Section III. Statistical Models
A DifferenceCumExponential Type Efficient Estimator of Population Mean
INTRODUCTION
Precision is the reason for most of the research work in statistical theory and computation. Auxiliary information usually enhances the precision in survey sampling. It can be used at the selection stage or in estimation or on both. In the literature, auxiliary information is used by authors to construct various efficient estimators of different population parameters. The ratio estimator works better when there is a positive correlation between study and auxiliary variables, while the product estimator performs well in case of the negative correlation. Generally, a linear regression estimator is better than both ratio and product estimators. The equality in the efficiencies is obtained if the intercept of the regression line of у on x is zero. Since the regression estimator either performs better or equal to ratio and product estimators, it is convenient to use linear regression estimator than classical ratio and product estimator to gain in the efficiency. Many authors, such as [116], etc., worked on ratio and product type estimator to get the better results of such type estimator. Further, to get a better result, researchers worked on difference and exponential type estimators and showed its usefulness. Some of them are [1720], etc.
Motivated by Rao [21] and Ekpenyong and Enang [3], in this paper we propose a differencecumexponential type estimator and studied their properties. In section 13.2, we made a review of the existing estimator related to the proposed estimator. Later, we suggest a difference cum exponential type estimator and derive the expression for the bias and mean squared error and given the conditions for minimum mean square error. Last, we do practical analysis by a simulation study and reallife populations to illustrate the performance of the proposed estimator.
NOTATIONS AND LITERATURE REVIEW
Let a finite population u = {U_{[},U_{1},..,U_{N}} of size TV. A sample of size n is taken under simple random sampling without replacement technique. Let у be the study variable and x be the auxiliary variable, y, and x_{h} i = 1,2,..,7V be the observations of у and x at /"' unit in the population. We will use the following notations in the subsequent work.
_ i ^ ^{n}
Sample mean for study variable у = — > v,, sample mean for auxiliary variable
n ‘—‘M
1 " — 1 x^{1 }N
^{x=}~2^ population mean forstudy variable F =t; / , V/, population mean for
— 1 x^{1 }N У
auxiliary variable X = ~2^ ^{X}’’ P°P^{u}'^{at}i°^{n rat}*° Я = =, population variance for y,
. 1 X^{1 N} ~ 7 7 1 X^{1} '^{V} ~7
S;:= ——2^ ()’i F)',populationvarianceforA,5; = ——(•*/ ЗО’.рор
1 x^{1 N} — — S, 5, S„.
ulation covariance S,_{v} => (у,  У)(jc,  X), C_{v} = —, C, = —. p = ——,
" ^{y} Y X S_{x}S_{y}
p,x=c,v = pc,c_{v}, к_{ух} = hL,f=±x=IzL'
S_{x} R N n
Some estimators of population mean and their mean square error (MSE) from literature are presented here.
Sample mean у is the usual unbiased estimator of population mean with variance.
The classical ratio estimator to estimate the population mean F w'hen the population mean of auxiliary variable .v is known, defined as
The MSE of the classical ratio estimator is
The expression for linear regression estimator is written as
and MSE of regression estimator is given as
In 1991, Bahl and Tuteja [17] suggested the ratio type exponential estimator as and obtained MSE as
Rao [21] proposed a difference estimator as where h_{{} and h_{2} are suitable constants.
MSE of the defined estimator for optimum values of /г, and h_{2} is obtained as Further, Grover and Kaur [22] suggested an exponential type estimator as
where b_{t} and b_{2} are suitably chosen constants.
For the optimum values of fr, and b_{2}, the MSE of Ток is derived as
Ekpenyong and Enang [3] proposed two exponentialratio type estimators as and
The MSE of T_{PR} for optimum values of a_{:} and a_{2} are calculated as Also, the MSE of T_{PR1} for optimum values of 8, and S_{2} is given as
SUGGESTED ESTIMATOR
To get a better and efficient estimate of population mean Y when X is known, we suggest a differencecumexponential type estimator as
where к, t are suitably chosen constants and 8(* 0) is either constant or any parametric value such as p_{vv}, C_{x}, p, etc.
To derive the bias and MSE of T_{s}, the error terms are defined as
with £(г,) = 0, £(e_{2}) = 0 and £(er) = X.C^, £(e_{2}) = A.C?, £(££_{2}) = C_{xy}.
Express T_{s} in terms of errors. Eq. (13.16) becomes
Assuming e, < 1, i = 1,2, expand expression in terms of e’s and ignoring terms having e’s degree greater than two to get first order approximation, we have
On simplifying, we get
To get the bias of T_{s} to the first order of approximation, take expectation on both sides of Eq. (13.17), we have
By squaring Eq. (13.17) on both sides and terminating the terms having e’s degree more than two, we have
To get the MSE of T_{s} to the first order of approximation, take expectation on both sides of Eq. (13.19), we have
or
or
where,
To minimize MSE(T_{s}), we have to differentiate MSE(T_{s}) in Eq. (13.22) partially with respect to к and t, and equating to zero, we get
and
On solving simultaneous Eqs. (13.23) and (13.24) for к and t, the optimal values of к and t are obtained as
The minimum MSE(T_{s}) can be obtained by using the optimum values of к and t from Eq. (13.25) in Eq. (13.22), as
SPECIAL CASES
In this section, we do consider the different estimators with minimum of MSE(T_{s}) by assigning the different values to 8 to show the variability of the proposed estimator.
13.4.1 For 5 = 1
For 6 = 1, the proposed estimator T, becomes
with
where,
13.4.2 For 8 = Cx
For 6 = C_{x}, the estimator T_{s} becomes
The optimum MSE of T_{s{2)} can be obtained as where,
13.4.3 For 6 = (З_{у}*
For 8 = p_{v}._{t}, the estimator T_{s} becomes
The minimum MSE of T_{v(3)} is where,
13.4.4 For 8 = p
For 8 = p, the estimator T_{s} becomes
The minimum MSE of T_{sW } where,
EFFICIENCY COMPARISON
The proposed estimator T_{s} will be more efficient than other estimators if minimum MSE(T_{s}) is less than MSEs of the other estimators.
For efficiency comparison, first we express the MSEs of the estimators у, T_{R}, T_{RE}, T_{expR}, Trao, T_{gk}, T_{PRl} and T_{PR2} in terms of £‘s, we have
where,
The estimator T, will be more efficient than estimators y, T_{R}, T_{RE}, T_{afR}, T_{rao}. T_{GK}, 7>_{R1 }and T_{PR2} whenever the following conditions satisfied,
or
or
or
or
or
or
or
or
EMPIRICAL STUDY
In this section, we have considered some reallife populations to show the efficiency of the proposed estimator over existing estimators. The description of the seven different population parameters are stated in Table 13.1.
PRE of estimators are calculated by
It is observed from Table 13.2 that the proposed estimator 7, for different values of 8 i.e. 8 = 1, C,, руд and p performs efficiently than the usual unbiased estimator у, ratio estimator T_{R}, linear regression estimator T_{RE}, ratio type exponential estimator T_{expR}, Rao’s difference type estimator T_{RM}>, Grover and Kaur’s estimator T_{GK}, Ekpenyong and Enang’s estimators 7>_{R1} and T_{PR2}, respectively, for different populations. It is worth to mention that the proposed estimator T_{s{3)} is more efficient than the other considered estimator(s) for Population 1 and 3. The estimator T_{s{2)} is efficient for Populations 2, 4, 5, 6 and 7. Also, it is worth to mention that correlation coefficients in all considered cases are different with least in Population6 as p = 0.2522 to highest in Population7 as p = 0.9410 still the results are good in all cases. This shows that the proposed estimator performs better irrespective of the correlation coefficient is mild, moderate or high.
TABLE 13.1
Population Parameters
Population Source 
Study Variable у 
Auxiliary Variable x 
N 
n 
Y 
X 
c, 
C, 
P 
1. Steel and Torrie [23] 
Logarithm of leaf burn in seconds 
Chlorine percentage 
30 
6 
0.6860 
0.8077 
0.7001 
0.7493 
0.4996 
2. Maddala [24] 
Per capita consumption of veal 
Price of veal per pound 
16 
4 
7.6375 
75.4343 
0.2278 
0.0986 
0.6823 
3. Kadilar and Cingi [25] 
Apple production amount 
Number of apple trees 
204 
50 
966 
26441 
2.4739 
1.7171 
0.71 
4. Srivstava, Srivastava and Khare [26] 
Weight of children 
Midarm circumference of children 
55 
30 
17.08 
16.92 
0.12688 
0.07 
0.54 
5. Cochran [27] 
Food cost 
Income of the family 
10 
4 
101.1 
58.8 
0.1449 
0.1281 
0.6515 
6. Cochran [27] 
Food cost 
Income of the family 
33 
16 
27.49 
72.545 
0.3685 
0.1458 
0.2522 
7. Murthy [28] 
Fixed capital 
Output of factory 
80 
20 
11.264 
51.826 
0.750 
0.354 
0.9410 
TABLE 13.2
Comparison of Estimators Through PRE with Respect to у
Estimators 
PRE(., у) 

Source1 
Source2 
Source3 
Source4 
Source5 
Source6 
Source7 

100 
100 
100 
100 
100 
100 
100 

92.10 
167.59 
201.55 
141.13 
158.82 
104.49 
298.97 

133.26 
187.10 
201.65 
141.16 
173.74 
106.79 
873.21 

133.04 
133.06 
159.32 
128.50 
161.43 
106.45 
163.52 

139.79 
188.07 
210.89 
141.18 
174.06 
107.22 
875.32 

142.72 
188.16 
213.41 
141.19 
174.17 
107.24 
876.48 

149.47 
158.72 
176.87 
128.03 
150.25 
106.05 
361.28 

154.55 
189.45 
242.00 
141.21 
175.01 
107.32 
925.56 

181.95 
1.177.37 
407.76 
597.24 
227.90 
1,138.57 
3,204.70 

205.89 
2.691.97 
256.52 
1,338.97 
478.13 
2,223.96 
6,277.01 

256.57 
2,521.67 
1,014.39 
602.37 
193.86 
2,036.63 
5.744.44 

239.09 
1.529.56 
515.67 
856.18 
293.71 
2,017.26 
3,393.64 
SIMULATION
Here we execute a simulation using R software to show the performance of proposed estimators over others by calculating percent relative efficiency of all estimators concerning у as defined in Eq. (13.39). For this, we have generated four hypothetical populations of size N = 5000 and sample size n = 200. Consider a linear relationship between study variable у and auxiliary variable л with slop 0.5 and drift 1. We also uses an error factor in the model Y = 1 + 0.5X by random t, which is independent of X, follows N(0,1). We have considered four cases, in the first three, the auxiliary variable X follows normal distribution as N(2,0.5), N(3,1) and N(4,1.5). In the latter case, X is taken from Uniform distribution L/(0,1). To make results more accurate, we have replicated it 10,000 times.
From Table 13.3, it is envisaged that the proposed class of estimators 7*(i)> T_{S}(Y,, T_{S}0) and 7j(4) performs efficiently in all the cases, which also supports the result obtained in numerical study in section 6.
CONCLUSION
In this paper, we propose a differencecumexponential type estimator for the estimation of population mean Y of the study variable у when auxiliary information is available. The bias and MSE formulae of the proposed estimator are obtained and compared with that of the usual unbiased estimator y, classical ratio estimator T_{R}, regression estimator T_{R}E, exponential estimator T_{expR}, difference estimator Grover and Kaur’s estimator T(;K > Ekpenyong and Enang’s estimator TpR and TpR2 We have examined the performance of the proposed estimator by considering seven different reallife populations and also by a simulation study. It is interesting
TABLE 13.3
PRE of Estimators with Respect to у
Estimators 
PRE(., у) 

Case1 
Case2 
Case3 
Case4 

100 
100 
100 
100 

98.3515 
109.3995 
132.2530 
76.2459 

105.3458 
122.7273 
152.1649 
101.9720 

105.3127 
122.2299 
148.0307 
97.2554 

105.4733 
122.8233 
152.2483 
102.2855 

105.4813 
122.8404 
152.2759 
102.3278 

105.9562 
119.7086 
139.3933 
111.1496 

105.5044 
122.9139 
152.4357 
102.3529 

770.9312 
326.5865 
222.2267 
404.7270 

1.346.4636 
529.0413 
351.6292 
522.1004 

1.126.0685 
468.4702 
319.4904 
555.5838 

1.376.1330 
487.9484 
295.5479 
726.9252 
to remark that the proposed class of estimators performs efficiently than the other considered estimators. Thus, our conclusion is to recommend the proposed estimator for future study.
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