1. Introduction

In many practical situations, the more appropriate measure of location is mode rather than mean and median. For example, it is useful in finding the ideal size as in studies relating to marketing, trade and industry and also to find the ideal size such as in the manufacturing of shoes or ready-made garments, etc. In addition, the mode is also useful when dealing with skewed distributions such as income, drugs, AIDS etc. The impact of the proposed research is expected to be useful for scientists in Sociology, Psychology, Demography, Business and Economics, where the mode value is routinely being used in practice.

Sharma et al (2016) established the new parametric relationship for population mode as

where is unknown constant to be determined for the given population. They proposed three estimators of under the different situations as

and

Where and are unknown constants, whose values are determined by minimizing the MSE’s of respective estimators and Here the estimator uses no information on auxiliary variable which is highly correlated with whereas and uses the known information of which are of ratio and product type estimators respectively.

In the present paper, we propose a class of estimators of population mode using the new parametric relationship for population mode when the population mean of the auxiliary variable is known. Asymptotic expression for the bias and MSE of any estimators of the proposed class and also their minimum values are obtained upto to terms of order The results are also illustrated numerically.

2. Notations and Expectations

Let a simple random sample of size is drawn from a population of size by using simple random sampling without replacement (SRSWOR). Without loss of generality, we ignore the finite population correction (fpc) throughout the paper.

Let and be the values of study and auxiliary variables for unit in the population respectively and the corresponding small letters and denote the sample values.

Taking,

Obviously

Defining,

For the given SRSWOR, we have the following expectations,

and up to terms of order

3. Proposed Class of Estimators

In this section, we generalize the mean per unit, ratio-type and product-type estimators of population mode given by Sharma et al. (2016).

Whatever be the sample chosen, let assume values in a bounded closed convex subset of the one-dimensional real space containing the value ‘1’. Let be a function of such that

and it satisfies the following conditions:

(i) The function is continuous and bounded in

(ii) The first and second order partial derivatives of exist and are continuous and bounded in

We propose the class of estimator of the population mode as

(3.1)

To find the biases and of estimators of class we expand the function about the value ‘1’ in second-order Taylor’s series, we get

where and

After writing the above function in terms of and then taking the expectations given in section 2, upto terms of order we get,

(3.2)

(3.3)

where

Here and are unknown constants whose values are determined by minimizing

To obtain the minimum value of we differentiate (3.3) w.r.t. and then equating to zero, we get,

and

Solving above quadratic equations for and we get two pairs of solutions and where,

Clearly, the only feasible pair of values is

Substituting the values of pair in (3.3), we get,

We can construct the large number of estimators belonging to the proposed class Here it should be noted that the efficient use of estimators of the proposed class requires the optimum values of constants and which are further functions of unknown population parameters. However, if it is possible to guess accurately the values of such parameters either through past experience or through a pilot sample survey and the value of optimum constants so obtained by using these guessed values of parameters are close enough to the optimum values of constants, then the resulting estimators will be better than the convention estimators. Srivastava and Jhajj (1983) have shown that upto the first order of approximation, the estimators of the class with estimated values of optimum parameters obtained by their respective consistent estimators, attain the same optimum MSE of class based on optimum values upto the first order of approximation. Therefore, the presence of unknown population parameters in optimum values of constants will not create any problem for practical use of the proposed class

3.1. Special Case of Bivariate Normal Population

Let then we have if is odd. Also, and

Using these values, we get,

4. Numerical Illustrations

To illustrate the result numerically, we have made computations for 12 populations taken from literature by using Microsoft Excel 2010.

The source of the populations, the nature of the variables, the values of and are listed in Table 1.

Table 1. Description of populations

The efficiencies of proposed estimators are given in Table 2.

Table 2. MSE’s of up to terms of

From Table 2, it has been clarified that upto terms of order is less than and which implies that the proposed estimator is more efficient than the existing ones. Further, is also smaller than which shows that at optimum value of the proposed class estimators is more stable than the and

5. Conclusions

A general class of estimators of population mode has been proposed and the lower bound for the MSE has been obtained for the class of estimators. Further, the empirical study shows that is less than which implies that the optimum estimator is more stable than and