Section III. Statistical Models

A Difference-Cum-Exponential 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 [1-16], 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 [17-20], etc.

Motivated by Rao [21] and Ekpenyong and Enang [3], in this paper we propose a difference-cum-exponential 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 real-life populations to illustrate the performance of the proposed estimator.

NOTATIONS AND LITERATURE REVIEW

Let a finite population u = {U[,U1,..,UN} 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 xh 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°Pu'atn 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), Cv = —, C, = —. p = ——,

" y Y X SxSy

p,x=c,v = pc,cv, кух = hL,f=±x=IzL'

Sx 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 h2 are suitable constants.

MSE of the defined estimator for optimum values of /г, and h2 is obtained as Further, Grover and Kaur [22] suggested an exponential type estimator as

where bt and b2 are suitably chosen constants.

For the optimum values of fr, and b2, the MSE of Ток is derived as

Ekpenyong and Enang [3] proposed two exponential-ratio type estimators as and

The MSE of TPR for optimum values of a: and a2 are calculated as Also, the MSE of TPR1 for optimum values of 8, and S2 is given as

SUGGESTED ESTIMATOR

To get a better and efficient estimate of population mean Y when X is known, we suggest a difference-cum-exponential type estimator as

where к, t are suitably chosen constants and 8(* 0) is either constant or any parametric value such as pvv, Cx, p, etc.

To derive the bias and MSE of Ts, the error terms are defined as

with £(г,) = 0, £(e2) = 0 and £(er) = X.C^, £(e2) = A.C?, £(£|£2) = Cxy.

Express Ts 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 Ts 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 Ts to the first order of approximation, take expectation on both sides of Eq. (13.19), we have

or

or

where,

To minimize MSE(Ts), we have to differentiate MSE(Ts) 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(Ts) 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(Ts) 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 = Cx, the estimator Ts becomes

The optimum MSE of Ts{2) can be obtained as where,

13.4.3 For 6 = (Зу*

For 8 = pv.t, the estimator Ts becomes

The minimum MSE of Tv(3) is where,

13.4.4 For 8 = p

For 8 = p, the estimator Ts becomes

The minimum MSE of TsW where,

EFFICIENCY COMPARISON

The proposed estimator Ts will be more efficient than other estimators if minimum MSE(Ts) is less than MSEs of the other estimators.

For efficiency comparison, first we express the MSEs of the estimators у, TR, TRE, TexpR, Trao, Tgk, TPRl and TPR2 in terms of £‘s, we have

where,

The estimator T, will be more efficient than estimators y, TR, TRE, TafR, Trao. TGK, 7>R1 and TPR2 whenever the following conditions satisfied,

or

or

or

or

or

or

or

or

EMPIRICAL STUDY

In this section, we have considered some real-life 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 TR, linear regression estimator TRE, ratio type exponential estimator TexpR, Rao’s difference type estimator TRM>, Grover and Kaur’s estimator TGK, Ekpenyong and Enang’s estimators 7>R1 and TPR2, respectively, for different populations. It is worth to mention that the proposed estimator Ts{3) is more efficient than the other considered estimator(s) for Population 1 and 3. The estimator Ts{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 Population-6 as p = 0.2522 to highest in Population-7 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

Mid-arm 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)

Source-1

Source-2

Source-3

Source-4

Source-5

Source-6

Source-7

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)> TS(Y,, TS0) 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 difference-cum-exponential 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 TR, regression estimator TRE, exponential estimator TexpR, 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 real-life populations and also by a simulation study. It is interesting

TABLE 13.3

PRE of Estimators with Respect to у

Estimators

PRE(., у)

Case-1

Case-2

Case-3

Case-4

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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