Towards the Efficiency of the Ratio Estimator for Population Median in Survey Sampling

Matthew Joshua Iseh

I. INTRODUCTION

Most times in survey sampling, some researchers do not take into consideration the tool that will be most appropriate in the measure of location. As a result, users of Statistics often go for the mean or total which has been widely discussed in the finite population sampling literature unlike the median which is more complicated to deal with since it has to do with ordered data. However, it has been established that the median unlike the mean performs better when the considered variables are from a highly skewed distribution. In surveys involving the estimation of income, expenditure, scores, etc., it is very reasonable to assume that the population median unlike the population mean is known, hence the possibility of incorporating auxiliary information in the formulation of such estimators.

So far, the shortcoming of the existing estimators is that, while some of these estimators are less biased with large mean square error (MSE), others are highly biased with less MSE. Based on these developments as a benchmark, this study proposes a separate ratio exponential-type estimator in simple random sampling and two phase sampling schemes with two auxiliary variables that will be more precise with greater gains in efficiency to estimate the median for finite population.

a) Notations

Consider a finite population $U = \{u_{1}, u_{2}, \ldots, u_{N}\}$ with size N. Let Y, X, and Z be the study and auxiliary variables respectively. Let $y_{i}$ represents the samples of the interest variable and $x_{i}$ and $z_{i}$ represents the samples of the auxiliary variables known for every unit in the population for the $i^{th}$ element drawn under SRSWOR. Let $f_{Y}(M_{Y}), f_{X}(M_{X})$ , and $f_{Z}(M_{Z})$ represent the density functions of the random variables with $\widehat{M}_{y}, \widehat{M}_{x}$ , and $\widehat{M}_{z}$ being the samples from the population median $M_{Y}$ , $M_{X}$ and $M_{Z}$ respectively, with correlation coefficient $\rho_{M_{Y}M_{X}} = 4(P_{11} - 0.25)$ , where $P_{11} = P(Y \leq M_{Y} \cap X \leq M_{X})$ , $\rho_{M_{Y}M_{Z}} = 4(P_{11} - 0.25)$ , where $P_{11} = P(Y \leq M_{Y} \cap Z \leq M_{Z})$ , and $\rho_{M_{X}M_{Z}} = 4(P_{11} - 0.25)$ , where $P_{11} = P(X \leq M_{X} \cap Z \leq M_{Z})$ , (considering the continuous distributions of all variables y, x, and x with their marginal densities respectively as $N \to \infty$ ).

For large sample approximations, the following are obtainable:

where, it is also assumed that the distribution function $f_{Y}(M_{Y})$ , $f_{X}(M_{X})$ , and $f_{X}(M_{Z})$ are nonnegative.

II. EXISTING ESTIMATORS UNDER SIMPLE RANDOM SAMPLING

This section considers some existing estimators in simple random sampling in estimating population median and the expression of bias and MSE up to the first order approximation as follows;

A: The median estimator (per unit) due to Gross (1980) is given by

B: The median estimator (classical ratio) by Kuk and Mak (1989) is given by

Notes

C: The median estimator (exponential ratio) following Bahl and Tuteja (1991) is given by

D: The median estimator (chain ratio-type) by Kadilar and Cingi (2003) is given by

E: The median estimator (product-type) following Robson (1957) and Murthy (1964) is given by

F: The median estimator (exponential product-type) following Bahl and Tuteja (1991) is given by

Notes

G: The median estimator (alternative exponential) due to Enang et.al. (2016) is given by

H: Shabbir and Gupta (2017) suggested generalized difference-type estimator for population median as

where $m_{1}$ and $m_{2}$ are unknown constants whose values are to be determined.

Let $a$ and $b$ are defined to be unknown population parameters and $\alpha_{1}$ , $\alpha_{2}$ and $\gamma$ are scalar quantities which can take different values like $\alpha_{1} = b = 0$ and $\alpha_{2} = a = \gamma = 1$ . At the optimum values of $m_{1(opt)} = \frac{1 - \frac{1}{2}\lambda M_X^2}{1 + \lambda M_Y^2(1 - \rho_{M_{YM_X}}^2)}$ and

The expressions for the bias and the mean square error up to the first order of approximation are as follows:

I: Baig, Masood and Tarray (2019) suggested an improved class of difference-type estimators for population median using two auxiliary variables

Where $m_{1}$ and $m_{2}$ are unknown constant.

Notes

where $m_{1(opt)} = \frac{M_Y C_{M_Y} (\rho_{M_X M_Z} \rho_{M_Y M_Z} - \rho_{M_Y M_X})}{M_X C_{M_X} (1 - \rho_{M_{X M_Z}}^2)}$ and

III. THE PROPOSED ESTIMATOR UNDER SIMPLE RANDOM SAMPLING WITH ONE

Squaring both sides of (11), retaining terms to the second-degree and taking expectations, the MSE of $\widehat{M}_{st}(\alpha)$ to the first order of approximation is obtained as;

minimizing (13) with respect to $\alpha$ gives

a) A Case of Two Variables under Simple Random Sampling

Notes

Minimizing (18) with respect to $\alpha$ gives $\alpha = \frac{2C_{M_X}^2 + 2C_{M_X}C_{M_Y}\rho_{M_XM_Y} - C_{M_X}C_{M_Z}\rho_{M_XM_Z}}{4C_{M_X}^2}$

Substituting into (18) gives

And the minimum bias given as

b) Application

The bias and MSE values of the existing and proposed estimators are computed using two different populations under simple random sampling. The percent relative efficiencies of the estimators are obtained as follows:

where $MSE(\widehat{M}_{G})$ is the MSE of classical median estimator while $MSE(.)$ denotes the MSE of estimators mentioned here. The population statistics and the results of analyses are shown as follows:

Population 1: Let Population 1: Let y; x and z respectively be the number of fish caught by the marine recreational fisherman in years 1995, 1994 and 1993 in USA given by Singh (2003a)

Notes

Population 2: Let y as the U.S. exports to Singapore in billions of Singapore dollars, x as the money supply figures in billions of Singapore dollars and z is the local supply in U.S. dollars given by Aczel and Sounderpandian (2004).

Table 1: Results for Simple Random Sampling

Estimator	Population I			Population II
Estimator	Absolute bias	MSE	PRE	Absolute bias	MSE	PRE
$\widehat{M}_{G}$	0	565443.6	100	0	1.23	100
$\widehat{M}_{R}$	246.3	988372.8	57.2	0.08	0.82	150
$\widehat{M}_{ER}$	87.27	627420.2	90.1	0.01	0.72	170.8
$\widehat{M}_{CR}$	373.78	3307296	17.1	0.59	9.31	13.2
$\widehat{M}_{P}$	42.32	1338418	42.2	0.17	4.06	30.3
$\widehat{M}_{EP}$	14.98	802442.8	70.5	0.05	2.34	52.6
$\widehat{M}_{AE}$	408.87	552636.1	102.3	0.03	0.69	178.3
$\widehat{M}_{PP}^{G}$	3373.88	491568.8	115.0	1.54	0.52	236.5
$\widehat{M}_{P}^{I}$	46089.73	502378.1	112.6	0.48	0.31	396.8
$\widehat{M}_{srs}(\alpha)$	144.55	552636.1	102.3	0.15	0.69	178.3
$\widehat{M}_{srs}^{*}(\alpha)$	207.98	520612.8	108.6	0.13	0.38	323.7

c) Median Estimators under Two Phase Sampling

Consider a finite population with Nunits $U = \{u_{1}, u_{2},..., u_{N}\}$ . The $i^{th}$ unit of the population values for the study variable y, auxiliary variables x and z are $y_{i}$ ; $x_{i}$ and $z_{i}$ respectively. Under two-phase sampling design, a sample of size $n'$ is drawn using simple random sampling without replacement at first phase and the values on x and z are obtained on the units of the sample. In second phase, a simple random sampling without replacement criteria is used for drawing sample of size n from the first phasesample and the values on the variables y; x and z are taken on selected units. To obtain the properties of the proposed median estimator under two-phase samplingscheme the following existing estimators are summarized as;

i) Singh, Joarder, and Tracy suggested a ratio estimator for median in two phase Sampling

Singh, Singh, and Upadhyay (2007) studied a ratio-type estimator of median using two auxiliary variables

Notes Where

ii) Jhajj, Kaur, and Jhajj (2016) defined ratio-exponential-type estimator as

Notes

iii) Baig, Masood and Tarray (2019) suggested an improved class of difference-type estimators for population median under two phase sampling with two auxiliary variables

where $m_{1}$ and $m_{2}$ are unknown constant.

d) Proposed Median Estimator Under Two Phase Sampling

Notes

年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 的 年 年 年 年 年

年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 年 的 年 年 年 年 年

For optimum value of the MSE of $\widehat{M}_{srs}^{D}(\alpha)$ , the value of $\alpha$ is given as $\alpha = \frac{k_{1} + 1}{2}$ , then (27) becomes

And the optimum bias becomes

e) Numerical study under two-phase sampling

Here three different populations will be considered to validate the theoretical claims of both the existing and the proposed estimators. The population statistics and the results of analyses are shown as follows:

Population 3: Let Population 1: Let y; x and z respectively be the number of fish caught by the marine recreational fisherman in years 1995, 1994 and 1993 in USA given by Singh (2003a).

$\rho_{M_X M_Z} = 0.1431; \rho_{M_Y M_Z} = 0.3166; f_Y(M_Y) = 0.00014; f_X(M_X) = 0.00014; \text{and } f_Z(M_Z) = 0.00013.$

Population 4: Let y as the U.S. exports to Singapore in billions of Singapore dollars, x as the money supply figures in billions of Singapore dollars and z is the local supply in U.S. dollars given by Aczel and Sounder pandian (2004).

Population 5: Let y be the district-wise tomato production (tones) in 2003, x as a district-wise tomato production (tones) in 2002 and z as a district-wise tomato production (tones) in 2001 given by MFA (2004).

From the numerical study of three real life data sets, the following remarks are deduced;

Table 2: Results for Two Phase Sampling N/B: AB=Absolute bias

Estimator	Population III			Population IV			Population V
Estimator	AB	MSE	PRE	AB	MSE	PRE	AB	MSE	PRE
$\widehat{M}_{G}$	0	346606	100	0	1.23	100	0	64795.1	100
$\widehat{M}_{SA}$	0	729125.3	47.54	0.00	1.89	65.08	0	146126.9	44.3
$\widehat{M}_{S}$	58.72	391934.3	88.43	0.35	-85.12	-1.45	8.49	69573.62	93.1
$\widehat{M}_{YH}$	259.20	363711.8	95.30	0.09	-85.60	-1.44	286.51	70062.2	92.5
$\widehat{M}_{P}^{I}$	32818.5	294885.2	117.54	1.21	0.01	12300	894.49	65213.1	99.4
$\widehat{M}_{srs}^{D}(\alpha)$	8.20	310481.2	111.64	0.07	0.29	424.14	8.57	84943.4	76.3

IV. DISCUSSION

From the result in Table 1, it is observed that the proposed estimator $\widehat{M}_{srs}^{*}(\alpha)$ has outperformed other existing estimators considered in this study and as a result considered the more preferred estimator with respect to this set of data. However, the existing estimator $\widehat{M}_P^I$ had greater efficiency but heavily biased in both population I and II respectively. Again, as shown in Table 2, the proposed estimator $\widehat{M}_{srs}^{D}(\alpha)$ has an overwhelming performance in terms of higher gains in efficiency and minimum bias as compared to the existing estimators considered in this study. This superiority in the gain in efficiency and minimum bias of the proposed estimator is because of the endearing properties of the separate ratio product-type estimator. However, $\widehat{M}_P^I$ , notice ably from population III, IV, and V had slight gain in efficiency over the proposed estimator but highly biased, and not recommended for estimation of population median.

It is important to note that, although the existing estimator $\widehat{M}_{SA}$ is asymptotically unbiased from the sets of data considered in this study, it was less efficient, and should not form a bet for median estimation. It becomes imperative to seek for an estimator that is less bias with minimum MSEthat will enhance the improvement of the estimation of the median of a finite population of which the proposed estimator has bridged the gap.

V. CONCLUSION

This study was towards formulating an improved exponential-type estimator for population median using both simple random and two-phase sampling with two auxiliary variables. The bias and mean square error were obtained. From the theoretical derivations and numerical illustrations, it is evident to say that among the existing and proposed estimators, the proposed estimators under simple random sampling and two-phase sampling exhibit superior performance both in negligible bias and in gain in efficiency. It suffices to conclude that the proposed estimators perform better than existing estimators considered in this work in estimating population median when the population median of the auxiliary variables are known and positively correlated with the study variable. However, though the suggested estimator performed poorly with less efficiency in population V, probably because of the data type, it was still shown to have less bias compared to other estimators of its class. The idea of using separate ratio-product exponential-type estimator in this case has really paid off in improving on the efficiency of the median estimator under stratified random sampling. In addition, the proposed estimator will be suitable and highly recommended when the variable considered is from a distribution that is highly skewed.