SYSTEMATIC SAMPLING FOR NON-LINEAR TREND IN MILK YIELD DATA

Transcription

1 Joural of Reliability ad Statistical Studies; ISS (Prit): , (Olie): Vol. 7, Issue (04): SYSTEMATIC SAMPLIG FOR O-LIEAR TRED I MILK YIELD DATA Tauj Kumar Padey ad Viod Kumar Departmet of Mathematics, Statistics ad Computer Sciece, G.B. Pat Uiversity of Agriculture ad Techology, Patagar E Mail: taujumar.63@gmail.com, viod_umarbcb@yahoo.com, Received July, 03 Modified May 5, 04 Accepted May 3, 04 Abstract The preset paper utilizes systematic samplig procedures for mil yield data exhibitig some o-liear treds. The best fitted mathematical forms of o-liear tred preset i the mil yield data are obtaied ad the expressios of average variaces of the estimators of populatio mea uder simple radom, usual systematic ad modified systematic samplig procedures have bee derived for populatios showig o-liear tred. A comparative study is made amog the three samplig procedures for five data sets by calculatig average variaces usig best fitted tred equatios. Usual systematic samplig is foud more precise tha simple radom ad modified systematic samplig for four data sets whereas modified systematic samplig is foud better tha the other two procedures for oe data set. Estimates of parameters for these tred fuctios are obtaied with the help of SAS software. Key Words: Simple Radom Samplig, Usual Systematic Samplig, Modified Systematic Samplig, o-liear Tred, Average Variaces, SAS Software.. Itroductio Systematic samplig ca be treated as stratified samplig cosistig of strata, each stratum cosistig of uits ad we select oe uit from each stratum which is located at the same relative positio i each stratum. Systematic samplig is also equivalet to cluster samplig by selectig oe of the clusters cosistig of uits. We select each of the clusters at radom. The radom start i correspods i th cluster which cosists of the uits with labels i (j-); j =,,3,,. Let y ra, y sys ad y mod be the estimators of populatio mea Y uder simple radom samplig, systematic samplig ad modified systematic samplig respectively. The correspodig variace formulae of the estimators for the fiite populatio are ow to be σ yra = ( ) y i Y σ ysys = y i Y σ ymod = y i, Y

2 58 Joural of Reliability ad Statistical Studies, Jue 04, Vol. 7(), Where y i ad y i are i th sample meas uder systematic ad modified systematic samplig procedures resspectively. The presece of liear, polyomial or parabolic tred i the populatio maes the usual systematic samplig less efficiet tha other samplig schemes. I order to overcome this problem, Cochra (946) studied the depedece of the performace of systematic samplig o the structure of the populatio ad itroduced the so-called super populatio models. To elimiate the effect of the liear tred, Yates (948) suggested some ed correctios; Sethi (965) itroduced a balaced systematic samplig; Sigh et al. (968) proposed a modified method ad Sigh ad Garg (979) provided a balaced radom samplig. Agarwal ad Jai (988) proposed some improved methods of elimiatig liear tred. They also proposed some modified methods of elimiatig quadratic tred. Uthayaumara (998) claimed that his samplig method is more efficiet tha circular systematic samplig whe the populatios exhibit liear ad parabolic treds. Modified systematic samplig method of Sigh et al. (968) cosists of selectig pair of uits equidistat from both eds of the populatio. The uits selected i the i th cluster will be i j, i j ; = 0,,,, for eve j i j, i j, i ; j = 0,,,, 3 for odd The sample mea is equal to the populatio mea for eve without applyig ed correctios. Ashutosh (995) derived the expressios for the average variaces uder simple radom samplig, systematic samplig ad modified systematic samplig for the populatio havig a geeral tred. The average variaces were derived uder the coditio that radom elemet has o serial correlatio, i.e. there is o correlatio betwee the values of e i ad e j. The average variaces with this coditio are as follows: E σ yra = ( ) E f(i) f(i) σ () E σ ysys = E E j = f i j f i σ () E σ ymod = E f i j f( i j ) f i σ (3) Whe is eve ad

3 Systematic Samplig for o-liear Tred i Mil Yield Data 59 3 E σ ymod = E f i j f( i j ) f i f i σ (4) Whe is odd. I the preset paper systematic samplig procedures are used for mil yield data exhibitig some o-liear treds. The best fitted mathematical forms of oliear tred preset i the mil yield data are obtaied ad the expressios of average variaces of the estimators of populatio mea uder simple radom, usual systematic ad modified systematic samplig procedures have bee derived for populatios showig o-liear tred. A comparative study is made amog the three samplig procedures for five data sets by calculatig average variaces usig best fitted tred equatios.. Average Variaces for o-liear Treds Here, we have cosidered two cases whe populatio have a positively sewed tred characterised by a iverse term model. Usig equatios (), (), (3) ad (4), the expressios for average variaces of estimates of populatio mea uder simple radom samplig, systematic samplig ad modified systematic samplig are derived by taig ito accout the absece of radom compoet i.e. σ = 0. Case I Whe f i is of the form f i = a bi c i d i e i ; i =,, 3,, a, b, c, d are costats, the usig (), we get σ yra = a bi c i d i a bi c i d i After simplificatio, we get σ yra = ( ) b bd bc cd Q P c bd( ) Q d S cdr c P d Q bc (5)

4 60 Joural of Reliability ad Statistical Studies, Jue 04, Vol. 7() Where i = P, i = Q, i 3 = R ad i 4 = S Similarly usig (), we get σ ysys = c d (ij ) (ij ) } j = {a b(i j ) a bi c i d i After simplificatio, we get the followig expressio for average variace uder systematic samplig σ ysys = b P i P c Q i Q d Where j = = P i, ij ij P iq i PQ cd i P i P i Q i Q j = = Q i ad P i = P ad Q i = Q Usig (3) ad (4), we get the followig expressios, Whe is eve bc bd (6) σ ymod = c a b i j (i j) c ( i j ) d i j d a b i j i j a bi c i d i Which after simplificatio reduces to

5 Systematic Samplig for o-liear Tred i Mil Yield Data 6 Where Whe is odd σ ymod = σ ymod = P i P i j i j P i Q i PQ = P i ad c Q i Q cd d i j i j = Q i such that P i = P ad Q i = Q c a b i j (i j) c ( i j ) c i a bi c i d i Which after simplificatio reduces to σ ymod = b Where (7) d a b i j i j d a b i i j d i P i P c Q i Q P i Q i PQ cd i i d P i P Q i Q (8) bc bd

6 6 Joural of Reliability ad Statistical Studies, Jue 04, Vol. 7() Q i = 3 ij i j such that P i = P Case II Whe f i is of the form ad i Q i = Q f i = a bi ci d i e i ; i =,, 3,, The expressios for average variaces uder differet samplig procedures are give by σ yra = b 8 c 80 Q P d 6 3 P 3 σ ysys = b cd bc P bd (9) c P i P d bc i P i P 6i 3 6i P i P cd 0 bd

7 Systematic Samplig for o-liear Tred i Mil Yield Data 63 σ ymod = Where P = i P i, Q = i, R = P d Z ij i 3, S = P i c ( )( ) P cd 6 i 4, P i = j =, (i j ) () Q i = j = (i j ) ad Z ij = i j 4ij i j Case 3 A compariso of systematic samplig ad modified systematic samplig procedures whe the tred fuctio is quadratic may be useful. If we have, f i = a bi ci e i () I the absece of radom compoet the average variaces uder radom samplig, usual systematic samplig ad modified systematic samplig derived by Ashutosh (995) ad Sigh et al (968) are give by the followig formulae: E σ yra = E σ ysys = ( ) ( )( ) b ( ) bc 6 b ( )( )(8 )( ) c 80 (3) c ( )( ) bc 6 (4)

8 64 Joural of Reliability ad Statistical Studies, Jue 04, Vol. 7() E σ ymod = c (5) 3. Illustratios The mil yield data used for the study have bee collected from Istructioal Dairy Farm of G.B. Pat Uiversity of Agriculture ad Techology, Patagar. The data relate to mil yields of four breeds of cows (two brads S6 ad S9 of Sahiwal cows ad two brads X4 ad X05 of crossbred cows) ad oe breed (Murrah brad o. M5) of buffaloes over oe lactatio period durig These breeds are chose because the lactatio records show mostly the assumed o-liear treds. The records are i Litre uits. For simplicity, we have ot tae ito accout yield records durig the colostrum period. Tables,, 3, 4 ad 5 show the mil yield data for the abovesaid cows ad buffaloes Table : Horizotally Day wise Mil Yield Data (i Litres) for S-6 brad of Sahiwal Cows for 56 days from the date (0/03/0) of calvig

9 Systematic Samplig for o-liear Tred i Mil Yield Data Table : Horizotally Day wise Mil Yield Data (i Litres) for S-9 brad of Sahiwal Cows for 5 days from the date (6/0/0) of calvig Table 3: Horizotally Day wise Mil Yield Data (i Litres) for X-4 brad of Crossbred Cows for 80 days from the date (5/07/0) of calvig

10 66 Joural of Reliability ad Statistical Studies, Jue 04, Vol. 7() Table 4: Horizotally Day wise Mil Yield Data (i Litres) for X-05 brad of Crossbred Cows for 40 days from the date (/03/0) of calvig

11 Systematic Samplig for o-liear Tred i Mil Yield Data Table 5: Horizotally Day wise Mil Yield Data (i Litres) for M-5 brad of Murrah Buffaloes for 64 days from the date (/07/0) of calvig 4. Results ad Discussio (i) For data set I (Sahiwal Cows Brad o. S6), the best fitted tred equatio is y i = i i i with the highest value (0.7545) of R ad miimum value ( ) of S.S.R. I this case, usual systematic samplig is better with miimum average variace (0.003) tha simple radom ad modified systematic samplig. Moreover, Sahiwal Cows Brad o. S6 give mil yield of 7. Litres/day o a average durig its lactatio period of 56 days. (ii) For data set II (Sahiwal Cows Brad o. S9), the best fitted tred equatio is y i = i i i with the highest value (0.795) of R ad miimum value (99.54) of S.S.R. I this case, modified systematic samplig (whe = 8 ad = 9) is better with miimum average variace (0.0034) tha simple radom ad usual systematic samplig. Moreover, Sahiwal Cows Brad o. S9 give mil yield of 8.39 Litres/day o a average durig its lactatio period of 5 days. (iii) For data set III (Crossbred Cows Brad o. X4), the best fitted tred equatio is y i = i 0.000i i with the highest value (0.6407) of R ad miimum value (473.4) of S.S.R. I this case, usual systematic samplig (whe = 0 ad = 4) is better with miimum average variace (0.006) tha simple radom ad modified systematic samplig. Moreover, Crossbred Cows Brad o. X4 give mil yield of.9 Litres/day o a average durig its lactatio period of 80 days. Further, it is also foud that by reducig samplig iterval of 4 days to 0 days, efficiecy of usual systematic samplig scheme icreases, whereas the efficiecy of modified systematic samplig decreases. Thus, it may be cocluded that for data set III, it may be better to use samplig iterval of 0 days rather tha 4 days. (iv) For data set IV (Crossbred Cows Brad o. X05), the best fitted tred equatio is y i = i i i with the highest value (0.8595) of R ad miimum value (84.349) of S.S.R. I this case, usual systematic samplig is better with miimum average

12 68 Joural of Reliability ad Statistical Studies, Jue 04, Vol. 7() variace (0.0055) tha simple radom ad modified systematic samplig. Moreover, Crossbred Cows Brad o. X05 give mil yield of 7.05 Litres/day o a average durig its lactatio period of 40 days. (v) For data set V (Murrah Buffaloes Brad o. M5), the best fitted tred equatio is y i = i i i with the highest value (0.687) of R ad miimum value ( ) of S.S.R. I this case, usual systematic samplig is better with miimum average variace (0.0036) tha simple radom ad modified systematic samplig. Moreover, Murrah Buffaloes Brad o. M5 give mil yield of 5.4 Litres/day o a average durig its lactatio period of 64 days. Refereces. Agrawal, M.C. ad Jai, irmal (988). Compariso of some samplig strategies i the presece of a tred, Jour. Id. Soc. Ag. Statistics, 40, p Ashutosh (995). O some cotributio to systematic samplig, Upublished Ph.D. Thesis, Rohilhad Uiversity, Bareilly, Idia. 3. Bellhouse, D.R. ad Rao, J..K. (975). Systematic samplig i the presece of a tred, Biometria, 6, p Cochra, W.G. (946). Relative accuracy of systematic ad stratified radom samples for a certai class of populatio, A. Math. Stat., 7, p Cochra, W.G. (985). Samplig Techiques, Fourth Wiley Easter Reprit, Oct. (985), Idia. 6. Sigh, D. ad Chaudhary, F.S. (986). Theory ad Aalysis of Sample Survey Desigs, Wiley Easter Limited, Idia. 7. Leu, C.H. ad Kao, F.F. (006). Modified balaced circular systematic samplig, Statistics ad Probability Letters, 76, p Murthy, M.. (967). Samplig Theory ad Methods, Statistical Publishig Society, Calcutta, Idia. 9. Sethi, V.K. (965). O optimum pairig of uits, Sahya, B-7, p Sigh, D., Jidal, K.K. ad Garg, J.. (968). O modified systematic samplig, Biometria, 55, p Sigh, P. ad Garg, J.. (979). O balaced radom samplig, Sahya, C- 4, p Uthayaumara,. (998). Additioal circular systematic samplig methods, Biometrica, 40, p Yates, F. (948). Systematic Samplig, Phil. Tras. Roy. Soc. Lodo, A 4, p Wu, C.F.J. (984). Estimatio i systematic samplig with supplemetary observatios. Sahya, B-46, p