.96	958.26
y¯_4	1282.01	1024.46	824.15
y¯_5	1278.03	1021.27	821.59
y¯_6	1275.16	1018.37	819.03
y¯_7	1274.17	1017.59	818.40
y¯_8	1462.31	1181.19	950.65
y¯_m	1136.84	921.83	748.56

Table 2

Percent relative efficiency (PRE’s) of the proposed estimator with the existing estimators

Estimators	Percent relative efficiency (PRE’s)
	n′ = 60, n = 30	n′ = 65, n = 35	n′ = 70, n = 40
y¯_0	100.00	100.00	100.00
y¯_1	211.87	210.48	211.87
y¯_2	211.99	210.60	211.99
y¯_3	240.41	239.09	240.41
y¯_4	279.54	278.42	279.54
y¯_5	280.41	279.29	280.41
y¯_6	281.04	280.09	281.28
y¯_7	281.26	280.30	281.50
y¯_8	244.79	303.05	376.54
y¯_m	314.87	388.31	478.19