Estimation of ratio of two population means using ratio type estimators in stratified sampling under non – Response

Introduction

The applications of ratio of two population means are in calculation of per capita income (socio-economics field), per hectare production of crops (agriculture field), per kilometre population density (population study), growth index rate, in field of education and medical etc. Various research works have been done for estimation of ratio of two population means using auxiliary variable in simple random sampling technique by Singh [1] , Tripathi [2,3] , Upadhyaya and Singh [4] , Srivastava et al [5,6] , Singh et al [7,8] , and Singh and Singh [9] . But in some cases and situations, it has been seen that estimators in stratified sampling are more efficient than those of simple random sampling. Tailor et al [20] have been suggested a ratio estimator of two population means using auxiliary information in stratified sampling. Due to the problem of non-response, we unable to find the information of all selected units in the sample. Thus, to handle this problem of non-response, Hansen and Hurwitz [10] developed a technique using sub sample from non-respondent. Using this technique, various researchers have been studied the estimation of ratio two population means using auxiliary variables under non-response Khare and Pandey [11] , Khare and Sinha [12][13][14][15] , Khare et al [16,17] , and N. kumar and Patel [18,19] .

Sampling strategy and estimation methodology

Let the heterogeneous population of size N is divided into k homogeneous strata of size Ni (i =1,2,3,……..,k) such that N(=N1+N2+N3+……..+NK) be the total population size. Then we can select a sample of size ni( i=1,2,3,………,k)from each one of the strata using SRSWOR. Such that n(=n1+n2+n3+……..+nk)be total sample size. In case of non-response, it is seen that ni1 units provide information but ni2 units be nonrespondent of the sample size ni(<Ni). By extensive efforts the data are later obtained from a random sample of ui2 of ni2 units such that ni2 = Kiui2 (Ki>1), Where, 1/Ki be the sampling fraction among non-respondents in the ith stratum (i = 1,2,…..,k). Then using Hansen-Hurwitz 10 technique, the sample mean of jth character,taken from the ith stratum,of study variable Y in case of non-response is given by

; unbiased estimator of 𝑌 ̅ 𝑗𝑖 in ith stratum for jth characteristics and 𝑦 ̅ 𝑗𝑛 𝑖1 and 𝑦 ̅ 𝑗𝑢 𝑖2 are the means based on ni1 units for response group and ui2 units of sub sample of non-response group respectively for jth character in ith stratum.

𝑁 𝑖2 ℎ=1

Also, Wi2 = Ni2/Ni ; non-response rate in the ith stratum.

Here, we have proposed a class of estimators for ratio of two population means using stratified sampling technique based on nonresponse. In order to efficiency comparison, the relative bias and mean square error of proposed estimator are compared with existing estimators. .

In order to derive bias relative and mean square error of above estimator, Let us assume, 𝑦 ̅ 𝑗𝑠𝑡 * = 𝑌 ̅ 𝑗 (1 + εj); such that-E(εj) = 0; j = 1,2

Then, expressing the estimator 𝑅 𝑠𝑡 * in term of εj ,s , we get-

Expanding the above expression and neglecting 3 rd and higher degree of terms, we get-

𝑁 𝑖 ℎ=1

The ratio type estimator 0f population mean in stratified sampling, proposed by Tailor et al 20 , is given as

Adopting above idea of Tailor, we proposed the conventional (when complete information on X, but incomplete information on Y) and alternative estimator (when incomplete information on X and Y both) i.e. Z1 and Z2 respectively for ratio of two population means in stratified sampling in case of non-response, given by

The proposed class of estimators After combining and generalizing, the above proposed estimators we suggest a class of estimators of ratio of two population means in stratified sampling in case of non-response, given by Also, E(𝜀

Then, expressing the estimator 𝑅 𝑠𝑡 * in term of εj ,s and εm ,s we get-

Expanding the above expression and neglecting 3 rd and higher degree of terms, we get-

To obtain optimum α ( 𝛼 ̂) and optimum β (𝛽 ̂), we have to minimize M.S.E.(Gst) with respect to α and β respectively. In this order, differentiating it w.r.t. α and β respectively and equating derivatives to zero. The M.S.E.(Gst) is minimum when - Thus, in condition Ast > 0 < A * st ; the proposed estimator Gst is always efficient than the estimator G2s t. [3] On Comparing section 4.2 with 6.2; the proposed estimator Gst will be more efficient than G3s t iff M.S.E.

Obviously, terms R 2 , (𝐴 𝑠𝑡 -𝐵 𝑠𝑡 ) 2 , and (𝐴 𝑠𝑡 * -𝐵 𝑠𝑡 * ) 2 be always positive.

Thus, in condition Ast > 0 < A * st ; the proposed estimator Gst is always efficient than the estimator G3s t. [4] On Comparing section 4.4 with 7.2; the proposed estimator Gst will be more efficient than G4s t iff M.S.E.

Obviously, terms R 2 , (𝐴 𝑠𝑡 + 𝐵 𝑠𝑡 ) 2 , and (𝐵 𝑠𝑡 * ) 2 be always positive.

Thus, in condition Ast > 0 < A * st ; the proposed estimator Gst is always efficient than the estimator G4s t. [5] On Comparing section 4.4 with 8.2; the proposed estimator Gst will be more efficient than G5s t iff M.S.E. Obviously, terms R 2 , (𝐴 𝑠𝑡 -𝐵 𝑠𝑡 ) 2 , and (𝐵 𝑠𝑡 * ) 2 be always positive.

Thus, in condition Ast > 0 < A * st ; the proposed estimator Gst is always efficient than the estimator G5s t. In each efficiency situation, it has been seen that if Ast > 0 < A * st then, the proposed estimator Gst is always efficient than each family of estimators i.e. Gqst (q = 1,2,…..,5). family of estimators i.e. Gqst (q = 1, 2,…..,5). Due to unavailability of data based on ratio of two population means in stratified sampling under non-response, we are unable to compare to the proposed estimator with existing estimators empirically.

Conclusion

In present paper, we have proposed a class of ratio type estimators for ratio of two population means in stratified sampling under non-response and derive and discuss their properties. After comparing their mean square errors, we have conclude that the proposed estimator in optimum case provides better estimate than usual conventional and alternative estimators in any condition.

References

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