Rajesh Singh, Pankaj Chauhan, Nirmala Sawan, Florentin Smarandache AUXILIARY INFORMATION AND A PRIORI VALUES IN CONSTRUCTION OF IMPROVED ESTIMATORS Estimators PRE (.,S2) y Population I Population II s2 100 100 y t 223.14 228.70 1 t 235.19 228.76 2 t (optimum) 305.66 232.90 r t (optimum) 305.66 232.90 p PRE of different estimators of S2 with respect to s2 y y 2007 Auxiliary Information and a priori Values in Construction of Improved Estimators Rajesh Singh, Pankaj Chauhan, Nirmala Sawan School of Statistics, DAVV, Indore (M. P.), India Florentin Smarandache Department of Mathematics, University of New Mexico, Gallup, USA Renaissance High Press 2007 1 In the front cover table the percent relative efficiency (PRE) of s2,t ,t ,t (in y 1 2 r optimum case) and t (in optimum case) are computed with respect tos2. p y This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) http://wwwlib.umi.com/bod/basic Copyright 2007 by Renaissance High Press (Ann Arbor) and the Authors Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm Peer Reviewers: Prof. Mihaly Bencze, Department of Mathematics, Áprily Lajos College, Braşov, Romania. Dr. Sukanto Bhattacharya, Department of Business Administration, Alaska Pacific University, U.S.A. Prof. Dr. Adel Helmy Phillips. Ain Shams University, 1 El-Sarayat st., Abbasia, 11517, Cairo, Egypt. (ISBN-10): 1-59973-046-4 (ISBN-13): 978-1-59973-046-2 (EAN): 978159973062 Printed in the United States of America 2 Contents Preface ………………………………………………………………………………….. 4 1. Ratio Estimators in Simple Random Sampling Using Information on Auxiliary Attribute ………………………………………………………………………………… 7 2. Ratio-Product Type Exponential Estimator for Estimating Finite Population Mean Using Information on Auxiliary Attribute …………………………………… 18 3. Improvement in Estimating the Population Mean Using Exponential Estimator in Simple Random Sampling ……………………………………………………………. 33 4. Almost Unbiased Exponential Estimator for the Finite Population Mean ……... 41 5. Almost Unbiased Ratio and Product Type Estimator of Finite Population Variance Using the Knowledge of Kurtosis of an Auxiliary Variable in Sample Surveys ………………………………………………………………………………… 54 6. A General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s) …………………………………62-73 3 Preface This volume is a collection of six papers on the use of auxiliary information and a priori values in construction of improved estimators. The work included here will be of immense application for researchers and students who employ auxiliary information in any form. Below we discuss each paper: 1. Ratio estimators in simple random sampling using information on auxiliary attribute. Prior knowledge about population mean along with coefficient of variation of the population of an auxiliary variable is known to be very useful particularly when the ratio, product and regression estimators are used for estimation of population mean of a variable of interest. However, the fact that the known population proportion of an attribute also provides similar type of information has not drawn as much attention. In fact, such prior knowledge can also be very useful when a relation between the presence (or absence) of an attribute and the value of a variable, known as point biserial correlation, is observed. Taking into consideration the point biserial correlation between a variable and an attribute, Naik and Gupta (1996) defined ratio, product and regression estimators of population mean when the prior information of population proportion of units, possessing the same attribute is available. In the present paper, some ratio estimators for estimating the population mean of the variable under study, which make use of information regarding the population proportion possessing certain attribute are proposed. The expressions of bias and mean squared error (MSE) have been obtained. The results obtained have been illustrated numerically by taking some empirical populations considered in the literature. 2. Ratio-Product type exponential estimator for estimating finite population mean using information on auxiliary attribute. 4 It is common practice to use arithmetic mean while constructing estimators for estimating population mean. Mohanty and Pattanaik (1984) used geometric mean and harmonic mean, while constructing estimators for estimating population mean, using multi-auxiliary variables. They have shown that in case of multi-auxiliary variables, estimates based on geometric mean and harmonic means are less biased than Olkin’s (1958) estimate based on arithmetic mean under certain conditions usually satisfied in practice. For improving the precision in estimating the unknown mean Y of a finite population by using the auxiliary variable x, which may be positively or negatively correlated with y with known X; the single supplementary variable is used by Bahl and Tuteja (1991) for the exponential ratio and product type estimators. For estimating the population mean Y of the study variable y, following Bahl and Tuteja (1991), a ratio- product type exponential estimator has been proposed by using the known information of population proportion possessing an attribute (highly correlated with y) in simple random sampling. The proposed estimator has an improvement over mean per unit estimator, ratio and product type exponential estimators as well as Naik and Gupta (1996) estimators. The results have also been extended to the case of two-phase sampling. 3. Improvement in estimating the population mean using exponential estimator in simple random sampling. Using known values of certain population parameter(s) including coefficient of variation, coefficient of variation, coefficient of kurtosis, correlation coefficient, several authors have suggested modified ratio estimators for estimating population meanY. In this paper, under simple random sampling without replacement (SRSWOR), authors have suggested improved exponential ratio-type estimator for estimating population mean using some known values of population parameter(s). An empirical study is carried out to show the properties of the proposed estimator. 4. Almost unbiased exponential estimator for the finite population mean. Usual ratio and product estimators and also exponential ratio and product type estimators suggested by Bahl and Tuteja (1991) are biased. Biasedness of an estimator is disadvantageous in some applications. This encouraged many researchers including 5 Hartley and Ross (1954) and Singh and Singh (1992) to construct either estimator with reduced bias known as almost unbiased estimator or completely unbiased estimator. In this paper we have proposed an almost unbiased ratio and product type exponential estimator for the finite population mean Y. It has been shown that Bahl and Tuteja (1991) ratio and product type exponential estimators are particular members of the proposed estimator. Empirical study is carried to demonstrate the superiority of the proposed estimator. 5. Almost unbiased ratio and product type estimator of finite population variance using the knowledge of kurtosis of an auxiliary variable in sample surveys. In manufacturing industries and pharmaceutical laboratories sometimes researchers are interested in the variation of their product or yields. Using the knowledge of kurtosis of auxiliary variable Upadhyaya and Singh (1999) have suggested an estimator for population variance. In this paper following the approach of Singh and Singh (1993), we have suggested almost unbiased ratio and product type estimator for population variance. 6. A general family of estimators for estimating population variance using known value of some population parameter(s). In this paper, a general family of estimators for estimating the population variance of the variable under study; using known values of certain population parameter(s) is proposed. It has been shown that some existing estimators in literature are particular member of the proposed class. An empirical study is caring out to illustrate the performance of the constructed estimator over other. The Authors 6 Ratio Estimators in Simple Random Sampling Using Information on Auxiliary Attribute Rajesh Singh, Pankaj Chauhan, Nirmala Sawan, School of Statistics, DAVV, Indore (M.P.), India ([email protected]) Florentin Smarandache Chair of Department of Mathematics, University of New Mexico, Gallup, USA ([email protected]) Abstract Some ratio estimators for estimating the population mean of the variable under study, which make use of information regarding the population proportion possessing certain attribute, are proposed. Under simple random sampling without replacement (SRSWOR) scheme, the expressions of bias and mean-squared error (MSE) up to the first order of approximation are derived. The results obtained have been illustrated numerically by taking some empirical population considered in the literature. AMS Classification: 62D05. Key words: Proportion, bias, MSE, ratio estimator. 7 1. Introduction The use of auxiliary information can increase the precision of an estimator when study variable y is highly correlated with auxiliary variable x. There exist situations when information is available in the form of attributeφ, which is highly correlated with y. For example a) Sex and height of the persons, b) Amount of milk produced and a particular breed of the cow, c) Amount of yield of wheat crop and a particular variety of wheat etc. (see Jhajj et al., [1]). Consider a sample of size n drawn by SRSWOR from a population of size N. Let y i and φ denote the observations on variable y and φ respectively for ith unit (i=1,2,....N). i Suppose there is a complete dichotomy in the population with respect to the presence or absence of an attribute, say φ, and it is assumed that attribute φ takes only the two values 0 and 1 according as φ = 1, if ith unit of the population possesses attribute φ i = 0, otherwise. N n Let A=∑φ and a =∑φ denote the total number of units in the population and i i i=1 i=1 A a sample respectively possessing attribute φ. Let P= and p= denote the proportion N n of units in the population and sample respectively possessing attributeφ. Taking into consideration the point biserial correlation between a variable and an attribute, Naik and Gupta (1996) defined ratio estimator of population mean when the 8 prior information of population proportion of units, possessing the same attribute is available, as follows: ⎛P⎞ t = y⎜ ⎟ (1.1) NG ⎜⎝p⎟⎠ here y is the sample mean of variable of interest. The MSE of t up to the first order of NG approximation is ⎛1−f ⎞[ ] MSE(t )=⎜ ⎟S2 +R2S2 −2R S (1.2) NG ⎝ n ⎠ y 1 φ 1 yφ where f = n , R = Y , S2 = 1 ∑N (y −Y)2 , S2 = 1 ∑N (φ −P)2 , N 1 P y N−1 i φ N−1 i i=1 i=1 S = 1 ∑N (φ −P)(y −Y). yφ N−1 i i i=1 In the present paper, some ratio estimators for estimating the population mean of the variable under study, which make use of information regarding the population proportion possessing certain attribute, are proposed. The expressions of bias and MSE have been obtained. The numerical illustrations have also been done by taking some empirical populations considered in the literature. 2. The suggested estimator Following Ray and Singh (1981), we propose the following estimator y+b (P−p) t = φ P=R*P (2.1) 1 p s y+b (P−p) ⎛ 1 ⎞ n where b = yφ , R* = φ , s2 =⎜ ⎟∑(φ −p)2 and φ s2 p φ ⎝n−1⎠ i φ i=1 s =⎜⎛ 1 ⎟⎞∑n (φ −p)(y −Y). yφ ⎝n−1⎠ i i i=1 9