Improved ratio-cum-product estimators of finite population mean using known parameters of two auxiliary variates in double sampling

Introduction

This paper considers the problem of estimation of finite population mean in double sampling. In this paper, two ratio-cum-product estimators of finite population mean, using known coefficient of variation and coefficient of kurtosis of two auxiliary variates, have been suggested. Suggested estimators have been compared with usual unbiased estimator, classical ratio and product estimators in double sampling and double sampling versions of Singh (1967) [3] and Upadhyaya and Singh (1999) estimators. To judge the performance of the suggested estimators over other considered estimators an empirical study also has been carried out. Let us consider a finite population: Population covariance between the auxiliary variate 1

Cochran (1940) [2] envisaged classical ratio estimator for estimating the population mean Y when study variate y and auxiliary variate 1

x are positively correlated as

In case of negative correlation between the study variate y and the auxiliary variate 2

x , the classical product estimator was given by Robson (1957) [6] as

Assuming that the population mean 1 X and coefficient of variation

When the correlation coefficient between the study variate y and auxiliary variate 2

x is negative product type estimator using coefficient of variation

Singh et al. (2004) [10] defined ratio and product type estimators using coefficient of kurtosis

Upadhyaya and Singh (1999) utilized both coefficient of kurtosis as well as coefficient of variations of auxiliary variates and suggested two ratio and two product type estimators of population mean Y as

Singh (1967) [3] utilized information on known population means 1 X and 2 X of auxiliary variates 1 x and 2

x respectively and envisaged a ratio-cum-product estimator of population mean Y as (1.11) The problem of estimating the population mean Y of y when the population means 1 X and 2 X of 1 x and 2 x respectively are known, has been discussed by many researchers including Singh and Tailor (2005) [9] , Tailor and Tailor (2008) [12] , Tailor et al.

(2011a) [13] , Tailor (2012) [11] and many others. When information is not available on 1

X and 2 X in advance, double sampling procedure is used. The standard double sampling procedure is described as (i) a first phase sample 1 S of fixed size n' is drawn form U to observe only 1

x and 2 x to estimate 1 X and 2 X respectively then (ii) a second phase sample 2 S of fixed size n is drawn either from 1 S from first phase sample or directly from the population.

These two cases may be designated as Case I: As a sub sample from the first phase sample and Case II: Drawn independently to the first phase sample.

In double sampling, the usual ratio and product estimators of population mean Y are respectively defined as

x and 2 x are sample means based on second phase sample of size n whereas x , which are unbiased estimates of population means 1 X and 2 X respectively.

In double sampling, Sisodia and Dwivedi (1981) ratio type and Pandey and Dubey (1988) [4] product type estimators of population mean Y are defined as

In double sampling, Singh et al. (2004) [10] ratio and product type estimators of population mean Y are defined as

(1.17) Double sampling versions of Upadhyaya and Singh (1999) ratio type estimators are

In double sampling, Singh (1967) [3] ratio-cum-product estimator RP Y ˆ of population mean Y is expressed as

The biases and mean squared errors x as

Suggested Ratio-Cum-Product Estimators

To obtain the bias and mean squared error of

 also may not be available due to lack of information on population mean of auxiliary variates. In this type of situation, estimators of

Similarly the biases and mean squared errors of the suggested estimator

Efficiency Comparisons

Efficiency Comparisons of

It is well known that under simple random sampling without replacement (SRSWOR) variance of unbiased estimator y is defined as [10] ratio type estimator [10] product type estimator [3] ratio-cum-product type estimator

Efficiency Comparisons of

From (1.56), (1.57), (1.58), (1.59), (1.60), (1.61), (1.62) and (2.6) it is observed that the suggested estimator

), would be more efficient than

(ii) Double sampling ratio estimator

(iii) Double sampling product estimator

(iv) Sisodia and Dwivedi (1981) double sampling ratio type estimator

(v) Pandey and Dubey (1988) [4] product estimator in double sampling

(vi) Singh et al. (2004) [10] ratio type estimator in double sampling

(vii) Singh et al (2004) [10] product type estimator in double sampling

(viii) Upadhyaya and Singh (1999) double sampling ratio type estimator

(ix) Upadhyaya and Singh (1999) double sampling ratio type estimator

(x) Singh (1967) [3] double sampling ratio-cum-product estimator

(ii) double sampling ratio estimator

(iii) double sampling product estimator

(iv) ) Sisodia and Dwivedi (1981) ratio type estimator in double sampling

(v) Pandey and Dubey (1988) [4] product type estimator in double sampling

(vi) Singh et al (2004) [10] ratio type estimator in double sampling

(vii) Singh et al (2004) [10] product type estimator in double sampling

(viii) Upadhyaya and Singh (1999) double sampling ratio type estimator

(xi) Upadhyaya and Singh (1999) double sampling ratio type estimator

.4.9) (x) Singh (1967) [3] double sampling ratio-cum-product estimator [4] , Singh et al. (2004) [10] , Upadhyaya and Singh (1999) and Singh (1967) [3] .

Empirical Study

To analyze the performance of the suggested estimator ) ( 1 ˆd T Y of population mean Y in double sampling in comparison to other considered estimators, a natural population data set is being considered. We have computed Percent relative efficiencies (PREs) of

ˆd T Y

has lower percent relative efficiency as compared to other estimators. It may be due to not satisfying conditions obtained in section 2.6. Hence there is need to explore another data set where conditions obtained in section 2.6 are satisfied. Finally, it can be concluded that suggested estimators can be used for the estimation of population mean when conditions obtained in sections 3.1, 3.2, 3.4 and 3.5 are satisfied.

References

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