A class of estimators using auxiliary information in two-stage sampling for two phases of sample selection

Introduction

Consider a finite population 𝑈, partitioned into 𝑁 clusters called primary sampling units (PSU) denoted by 𝑈 1 , 𝑈 2 , … . , 𝑈 𝑁 such that the number of secondary sampling units (SSU) in 𝑈 𝑖 is 𝑀 𝑖 and 𝑀 = ∑ 𝑀 𝑖 𝑁 𝑖=1

. Let 𝑦 𝑖𝑗 and 𝑥 𝑖𝑗 respectively be the values of the study variable 𝑦 and an auxiliary variable 𝑥 for the jth SSU of 𝑈 𝑖 (𝑗 = 1, 2, … , 𝑀 𝑖 ; 𝑖 = 1, 2, … , 𝑁). Define To estimate 𝑌 ̅ , we consider a two-stage sampling design. At stage one, a sample 𝑠 of 𝑛 PSUs is drawn from 𝑈. Then, for every 𝑖 ∈ 𝑠, a sample 𝑠 𝑖 of 𝑚 𝑖 SSUs is drawn from 𝑈 𝑖 at the second stage. Assume that units at each stage are sampled according to the design simple random sampling without replacement (SRSWOR). Let us define the following sample means: Sampling theory provides many successful attempts on the use of auxiliary information about the population for improving precision of the estimates under a given sampling design. However, precision of an estimator in a two-stage sampling can be improved considerably when the kind and extent of available auxiliary information at different stages/levels are effectively utilized. According to the nature of the available auxiliary information, three different cases can be taken into consideration viz., (i) The values of 𝑋 𝑖 or 𝑋 ̅ 𝑖 i.e., the cluster totals or means of the auxiliary variable is available for all PSUs in 𝑈 so that a correct value of 𝑋 or 𝑋 ̅ is available, (ii) The values of 𝑥 i.e., 𝑥 𝑖𝑗 's can be observed for all SSUs in 𝑈 so that a correct value of 𝑋 or 𝑋 ̅ is available, and (iii) The values of 𝑥 can be observed for SSUs in 𝑈 𝑖 , 𝑖 ∈ 𝑠, i.e., for the selected SSUs so that a correct value of 𝑋 𝑖 or 𝑋 ̅ 𝑖 for 𝑖 ∈ 𝑠 can be available [see, for example, Sarndal, Swensson and Wretman (1992)] [13] .

Many two-stage sampling estimation techniques consider case (ii) and use advance knowledge on overall population mean or total of the auxiliary variable. But these techniques not only need up-to-date lists of all SSUs of the entire population but also do not fully exhaust the information contents of the auxiliary variable. Cases (i) and (iii) have many practical applications because they ordinarily use auxiliary information that are easily obtained at the two different stages of the survey operation. Combination of cases (i) and (iii) are also considered in many traditional texts [cf., Hansen, Hurwitz, and Madow (1953) [2] , Murthy (1977)] [4] and journal articles [cf., Smith (1969) [14] , Sahoo and Swain (1983) [11] , Sahoo (1987) [5] , Zheng and Little (2004) [16] , Kim et al.

(2009)] [3] . Because, in many surveys cluster means or totals of 𝑥 for the selected clusters are usually known or can be known easily or cheaply. Therefore, a survey statistician can take the advantage of utilizing this type of more extensive available information while developing efficient estimation techniques. As an example to indicate how such a situation arises, we may refer to a crop survey conducted to estimate the yield rate of a crop in a district with blocks (cluster of villages) as the PSUs and villages as the SSUs. If 𝑦 and 𝑥 represent respectively yield of the crop and area under cultivation, then information on mean area under cultivation (𝑋 ̅ 𝑖 ) for the 𝑖th selected block can be obtained easily from the block records and information on the mean cultivated area for the entire district i.e., 𝑋 ̅ can be easily available from the district records. With the spirit of utilizing such type of extensive auxiliary information on 𝑥, Sahoo and Panda (1997) [7] and Sahoo et al. (2009) [6] formulated some efficient estimation methods under two-stage sampling. Similar problems, with the involvement of two auxiliary variables, are also considered in Sahoo and Panda (1999) [8] , Sahoo et al. (2005, 2011) [9,10] . Sahoo and Panda (1997) [7] developed a class of estimators that needs advance knowledge on the overall population mean 𝑋 ̅ as well as the cluster means 𝑋 ̅ 𝑖 , 𝑖𝜖𝑠. But, in many surveys, no such extensive information is available at the outset. The common procedure in such cases is to use a two-phase sampling or sampling followed by sub-sampling. The main objective of this paper is to construct and study a general class of estimators in two-phase sampling scheme for estimating the population mean 𝑌 ̅ when one fails to gather prior knowledge on 𝑋 ̅ 𝑖 and 𝑋 ̅ . SSUs used to gather information on 𝑥. b) In the second phase, a sub-sample 𝑠 (𝑠 ⊂ 𝑠 ′ ) of 𝑛 PSUs is selected out of 𝑛 ′ PSUs selected in the first phase sample 𝑠 ′ and then in 𝑈 𝑖 , 𝑖 ∈ 𝑠, a sub-sample 𝑠 𝑖 (𝑠 𝑖 ⊂ 𝑠 𝑖 ′ ) of 𝑚 𝑖 SSUs is selected out of the 𝑚 𝑖 ′ SSUs selected in the first phase sample 𝑠 𝑖 ′ . The study variable 𝑦 is then observed for the SSUs selected in the second phase sample.

Two

We now define the following sample means in the context of two-phase sampling:

The proposed class of estimators

For 𝑖th selected PSU 𝑈 𝑖 , 𝑖 ∈ 𝑠 ′ , and for given 𝑠 𝑖 ′ and 𝑠 𝑖 (𝑠 𝑖 ⊂ 𝑠 𝑖 ′ ⊂ 𝑈 𝑖 ), consider a class of estimators for 𝑌 ̅ 𝑖 defined by ℓ 𝑖 = ℊ 𝑖 (𝑦 ̅ 𝑖 , 𝑥̅ 𝑖 , 𝑥̅ 𝑑𝑖 ), where ℊ 𝑖 (𝑦 ̅ 𝑖 , 𝑥̅ 𝑖 , 𝑥̅ 𝑑𝑖 ) is a known function of 𝑦 ̅ 𝑖 , 𝑥̅ 𝑖 and 𝑥̅ 𝑑𝑖 such that ℊ 𝑖 (𝑌 ̅ 𝑖 , 𝑋 ̅ 𝑖 , 𝑋 ̅ 𝑖 ) = 𝑌 ̅ 𝑖 and admits the following regularity conditions: a) (𝑦 ̅ 𝑖 , 𝑥̅ 𝑖 , 𝑥̅ 𝑑𝑖 ) is considered as a point in a bounded and closed convex subspace 𝒮 of the three-dimensional real space containing the point (𝑌 ̅ 𝑖 , 𝑋 ̅ 𝑖 , 𝑋 ̅ 𝑖 ), and b) The function ℊ 𝑖 (𝑦 ̅ 𝑖 , 𝑥̅ 𝑖 , 𝑥̅ 𝑑𝑖 ) is continuous having the first and second order partial derivatives which are also continuous in 𝒮.

Further, for given 𝑠 ′ and 𝑠 (𝑠 ⊂ 𝑠 ′ ⊂ 𝑈), define ℓ 𝑠 = 1 𝑛 ∑ 𝑢 𝑖 𝑖𝜖𝑠 ℓ 𝑖 , and let ℊ(ℓ 𝑠 , 𝑥̅ 𝑑 , 𝑥̅ 𝑑 ′ ) be a known function of ℓ 𝑠 , 𝑥̅ 𝑑 and 𝑥̅ 𝑑 ′ such that ℊ(𝑌, ̅ 𝑋, ̅ 𝑋 ̅ ) = 𝑌 ̅ and admits regularity conditions in 𝒮. Then, the proposed class of estimators of 𝑌 ̅ is defined byℓ 𝑔 = ℊ(ℓ 𝑠 , 𝑥̅ 𝑑 , 𝑥̅ 𝑑 ′ ). Here ℓ 𝑖 = ℊ 𝑖 (𝑦 ̅ 𝑖 , 𝑥̅ 𝑖 , 𝑥̅ 𝑑𝑖 ) may be a linear or a nonlinear function of 𝑦 ̅ 𝑖 , 𝑥̅ 𝑖 and 𝑥̅ 𝑑𝑖 within the framework of the mentioned regularity conditions. Therefore, for analyzing properties of ℓ 𝑖 in a simpler way, we use the Taylor linearization technique [cf., Sarndal, Swensson and Wretman (1992)] [13] to approximate ℓ 𝑖 by a more easily handled linear function so that approximate expressions for the mean and variance, under the sampling design in cluster 𝑖, can be easily found. Hence, considering the first order Taylor approximation of ℊ 𝑖 (𝑦 ̅ 𝑖 , 𝑥̅ 𝑖 , 𝑥̅ 𝑑𝑖 ), expanding around the point (𝑌 ̅ 𝑖 , 𝑋 ̅ 𝑖 , 𝑋 ̅ 𝑖 ) and neglecting the remainder term, we obtain

Where ℊ 𝑖0 , ℊ 𝑖1 and ℊ 𝑖2 are respectively the first order partial derivatives of ℊ 𝑖 with respect to 𝑦 ̅ 𝑖 , 𝑥̅ 𝑖 and 𝑥̅ 𝑑𝑖 about (𝑌 ̅ 𝑖 , 𝑋 ̅ 𝑖 , 𝑋 ̅ 𝑖 ).

Let 𝐸 1 , 𝐸 2 (𝑉 1 , 𝑉 2 ; 𝐶𝑜𝑣 1 , 𝐶𝑜𝑣 2 ) denote the expectation (variance, covariance) operators over repeated sampling in the first and second stages respectively; by 𝐸 (𝑉 or 𝐶𝑜𝑣) we denote the overall expectation (variance or covariance). Thus, noting that ℊ 𝑖0 = 1 and ℊ 𝑖1 = -ℊ 𝑖2 , to a first order of approximation, we have 𝐸 2 (ℓ 𝑖 ) ≈ 𝑌 ̅ 𝑖 , and

After a considerable simplification, we get

. Let ℊ 0 , ℊ 1 and ℊ 2 be respectively the first order derivatives of the function ℊ with respect to ℓ 𝑠 , 𝑥̅ 𝑑 and 𝑥̅ 𝑑 ′ around (𝑌, ̅ 𝑋, ̅ 𝑋 ̅ ). For simplicity, once again considering first-order Taylor linearization and noting that ℊ 0 = 1 and ℊ 1 = -ℊ 2 , ℓ 𝑔 is approximated by the following serviceable linear form:

This linearized estimator ℓ 𝑔 looks mysterious, because of that 𝑔 is a function of ℓ 𝑠 , 𝑥̅ 𝑑 and 𝑥̅ 𝑑 ′ , and on the other hand in the light of (1), ℓ 𝑠 is also a function of 𝑥̅ 𝑑 . Hence, ℓ 𝑔 is a composite function whose approximate variance is given by

Using these conditional operators, we find, after simplification and rearrangement of terms,

Finally, we derive formula for the approximate variance of ℓ 𝑔 (in a compact form) as

is the variance of 𝑦 ̅; 𝜈 𝑖 = 𝑢 𝑖

Minimization of 𝑉(ℓ 𝑔 ) in ( 9) with respect to ℊ 1 and ℊ 𝑖1 leads to their unique optimum values as ℊ 1

(opt) = -𝛽 and ℊ 𝑖1 (opt) = -𝛽 𝑖𝑦𝑥 .

When these optimum values are inserted into (9), we obtain a minimum approximate variance (which may be called approximate minimum variance bound (MVB) of the class) as min𝑉(ℓ 𝑔 ) = 𝑉(𝑦 ̅) - ⁄ . One can easily verify that the estimators ℓ 𝑅 , ℓ 𝑅𝐺1 and ℓ 𝑃 are two-phase sampling extensions of the chain ratio, chain regression and chain product estimators of Murthy (1977) [4] , Sahoo (1987) [5] , and Sahoo and Swain (1986) [12] . Das and Sahoo (2015) [1] also evaluated the performance of ℓ 𝑅𝐺1 relative to other regression-type estimators.

4.2

When the available auxiliary information within the selected PSUs only is taken into consideration (i.e., only case (iii) is considered), then ℓ 𝑔 reduces to ℓ 𝑠 , producing a family of separate variety of estimators whose approximate variance structure is

The class of estimators produced by ℓ 𝑠 constitutes a subclass of estimators produced by ℓ 𝑔 . Two estimators ℓ 𝑅 (𝑠) = [2] . The minimum variance bound and the MVB estimator of ℓ 𝑠 are given by min𝑉(ℓ 𝑠 ) = 𝑉(𝑦 ̅) -

4.3

When the available auxiliary information about the clusters as well as about the population are utilized (i.e., only case (i) is considered), then ℓ 𝑔 reduces to ℓ 𝑐 , a class of estimators (of course a subclass of ℓ 𝑔 ) defined by

Now we see that three estimators defined by ℓ 𝑅

′ are direct byproducts of ℓ 𝑐 , where ℓ 𝑅 (𝑐) is the Smith's (1969) [14] ratio estimator under our two-phase sampling design.

The approximate MVB of ℓ 𝑐 and the corresponding MVB estimator are

and ℓ 𝑅𝐺 (𝑐) = 𝑦 ̅ -𝛽(𝑥̅ 𝑑 -𝑥̅ 𝑑 ′ ).

4.4

If 𝑠 𝑖 = 𝑠 𝑖 ′ ∀ 𝑖 and 𝑠 = 𝑠 ′ i.e., if we consider two-stage sampling with single phase selection at each stage, then ℓ 𝑔 defines the class of estimators studied earlier by Sahoo and Panda (1997) [7] .

4.5

When 𝑠 𝑖 = 𝑠 𝑖 ′ = 𝑈 𝑖 ∀ 𝑖, ℓ 𝑔 generates a system of estimators for two-phase cluster sampling. On the other hand, when 𝑠 = 𝑠 ′ = 𝑈, we get a system of separate estimators for two-phase stratified random sampling with PSUs as strata.

Efficiency of the class

In order to study efficiency aspect of the formulated estimation technique of the utilization of auxiliary variable at different stages, we consider two cases. In the first case, we compare efficiency of ℓ 𝑔 with that of 𝑦 ̅, the traditional estimator incorporating no auxiliary variable, and in the second case we compare with that of the traditional class of estimators suggested by Srivastava (1980) incorporating 𝑥.

5.1

We note from (9) that 𝑉(ℓ 𝑔 ) ≤ 𝑉(𝑦 ̅) i.e., an estimator of ℓ 𝑔 is more efficient than 𝑦 ̅ if 𝛽 𝑖𝑦𝑧 < -

These sufficient conditions, although strongly dependent on the selection and nature of the functions ℊ 𝑖 and ℊ, clearly indicate that there is scope for improving upon the direct estimation strategy that considers no auxiliary variable through the suggested estimation strategy that considers one auxiliary variable. From (11) it is also clear that for optimum choices of ℊ 𝑖1 and ℊ 1 , the later strategy always yields higher efficiency gain over the former one.

5.2 Srivastava's (1980) class of estimators for the adopted two-phase sampling mechanism, may be defined by

Where (𝑦 ̅, 𝑥̅ , 𝑥̅ 𝑑 ′ ) 𝜖 𝒮 and the function 𝒽(𝑦 ̅, 𝑥̅ , 𝑥̅ 𝑑 ′ ) involved in composing the class admits regularity conditions as stated in section 3 such that 𝒽(𝑌 ̅ , 𝑋 ̅ , 𝑋 ̅ ) = 𝑌 ̅ . As is discussed earlier, here we also have 𝒽 0 = 1 and we assume that 𝒽 1 = -𝒽 2 , where 𝒽 0 , 𝒽 1 and 𝒽 2 are respectively the first order partial derivatives of 𝒽(𝑦 ̅, 𝑥̅ , 𝑥̅ 𝑑 ′ ) with respect to 𝑦 ̅, 𝑥̅ and 𝑥̅ 𝑑 ′ about (𝑌 ̅ , 𝑋 ̅ , 𝑋 ̅ ). Hence, an approximate expression of 𝑉(𝓉 ℎ ), derived through the Taylor linearization method, is given by

From ( 9) and ( 17) one can easily check that 𝑉(ℓ 𝑔 ) ≤ 𝑉(𝓉 ℎ ) i.e., an estimator of ℓ 𝑔 is more efficient than an estimator of 𝓉 ℎ when

(𝒽 1 + ℊ 𝑖1 ) ∀ 𝑖 and 𝛽 ≥ -

After simplification and rearrangement of terms in (17), an alternative expression for 𝑉(𝓉 ℎ ) is given by

Numerical Study

From the various results derived in the preceding sections we note that an analytical comparison to test the effectiveness of different estimation methods discussed here is not very much useful. One way to avoid this difficulty is to consider a numerical comparison as a counterpart to the analytical comparison. Here we carry out a numerical study to illustrate the performance of different estimators using data on two populations. Population I: It consists of strip-wise complete enumeration data on timber volume (= 𝑦) and length (= 𝑥) for 176 strips (SSUs) divided into 10 (= 10) blocks (PSUs) of the Black Mountain Experimental Forest given in Murthy (1977, p.131) [4] . For this population, we select 𝑛 ′ = 6, 𝑛 = 3, 𝑚 𝑖 ′ = 5, 5, 5, 5, 4, 5, 4, 5, 8 and 8, and 𝑚 𝑖 = 2, ∀𝑖 ∈ 𝑠. Population II: This population is the MU284 population available in Sarndal, Swensson and Wretman (1992, p.652 [13] , Appendix B). It consists of 284 municipalities (SSUs) divided into 50 clusters (PSUs) with two variables 1985 population (= 𝑦) and 1975 population (= 𝑥). Here, we consider 𝑛 ′ = 20, 𝑛 = 10, 𝑚 𝑖 ′ = 3, ∀𝑖 ∈ 𝑠 ′ , and 𝑚 𝑖 = 2, ∀𝑖 ∈ 𝑠.

The estimators under consideration are the four MVB estimators viz., ℓ 𝑅𝐺 , ℓ 𝑅𝐺 (𝑠) , ℓ 𝑅𝐺 (𝑐) and 𝓉 𝑅𝐺 (ℎ) ; three ratio-type estimators: ℓ 𝑅 , ℓ 𝑅 (𝑠) and ℓ 𝑅 (𝑐) ; and two regression-type estimators: ℓ 𝑅𝐺1 and ℓ 𝑅𝐺1 (𝑐) already defined in section 4. We also include a ratio-type estimator 𝓉 𝑅 (ℎ) = 𝑦 ̅ 𝑥̅ 𝑑 ′ 𝑥̅ and a regression-type estimator 𝓉 𝑅𝐺1 (ℎ) = 𝑦 ̅ -𝛽 𝑏𝑦𝑥 (𝑥̅ -𝑥̅ 𝑑 ′ ) as some specific cases of 𝓉 ℎ in our numerical comparison. We do not touch product or product-type estimators as in the two populations 𝑦 is positively correlated with 𝑥.

Approximate variance expressions of the ratio-type estimators ℓ 𝑅 , ℓ 𝑅 (𝑠) , ℓ 𝑅 (𝑐) and 𝓉 𝑅 (ℎ) can be easily obtained from ( 9), ( 12), (14) and 𝑋 ̅ . However, the approximate variance expressions of the regression-type estimators ℓ 𝑅𝐺1 , ℓ 𝑅𝐺1 (𝑐) and 𝓉 𝑅𝐺1 (ℎ) , in the simplified form, are as follows:

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