COMPARISON OF RATIO ESTIMATORS WITH TWO AUXILIARY VARIABLES K. RANGA RAO. College of Dairy Technology, SPVNR TSU VAFS, Kamareddy, Telangana, India

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1 COMPARISON OF RATIO ESTIMATORS WITH TWO AUXILIARY VARIABLES K. RANGA RAO College of Dairy Technology, SPVNR TSU VAFS, Kamareddy, Telangana, India Abstract: Many estimators of the population parameters are constructed using known auxiliary variables. The classical well known ratio estimator is one of them. Ratio estimators are frequently employed in sample surveys when estimating the population mean Y of a variate Y with the help of the known population means of a correlated auxiliary variables. Various improvements of this ratio estimator have been considered by many authors. Some estimators use two auxiliary variables. Some other use groups of estimators are composite. In case two auxiliary variables are known, the ratio-cum-product estimator may be used. It is well known that when the auxiliary information is to be used at the estimation stage, the ratio, product and regression estimators are widely utilied in many situations. There are no analytical procedures to compare the ratio estimators, since their mean square errors are approximated up to some extent. Theoretically, it is hard to compare the bias, mean square error, skewness and kurtosis of the estimators over the other estimators. However, we can compute these of the estimators using Monte Carlo simulation. This method is suitable when the theoretical comparisons fail. In this paper, some ratio estimators with two auxiliary variables available in literature are reviewed and their efficiencies are compared by simulation for different distributions with known correlation coefficients. The results show that the simulation method is more appropriate when there is no closed expression for the bias and mean squared error of the estimators. Keywords: Ratio estimator, Auxiliary variable and Simulation. InterStat 015 Page 1

2 1. Introduction In sample surveys it is usual to make use of information on auxiliary variables to obtain improved designs and more efficient estimators. It is well known that when the auxiliary information is to be used at the estimation stage, the ratio, product and regression estimators are widely utilied in many situations. Theoretically, it has been established that, in general, the regression estimator is more efficient than the ratio and product estimators except when the regression line of the character under study on the auxiliary character passes through the neighbourhood of the origin. In this case the efficiency of the estimators is almost equal. However, due to the stronger intuitive appeal, statisticians are more inclined towards the use of ratio and product estimators. Perhaps that is why an extensive work has been done in the direction of improving the performance of these estimators. In sampling literature, when a single auxiliary variable is available, many estimators have been proposed which, under some realistic conditions, are more efficient than the sample mean, the ratio and product estimators and as efficient as the regression estimator in the optimum case. Suppose the population mean of x, vi. X is known. Suppose further that the random sample of, say, sie n taken from the population is enumerated for both x and y, if x and y are the sample means of x and y, then as an estimator of the population mean, Y, of y, we may consider y R, rather than y itself, where y R y = X = RX ˆ, (1.1) x where y R ˆ = r 0 =, (1.) x InterStat 015 Page

3 being the estimator of the population ratio Y X. Although this estimator y R is biased, one can see that it is almost unbiased for large n and that the variance may, indeed, be smaller than the variance of y in some situations. This paper is concerned with the problem of estimating the population ratio using two auxiliary variables. When two or more auxiliary variables are available many estimators may be defined linking together different estimators, such as ratio, product or regression, each one utiliing a single variable. Olkin (1958) suggested the use of information on more than one supplementary character, positively correlated with the variable under study, using a linear combination of ratio estimators based on each auxiliary variable separately. Singh (1967) gave a multivariate expression of product estimator, while Raj (1965) suggested a method of using multi-auxiliary variables considering a linear combination of single difference estimators. Many other contributions are present in sampling literature when two auxiliary variables are available some estimators obtained from the ratio cum product estimators developed by Singh (1965, 1967). estimators Estimation of R can also be made in a different way by employing alternative Λ Y and Λ X respectively in terms of an auxiliary variable, having high degree of associations with y and x. For example, in a labour force survey, the survey statistician may be interested in to estimate the proportion of female workers employed in industry, number of female workers and female population for a sampling unit as y, x and respectively. In this context, possibility of creating estimators for R has been indicated by Tripathi (1980) who defined an estimator by considering difference estimators for Y and X in terms of the auxiliary variable. In their text, Sarndal, Swenson and Wretman (1991) also pointed out that if the presence of such an auxiliary variable improves the accuracy of one or both means, efficiency can also gained in the estimation of R. The authors encouraged the use of InterStat 015 Page 3

4 regression estimators for this purpose. However, this approach encourages us to estimate R by a generalied estimator of the form Rˆ = Y ˆ X ˆ which is able to produce many estimators.. Some Ratio Estimators with Two Auxiliary Variables Let P = {1,,...N} be a finite population of N elements, Y be the study variable and X and Z two auxiliary variables. Assume that the mean,y, of the variable of interest is unknown. Let a sample of sie n be drawn from the population according to the simple random sampling without replacement (srswor) and let y, x and be the sample means of the variables Y, X and Z. Supposing that the population mean Z of is known, let us consider Λ Y = y Λ x [ 1+ θ ( c c )]Z and X = 1+ ( c c ) [ θ x ]Z as Tin s (1965) type AUR estimators for Y and X respectively, where 1 1 θ =, c = s /, cx = s x / x and n N c = s /.Then Λ R reduces to an estimator defined by r 1 = r 0 [ 1+ θ ( c c )] [ 1+ θ ( c c )] x (.1) another estimator [ 1+ θ c ] r = r0, for R. (.) [ 1+ θ c ] x InterStat 015 Page 4

5 In many surveys, information on auxiliary variates which are highly correlated with the variable under study is readily available and can be used for improving sampling design. Ratio estimators and linear regression estimators make use of auxiliary information for increasing precision. It was seen that the ratio estimator provides a precise estimate of the population mean if regression is linear and the line passes through the origin. When regression is linear and line does not go through the origin, it is better to use estimators based on linear regression. In other words, if the study variate is approximately a constant and a multiple of the auxiliary variate, it is more precise to estimate the population mean or total by fitting a linear regression. Such an estimator is called regression estimator. We will discuss the applicability of regressions estimators in ratio estimation. Regression estimators are used to estimate the population means individually and by taking the ratio of these estimators, one can get the following ratio estimator. Employing regression estimators ( Z ) and x b ( Z ) y b for Y and X respectively explained in terms of, the estimator of R is then x ( Z ) ( Z ) y b r3 =. (.3) x b x It may be mentioned here that this estimator was considered by Tripathi (1980). Chand (1975) and Sukhatme and Chand (1977) proposed a technique of changing the available information on auxiliary characteristics with the main characteristic. This technique does not involve unknown weights like Olkin(1958), Raj (1965) and Rao and Mudholkar (1967) estimators and at the most assume knowledge of population mean of auxiliary character least correlated with the main character. Kiregyera (1980, 1984) also proposed some chain type ratio and regression estimators based on two auxiliary variables. InterStat 015 Page 5

6 Many estimators of the population parameters are constructed using known auxiliary variables. The classical well known ratio estimator is one of them. Various improvements of this ratio estimator have been considered by many authors. Some estimators use two auxiliary variables. Some other use groups of estimators are composite. In case two auxiliary variables are known, the ratio-cum-product estimator may be used (Singh 1965). This estimator behaves similarly as simple ratio estimator. Singh s (1965, 1967) ratio cum product estimators given by r 4 = r0 Z / (.4) and r 5 = r0 / Z (.5) The product method of estimation is generally used when the study variable Y is negatively correlated with an auxiliary characteristics X whose population mean is assumed to be known. In order to improve the efficiency of product estimation, sometimes producttype estimators are used which are developed by mixing product estimator with usual mean estimator. Kadilar and Cingi (004) and Sahoo and Sahoo (1993) used a second auxiliary variable Z closely related to X and suggested different improved competitive estimators, assuming the availability of information on for all units of the population. Several researchers have established many classes of ratio and product-type estimators in the past that reduce the bias and the mean square error by improving the auxiliary variables. The above review clearly motivates the need for a comparative study of these estimators on the basis of their design based properties like biasedness, efficiency, and InterStat 015 Page 6

7 approach to normality (asymmetry) etc., before we decide to use any one of them in practice. It is however difficult to investigate analytically the behaviour of the estimators. Because the results derived through Taylor s lineariation method [e.g., Tin (1965)] are not only in asymptotical forms but also very much complicated to lead to make a good choice among different estimators. 3. Empirical Study First we generate the uniform random numbers and then normal random numbers with specified parameters is carried out to define the following populations of sie 000. The generation of the populations involves the following two steps based on the algorithm proposed by Rao (009). Population I: Step 1: Generate three independent random variables Z, Z 1 and Z from N( 3,4) distribution using Box-Muller method. Step : Using steps, 3, and 4 of algorithm given by Rao (009), generate the population-i of the triplet ( Y X, Z ),. Population II: Step 1: Generate three independent random variables Z, Z 1 and Z using Box-Muller method with Z ~ N (3,4), Z ~ (4,9) and Z ~ (5,16) 1 N N distributions. Step : Using steps, 3, and 4 of algorithm given by Rao (009), generate the population- II of the triplet ( Y X, Z ),. InterStat 015 Page 7

8 From the above two populations, 1000 simple random samples without replacement of sie n=10, 30 and 50 are drawn and for each of the sample, the 6 ratio estimators are computed. For each ratio estimator, the estimate i Rˆ, standard error ( ) SE ˆ, relative bias RB( R ˆ i ), skewness β 1 and kurtosis β are computed using the simulation method and presented against the sample sie n= 10, 30, 50 and the correlations between the variables in the following tables. Results are reported up to two decimals in the tables, but the conclusions are drawn from the original values. R i Table 1. Comparison of ratio estimators using two auxiliary variables from the Population-I Correlation Coefficient r yx =0.7 r x =0.80 r =0.90 Sample Sie (n) Ratio Estimator Rˆi S. E. ( ˆ ) ( R i ) R i RB ˆ 1 β β r r r r r r r r r r r r r r r r r r InterStat 015 Page 8

9 Table. Comparison of ratio estimators using two auxiliary variables from the Population-II Correlation Coefficient r yx =0.48 r x =0.66 r =0.7 Sample Sie (n) ( ) ( ) Ratio Estimator Rˆi S E. Rˆ i RB R i β1 β r r r r r r r r r r r r r r r r r r Conclusion It is observed from the above tables, from the population-i, the estimators r 1 & r are more efficient than the other estimators when the sample sie is small, whereas the estimator r 3 is more efficient estimator than the other estimators for large samples. The estimator r 3 is relatively, less biased as compared with other estimators for the small samples, whereas the estimators r 1 & r are less biased as compared with other estimators for the large samples under population-i. InterStat 015 Page 9

10 Similarly, in the second population, relatively best estimators are r 1 & r for small samples and r 3 is for large samples. The estimators r 1 and r are less biased under the second population. From the above study, it is observed that the Tin s (1965) type AUR estimators with two auxiliary variables (r 1 & r ) are more efficient for small samples, whereas the Tripathi (1980) estimator (r 3 ) is more efficient for large samples. It is also observed that the variation of the auxiliary variables resulting in the performance of the estimators when the small samples are selected. From the coefficients of skewness and kurtosis, it is observed that the estimators slightly deviating from the normal distributions whenever the variances of the auxiliary variables are different (Population-II), whereas in population-i, empirically observed that the considered estimators follows asymptotic normal distribution. References: 1. Chand, L. (1975), Some ratio-type estimators based on two or more auxiliary variables, Unpublished Ph. D. Dissertation, Iowa State University, Ames, Iowa.. Kadilar, C. and Cingi, H. (004), Estimator of a population mean using two auxiliary variables in simple random sampling, Int. J. Math., 4, Kiregyera, B. (1980), A chain ratio-type estimator in finite population double sampling using two auxiliary variables, Metrika, 7, Kiregyera, B. (1984), Regression type estimators using two auxiliary variables and the model of double sampling from finite populations, Metrika,31, Olkin, I. (1958), Multivariate ratio estimation for finite population, Biometrika, 45, InterStat 015 Page 10

11 6. Raj, D. (1965), On a method of using multi-auxiliary information in sample surveys, Jour. Amer. Stat. Assoc., 60, Rao, P.S.R.S. and Mudholkar, G.S. (1967), Generalied multivariate estimations for the mean of finite populations, Jour. Amer. Stat. Assoc.,6, Rao, K. R. (009), Comparison of ratio estimators using Monte Carlo Simulation, Unpublished Ph.D. Thesis, Department of Statistics, Osmania University, Hyderabad. 9. Sahoo, L.N. (1987a), On a class of almost unbiased estimator for population ratio, Statistics, 18, Sarndal, C.E., Swensson, B. and Wretman, J. (1991), Model assisted survey sampling, Springer Verlag, New York. 11. Singh, M.P. (1965), On the estimation of ratio and product of the population parameters, Sankhya,7, Singh, M.P. (1967), Ratio cum product method of estimation, Metrika,1, Sukhatme, B.V. and Chand, L. (1977), Multivariate ratio-type estimators, Proceedings of American Statistical Association, Social Statistics Section, Tin, M. (1965), Comparison of some ratio estimators, Jour. Amer. Stat. Assoc.,60, Tripathi, T.P. (1969), A regression type estimator in sampling with PPS and with replacement, Aust. Jour. Stat.,11, Tripathi, T.P. (1980), A general class of estimators for population ratio, Sankhya,C 4, InterStat 015 Page 11