Two phase stratified sampling with regression method of estimation using two auxiliary variables

Introduction

Sometimes a sampler would select a large sample of units to collect information on certain variables and then select a relative smaller sample to collect information on study variable. This is the problem of two-phase sampling. Two-phase sampling is a powerful technique which was firstly introduced by Neyman (1934) for the stratification purpose. In two-phase sampling, regression and ratio estimation techniques are used to estimate the finite population mean. Regression estimator is considered to be more useful than ratio estimator except when regression line does not pass through origin otherwise these two estimators have almost same significance and analyst has to decide intuitively. Auxiliary information can be used to improve the efficiency of the estimator of a particular population parameter. The effectiveness of the estimation procedure with the use of auxiliary information closely depends upon the method in which the estimator has been proposed. Several authors have worked on different aspects of two-phase sampling using auxiliary information. They include Hansen and Hurwitz (1943) [6] , Spurr (1952), Freese (1962), Mohanty (1967) [5] , Tripathy (1970), Tripathy(1978), Srivastava and Jhajj (1981) [8] , Chand (1975) [2] , Sen (1978), Kiregyera (1980, 1984) [3,4] , Sahoo and Sahoo (1993) [9] , Sahoo et al. (1994), Singh (2001) [10] , Samiddin and Hanif (2007) [7] , Singh et al. (2011). Using multiple auxiliary variables can improve the efficiency of the estimator during estimation procedure i. e. for instance when 𝑋 ̅ is not known, then we may use an additional auxiliary variable z with known population mean 𝑍 ̅ . In this case it is assumed that the variable z is also correlated with y than x, hence, the need for this regression estimator in two-phase stratified sampling with two auxiliary variables.

Material and Method

There are two types of two-phase sampling design. In the first type, the auxiliary variable does not depend on the measurements, but is purely an indicator variable showing the stratum to which the variable of interest is to be allocated. This type is termed as two-phase sampling for stratification. In the second type, the relationship between the auxiliary variables and the variables of interest is described by means of ratio or regression. In this case, the design is termed as two-phase sampling with ratio estimators and two-phase sampling with regression estimators. Pradhan (2000) considered a regression estimator under two phase stratified sampling scheme given by

Where 𝛽 ℎ is the regression coefficient between x and y for the h-th stratum. Khan and Al-Hossain (2016) gave a difference-type estimator for population mean under two phase sampling as

Where 𝑘 1 and 𝑘 2 are unknown constant

Sampling Design

Let a large sample 𝑠 1 of fixed size 𝑛 ′ be drawn from a population of size N and is classified into different strata with 𝑛 ℎ ′ units falling in the ℎ 𝑡ℎ stratum 𝑠 1ℎ (h = 1, 2, L) where

. In the second phase a SRSWOR subsample 𝑠 2ℎ of size 𝑛 ℎ ′′ is drawn from 𝑛 ℎ ′ independently for each h and the study variable y is observed. We also assume that at the second phase a constant proportion of units

′ is sampled from the h-th stratum of the initial sample. Note that the auxiliary variables x and z are observed at the first phase.

The proposed Estimator 4.1 The Regression Estimator

The proposed unbiased estimator for population mean of two-phase stratified sampling with two auxiliary variables is given as

Where 𝛽 𝑦𝑥 is the regression coefficient between y and x for the ℎ 𝑡ℎ stratum, while 𝛽 𝑦𝑧 is the regression coefficient between y and z for the ℎ 𝑡ℎ stratum.

Proof of Unbiasedness

Variance of the Estimator

Unbiased Estimator of V (𝒚 ̅ ̅ 𝟐𝒓𝒆𝒈 )

The unbiased estimator of V (𝑦 ̅ ̅ 2𝑟𝑒𝑔 ) is given as

Where 𝜌 ̂𝑦𝑥 𝑎𝑛𝑑 𝜌 ̂𝑦𝑧 are estimated correlated coefficient of y and x; and y and z respectively. Proof

We can see that it has unbiased estimator

Optimum Allocation

Consider the cost function

Where 𝑐 ′ is the cost per unit in the first phase sample and 𝑐 ℎ ′′ is the cost per unit in the second phase sample. Since 𝑛 ℎ ′′ is a random variable the expected cost is

It is required to find 𝑛 ′ and 𝑔 ℎ so as to minimize V (𝑦 ̅ ̅ 2𝑟𝑒𝑔 ) for a given expected cost. This is the same as to minimize the product

With respect to 𝑔 ℎ and it exist if and only if

Hence the optimum value of 𝑔 ℎ is

Substituting the optimum value of 𝑛 ′ which is obtain from the expected cost in (5.1 ) and the optimum value of 𝑔 ℎ into the variance expression in (4.2 ) the optimum variance is obtained as

(5.2) Let the expected cost of the experiment be C*=60 and the cost for each unit of the sample at the second phase be 𝐶 ℎ ′′ =0.5 while 𝐶 ′ = 0.15. Hence for SRS, a sample of size n=100 is permissible.

Numerical Illustration

Now

References

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