chapter 5

THE MOST RECENT STUDIES IN SCIENCE AND ART

CHAPTER 5

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NOVEL TYPE ESTIMATORS BASED ON DIFFERENT MEANS IN STRATIFIED RANDOM SAMPLING

Tolga ZAMAN, Vedat SAGLAM, Kamil ALAKUS INTRODUCTION Let an N sized population be divided to L stratum. The estimation of the sum of Y and X populations in stratified random sampling, which units of units in each stratum is chosen using simple random sampling, and the usage of them in the ratio estimation is called combined estimator. In this case sample means of the variables are as follows: Let an N sized population be divided to L stratum. The estimation of the sum of Y and X populations in stratified random sampling, which nh units of Nh units in each stratum is chosen using simple random sampling, and the usage of them in the ratio estimation is called combined estimator. In this case sample means of the variables are as follows:

And population means are as follows:

In a stratified population, is the real value of population mean and is the estimation of it from the sample. It is shown below. Real value is obtained by

and if we replace

with

, then,

If , combined estimator of population mean belonging to y is obtained as below.


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Ratio of the variables is

the estimator of R is

and if (9) equation is placed in (7) equation then

Let as first examine the variance of estimator variance of estimator or random variable .

Here, if

in order to calculate the

is assumed and both sides are squared, then it is

Now let us define new variable. Stratified sampling and population means of this are as follows.


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If (18) equation is placed in (14) equation, then it is the variance of calculated as below.

is

Be careful that obtained equation (20) is actually equal to numerator of equation (12). In this regard, variance of is

Using equation (21), the variance of estimator

is calculated as below

(Cochran, 1977). And, the amount of bias of ratio estimator

is

(Çıngı, 2009). PROPOSED ESTIMATOR In equation (7), new estimators are given geometric writing harmonic and quadratic means instead of arithmetic mean under the assumption of . Estimator Based on Geometric Mean Definition 1. Let x1,x2,....xn be the population values belonging to auxiliary variate. Then Geomeric Mean is defined as

In equation (7), if obtained,

is replaced with equation (24) then estimator below is


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Estimated value of this estimator is

or

Its variance is

or

Its amount of bias is

or

Estimator Based on Harmonic Mean Definition 2. Let x1,x2,....xn be the population values belonging to auxilliary variate. Then Harmonic Mean is defined as




or


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Its variance is

or


or

Estimator Based on Quadratic Mean Definition 3. Let x1,x2,....xN be the population values belonging to auxiliary variate. Then Quadratic Mean is defined as




or


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Its variance is

or


or

EFFICIENCY COMPARISON Comparison of The Estimators Inequations below are available in order to compare arithmetic mean with other means (Shahbazov, 2005). Let x1,x2,....xN be the population values belonging to auxiliary variate. Then inequations below are valid.


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Comparison of The Means Using inequations (48), (50) and (52), inequations below can be given.

Comparison of The Amount of The Bias

Comparison of The Variances Again, using inequations (48), (50) and (52), inequations below can be given.

Equation (62) above is not a desired case. That’s why ignored. In this regard using estimator below

estimator will be

below is obtained, According to these results, and estimators can be suggested in stratified random sampling when population values belonging to auxiliary variate is positive and known. Comparison Criteria It is the estimated value of the square of the difference between an estimator and its parameter value and it is expressed as below

This expression can be examined mathematically as follows. In equation (65), if ) is added and subtracted once, then

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If this expression is solved, then

(Çıngı, 2009). NUMERICAL ILLUSTRATION Empirical Study In this section, Zaman and Yilmaz’s (2017) data is used. Proposed estimators and combined ratio estimators in the scope of the study are applied to the data set belonging to crop yield (as the interest of the variate) and cultivation area (as the auxiliary variate) of 1050 countries in the World (Resource URL1: World databank). Firstly, assuming that cultivation areas and yield in these areas belonging to different continents in the world will vary, the world divided into strata. Samples are taken from each of these strata (countries) using Neyman allocation method (Cochran, 1977),

Estimation and variance values of the newly proposed estimators are given in Table 2. Table 1. Data Statistics

Table 2. Ratio estimator values, the amount of bias and variance values


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Table 3. MSE values of ratio estimators

We have computed the sample size in stratum h. A sample of n=408 units is selected from a population of 1050 units with d= 0.05 error margin and %99 confidence level (Çıngı, 2009). 4 stratum is taken for this data (1: Europe, 2: America, 3: Africa, 4: Asia). Then, using stratified random sampling, MSE values of combined and proposed ratio estimators are calculated using equation (67). Finally, these estimations are compared with each other according to their MSE values. In Table 1, data statistics of population, stratum and sample size are shown. In Table 2, estimation values, the amount of bias and variance estimation values are given and in Table 3, the mean square of error values are given. Ratio estimator based on the harmonic mean is smaller than the one based on geometric mean and combined ratio estimator, respectively. Simulation Study Suppose population mean and standard deviation of auxiliary variate is μ1=μ2=μ3=10; σ1=2, σ2=2.5 and σ3=3 for each stratum and let us assume that a sample of n=90 units is selected from a population whose population mean and standard deviations are μ1=μ2=μ3=5; σ1=2, σ2=2.5 and σ3=3. Figuration of population 1: Let it be n1=n2=n3=30; as n=90, n= (n1,n2,n3 ) Figuration of population 2: Let it be n1=65, n2=25, n=90, n= (n1,n2 ) Table 4. MSE values of ratio estimators

In Table 4, for both population indicators, ratio estimators which are proposed based on both harmonic mean and geometric mean have smaller MSE values than combined ratio estimator. If we evaluate MSE values of ratio estimators all together, we see that ratio estimator based on the harmonic mean is followed by the ones based on geometric mean and combined ratio estimator, respectively.

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CONCLUSION In this study, a new ratio estimator is formed by adapting the estimators, which are proposed based on simple random sampling in Sağlam et al. (2016), to stratified random sampling. Using this value variance values of proposed estimators are theoretically compared with combined ratio estimators and it is found that variance values of ratio estimators based on geometric and harmonic mean are smaller or equal. Also, its MSE values are obtained. When the obtained results are discussed it is seen that the variance of the estimator, proposed by harmonic mean, is the smallest. Also, all these results are supported by a simulation study and a numerical example. REFERENCES 1. W. G, “Cochran Sampling Techniques”, John Wiley and Sons, New-York, 1977. 2. H. Çıngı, “Örnekleme kuramı”. Hacettepe Üniversitesi. Fen Fakültesi Basımevi, Beytepe, Ankara, 2009 3. V. Sağlam, T. Zaman, E. Yücesoy, M. Sağır, “Estimators Proposed by Geometric Mean, Harmonic Mean and Quadratic Mean”. Science Journal of Applied Mathematics and Statistics. 4(3): 115-118,2016. 4. A.Shahbazov, “Introduction to Probability Theory”, Birsen Yayınevi Ltd. Şti. Kod No: Y. 0029, ISBN: 975-511-415-7, İstanbul, 2005. 5. T. Zaman, S. Yilmaz, “Review of some ratio estimators in stratified random sampling”. Journal of Mathematical and Computational Science, 7(2), 364 2017 6. URL-1. http://data.worldbank.org/ (Accessed Date: 23.10.2016)