Estimation of Current Population Variance in Two Successive Occasions

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1 ISSN Journal of Statitic Volume 7, 00, pp Etimation of Current Population Variance in Two Succeive Occaion Abtract Muhammad Azam, Qamruz Zaman, Salahuddin 3 and Javed Shabbir 4 The problem of etimation of the population variance for the current occaion baed on the ample elected over two ucceive occaion ha been conidered. Expreion for the Optimum etimator and it variance have been derived for both the occaion. Alo the expreion ha been derived when no information ha been collected on the firt occaion. Expreion for the Optimum fraction of unmatched obervation ha been derived. The value of Optimum matched and unmatched fraction have been tabulated. The gain in efficiency of the propoed etimator i compared to the etimator without uing the information collected on the firt occaion. An empirical tudy ha alo been conducted to tudy the performance of the propoed trategy. Keyword Variance etimator, Sampling on ucceive occaion, Optimum fraction, Gain in efficiency matching. Introduction The theory and practice of urveying the ame population at different point of time, technically called Repetitive ampling, Rotational ampling or ampling over ucceive occaion have been given coniderable attention by ome urvey tatitician. The main objective of ampling on ucceive occaion i to Department of Medical Statitic, Informatic and Health Economic, Medical Univerity of Innbruck, Autria. Department of Statitic, Univerity of Pehawar, Pakitan 3 Department of Statitic, Quaid-e-Azam Univerity, Ilamabad, Pakitan alahuddin_90@yahoo.com 4 Department of Statitic, Quaid-e-Azam Univerity, Ilamabad, Pakitan

2 Etimation of Current Population Variance in Two Succeive Occaion etimate ome population parameter (like total, mean, ratio, variance etc.) for the mot recent occaion a well a change in thee parameter from one occaion to the next (Mukhopadhyay, 998). 55 Jeen (94) wa the firt author who conidered the problem of ampling on two ucceive occaion by uing the information collected on the previou occaion to improve the current etimate. Thi technique wa further extended by Eckler (955), Patteron (950), Rueda et al. (008), Tikkiwal (95), Yate (960) and many other. The problem of etimating the current population variance in ucceive ampling wa conidered by Sud et al. (00). The theory of etimation of the population ratio of two character over two occaion ha been conidered by Rao and Mudholkar (967), Okafor and Arnab (987), Arte et al. (00) and many other. The aim of preent work i to propoe an etimator, which i bet linear combination of available ample variance, baed on ucceive ampling cheme on two occaion, where a fraction of the previouly collected ample i retained and a new ample i drawn with SRSWOR trategy from the population at the econd occaion. The gain in efficiency of the propoed etimator ha been obtained over ample variance etimator without uing the information available on the firt occaion and the Optimum variance etimator of the current occaion, which i the combination of the ample variance of the matched and unmatched portion of the ample at the econd occaion.. Sampling Strategy Let a character under tudy be denoted by x and y on the firt and the econd occaion repectively. Suppoe a population of ize N, which i ampled over two occaion. Aume that the ize of the population remain unchanged but value of unit change over occaion. Let a ample of ize n be elected uing SRSWOR cheme at the firt occaion. Out of thi ample, let m unit be retained on the econd occaion, while a freh ample of the ame ize u = (n-m) drawn on the econd occaion from the remaining (N-n) unit of the population, o that the total ample ize at the econd occaion i alo n i.e. n = m + u. Aume that the ize of the population i large, o we ignore fpc. Briefly, the ampling trategy may be illutrated a follow (Table ):

3 56 Azam, Zaman, Salahuddin and Shabbir Table : Sampling Strategy Occaion Sample Unit Total Sample Size x x x x x x x x ( n) y n x x x x x ( m) x m m x x x x x ( m)ym x x x ( u)yu n m u 3. The Propoed Etimator and it Propertie We are intereted to etimate the current population variance S (for the econd occaion) by a linear combination of available ample variance baed on: ˆ (3.) u m m u where the contant,,, and the matching fraction are to be determined, o a to minimize the variance. Alo S and S are the population variance of X on the firt occaion and the population variance of the tudy variable Y on the econd occaion repectively. Similarly m and u are the ample variance of matched and unmatched unit on the firt occaion while m and u are the ample variance of matched and unmatched unit on the econd occaion repectively. By following Cochran (977), Sukhatame et al. (984) and Sukhatame and Sukhatame (970) and we have: E E and E E u m S So, we get: ˆ S S u m S E (3.) Here, ˆ i an unbiaed etimator of ubject to 0 and.

4 Etimation of Current Population Variance in Two Succeive Occaion Hence, u m m ˆ (3.3) Define: u 57 Let S e, u u S S, m m S e e S, u u S e S. m m S Therefore, E( e ) E( e ) 0, ( i, ), iu im e e Ee e Ee e 0 E, m u m u u u E e u, E e m, E eu, u m u where and r r r/ / 0 0 N r X i X Y i Y i r. N Solving (3.), we get: e S e S e S e ˆ S u m m u or ˆ S e e S e S e (3.4) u m m u S So (3.4) become: A B B ˆ C V (3.5) n n n n where, and S S A S B S C.

5 58 Azam, Zaman, Salahuddin and Shabbir Alo, m/ n and u/ n. ˆ. The neceary condition for minimum i that the firt derivative of the function be equal to zero. We obtain the Optimum/bet value of and by minimizingv ˆ V 0 and ˆ V 0 A C 0 (3.6) B B C 0 (3.7) Solving (3.6) and (3.7), we obtain the Optimum value of and, which are obtained through neceary condition tated above. Thee equation alway atify uch relationhip. BC AB C AB AB and AB C Subtituting the Optimum value of and in (3.4) and (3.5), we get: BC AB AB AB AB ˆ ( ) ( ) u m m u (3.8) AB C AB C AB C and 3 A BC A B B AB C AB C V ˆ n AB C n AB C n AB C n AB C or B( AB AB) V ˆ (3.9) n( AB C ) If 0, i.e. complete matching or =, = 0 i.e. no matching. We obtained the ame variance in both cae i.e.

6 Etimation of Current Population Variance in Two Succeive Occaion 4 B S ( 40 ) V ˆ (3.0) n n It mean, we get the ame preciion either by keeping the ame ample or by changing it on every occaion. Alo note that an etimate that the firt occaion i given by (3.8) jut by interchanging A and B and the occaion. AC AB AB AB AB ˆ ( ) ( ) u m m u (3.) AB C AB C AB C V ˆ, we differentiate (3.) w. r. t., then equating zero to get optimum value of i.e. Thi can be achieved if we have data on both occaion. To get optimum 59 AB A B C AB opt (3.) C or opt (3.3) C AB Subtituting Optimum value of in (3.9), we get B AB A B C AB V ˆ opt (3.4) n AB Subtituting the value of A, Band C in (3.4), we get 4 S ( 40 ) V ˆ opt (3.5) n However, if only the etimate uing information collected on the econd occaion, then we have etimator of population variance a:

7 60 Azam, Zaman, Salahuddin and Shabbir ˆ (3.6) m u Such that. Alo, we find that: ˆ V V opt if ˆ B AB n (3.7) A B C AB AB B n or C AB 0 Becaue, A, B and C are all poitive quantitie. Now, we compute the gain in the preciion G of the propoed etimator over an etimate uing no information collected on the firt occaion i.e. V AB C G (3.8) V ˆ AB C and V ˆ AB Gopt (3.9) V ˆ AB A B C AB opt or. C AB G opt Let C ( 00) K. (3.0) AB ( )( ) We oberve that, in mot of the cae, 0 K. If the value of K i not real, then the Optimum value of a well a variance will be inadmiible. We obtained value of opt, optand Goptfor different value of K in the Table below.

8 Etimation of Current Population Variance in Two Succeive Occaion 6 Table : Computation of Fraction of Unmatched Unit and Gain In Table, G 4. Empirical Study K opt opt G % % 00 for all value of K. In the following ection the gain in the propoed etimator (3.) i calculated relative to uual variance etimator in the following example. Example : The data ha been taken from Singh and Singh (996) containing 58 univerity profeor. A random ample of 5 unit ha been taken and value have been calculated. The performance of uggeted etimator ha been expreed in Table 3 and 4. Example : The data on electricity, total production of 06 countrie for two year (998 and 999) a under ha been taken from Indutrial Commodity Statitic Yearbook 000 (United Nation, 003). A random ample of 5 unit ha been taken and value have been calculated. The performance of uggeted etimator ha been expreed in Table 5 and 6. Example 3: The data about the Ditrict-Wie Production of Fruit of 07 ditrict for the year 00-0 and ha been taken from Minitry of Food,

9 6 Azam, Zaman, Salahuddin and Shabbir Agriculture and Livetock (Economic Wing), Ilamabad. A random ample of 30 unit ha been taken and value have been calculated. The performance of uggeted etimator ha been expreed in Table 7 and Concluding Remark In the preent tudy, we propoed an etimator dicued in Section for the etimation of one of important tatitical meaure i.e. variance uing two ucceive occaion. The propoed etimator i unbiaed under certain condition. It i proved theoretically that the propoed etimator i more efficient than the uual unbiaed variance etimator ˆ under certain condition provided in equation 3.7 (which i alway true). An empirical tudy i alo conducted to validate the performance of the propoed etimator and it i oberved in three data et that the propoed etimator perform better than uual variance etimator for etimating the population variance. Table 3: Decriptive Statitic Baed on Example ˆ Value n u m u 3.63 ˆ ˆ ˆ ˆ.948 ˆ ˆ ˆ ˆ ˆ ˆ K ˆopt Table 4: Bia, MSE and RE (%) of Etimator Baed on Example Etimator Bia MSE RE ˆ ˆ ˆ

10 Etimation of Current Population Variance in Two Succeive Occaion 63 Table 5: Decriptive Statitic Baed on Example n 5 m Value u.9e+09 u 5.46E+09 ˆ 0 6.5E ˆ ˆ ˆ ˆ 3.78E+8 ˆ ˆ E+8 ˆ -3.99E- ˆ 3.8E+8 ˆ E- 40 ˆopt K Table 6: Bia, MSE and RE (%) of Etimator Baed on Example Etimator Bia MSE RE ˆ E E ˆ Table 7: Decriptive Statitic Baed on Example 3 n 30 m 5 Value u u 6.406E+09 ˆ ˆ ˆ ˆ ˆ.66E+0 ˆ ˆ 04.60E+0 ˆ E-

11 64 Azam, Zaman, Salahuddin and Shabbir ˆ 40.09E+0 ˆ 0.5 ˆopt K Reference Table 8: Bia, MSE and RE (%) of Etimator Baed on Example 3 Etimator Bia MSE RE ˆ E ˆ. Arte, R., Eva, M. and Garcia, A. (00). Etimation of current population ratio in ucceive ampling. Journal of Indian Society of Agricultural Statitic, 54 (3), Cochran, W. G. (977). Sampling Technique. Third Edition, John Wiley and Son, New York. 3. Eckler, A. R. (955). Rotation ampling. Annal of Mathematical Statitic, 6, Jeen, R. J. (94). Statitical invetigation of a ample urvey for obtaining farm fact. Iowa Agricultural Experiment Station Reearch Buliten, 304, Mukhopadhyay, P. (998). Theory and Method of Survey Sampling. Prentice Hall of India, New Delhi. 6. Okafor, F. C. and Arnab, R. (987). Some trategie of two tage ampling for etimating population ratio over two occaion. Autralian Journal of Statitic, 9 (), Patteron, H.D (950): Sampling on ucceive occaion with partial replacement of unit. Journal of Royal Statitical Society-B,, Rao, P. and Mudholkar, G. (967). Generalized multivariate etimator for the mean of finite population parameter. Journal of American Statitical Society, 6, Rueda, M., Munoz, J. and Arco, A. (008). Succeive ampling to etimate quantile with P-auxiliary variable. Quality and Quantity, 4,

12 Etimation of Current Population Variance in Two Succeive Occaion 0. Singh, R. and Singh, M. N. (996). Element of Survey Sampling. Vol. 5, Kluwer Academic Publiher, The Netherland.. Sud, U. C., Srivatava, A. K. and Sharma, D. P. (00). On the etimation of population variance in repeat urvey. Journal of Indian Society of Agricultural Statitic, 54(), Sukhatame, P. V. and Sukhatame, B. V. (970). Sampling Theory of Survey with Application. nd Edition, New Delhi, India. 3. Sukhatame, P. V., Sukhatame, B. V., Sukhatame, S. and Aok, C. (984). Sampling Theory of Survey with Application. Iowa State Univerity Pre, U.S.A. 4. Tikkiwal, B. D. (95). Theory of Succeive Sampling, Unpublihed Diploma Diertation. Intitute of Agricultural Reearch Statitic, New Delhi, India. 5. UNO. (003): Indutrial Commodity Statitic Yearbook: United Nation, New York. 6. Yate, F. (960): Sampling Method for Cenue and Survey. Charle Griffin and Company, London. 65