Introduction

Statistically, the coefficient of variation is used to make a comparison between two or more things with different units or dimensions, for example, the weight and height of individuals, are in kilograms (kg) and centimeters (cm). Auxiliary variables are often used in sample surveys to obtain improved precision in the estimate of the population characteristics of the study variable. This information may be used at both the design and estimation stages. Ratio, regression, and product method of estimation are used in this setting.

For appraising population parameters of the study variable using the information on auxiliary variables, several researchers ^{3,4,9,11,12,13,15,16 and 17} have performed a considerable amount of work. Those of them ceased to emphasize the difficulty of estimating the coefficient of variation for a long time. But some still do, like ¹ first suggested the estimator for the coefficient of variation when samples were selected using SRSWOR while other work includes ^{2,6,7,8,10,18,19} and ²⁰.

In this current study, a class of logarithmic-product-cum-ratio type estimators for the estimation of the coefficient of variation have been proposed. The proposed class of estimators is expected to produce an estimate nearer to the accurate population coefficient of variation.

Methodology

Let us consider a simple random sample n drawn from the given population of N units. Let the value of the study variable Y and the auxiliary variable X for the i^th units ( i=1, 2,3, 4, …, N) of the population be denoted by Y_i  and X_i for the i^th unit in methods: population means of the auxiliary variables, population variance of the study variable, population variance of the auxiliary variable, population covariance of the auxiliary and study variable, p denotes the correlation coefficient between X and Y.

Existing Estimators

The sample coefficient of variation is given as:

The variance is given by:

²ratio estimator of population coefficient of variation of the study variable utilizing information on the population mean.

They also proposed a ratio estimator of the population coefficient of variation of the study variable utilizing information on the population variance.

¹⁸the following estimators for the coefficient of variation based on information on a single auxiliary variable, the population mean, as

The MSE expressions of the estimators are given by:

¹⁸the following estimators for coefficient of variation based on information on single auxiliary variable, population variance, as

The mean square error (MSE) expressions of the estimators are given by:

²⁰logarithmic ratio type estimator for the estimation of population coefficient of variation when the natural logarithm of the auxiliary variable is known as:

The mean square error (MSE) of (19) is given by:

Proposed Estimator

Motivated by the work of ²⁰, we suggested a logarithmic-product-cum-ratio type estimator for estimating the coefficient of variation as;

The above estimator is defined under the assumptions that,

Such that

Derivation of the proposed estimator T_am

Expressing (21) in terms of error terms defined in section 1, we have,

Expand (22) using law of logarithm, we obtained (23)

Simplifying (23) by factorizing Ln(?) and Ln(S_x²) from the numerator and denominator of the expression in the last two brackets of (23), we have obtained (24)

By expanding Ln(1 + e₃), (1 + e₂)^1/2 , Ln(1 + e₁) and (1 + e₀)^-1 up to first order of approximation we have,

Simplify, Subtract C_y from both sides and consider terms of degree one, we have

Square both sides of equation (28) and expand right hand side relation to first order of approximation we have,

Take expectation on both sides of equation (29) to obtain the MSE of the proposed estimator as:

Numerical analysis

 A numerical analysis was carried out to clarify the accomplishment of proposed estimator. Data used are from the book ⁵ and ¹⁴.

Dataset 1: X: Area under wheat in 1963, Y: Area under wheat in 1964

Dataset 2: X: Number of fish caught in year 1993, Y: Number of fish caught in year 1995

Table 1: MSE and Percentage Relative Efficiencies of proposed estimator and the existing estimators.

Table 1 above revealed that mean square error of the suggested estimator is minimal compared to those estimators considered in this study. This implies that the suggested estimator is ameliorated, more so, this is evidence, as the proposed estimator has a higher percentage of relative efficiency.

Results and Discussion

An efficient product-cum-ratio of finite population coefficient of variation is proposed. The attribute of the proposed estimator was secured. The result shows the mean square errors and percentage relative efficiencies of the suggested estimator and some existing work using the two datasets. The results revealed that the proposed estimator is more efficient than other existing estimators considered in the study.  

Conclusion

Currently, we suggested logarithmic-product-cum-ratio type estimator for estimating the coefficient of variation of the study variable, and this estimator employed information on the natural logarithm of the sample and population mean as well as the sample and population variance of the auxiliary variable. From numerical analysis, the results show that the proposed estimator is more efficient than other existing estimators considered in the study. Hence, the ameliorated proposed estimator is recommended for practical usage.

Conflict of Interest

The authors declare no conflict of interest.

Funding Sources

The authors received no financial support for the research, authorship and publication of this article.

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