Ratio-type Estimators in Stratified Random Sampling Using Auxiliary Attribute
Chapter One
AIM AND OBJECTIVES
The aim of this research work is to develop some ratio-type estimators under stratified random sampling scheme using auxiliary attributes that will produce more precise estimates than the conventional estimator.
The above aim is achieved through the following objectives;
- To linearly transform the sample mean of the variable of
- To transform the proportion of auxiliary attributes using auxiliary parameters like kurtosis, coefficient of variation and coefficient of Point-biserial
- To obtain the biases and mean square errors of the proposed estimators up to first order approximation using Taylors’
- To obtain the conditions for efficiency of the proposed estimators over the conventional
CHAPTER TWO
LITERATURE REVIEW
RATIO ESTIMATORS
The work incorporated in this dissertation is associated mainly with the study of efficiency of ratio-type estimators. Although, a significant contribution by the survey statisticians has been made towards the development of ratio and product estimators using the auxiliary information at the estimation stage; a review of only those items of dissertation work has been made in this section which have an immediate bearing on and relevance to the present work. However, the review presents only a proper orientation and perspective to the present work.
It was late 1930s and early 1940s when a few major centers emerged for research in and application of sampling methodology. One was the statistical laboratory at Ames, Iowa, another was the Bureau of Census in United States. At the same time major advances were occurred at a few other locations.
In the decade of 1940-1950, the contributions of Cochran (1942), Deming (1956), Hurvitz (1952), Hansen (1951), Jessen (1942), Madow (1953), Sukhatme (1947), Yates (1948) and others helped in lying the foundation of modern sampling theory. The work done during this period related mainly with the development of sampling and estimation procedures under different sets of assumptions about the population values.
In 1942, Cochran made a particular important contribution to the modern sampling theory by suggesting methods of utilizing the auxiliary information for the purpose of estimation of the population mean in order to increase the precision of the estimates (Cochran 1942).
Several well-known procedures use auxiliary information at the estimation stage. This is most commonly used way of utilizing auxiliary information which gives rise to some estimators that are known today, and in under certain conditions, these estimators are more efficient than the estimators based on simple random sampling. He used the ratio estimator of the form
y = y X
r x
; x ¹ 0
(2.1)
Where y and x are the sample means of the characteristics under study and auxiliary characteristics respectively based on a sample drawn under simple random sampling design and X is the population mean of the auxiliary characteristic X .
The aim of this method was to use the ratio of sample means of two characteristics which would be almost stable in sampling fluctuation and thus, would provide a better estimate of the true value. It has been a well-known fact that
yr is more efficient than the sample mean estimator, y , where no auxiliary information is used.
Contrary to the situation of ratio estimator, if variants Y and X are negatively correlated,
then the product estimator
yp given by
yp = X x
; X ¹ 0
(2.2)
which was proposed by Murthy (1964), has been observed to give higher precision than
y , the sample mean estimator.
The expression for the bias and mean square error (MSE) of
yr and
yp have been
derived by Cochran (1942) and Murthy (1964) respectively which are also available in the books by Cochran (1977), Jessen (1978) and Murthy (1967).
Since the ratio estimator was observed to be more precise than the usual sample mean estimator under different conditions, several researchers diverted their attention in the direction of modifying estimation procedure so that unbiased or less bias estimators could be obtained. The work of Hartley and Ross (1954) deserves special attention in this direction. Several other authors also proposed unbiased or almost unbiased ratio-type estimators. Beale (1962) and Tin (1965) proposed bias-adjusted ratio estimators which are equally efficient in finite population. Rao (1981a, 1981b), Rao and Webster (1966) and Rao and Beagle (1967) also suggested some almost unbiased ratio-type estimators.
Several authors have used prior value of certain population parameter(s) to find more precise estimates. Searls (1964) used coefficient of variation of study character at estimation stage. In practice this coefficient of variation is seldom known. Motivated by Searls (1964) work, Sen (1978), Sisodiya and Dwivedi (1981), and Upadhyaya and Singh (1984) used the known coefficient of variation of the auxiliary character for estimating population mean of a study character in ratio method of estimation. The use of prior value of coefficient of kurtosis in estimating the population variance of study character was first made by Singh et. al. (1973). Later used by Searls and Intarapanich (1990), Singh and Upadhyaya (1999), Singh (2003) and Singh et. al. (2004) in the estimation population mean of study character. Recently Singh and Tailor (2003) proposed a modified ratio estimator by using estimator by using the known value of correlation coefficient.
CHAPTER THREE
MATERIALS AND METHODOLOGY
INTRODUCTION
Sampling technique adopted in this research work is stratified random sampling for estimating population mean for random variable y using parameters of auxiliary information (attribute). This enables the researchers to obtain ratio estimators, their mean square errors and biases, as well as identifying the estimators with least mean square error and bias. The estimator with least mean square error and bias is preferred and identified as the most efficient estimators among the ratio estimators defined in the work.
DATA USED FOR THE ANALYSIS
The data used for the analysis was taken from Students Pre-Medical Registration, Usmanu Danfodiyo University, Sokoto (2011/2012 Session). The height and sex of students were taken according to their respective faculties. The data is shown in the Appendix I.
CHAPTER FOUR
EMPIRICAL STUDY
PREAMBLE
The information on 1500 students taken from Students Pre-Medical Registration, Usmanu Danfodiyo University, Sokoto (2011/2012 Session) was used as data for empirical study. The height of the student is the variable of interest and their gender was used as auxiliary attribute (Male=1 and Female=0). The stratification is based on the faculties. By using Neyman allocation (Cochran, 1977),
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATION
SUMMARY
Ten ratio-type estimators are proposed in stratified random sampling using attribute as auxiliary information. The estimators were formed from the traditional estimator by linearly transforming the mean of study variable and used the known parameters values of auxiliary attribute (kurtosis, coefficient of variation and point-biserial correlation) to modified the population proportion. The bias and MSE of the proposed estimators were obtained up to first order approximation using Taylors’ expansion. The bias and MSE of the proposed estimators were compared with that of traditional estimator. The efficiency comparison is done both theoretically and empirically. The proposed estimators were also compared and the order of their preference is obtained. The properties of the proposed estimators were also stated based on the theoretical study. Formulae for determine sample sizes under various allocations (Optimum, Neyman and Proportion) for fixed cost and desired precision when applying the proposed estimators were also obtained.
CONCLUSION
In this study, ratio estimators are proposed for stratified random sampling using auxiliary information on attribute. The ten estimators proposed, nine of which use some known parameters values of the population proportion perform better and less bias than the traditional estimator. In addition, procedures for sample determination for Optimum, Neyman and Proportional allocations were derived for the estimators proposed.
RECOMMENDATION
The estimators
Tˆ* ,Tˆ* ,Tˆ* ,Tˆ* ,Tˆ* ,Tˆ* ,Tˆ* ,Tˆ* and
Tˆ* are recommended for use in any practical situation involving heterogeneous population and for which efficient and less bias estimate of population mean Y is needed.
REFERENCES
- Al odat, N. A. (2009): Modification in Ratio Estimator Using Rank Set Sampling.
- European Journal of Scientific Researc. 29 (2): 265-268.
- Beale, E.M.L. (1962): Some uses of Computers in Operation Research. Industrielle organization. 31: 51-52.
- Bedi, P. K. (1996): Efficient Utilization of Auxiliary Information at Estimation Stage. Biomet. Journal. 38 (8): 973-976.
- Chaudhuri, A. and Adrikari, A. K. (1979): Improving on the efficiencies of standard sampling Strategies through variate transformations in estimators for finite population means. Tech. Reports, ISI Calcutta.
- Cochran, W. G. (1942): Sampling Theory when the Sampling Units are of Unequal Sizes. Journ. American Statistical Association. 37: 199-212.
- Cochran, W. G. (1977): Sampling Techniques. New York: John Wiley and Sons.
- Dash, P., Dash, M., Mishra, S. and Pattanaik, P. S. (2011): Different Classes of Transformed Ratio and Product Estimators of Finite Population Mean in Stratified Sampling. Euro. Journal of Social Sciences. 19 (2): 227-232.
- Deming, W. E. (1956): Sample Design in Business Research. New York: John Wiley and Sons.
- Hansen, M. H. (1951): Sample Survey Methods and Theory. New York: John Wiley and Sons.
- Hartley, H. O. and Ross, A. (1954): unbiased Ratio Estimates. Nature. 174: 270-271.
- Hurvitz, D. G. (1952): Sampling and Field procedures of the Pittsburg Morbidity Survey. Public Health Reports. 67: 1003-1012.
