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17 5 16 1 2 2 2 3 4 4 5 5 7 5.1... 8 5.2... 9 6 10 1

1 (sample survey metod) 1981 4 27 28 51.5% 48.5% 5 10 51.75% 48.24% (complete survey ( ) ) (populatio) (sample) (parameter) (estimator) 1936 200 2 N U, s f = /N (samplig fractio) N s S (support) s (probability samplig) p(s) =P (s ), s S (1) (samplig desig) s S s (samplig witout replacemet) s S (samplig wit replacemet) 2

(1) θ ˆθ E (ˆθ) =θ (2) ˆθ θ (desig ubiased estimator) (2) (asymptotic desig ubiasedess) E (ˆθ θ)/ E (ˆθ θ) 2 0 (3) (asymptotic desig cosistecy) ( ) ˆθ θ P >ɛ 0, ɛ >0 (4) θ (3) (4), N V(ˆθ)/θ 2 0 (3) (4) i π i, i, j π ij π i = P (i s), π ij = P (i, j s) s S s S π i 1 (first-order iclusio probability) π ij 2 (secod-order iclusio probability) π ii = π i N π i =, N π ij =( 1)π i j i i Z i =1 Z i =0 Z i Z i Z j E (Z i )=π i, V(Z i )=π i (1 π i ). (5) cov (Z i,z j )=π ij π i π j (6) (populatio total), Y = i U y i, (populatio mea), µ = Y/N, y i i π i > 0 (1) Horvitz-Tompso (Horvitz- Tompso estimator), π (π-estimator) Ŷ HT y i /π i (7) 3

Y, ŶHT = N Z iy i /π i (5) Ŷ HT 1/π i (desig weigt) (6) ŶHT V(Ŷ HT )= N i,j=1 π ij π i π j π i π j y i y j (8) i y i (auxiliary iformatio) x i x X (ratio estimator) Ŷ R = y i/π i x X (9) i/π i x i =1 X = N ŶR Hájek (Hájek estimator) Ŷ Haj N y i/π i 1/π i (10) (omogeeous) π i π i y i Hájek Horvitz-Tompso (7) 3 (simple radom samplig) p(s) =1/ N C 1 (frame) 1 2 π i = N, π ij = N 1 N 1 (11) 4

(i j) (11) (7), (10) ŶHT = ŶHaj π (sample mea), ȳ = 1 s y i, N Ŷ sr = N y i = N ȳ (12) (8), (11) Ŷsr V(Ŷsr )=N 2 (1 f) σ2 σ 2 (populatio variace) (13) σ 2 = 1 N 1 N (y i µ) 2 (14) (13) 1 f (fiite populatio correctio) µ ȳ µ V(ȳ) =(1 f) σ2 σ 2 ˆσ 2 = 1 1 (y i ȳ) 2 (15) (13) ˆσ 2 σ 2 E (ˆσ 2 )=σ 2 (16) f, N, N ȳ µ, (15) (16) µ 1 2α ( ) ˆσ P µ ȳ ± z α 1 f 1 2α (17) z α 100α 4 N U (stratum) U 1,,U L (stratified samplig) 5

(stratified radom samplig) (stratificatio) U Y = Y Y U N U s ( =1,,L), s π Ŷ st = ŶHT = y i /πi (18) =1 Y y i π i U i 1 V(Ŷ st )= V(Ŷ HT ) V(Ŷ st )= =1 i,j=1 π ij π i π j π i π j y i y j (19) πij U i, j 2 f = /N = f (18) π i Ŷ sr st = N ȳ (20) ȳ µ = Y /N σ 2 = i (y i µ ) 2 /(N 1) (19) Ŷsr st V(Ŷsr st )= N 2 (1 f ) σ2 (21) (20) (sample allocatio), 1,, L (proportioal allocatio) f = /N = /N = f Ŷst sr V prop = 1 f w σ 2 (22) w = N /N (stratum weigt) (liear cost fuctio) C = c 0 + c (c 0 0,c > 0: ) C (21) 1,, L (optimum allocatio) (21) = N σ / c, =1,,L (23) 6

σ 2 N,σ 2 c c c = c (23) = N σ N σ, =1,,L (24) (24) (Neyma allocatio) Ŷ sr st (21) ( ) V opt = 1 2 w σ 2 1 N w σ 2 (25) 1/N σ 2 = w σ 2 + w (µ µ) 2 Ŷsr (13) V(Ŷsr )=V prop + 1 f w (µ µ) 2 (26) (26) µ (22) (25) V prop = V opt + 1 w (σ σ) 2 (27) σ = w σ σ V(Ŷsr ) V prop V opt 5 y x x y Y 7

(ratio estimator) x X X Y π ˆX HT Ŷ HT Y (ratio estimator) Ŷ R ŶHT ˆX HT X = y i/π i x i/π i X (28) 5.1 (28) Ŷ R = ȳ x X (29) x i Ŷ R ŶR (bias) E(Ŷ R ) Y = N 2 (1 f) (Rσx 2 X σ xy) (30) σ xy = i (y i µ y )(x i µ x )/(N 1), R = Y/X ŶR (31), V(Ŷ R )= N 2 (1 f) (σy 2 + R2 σx 2 2Rσ xy) (31) ρ> 1 σ x /µ x, 2 σ y /µ y ρ = σ xy σ x σ y V(ŶR) < V(Nȳ) ŶR Nȳ y i = βx i + ɛ i, i =1,,N (32) Ŷ R (BLUE; best liear ubiased estimator) ŶR l i y i (32), x i ɛ i,ɛ j E(ɛ i )=0,V(ɛ i ) x i (32) Y = βx + N ɛ i Ŷ R E(ŶR) =E(Y) =βx 8

(model ubiasedess) (x i,y i ) y i x i Ŷ R Ŷ Rt = rx + (N 1) (ȳ r x) ( 1)X r = 1 y i/x i 5.2 X (separate ratio estimator) Ŷ Rs = ȳ x X (33) (combied ratio estimator) Ŷ Rc = N ȳ N X (34) x X ŶRs ŶRc ŶRs ŶRc V(ŶRs) = V(Ŷ Rc ) = N 2 (1 f ) (σ 2 y + R 2 σ 2 x 2R σ xy ) N 2(1 f ) (σy 2 + Rσ2 x 2Rσ xy) R V(Ŷ Rs ) < V(Ŷ Rc ) 2 (mea squared error) V(ŶRs) = N σ d N, =1,,L σ d σd 2 N σd 2 =(N 1) 1 (y i R x i ) 2 9

6 (samplig error) (delta metod) (bootstrap metod) (jackkife metod) s s θ ˆθ i ˆθ i (jackkife pseudovalue) θ i = ˆθ ( 1)ˆθ i (35) } θ i = ˆθ ( 1) {ˆθ 1 f (ˆθ ˆθ i ) (36) f = /N f 0 (36) (35) ˆθ (jackkife estimator) J(ˆθ) = 1 θ i = ˆθ ( 1)ˆθ ( ) (37) ˆθ ( ) = 1 ˆθ i ˆθ V J (ˆθ) = 1 1 ( θ i J(ˆθ)) 2 1 = 1 (ˆθ i ˆθ ( ) ) 2 (38) θ θ = l θ (39) =1 l > 0 θ Y = Y µ = w µ (39) θ ˆθ ˆθ = l ˆθ (40) =1 10

ˆθ ˆθ θ (40) ˆθ V(ˆθ) = l 2 V(ˆθ ) =1 ˆθ V Js (ˆθ) = l 2 V J (ˆθ ) (41) =1 V J (ˆθ ) (38) (37) J(ˆθ ) J s (ˆθ) = l J(ˆθ ) (42) =1 [1] W. G. Cocra, Samplig Teciques, 3, Jo Wiley, 1977. [2] C.-E. Särdal, B. Swesso, J. Wretma, Model Assisted Survey Samplig, Spriger, 2003. [3] ( ),,, 2002. [4] ( ),,, pp.242 257, 1989. 11

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