Ratio Method to the Mean Estimation Using Coefficient of Skewness of Auxiliary Variable Zaizai Yan1,2 and Bing Tian1 1
2
Science College of Inner Mongolia university of technology, Hohhot, Inner Mongolia, P.R. China, 010051
[email protected] Management College of Inner Mongolia University of Technology, Hohhot, Inner Mongolia, P.R. China, 010051
Abstract. This paper proposes some ratio-type estimators for population mean using known skewness coefficient of auxiliary variable. Theoretically, the expressions of mean square error(MSE) up to first order of approximation for all proposed ratio estimators are obtained. The efficiency between the proposed estimators and some known estimators is compared. The results are supported by an application with original data. Keywords: ratio estimator, mean square error, population mean, auxiliary variable, Monte Carlo simulation.
1
Introduction
Let U = (1, 2, · · · , N ) be a finite population, Y and X be the population means of the study variable y and auxiliary x respectively. It is assumed that these values X1 , X2 , · · · , XN of auxiliary variable x in the population are known. Consider a simple random sample of size n without replacement from population U . The classical ratio estimator for population mean Y of interest variable y is defined by y¯ yR = X (1) x ¯ with approximative MSE M SE(yR ) ∼ =
1−f 2 2 Y [Cy + Cx2 (1 − 2K)] n
(2)
S
n is the sampling fraction; Cy = Yy and Cx = SXx are the population where f = N coefficients of variation of study variable and auxiliary variable respectively. Here Cy K = ρ Cx , where ρ = Sxy /(Sx Sy ) is the correlation coefficient between study variable and auxiliary variable; Sy2 and Sx2 are the population variances of x and y respectively; Sxy is the population covariance between x and y. By using population coefficient of variation Cx of variate of auxiliary x, Sisodia and Dwivedi 1981 [2] suggested a modified ratio estimator for Y as
ySD = y
¯ + Cx X x ¯ + Cx
R. Zhu et al. (Eds.): ICICA 2010, Part II, CCIS 106, pp. 103–110, 2010. c Springer-Verlag Berlin Heidelberg 2010
(3)
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For large n, approximate MSE of this estimator is M SE(ySD ) ∼ = where α =
X X+Cx
1−f 2 2 Y [Cy + Cx2 α(α − 2K)] n
(4)
C
and K = ρ Cyx .
Singh and Kakran 1993 [10] developed ratio-type estimator for Y using the population coefficient of Kurtosis β2 of auxiliary variate as ¯ + β2 X y SK = y (5) x¯ + β2 with approximation to MSE 1−f 2 2 Y [Cy + Cx2 δ(δ − 2K)] M SE(ySK ) ∼ = n
(6)
X . where δ = X+β 2 Subsequently, various extensions of ratio or regression estimations have been introduced (see, for example, Roy 2003[8]; Kadilar and Cingi 2004 [3]; Jhajj, Sharma and Grover 2006[6]; Shabbir and Gupta 2007[7]; Abd-Elfattah et al. 2010 [1] and so on). Roy(2003) suggested a regression-type estimator in twophase sampling using two auxiliary variables. Shabbir and Gupta [7] proposed a ratio-type estimator by using the technique of Roy (2003). The proposed estimator has an improvement over mean per unit estimator as well as Jhajj, Sharma and Grover’s estimator [6] in simple random sampling and two phase sampling. Bacanli and Kadilar [9] proposed ratio estimators that can be used under unequal probability designs by adapting Horvitz-Thompson estimators to ratio estimators in literature. Kadilar and Cingi 2004 [3] suggested the following ratio estimators for the population mean Y of the variate of interest y:
y − b(x − X) ¯ X x y − b(x − X) ¯ = (X + Cx ) x + Cx y − b(x − X) ¯ = (X + β2 ) x + β2
y KC1 =
(7)
y KC2
(8)
y KC3
(9)
where b = sxy /s2x denotes the sample regression coefficient; s2x is the sample variance of the auxiliary variate; sxy is the sample covariance between x and y. In[3], the mean square error (MSE) equations of these ratio estimators were given by ) ∼ (10) M SE(y = 1−f [R2 S 2 + S 2 (1 − ρ2 )] KC1
M SE(yKC2 ) ∼ = ∼ )= M SE(y KC3
where RKC2 =
Y X+Cx
x n 1−f 2 2 n [RKC2 Sx 1−f 2 2 n [RKC3 Sx
and RKC3 =
Y . X+β2
y
+ Sy2 (1 − ρ2 )]
(11)
− ρ )]
(12)
+
Sy2 (1
2
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Kadilar and Cingi [4] proposed some estimators combining ratio estimators in (7), (8) and (9), and the optimal values of weights are given. Above improvements in estimating the population mean, the population coefficient of variation and the population coefficient of the kurtosis, of the auxiliary variate, have been used respectively, but so far as we know, the use of skewness coefficient of auxiliary variable for improvement in estimating the population mean has not been reported. Motivated by above authors, we develop a new class of ratio-type estimators using skewness coefficient of auxiliary variable and obtain the approximate MSE equations of these new estimators. Theoretically, we compare the efficiencies, based on MSE equations, between the proposed estimators and classical ratio estimators. Numerically, Based on Monte Carlo experiment, we also discuss comparisons among all the suggested estimators, by an application with original data.
2
Suggested Estimators
We propose the estimators using the skewness coefficient of auxiliary variable as follows: ¯ + β1 )/(¯ x + β1 )] (13) t1 = y[(X
N
¯ + β1 )/(β2 x ¯ + β1 )] t2 = y[(β2 X
(14)
¯ + β2 )/(β1 x ¯ + β2 )] t3 = y[(β1 X
(15)
¯ + β1 )/(Cx x t4 = y[(Cx X ¯ + β1 )]
(16)
¯ + β1 )/(¯ x + β1 )] t5 = [y − b(x − X)][(X
(17)
¯ + β2 )/(β1 x ¯ + β1 )] t6 = [y − b(x − X)][(β1 X
(18)
N
(X −X)3
i i=1 where β1 = (N −1)(N −2)S 3 is the population skewness coefficient of auxiliary variable x. Similar to the methods in [3], We can obtain MSEs of the proposed estimators as follow:
M SE(t1 ) ∼ =
(19)
M SE(t2 )
(20)
M SE(t3 ) M SE(t4 ) M SE(t5 ) M SE(t6 ) where η1 =
X , X+β1
(21) (22) (23) (24)
Xβ2 XCx Y 1 , η3 = XβXβ+β , η4 = XC ,Rt5 = X+β , Xβ2 +β1 1 2 x +β1 1 Cy ρ Cx . The proofs of MSEs of the proposed six estimators
η2 =
Y and K = Rt6 = β X+β 1 2 are omitted.
1−f 2 2 2 n Y [Cy + Cx η1 (η1 − 2K)] 1−f 2 ∼ = n Y [Cy2 + Cx2 η2 (η2 − 2K)] 2 2 2 ∼ = 1−f n Y [Cy + Cx η3 (η3 − 2K)] 2 2 2 ∼ = 1−f n Y [Cy + Cx η4 (η4 − 2K)] 1−f 2 2 ∼ Sx + Sy2 (1 − ρ2 )] = n [Rt5 2 2 2 2 2 ∼ = 1−f n [Rt6 β1 Sx + Sy (1 − ρ )]
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Efficiency Comparisons
In this section, we compare MSEs of proposed estimators, given in (19)∼ (24), with the MSE of the classical ratio estimator, given in (2). We obtain the folC lowing results by these comparisons: When β1 > 0 and 1 + ηi − 2ρ Cxy > 0, the proposed estimators ti (i=1,2,3,4) are more efficient than the classical estimator y R under large sample size. We compare MSE of traditional ratio estimator with MSE of proposed ratio estimator using Eqs. (2) and (23) as follows: M SE(t5 ) < M SE(¯ yR ) 1−f 2 2 n (Rt5 Sx
+ Sy2 − ρ2 Sy2 ) < R2t5 R2
1−f 2 2 n (R Sx C2
+ Sy2 − 2RSxy )
C
− 1 < ρ2 Cy2 − 2ρ Cyx x
¯ 2 Cy 2 X < −1 ¯ Cx X + β1
(25)
As far as the case of large sample size, when the condition (25) is satisfied, proposed ratio estimator t5 given in (17) is more efficient than the traditional ratio estimator y¯R given in (1). Similarly, we can get the condition that
¯ 1 2 Cy 2 Xβ < −1 ¯ Cx Xβ1 + β2
(26)
When the condition (26) is satisfied, t6 is more efficient than y R to large sample size.
4
Numerical Illustration
In order to gain the magnitude in efficiency of various estimators suggested here over the classical ratio estimator and simple estimator, we attempt random simulation to study the performance of the new estimators using the same data, concerning the level of apple production and number of apple trees, as in [3],[4] and [5], numerically. We have applied our proposed and other ratio estimators on the data of apple production amount (as interest of variate) and number of apple trees (as auxiliary variate) in 106 villages of Aegean Region in 1999 (Source: Institute of Statistics, Republic of Turkey). In Table 1, we observe the statistics about the population. Regarding the proposed six estimators and pre-exist estimators, we do not calculate the MSE values based on approximate mean square formulas in this paper. Since the mean square error values listed by Kadilar and Cingi (2004) were seriously overestimated under small sample size. For example, when sample size is 20, the MSE of estimator proposed (1993) by Singh and Kakran is 2568047 based on the approximate formula (6), and based on 50000 times Monte Carlo simulation, its MSE is 1544195 given in following Table 2. As for to the accuracy
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Table 1. Data statistics N = 106 n = 20, 30, 40 Sy = 11551.53 Cx = 2.10 Syx = 568176176
Y¯ = 2212.59 ¯ = 27421.70 X Cy = 5.22 β2 (x) = 34.57
R = 0.0807 ρ = 0.856 Sx = 57460.61 β1 (x) = 5.12
Table 2. MSE values of the proposed and the pre-existing ratio estimators
simple estimator classic ratio estimator Sisodia-Dwivedi (1981) Singh-Kakran (1993) Kadilar-Cingi 1 (2004) Kadilar-Cingi 2 (2004) Kadilar-Cingi 3 (2004) proposed estimator t1 proposed estimator t2 proposed estimator t3 proposed estimator t4 proposed estimator t5 proposed estimator t6
M SE values n = 20 5368785 1541957 1542092 1544195 947308 947186.4 947011 1542289 1541966 1542394 1542115 947011 946917
n = 30 3200826 1247204 1247290 1248628 741511.1 741485.3 741448 1247415 1247210 1247482 1247305 741448 741428
n = 40 2094776 953957 954013.1 954882.2 616916.3 616912.5 616907 954094.2 953961 954137.7 954022.5 616907 616904.1
of MSE based on 50000 times Monte Carlo simulation, we may examine by using the estimator y¯, here its simulation variance is 5368785 and the true variance 2 is 5413042 computed by the formula V (¯ y ) = 1−f n Sy . The ratio value of two variances is 0.991824. Then the difference is very small. A sample of size n is selected from the population, the sampling design takes the simple random sampling without replacement. The relative mean square errors of the proposed and pre-exist estimators are calculated. Furthermore, we proceed the comparison of the proposed procedure and other ratio procedures. For each of the proposed estimators, the process is repeated B = 50000 times and for different sample sizes, n = 20, 30, 40 in Monte Carlo experiments. Here we give a measure to study efficiency. This measure could be the RRMSE, which is given(for an estimator θˆ and B = 50000 samples) by RRM SE =
ˆ 1/2 [M SE(θ)] θ
(27)
where the simulated mean square error is given by B ˆ = 1 M SE(θ) (θˆ(i) − θ)2 B i=1
(28)
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simple estimator classic ratio estimator Sisodia-Dwivedi (1981) Singh-Kakran (1993) Kadilar-Cingi 1 (2004) Kadilar-Cingi 2 (2004) Kadilar-Cingi 3 (2004) proposed estimator t1 proposed estimator t2 proposed estimator t3 proposed estimator t4 proposed estimator t5 proposed estimator t6
RRM SE values n = 20 n = 30 1.047216 0.808592 0.561222 0.504739 0.561246 0.504757 0.561629 0.505027 0.439890 0.389186 0.439862 0.389179 0.439821 0.389169 0.561282 0.504782 0.561223 0.504740 0.561301 0.504795 0.561250 0.504760 0.439821 0.389169 0.439799 0.389164
n = 40 0.654135 0.441431 0.441444 0.441645 0.354986 0.354985 0.354983 0.441463 0.441432 0.441473 0.441446 0.354983 0.354983
Table 4. The percent relative efficiencies of the proposed estimators ti , i = 1, · · · , 6 with respect to y¯
simple estimator classic ratio estimator proposed estimator t1 proposed estimator t2 proposed estimator t3 proposed estimator t4 proposed estimator t5 proposed estimator t6
variance ratio values n = 20 n = 30 100.00 100.00 28.7208 38.9651 28.7270 38.9717 28.7209 38.9653 28.7289 38.9738 28.7237 38.9682 17.6392 23.1643 17.6375 23.1636
n = 40 100.00 45.5398 45.5464 45.5400 45.5484 45.5429 29.4498 29.4496
where θˆ(i) is the estimate of θ on the ith sample. To compute MSEs and RRMSEs of these estimators, we take all 50000 samples, say as ξ1 , ξ2 , · · · , ξ50000 . Under each sample ξα , the values of the proposed six estimators and the pre-exist estimators are calculated respectively. To compare the results in this paper with the results given by Kadilar and Cingi (2004), the mean square error values based on B times simulations of the proposed and the pre-existing ratio estimators are listed in Table 2. Furthermore, the relative squared errors (RRMSE) of the proposed six estimators and pre-exist estimators are calculated and listed in Table 3. y ), of We also have computed the percent relative efficiencies, 100 × V (ti )/V (¯ ti , i = 1, 2 · · · , 6 with respect to usual simple estimator y¯ and displayed in Table 4. From Table 2, 3 and 4, it can be concluded that all proposed estimators ti , i = 1, 2 · · · , 4, 5 are more efficient than both the usual unbiased estimator y¯ and ratio estimator y¯R . The proposed estimators possess approximately same
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precision with pre-exist estimators suggested by Sisodia and Dwivedi [2], Singh and Kakran [10] and Kadilar and Cingi[3]. Hence, just as coefficient of the kurtosis and coefficient of variation of the auxiliary variate may be applied to improve estimation of population mean, so may be the skewness coefficient of auxiliary variable.
Acknowledgement This work was supported by National Natural Science Foundation of China (10761004), Talent Development Foundation of Inner Mongolia(2007) and Natural Science Foundation of Inner Mongolia (2009MS0107). The authors thank two referees for several suggestions, which have helped to improve the article.
References 1. Abd-Elfattah, A.M., EI-Sherpieny, E.A., Mohamed, S.M., Abdou, O.F.: Improvement in estimating the population mean in simple random sampling using information on auxiliary attribute. Applied Mathematics and Computation 215, 4198–4202 (2010) 2. Sisodia, B.V.S., Dwivedi, V.K.: A modified ratio estimator using coefficient of variation of auxiliary variable. Jour. Ind. Soc. Agr. Stat. 33, 13–18 (1981) 3. Kadilar, C., Cingi, H.: Ratio estimators in simple random sampling. Applied Mathematics and Computation 151, 893–902 (2004) 4. Kadilar, C., Cingi, H.: Improvement in estimating the population mean in simple random sampling. Applied Mathematics Letters 19, 75–79 (2006) 5. Kadilar, C., Cingi, H.: Ratio estimators in stratified random sampling. Biometrical Journal 45(2), 218–225 (2003) 6. Jhajj, H.S., Sharma, M.K., Grover, L.K.: A family of estimators of population mean using information on auxiliary attribute. Pak. J. Statist. 22(1), 43–50 (2006) 7. Shabbir, J., Gupta, S.: On estimating the finite population mean with known population proportion of an auxiliary variable. Pak. J. Statist. 23(1), 1–9 (2007) 8. Roy, D.C.: A regression-type estimator in two-phase sampling using two auxiliary variables. Pak. J. Statist. 19(3), 281–290 (2003) 9. Bacanli, S., Kadilar, C.: Ratio estimators with unequal probability designs. Pak. J. Statist. 24(3), 167–172 (2008) 10. Singh, H.P., Kakran, M.S.: A modified ratio estimator using known coefficient of kurtosis of an auxiliary character. Revised version submitted to Journal of Indian Society of Agricultural Statistics, New Delhi, India (1993) 11. Rao, T.J.: On certain methods of improving ratio and regression estimators. Communications in Statistics: Theory and Methods 20(10), 3325–3340 (1991) 12. Cochran, W.G.: Sampling Techniques. John Wiley and Sons, New York (1977)
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Appendix: Computational Procedure Based on R Software rd
