Modified Systeatic Sapling in the Presence of Linear Trend Zaheen Khan, and Javid Shabbir Keywords: Abstract A new systeatic sapling design called Modified Systeatic Sapling (MSS, proposed by ] is ore general than Linear Systeatic Sapling (LSS and Circular Systeatic Sapling (CSS In the present paper, this schee is further extended for populations having a linear trend Expressions for ean and variance of saple ean are obtained for the population having perfect linear trend aong population values Expression for the average variance is also obtained for super population odel Further, efficiency of MSS with respect to CSS is obtained for different saple size LSS, CSS, MSS, linear trend, super population 000 AMS Classification: 6DO5 Introduction In survey sapling, Linear Systeatic Sapling (LSS is a coonly used design Generally, it is useful when population size N is a ultiple of saple size n, ie N = nk ( k is the sapling interval Thus, we have k saples each of size n However, LSS is not beneficial when population size N is not a ultiple of the saple size n, ie N nk Because in this case, LSS cannot provide a constant saple size n, thus, estiate of population ean (total is biased Therefore, Circular Systeatic Sapling (CSS was introduced by Lahiri in 95 (cited in, p39] Contrary to LSS, CSS is not advantageous when population size N is a ultiple of the saple size n, ie N = nk as in this case, CSS produces n replicates of k saples Further, in CSS, the nuber of saples also rapidly increase to N as copared to k saples of LSS To iprove the efficiency of systeatic sapling, researchers proposed several odifications in the selection procedure The considerable work is done by 4], 6] and 7] In the recent years, 8] proposed Diagonal Systeatic Sapling (DSS under the condition that n k as a copetitor of LSS Later, the condition n k for DSS have relaxed by 9] The generalization of DSS is suggested by 0] Soe odification in LSS are proposed by ], in which odd and even saple sizes are dealt separately Further odification on LSS is also proposed by ] Diagonal Circular Systeatic Sapling (DCSS proposed by 5] is an extension of DSS to Departent of Matheatics and Statistics, Federal Urdu University of Arts, Science and Technology, Islaabad, Pakistan Eail: zkurdu@gailco, Corresponding Auther Departent of Statistics, Quaid-i-Aza University, Islaabad 4530, Pakistan, Eail: javidshabbir@gailco
the circular version of systeatic sapling A note on DCSS has been proposed by 3] However, soe of these schees are applicable when N = nk while other can be used only when N nk A new systeatic sapling design called Modified Systeatic Sapling (MSS proposed by ], which is applicable in both situations, whether N = nk or N nk According to this design, first copute least coon ultiple of N and n, ie L, then find k,, s and k, k = L n, = L N, s = N k and k = k /] or k = N/n] is rounded off to integer Consequently, s = n, which eans that there are sets and each set contains s units in a saple Thus, in MSS the j th unit of the i th set of a saple of n units can be written as: ( y (r ij = r+(i k+(j k hn if hn < r+(i k+(j k (h+n for h = 0,, ; i =,, 3,, and j =,, 3,, s This sapling schee reduces to LSS if L = N or N = nk and CSS if L = N n, the detail is given below If N = nk then L = N, k = k, = and s = n Thus, Equation ( reduces to ( y (r j = r + (j k, j =,, 3,, n which is LSS Siilarly, if L = N n, then k = N, = n and s = So, Equation ( can be written as (3 y (r i = r + (i k hn if hn < r + (i k (h + N for h = 0,, ; i =,, 3,, n Which is CSS To study the characteristics of MSS, we use an alternative ethod by partitioning the total nuber of saples into different sets of siilar saples To develop an alternative ethod, let us assue that k can be written as k = qk + r, q and r are quotient and reainder respectively Further, we assue that w = if ( q and, w = ( q if ( q > In both cases, there are two types of partitioning, ie between saples and within saples(see detail in Subsections and When w = In this case partitioning between saples and within saples are given in the Subsections and Partitioning between saples In this case, k possible saples are ainly partitioned into two groups The first group consists of initial k ( k saples and second group contains last ( k saples However, in the second group, there are ( subgroups, each attains k saples If a rando nuber r is selected fro the first k units of a population, there is a possibility that it is selected fro the first group, ie k ( k or it is selected fro the ( subgroups of the second group, ie k ( uk < r k ( u k such that u =,,, (, integer u is selected corresponding to a rando nuber r
Saple article to present hjs class 3 Partitioning within saples Furtherore, in the first group, all s units of all sets in each saple are labeled as r + (i k + (j k, such that i =,,, and j =,,, s; while in the second group, all s units of the first ( u sets are labeled as r + (i k + (j k such that i =,,, ( u and j =,,, s and in each of the last u sets, first (s units are labeled as r + (i k + (j k such that i = ( u +,, ; j =,,, (s and last unit is labeled as r+(i k+(j k N such that i = ( u+,, and j = s When w = (n q > In this case partitioning between saples and within saples are given in the Subsections and Partitioning between saples In this case k saples are ainly partitioned into two groups, the first group consists of the nuber of saples in which r k (w k + r The econd group contains the nuber of saples in which r > k (w k + r The first group is further partitioned into ( w (w + subgroups in which, there are r nuber of saples in each of the first and the last subgroups, and k saples in each of the iddle ( w (w subgroups In each subgroup of the first group, corresponding to a rando nuber r, an integer u is picked in such a way that u = (w if r k ( u k, u = w, w +, w +,, ( w if k ( uk < r k ( u k and u = ( w+ if k ( uk < r k ( uk + r The second group consists of last (w k r = (k r + (w k saples, which is the cobination of the first (k r saples and the last (w sets of k saples These (w sets of k saples further partitioned in such a way that the first r of every k saples fors the first subgroup and the last (k r saples of every k saples together with the first (k r saples of this group fors the second subgroup However, when w =, then we have only (k r saples in the second group Partitioning within saples In each saple of the first group, all s units of the first ( u sets are labeled as r+(i k+(j k such that i =,,, ( u and j =,,, s, and in each of the last u sets, the first (s units are labeled as r + (i k + (j k such that i = ( u +,,, j =,,, (s and the last unit is labeled as r + (i k + (j k N such that i = ( u +,, and j = s In each saple of the first subgroup of the second group, all s units of the first (w x sets are labeled as r + (i k + (j k such that i =,,, (w x and j =,,, s; the units of iddle ( w + sets are labeled in such a way that, the first (s units of each set are labeled as r + (i k + (j k such that i (w x +,, ( x +, j =,,, (s and the last unit of each set is labeled as r + (i k + (j k N such that i (w x +,, ( x + and j = s; the units of the last (x sets are labeled in such a way that, the first (s units are labeled as r+(i k+(j k such that i ( x+,,, j =,,, (s and the last two units in each set is labeled as r+(i k+(j k N such that i ( x +,, and j = (s, s However, when s =, the units
4 in these (x sets are labeled as r + (i k + (j k N The possible values of x are, 3,, (w Note: If w =, then this set of saples does not exist In the second subgroup of the second group, all s units of the the first (w x sets are labeled as r + (i k + (j k such that i =,,, (w x and j =,,, s; The units of iddle ( w sets are labeled in such a way that the first (s units of each set are labeled as r + (i k + (j k such that i (w x+,, ( x and j =,,, (s, the last unit of each set is labeled as r + (i k + (j k N such that i (w x+,, ( x and j = s, the units of the last (x sets are labeled in such a way that, the first (s units are labeled as r +(i k +(j k such that i ( x+,,, j =,,, (s and the last two units in each set is labeled as r + (i k + (j k N such that i ( x +,, and j = (s, s However, when s =, the units in these (x sets are labeled as r + (i k + (j k N, the possible values of x are,, (w Mean and variance of MSS for population having linear trend The following linear odel of hypothetical population is to be considered as ( Y t = α + β t, t =,, 3,, N α and β are the intercept and slope of the odel respectively Mean of MSS The saple ean for both cases, ie w = and w > are given below (see detial in Appendix A Case (i when w = ( ȳ MSS = α + β Case (ii when w > r + (s k + ( k ], if r k ( k ] r + (s k + ( k u k, u =,,, ( if k ( uk < r k ( u k (3 ȳ MSS = α + β ] r + (s k + ( k u N, n u = (w if r k ( u k u = w, w +,, ( w if N ( uk < r k ( u k, u = ( w + if k ( k < r k ( uk ] + r r + (s k + ( k ( w + x N, n x =,, (w if k (w xk < r k (w xk ] + r r + (s k + ( k ( w + x N, n x =,, 3,, (w if k (w xk + r < r k (w xk + k
Saple article to present hjs class 5 If w =, the Equation (3 will reduce to (4 ȳ MSS = α + β ] r + (s k + ( k u N n u = (w if r k ( u k, u = w, w +,, ( w if N ( uk < r k ( u k, u = ( w + if k ( k < r k ( uk ] + r r + (s k + ( k ( w + x N, n x =,, 3,, (w if k (w xk + r < r k (w xk + k Unbiasedness of ȳ MSS (see detial in Appendix A The saple ean (ȳ MSS is an unbiased estiator of population ean (Ȳ N + (5 E(ȳ MSS = α + β = Ȳ Variance of ȳ MSS (see detial in Appendix A (i when w = (6 V (ȳ MSS = b (k + ( k(k k ] Note: In this case, if N = nk then L = N, so =, thus, V (ȳ MSS = b (k This is a variance of linear systeatic sapling (ii when w > (7 (ȳ MSS = b (k + ( k(k k +4w(w k 3k (3 w + k] 3 Yates corrected estiator Yates corrected estiator of population ean for MSS is derived below 3 Yates corrected estiator for MSS The corrected estiator ȳmss c of population ean using MSS is given by (3 ȳmss c = n λ l Y r + ] n l= Y rl + λ l Y rn, λ l and λ l are selected so that saple ean coincides with the population ean in the presence of linear trend for all choices of r,,, k Alternatively Equation (3 can be written as (3 ȳ c MSS = ȳ MSS + a l (r (Y r Y rn, a l (r = λ l n = λ l n
6 Under the odel given in (, the population ean is (33 Ȳ = α + β N + As entioned earlier, that there are two cases, ie (i w = and (ii w > First, we consider the Case (i 3 Case (i: when w = In this situation, a rando start r is selected fro k units such that r k ( k or r > k ( k If r k ( k, then l = and the last value of each saple is labeled r + ( k + (s k Thus, (3 becoes (34 ȳ c MSS = ȳ MSS + a (r ( Y r Y r+( k+(s k Under the linear odel given in (, we have ȳ MSS = α+β r + (s k + ( k ], Y r = α + β r and Y r+( k+(s k = α + β r + ( k + (s k Putting these values in (34, we have (35 ȳmss c = α + β r + ] ( k + (s k a(r ( k + (s k Coparing the coefficients of α and β in (33 and (35 and solving for a (r, we have r + ( k k a (r = ( k + (s k Putting a (r in (34, we have (36 ȳmss c = ȳ MSS + r + ( k k ( Yr Y r+( k+(s k ( k + (s k If r > k ( k, then l = and the last value of each saple is labeled r + ( k + (s k N Thus, (3 becoes (37 ȳ c MSS = ȳ MSS + a (r ( Y r Y r+( k+(s k N Under the linear odel (, we have ȳ MSS = α+β r + (s k ] + ( k u k, u =,,, ( ifk ( uk < r k ( u k, Y r = α+β r and Y r+(s k+( k N = α + β r + (s k + ( k N Putting these values in (37, we have (38 ȳmss c = α + β r + ] k (s k + ( k u a(r( k k Coparing the coefficients of α and β in (33 and (38 and solving for a (r, we have r (k + + ( k uk / (39 a (r =, ( k k u =,,, (, which are picked corresponding to a rando nuber r such that k ( uk < r k ( u k
Saple article to present hjs class 7 Putting a (r in (37, we have r ȳmss c (k + + ( k uk / = ȳ MSS + (30 ( k k ( Yr Y r+( k+(s k N 3 Case (ii: when w > As entioned earlier in Section, when s =, MSS becoes CSS (see ] Therefore, we focus the MSS for s > It is also entioned in Subsection, all k saples are partitioned into two groups The first group contains the saples r k (w k + r and the second group consist of the saples in which r > k (w k + r The corrected saple ean for each saple in the first group is siilar to the corrected saple ean found in Subsection 3, r > k ( k, because the pattern of saples in both situations is siilar Further, the weights assigned to the first and the last units of each saple in this group will be siilar to the weights given in (39, ie ȳ c MSS = ȳ MSS + a (r ( Y r Y r+( k+(s k N, r (k + + ( k uk /n a (r =, ( k k u = (w corresponding to a rando nuber r such that r k ( u k, u = w, w+,, ( w if k ( uk < r k ( u k and u = ( w + if k ( uk < r k ( uk + r Thus, (3 y c MSS = ȳ MSS + r (k++( k uk /n ( k k ( Y r Y r+( k+(s k N The second group having saples in which r > k (w k + r, and the first subgroup consists of the nuber of saples in which k (w xk < r k (w xk + r such that x =,, (w The Yates corrected estiator with l = 3 in (3, for the saples of the first subgroup can be written as (3 ȳ c MSS = ȳ MSS + a 3 (r ( Y r Y r+( k+(s k N Under a linear odel ( ȳ MSS = α + βr + (s k + ( k ( w + x k ], x =,, (w if k (w xk < r k (w xk + r, Y r = α + β r and Y r+(s k+( k N = α + βr + (s k + ( k N Putting these values in (3, we have (33 ȳmss r c = α + β + (s k + ( k ( w + x k ] +a 3 (r( k k Coparing the coefficients of α and β given in (33 and (33 and solving for a 3 (r, we have r (k + + ( k k ( w + x / (34 a 3 (r = ( k k
8 Putting a 3 (r in the corrected estiator given in (3, we have ȳmss c = ȳ (35 MSS + r (k ++( k k ( w+x / ( k k ( Y r Y r+( k+(s k N, x =,, (w, which are picked corresponding to a rando nuber r such that k (w xk < r k (w xk + r Siilarly, the second subgroup consists of the nuber of saples in which k (w xk + r < r k (w xk + k such that x =,,, (w The Yates corrected estiator with l = 4 in (3, for saples of this subgroup, can be written as (36 ȳ c MSS = ȳ MSS + a 4 (r ( Y r Y r+( k+(s k N Under the linear odel (, ȳ MSS = α+βr + (s k +( k ( w + x k ], x =,,, (w if k (w xk < r k (w xk + r, Y r = α + βr and Y r+(s k+( k N = α + β r + (s k + ( k N Putting these values in (36, we have (37 ȳmss r c = α + β + (s k + ( k ( w + x k ] +a 4 (r( k k Coparing the coefficients of α and β given in (33 and (37 and solving for a 4 (r, we have r (k + + ( k k ( w + x/ (38 a 4 (r = ( k k Putting a 4 (r in the corrected estiator given in (36, we have y (39 MSS c = ȳ MSS + r (k++( k k ( w+x/ ( k k ( Y r Y r+( k+(s k N, x =,,, (w, which are picked corresponding to a rando nuber r such that k (w xk < r k (w xk + r 4 Average variance In real life application, we hardly found such population exhibiting perfect linear trend Therefore, it is necessary to study the average variance of the corrected estiator under MSS using following super population odel (4 Y t = α + β t + e t, E(e t = 0, V (e t = E(e t = σ t g, Cov(e t, e v = 0, t v =,, 3,, N and g is the predeterined constant The average variance of ȳ (r MSS under odified systeatic sapling for population odeled by Y t = α + β t + e t is given by
Saple article to present hjs class 9 Case (i when w = (see detial in Appendix B (4 E V (ȳ (r MSS = σ r= χ (u, r k ( k k k ( u k + u= / N +k t= tg r=k χ ( uk+ (u, r N, χ (u, r = δ + (rrg + θ s i= j= r + (i k + (j k g +δ (rr + ( k + (s k g, χ (u, r = δ + u (rrg + θ s n j= (r + (i k + (j k g i= ( s + ( i= u+ j= (r + (i k + (j k g + r + (i k g +(s k N + δ (r (r + ( k + (s k N g, δ + l (r = a l(ra l (r + ( n N, δ l (r = a l(ra l (r ( n ( N and θ = n n N, such that l =, Case (ii When w > (see detial in Appendix B (43 ( E V ȳ (r MSS = σ + w u=w + w+ w k ( u k k u=w k ( u k r= χ (u, r r=k ( uk+ χ (u, r k ( u k u= w+ k (w xk+r + w x= r=k ( uk+ χ (u, r r=k (w xk+ χ 3 (x, r k (w xk+k + w x= r=k χ (w xk+r + 4 (x, r / ] N + k t= tg N, χ (u, r = δ + u (rrg + θ s n j= (r + (i k + (j k g + i= u+ ( s i= j= (r + (i k + (j k g +(r + (i k + (s k N g + δ (r (r + ( k + (s k N g, χ 3 (x, r = δ 3 + w x s (rrg + θ j= r + (i k + (j k g i= ( s + x+ i=w x+ j= r + (i k + (j k g +r + (i k + (s k N g ( s + i= x+ j= r + ( k + (j k g + s j=s r + (i k + (j k N g +δ3 (rr + ( k + (s k N g,
0 χ 4 (x, r = δ + 4 (rrg + θ w x i= ( s s j= r + (i k + (j k g + x i=w x+ j= r + (i k + (j k g +r + (i k + (s k N g ( s + i= x+ j= r + ( k + (j k g + s j=s r + (i k + (j k N g +δ4 (rr + ( k + (s k N g, δ + l (r = a l(ra l (r + ( n N, δ l (r = a l(ra l (r ( n ( N and θ = n n N, for l =, 3, 4 5 Epirical study Due to the coplex nature of the derived expressions, the average variances of MSS and CSS cannot be theoretically copared Therefore, in this paper, a coputer based efficiency coparison of MSS and CSS is ade nuerically under super population odel (4 The nuerical coparison has been ade for N =, N = 30, N = 50 and N = 78 As entioned earliar, if L = N then MSS reduces to LSS and if L = (N n then MSS becoes CSS Therefore, the choice of a saple size considered in this paper is based on the fact that N < L < (N n The relative efficiency of MSS over CSS is presented in Table under g = 0,,, 3 This table includes 40 different cobinations of N and n each at g = 0,, and 3 which are to be considered for efficiency coparison, and it is observed that CSS is not applicable for 4 cobinations Thus, we have 36 4 = 44 results of efficiency coparison and found that MSS is ore efficient than CSS in 35 cases Further, it is to be noted, whenever N n = ( n +, the efficiency of MSS over CSS is draatically increased 5 Natural Population We use the following natural population for efficiency coparison The results are given in Table Population : Source:, page8] Table reflacts that MSS is ore efficient than CSS 6 Conclusion Modified Systeatic Sapling (MSS is a ore general schee than LSS and CSS Because, when least coon ultiple of N and n is equal to lower extree, ie L = N, MSS coincides with LSS If it is equal to upper extree, ie L = (N n, then MSS coincides with CSS However, when L lies between these two extree values, ie N < L < (N n, MSS is advantageous over CSS In this case, the nuber of saples is considerably reduced in MSS as copared to CSS, ie iniu reduction is half of the saples Contrary to the CSS, the explicit expressions for ean and variance of ean are derived for population having perfect linear trend aong the population values Further, nuerical coparison is carried out in this paper clearly favors the use of MSS over CSS for population odeled by a super population odel with linear trend as well as for natural population
Saple article to present hjs class Table Rercent Relative Efficiency (PRE of MSS over CSS under linear trend N n g = 0 g = g = g = 3 N n g = 0 g = g = g = 3 6 3506 933767 70974 346609 78 4 747 3609 46405 47684 9 7385 530 7458 86989 8 9769 86 9578 59786 40000 59045 375565 540 9 9866 367 4684 454488 30 4 3593 650 8004 8574 0 39707 08739 8749 337633 8 463 37608 544687 7006 676695 398988 96993 87947 9 03060 3849 60844 70657 4 - - - - - - - - 5 90595 5753 048 34 4 03355 4453 88675 580 6 9878 33 5333 78994 50 4 565 4355 55795 5804 8 0543 05796 58754 78058 6 0443 3039 375 478 0 57844 45548 7806 9869 8 9789 5605 4667 56789 36637 63353 6076 345550 93959 3054 638 8458 75 4757 893 565 4 4363 84386 6900 30536 4 4967 60473 35995 39945 5 3396 975 37 5503 7 - - - - 6 93665 3668 8496 7408 8 9784 0399 93560 38668 8 7653 33939 66773 7540 30 - - - - 0 8394 3604 6068 69807 3 7980 034 3967 448077 889 93530 75040 35330 33 53748 3090 50355 655560 4 07 59836 3807 33959 34 964 546 33983 48954 36 565 48855 67344 79055 38 00656 707 80950 40473 The sybol (- indicates that CSS is not possible Table Percent Relative Efficiency (PRE of MSS over CSS for Population N = 80 Variance Eff = V (CSS V (MSS 00 n MSS CSS 6 48053500 483600 0084 373340 46858630 5585 4 337760 3784400 3663 5 80650 397650 40805 8 93660 50787490 7967 4 90840 378300 45353 5 7983309 9399580 4300 6 7470 8954 938 8 683677 549070 83566 Here, V (MSS = Variance of odified systeatic sapling and V (CSS = Variance of circular systeatic sapling Acknowledgeent: We would like to thank the two anonyous Referees whose coents helped to iprove the presentation of this paper
References ] Murthy, MN Sapling Theory and Method, Statistical Publishing Society, Calcutta, 967 ] Khan, Z Shabbir, J and Gupta, S A new sapling design for systeatic sapling, Counications in Statistics - Theory and Methods,4, 659-670, 03 3] Khan, Z Gupta, S and Shabbir, J A Note on Diagonal Circular Systeatic Sapling, Journal of Statistical Theory and Practice, 8, 439-443, 04 4] Madow, WG On the theory of systeatic sapling III, Annals of Matheatics and Statistics, 4, 0-06, 953 5] Sapath, S and Varalakshi, V Diagonal circular systeatic sapling, Model Assisted Statistics and Applications, 3 (4, 345-35, 008 6] Sethi, V K On optiu pairing of units, Sankhya B, 7, 35-30, 965 7] Singh, D Jindal, KK and Grag, JNOn odified systeatic sapling, Bioetrika, 55, 4-546, 968 8] Subraani, J Diagonal systeatic sapling schee for finite populations, Journal of the Indian Society of Agriculture Statistics, 53 (, 87-95, 000 9] Subraani, J Further results on diagonal systeatic sapling schee for finite populations, Journal of the Indian Society of Agricultural Statistics, 63 (3, 77-8, 009 0] Subraani, JGeneralization of diagonal systeatic sapling schee for finite populations, Model Assisted Statistics and Applications, 5, 7-8, 00 ] Subraani, J A Modification on linear systeatic sapling, Model Assisted Statistics and Applications, 8 (3, 5-7, 03a ] Subraani, J A further odification on linear systeatic sapling for finite populations, Journal of Statistical Theory and Practice, 7, 47-479, 03b
Saple article to present hjs class 3 Appendix A Mean and variance of MSS for population having linear trend The following linear odel of hypothetical population is to be considered (A Y t = α + β t, t =,, 3,, N, α and β are the intercept and slope of the odel respectively A Mean of MSS The saple ean for both cases, ie w = and w > are separately discussed below: Case (i when w = If r (k ( k, the ean, ȳ MSS can be written as ȳ MSS = α + β s w+ i= s r + (i k + (j k j= After siplification, we have ȳ MSS = α + β r + ] (s k + ( k If k ( uk < r k ( u k for u =,,,, then u ȳ MSS = α + β s s i= j= r + (i k + (j k + s i= u+ j= r + (i k + (j k ] + r + (i k + (s k N After siplification, we have ȳ MSS = α + β r + (s k + ( k u k ] Thus ȳ MSS is a piecewise function of r, ie r + (s k + ( k ] if r k ( k ] (A ȳ MSS = α + β r + (s k + ( k u k u =,,, ( if k ( uk < r k ( u k Case (ii when w > If r k (w k + r, then r ust belongs to any one of the three subgroups which have been discussed in Section Therefore, corresponding to a rando nuber r, an integer u is picked in such a way that u = (w if r k ( u k; u = w, w +, w +,, ( w if k ( uk < r k ( u k and u = ( w + if k ( uk < r k ( uk + r
4 (A3 For each subgroup, ȳ MSS can be written as u ȳ MSS = α + β s s i= j= r + (i k + (j k + s i= u+ j= r + (i k + (j k ] + r + (i k + (s k N After few steps, we have ȳ MSS = α + β r + (s k + ( k u k ] If w >, then it is also possible that k (w xk < r k (w xk+r, such that x =, 3,, w So, w x ȳ MSS = α + β s s i= j= r + (i k + (j k + x+ s i=w x+ j= r + (i k + (j k + r + (i k + (s k N + i= x+ s j= r + (i k + (j k + s j=s r + (i k + (j k N When s =, then Equation (A3 can be expressed as w x ȳ MSS = α + β s s i= j= r + (i k + (j k + x+ s i=w x+ j= r + (i k + (j k + r + (i k + (s k N + s ] i= x+ j=s r + (i k + (j k N Also, when s =, then Equation (A3 can be expressed as w x ȳ MSS = α + β s s i= j= r + (i k + (j k + x+ i=w x+ r + (i k + (s k N + i= x+ r + (i k + (s k N ] After siplifying of Equation (A3 for each case, ie s =, s = and s >, we have ȳ MSS = α + β r + (s k + ( k ( w + x k ] If k (w xk + r < r k (w xk + k, then w x ȳ MSS = α + β s s i= j= r + (i k + (j k + x s i=w x+ j= r + (i k + (j k (A4 + r + (i k + (s k N + i= x+ s j= r + (i k + (j k ] + s j=s r + (i k + (j k N
Saple article to present hjs class 5 When s =, then Equation (A4 can be expressed as w x ȳ MSS = α + β s s i= j= r + (i k + (j k + x s i=w x+ j= r + (i k + (j k + r + (i k + (s k N + i= x+ s j=s r + (i k + (j k N When s =, then Equation (A4 can be expressed as, w x ȳ MSS = α + β s s i= j= r + (i k + (j k + x i=w x+ r + (i k + (s k N + ] i= x+ r + (i k + (s k N After siplification of Equation (A4 for each case, ie s =, s = and s >, we have ȳ MSS = α + β r + (s k + ( k ( w + x k ] Thus, ean of MSS for above odel of hypothetical population with rando start r is given by: r + (s k + ( k u ] N n u = (w if r k ( u k, u = w, w +,, ( w if N ( uk < r k ( u k, u = ( w + if (A5 ȳ MSS = α + β k ( uk < r k ( uk + r r + (s k + ( k ( w + x ] N n, x =,, (w if k (w xk < r k (w xk + r r + (s k + ( k ( w + x ] N n, x =,, 3,, (w if k (w xk + r < r k (w xk + k If w =, then Equation (A5 reduces to r + (s k ] + ( k u N n u = (w if r k ( u k, u = w, w +,, ( w if N ( uk < r k ( u k, ȳ MSS = α+β u = ( w + if k ( uk < r k ( uk + r r + (s k ] + ( k ( w + x N n, x =,, 3,, (w if k (w xk + r < r k (w xk + k A Unbiasedness of saple ean ȳ MSS We have two cases: ]
6 Case (i when w = : Taking the expected value of (A, we have (k ( k E(ȳ MSS = k r= r + (s k + ( k ] + u= k ( u k r=(k ( uk+ r + (s k + ( k u k ] ] As sk = N, then (k ( k E(ȳ MSS = k r= α + β r + (N k + ( k ] + k ( u k u= r=(k ( uk+ α + β r + (N k + ( k ]] u k After a little algebra, we have N + E(ȳ MSS = α + β = Ȳ, which shows that ȳ MSS is an unbiased estiator of Ȳ Case (ii when w > : If w >, we take the expected value of (A5, we have w E(ȳ MSS = k ( u k k u=w r= α + β r + ((s k ] +( k u k + w k ( u k u=w r=k ( uk+ α + β ] +( k u k + w+ k ( uk+r u= w+ r=k ( uk+ α + β ] +( k u k + w k (w xk+r x= r=k (w xk+ α + β ] +( k ( w +xk + w k (w xk+k x= r=k (w xk+r + α + β ]] +( k ( w+xk r + ((s k r + ((s k r + ((s k r + ((s k But if w =, we take the expected value of (A5, given by w E(ȳ MSS = k ( u k k u=w r= α + β r + ((s k ] +( k u k + w u=w k ( u k r=k ( uk+ ] +( k u k + w+ u= w+ ] +( k u k + w x= α + β k ( uk+r r=k ( uk+ k (w xk+k r=k (w xk+r + +( k ( w+xk ]] r + ((s k α + β α + β r + ((s k r + ((s k
Saple article to present hjs class 7 After few steps, we have E(ȳ MSS = α + β k r k + wk + wr ] +kw k w kw + N + E(ȳ MSS = α + β (N + k (w ] +r (w wk(w + k(w ] N + (w (A6 E(ȳ MSS = α + β + (r k wk + k When w = ( q in (A6, we have ] N + ( q E(ȳ MSS = α + β + (r k ( qk + k, ] N + ( q E(ȳ MSS = α + β + (qk + r k k + k, N + ( q E(ȳ MSS = α + β + (qk + r k, N + E(ȳ MSS = α + β = Ȳ the above equation shows that ȳ MSS is unbiased estiator of Ȳ as k = qk+r Note: Putting w = in (A6, we also have N + E(ȳ MSS = α + β = Ȳ A The variance of ȳ MSS V (ȳ MSS = E(ȳ MSS Ȳ = k k r= (ȳ r(mss Ȳ (i when w = k ( k V (ȳ MSS = k ȳr(mss Ȳ + u= r= k ( u k r=k ( uk+ ȳr(mss Ȳ ] k ( k V (ȳ MSS = k r= α + β r + ((s k + ( k ] α + β N+ + k ( u k u= r=k ( uk+ α + βr + ((s k + ( k u k ] ] α + β N+ k ( k V (ȳ MSS = k r= β r + (( k k ] β + k ( u k u= r=k ( uk+ β r + (( k k ] ] u k β
8 After siplification, we have (A7 V (ȳ MSS = b (k + ( k(k k ], Note: If N = nk, then L = N, so =, thus V (ȳ MSS = b (k This is a variance of linear systeatic sapling (ii when w > If w >, then V (ȳ MSS will be expressed as: V (ȳ MSS = k w k ( u k u=w r= k ( u k r=k ( uk+ k ( uk+r u= w+ r=k ( uk+ k (w xk+r r=(k (w xk+ + w u=w + w+ + w x= + w x= ȳr(mss Ȳ ȳr(mss Ȳ ȳr(mss Ȳ ȳr(mss Ȳ k (w xk+k r=(k (w xk+r + ȳr(mss Ȳ ] If w =, then the ter w k (w xk+r x= r=(k (w xk+ ȳr(mss Ȳ will be oitted fro V (ȳ MSS w V (ȳ MSS = ( k ( u k k u=w r= α + β r + (s k +( k u k ] α + β N+ + w ( k ( u k u=w r=k ( uk+ α + β r + (s k +( k u k ] α + β N+ + w+ ( k ( uk+r u= w+ r=k ( uk+ α + β r + (s k +( k u k ] α + β N+ + w k (w xk+r x= r=(k (w xk+ α + β r + (s k +( k ( w + x k ] α + β N+ + w k (w xk+k x= r=(k (w xk+r + α + β r + (s k +( k ( w + x k ] ] α + β N+,
(A8 w V (ȳ MSS = k u k β + w u=w β Saple article to present hjs class 9 u=w ] u k + w+ u k k ( u k r= β r + (( k k k ( u k r=k ( uk+ u= w+ ] β k (w xk+r r=k (w xk+ + w x= ( w + x k + w x= β r + (( k k k ( uk+r r=k ( uk+ β r + (( k k β α + β ] k (w xk+k r=k (w xk+r + ( w + x k β ] r + (( k k α + β r + (( k k If w =, then Equation (A8 reduces to w V (ȳ MSS = k u k β + w u=w β u=w ] u k + w+ u k k ( u k r= β r + (( k k k ( u k r=k ] ( uk+ β u= w+ ] β k (w xk+k r=k (w xk+r + + w x= ( w + x k After siplification, we have r + (( k k k ( uk+r r=k ( uk+ β r + (( k k β ] ] α + β r + (( k k V (ȳ MSS = b (k + ( k(k k +4(w 3k( q w (k qk + (k k ] +k w 3k (3 w + k The ter (w 3k( q w (k qk + (k k] will be vanished in both situations, when w = or w = ( q So, we are left with (A9 V (ȳ MSS = b (k + ( k(k k +4w(w k 3k (3 w + k]
0 Appendix B Average variance In real life application, we hardly found such population exhibiting perfect linear trend, therefore, it is necessary to study the average variance of the corrected estiator under MSS using following super population odel (B Y t = α + βt + e t, E(e t = 0, V (e t = E(e t = σ t g, Cov(e t, e v = 0, t v =,, 3,, Nand g is a predeterine constant Under the above super population odel (B, the average variance expression of MSS is given below: Case (i when w = Consider that l th su of squares (SSl are given by (B SSl = (B3 ȳ (r MSS Ȳ ] = ȳmss + a l (r (Y r Y rn Ȳ ], l = if r k ( k and l = if r > k ( k When r k ( k, the expressions of ȳ MSS, Ȳ, Y rand Y rn under the odel Y t = α + β t + e t, can be expressed as: ȳ MSS = α + β r + (s k + ( k ] + s i= s j= e r+(i k+(j k, Ȳ = α + β N+ + N N t= e t, Y r = α + βr + e r α + β r + ( k + (s k + e r+( k+(s k and Y rn = Substituting these expressions in (B, we have ] SS = ȳ (r MSS Ȳ = s n i= j= e r+(i k+(j k +na (r ( e r e r+( k+(s k N N t= t] e Siilarly, if r > k ( k the expressions of ȳ MSS, Ȳ, Y r and Y rn under the super population odel Y t = α + β t + e t, can be written as ȳ MSS = α + β r + (s k ] + ( k u k u + s s i= j= e r+(i k+(j k + s ] i= u+ j= e r+(i k+(j k + e r+(i k+(s k N, Ȳ = α + β N+ + N N t= e t, Y r = α + βr + e r α + β r + ( k + (s k N + e r+( k+(s k N and Y rn = Thus, SS = ȳ (r MSS Ȳ ] = n + i= u+ u i= ( s s j= e r+(i k+(j k j= e r+(i k+(j k + e r+(i k+(s k N + na (r(e r e r+( k+(s k N N N t= t] e The average variance of the corrected saple ean can be written as: E V (ȳ (r MSS k ( k = k + u= r= E (SS k ( u k r=k E (SS ( uk+
Saple article to present hjs class under the assuptions of super population odel s E (SS = n i= j= E(e r+(i k+(j k + n a (r E(e r +E(e r+( k+(s k + na (r E(e r E(e r+( k+(s k + N N t= E(e t s nn i= j= E(e r+(i k+(j k ] +na (r E(e r E(e r+( k+(s k, E (SS = σ n i= s j= (r + (i k + (j k g +n a (r r g + (r + ( k + (s k g +na (r r g (r + ( k + (s k g s nn i= j= (r + (i k + (j k g +na (r r g (r + ( k + (s k g + N N t= tg ] (B4 E (SS = σ a (r ( a (r + ( n N r g ( + n n s N i= j= r + (i k + (j k g +a (r ( a (r ( n N r + ( k + (s k g + N N t= tg Siilarly, E (SS = u n i=( s + i= u+ + i= u+ +E(e r+(i k+(s k N s j= E(e r+(i k+(j k j= E(e r+(i k+(j k +E(e r+(i k+(s k N + n a (r E(e r +E(e r+( k+(s k N + na (r (E(e r E(e r+( k+(s k N u s nn i= j= E(e r+(i k+(j k ( s j= E(e r+(i k+(j k + na (r (E(e r E(e r+( k+(s k N + ] N N t= E(e t,
(B5 E (SS = u n i=( s s j= (r + (i k + (j k g + i= u+ j= (r + (i k + (j k g + (r + (i k + (s k N g + n a (r r g + (r + ( k + (s k N g +na (r r g (r + ( k + (s k N g u s j= (r + (i k + (j k g nn i= ( s + i= u+ j= (r + (i k + (j k g + (r + (i k + (s k N g + na (r r g (r + ( k + (s k N g + ] N N t= tg, E (SS = a (r ( a (r + ( n N r g ( n u s N n j= (r + (i k + (j k g + n i= ( s + i= u+ j= (r + (i k + (j k g + (r + (i k + (s k N g +a (r ( a (r ( n N ( r + ( k + (s k N g + N N t= tg ] Equations (B4 and (B5 can be written as: and E (SS = σ δ + (rrg + θ s i= j= r + (i k + (j k g +δ (rr + ( k + (s k g + N N t= tg E (SS = σ δ + u (rrg + θ s n i= j= (r + (i k + (j k g + ( s i= u+ j= (r + (i k + (j k g + (r + (i k + (s k N g +δ (r (r + ( k + (s k N g + N N t= ], tg δ + l (r = a l(ra l (r + ( n N, δ l (r = a l(ra l (r ( n ( N and θ = n n N, such that l =, Also (B6 E (SS = σ χ (u, r + N N t= tg and (B7 E (SS = σ χ (u, r + N N t= tg,
Saple article to present hjs class 3 χ (u, r = δ + (rrg + θ s i= j= r + (i k + (j k g +δ (rr + ( k + (s k g, and χ (u, r = δ + u (rrg + θ s n j= (r + (i k + (j k g i= ( s + i= u+ + (r + (i k + (s k N g j= (r + (i k + (j k g +δ (r (r + ( k + (s k N g Substituting the values of E (SS and E (SS in (B3, we have E V (ȳ (r MSS k ( k = σ k r= χ (u, r (B8 + k ( u k u= r=k χ ( uk+ (u, r / N +k t= tg N Case (ii when w > We can write ] (B9 SSl = ȳ (r MSS Ȳ = ȳmss + a l (r (Y r Y rn Ȳ ], l = if r k (w k + r, l = 3 if k (w xk < r k (w xk + r such that x =,, ( and l = 4 if k (w xk+r < r k (w xk+k such that x =,,, Furtherore, when r k (w k + r, we realize whether r k ( u k such that u = w, k ( uk < r k ( u k such that u = w, w +,, ( w or k ( uk < r k ( uk + r such that u = ( w + However, for each of these subgroups E (SS will be used Thus, the average variance of the corrected saple ean can be expressed as (B0 E V ( ȳ (r MSS ] w = k ( u k N u=w k ( u k + n w u=w + w+ r= E SS] r=k ( uk+ E SS] k ( uk+r u= w+ k (w xk+r + w x= + w x= r=k ( uk+ E SS] r=k E SS3] (w xk+ k (w xk+k r=k E SS4] (w xk+r + ] The E (SS is already obtained in case of w =, ie (B E (SS = χ (u, r + N N t= tg Now consider (B E (SS3 = E ȳ MSS + a 3 (r (Y r Y rn Y ]
4 Under the super population odel, we have ȳ MSS = α + β r + (s k + ( k ( w + x k w x + s n i= j= e r+(i k+(j k + x+ s i=w x+ j= e r+(i k+(j k + e r+(i k+(s k N + i= x+ s j= e r+(i k+(j k + s j=s e r+(i k+(s k N ], Ȳ = α + β N+ + N N t= e t, Y r = α + βr + e r β r + ( k + (s k + e r+( k+(s k and Y rn = α + Substituting these expressions in (B, we have E (SS3 = E n w x + x+ i=w x+ i= s s s j= e r+(i k+(j k j= e r+(i k+(j k + e r+(i k+(s k N + i= x+ j= e r+(i k+(j k + s + na (r(e r e r+( k+(s k N N N t= t] e j=s e r+(i k+(j k N Applying the assuption of super population odel, we have E (SS3 = w x n i= s s j= E(e r+(i k+(j k + x+ i=w x+ +E(e r+(i k+(s k N s j= E(e r+(i k+(j k + i= x+ j= E(e r+(i k+(j k + s j=s E(e r+(i k+(j k N +n a (r E(e r + E(e r+( k+(s k N +na (r E(e r E(e r+( k+(s k N nn w x i= + x+ i=w x+ +E(e r+(i k+(s k N s j= E(e r+(i k+(j k s j= E(e r+(i k+(j k s + i= x+ j= E(e r+(i k+(j k + s j=s E(e r+(i k+(j k N +na (r E(e r E(e r+( k+(s k N + N N t= E(e t ]
Saple article to present hjs class 5 E (SS3 = w x n i= s s j= (r + (i k + (j k g + x+ i=w x+ j= (r + (i k + (j k g + (r + (i k + (j k N g s + i= x+ j= (r + (i k + (j k g + s j=s (r + (i k + (j k N g +n a (r r g + (r + (i k + (j k N g +na (r r g (r + (i k + (j k N g w x s j= (r + (i k + (j k g nn i= s + x+ i=w x+ j= (r + (i k + (j k g + (r + (i k + (j k N g s + i= x+ j= (r + (i k + (j k g + s j=s (r + (i k + (j k N g +na (r r g (r + (i k + (j k N g + N N t= tg ], E (SS3 = a 3 (r ( a 3 (r + ( n N k (w xk+r r=k (w xk+ + w x= +(i k + (j k g ( s r g n ( n w x s N i= j= r + x+ i=w x+ j= r + (i k + (j k g +r + (i k + (s k N g ( s + i= x+ j= r + ( k + (j k g + s j=s r + (i k + (j k N g +a 3 (r ( a 3 (r ( n N r + ( k + (s k N g + N N t= tg, E (SS3 = δ + 3 (rrg + θ w x i= ( s s j= r + (i k + (j k g + x+ i=w x+ j= r + (i k + (j k g +r + (i k + (s k N g ( s + i= x+ j= r + ( k + (j k g + s j=s r + (i k + (j k N g +δ3 (rr + ( k + (s k N g + N N t= tg, δ 3 + (r = a 3(ra 3 (r + ( n N and δ 3 (r = a 3 (ra 3 (r ( n N
6 Also (B3 E (SS3 = χ 3 (x, r + N N t= tg, χ 3 (x, r = δ 3 + w x s (rrg + θ j= r + (i k + (j k g Siilarly, i= ( s + x+ i=w x+ j= r + (i k + (j k g +r + (i k + (s k N g ( s + i= x+ j= r + ( k + (j k g + s j=s r + (i k + (j k N g +δ3 (rr + ( k + (s k N g (B4 E (SS4 = χ 4 (x, r + N N t= tg, χ 4 (x, r = δ 4 + w x s (rrg + θ j= r + (i k + (j k g i= ( s + x i=w x+ j= r + (i k + (j k g +r + (i k + (s k N g ( s + i= x+ j= r + ( k + (j k g + s j=s r + (i k + (j k N g +δ4 (rr + ( k + (s k N g Putting E (SSl for l =, 3, 4 in (B0, we have ( E V ȳ (r w MSS = σ k ( u k k u=w r= χ (u, r + w k ( u k u=w r=k χ ( uk+ (u, r + w+ k ( u k u= w+ r=k χ ( uk+ (u, r + w k (w xk+r x= r=k χ (w xk+ 3 (x, r + w k (w xk+k x= r=k χ (w xk+r + 4 (x, r / N +k t= tg N ]