2009 6 Chinese Journal of Applied Probability and Statistics Vol.25 No.3 Jun. 2009 (,, 20024;,, 54004).,,., P,. :,,. : O22... (Credibility Theory) 20 20, 80. ( []).,.,,,.,,,,.,. Buhlmann Buhlmann-Straub ( [2])., 999 Frees, E.W., ( [3]).,,, ( [5]).,,,. [3],,,.,,, P, F. (066006) (200707MS33). 2007 2, 2008 4 4.
: 29 : n ( ),,. T, T +,. y ij i =, 2,, n, j =, 2,, T, y ij y ij = β + jβ 2 + α i + ɛ ij i =, 2,, n, j =, 2,, T, (.) β, β 2,. jβ 2,. β, β 2, α i i, ɛ ij, Var (α i ) = σα, 2 Var (ɛ ij ) = σɛ 2, α i N(0, σα), 2 ɛ ij N(0, σɛ 2 ), α i ɛ ij. i : Y i = X i β + U i α i + ɛ i i =, 2,, n, (.2) Y i = (y i, y i2,, y it ) 2, X i = (X i, X i2 ) =.., U i = T, T T, ɛ i = (ɛ i, ɛ i2,, ɛ it ), β = (β, β 2 ), Y i Cov (Y i ) = σα 2 T T + σɛ 2 I T = V i. (.3). 2., ( [3]). 2. Y N, E(Y ) = Xβ, Var (Y ) = V. W, E(W ) = λ t β, Var (W ) = σw 2. Cov (W, Y ) = Cov W Y, Y W W = E(W ) + Cov W Y V [Y E(Y )] = λ t β + Cov W Y V (Y Xβ).,.
292 : Y = Xβ + Uα + ɛ, (2.) Y =(y, y 2,, y T, y 2,, y nt ) =(Y, Y 2,, Y n), X = n X i, U =I n T, α = (α, α 2,, α n ), ɛ = (ɛ, ɛ 2,, ɛ T, ɛ 2,, ɛ nt ) = (ɛ, ɛ 2,, ɛ n ). T +, E(y i,t + α i ) = β + (T + )β 2 + α i, i =, 2,, n. (2.2) W i = β + (T + )β 2 + α i, i =, 2,, n, E(W i ) = β + (T + )β 2, j = i, Cov (W i, Y j ) = σ 2 α T, j i, Cov (W i, Y j ) = 0, [3] (3), W i W Y i = T i = E(W i ) + Cov Wi Y V i (Y i X i β) ( = T σ2 ) ( α T σ 2 ) α σɛ 2 + T σα 2 β + (T + ) 2(σɛ 2 + T σα) 2 β 2 + T σ2 α σɛ 2 + T σα 2 Y i, (2.3) 2. T y ij.. (.) ŷ i,t + = ( ξ) β + (T + )( ξ/2) β 2 + ξy i, i =, 2,, n, (2.4) ξ = T σ 2 α/( σ 2 ɛ + T σ 2 α), Y i = T T y ij..,,,., β, β 2, σ 2 α σ 2 ɛ,. 3.,. ( H T, H = T / ) T, H (T ) T, H HH = I T, H T = 0. H, H = (H 2,, H T ), H 2 = (/ 2, / 2, 0,, 0), H 3 = (/ 2 3, / 2 3, 2 2 3, 0,, 0),
: 293. Z i = H Y i, Z i = (z i, z i2,, z it ), Z i(2) = (z i2, z i3,, z it ). : ) Z i : Cov (Z i ) = diag(σɛ 2 + T σα, 2 σɛ 2,, σɛ 2 ), {z it : i =,, n, t =,, T }. 2) H T / T, z i = ( T ) / T y it, t= E(zi ) = T β + ( + T ) T 2 β 2. {z i : i =,, n},, ( ( N T β + + T ) β 2 ), σɛ 2 + T σα 2 = N(µ, στ 2 ) i =, 2,, n. 2 3) H T = 0, E(Z i(2) ) = HX i2 β 2, η = (η 2, η 3,, η T ) = HX 2 β 2, t = 2, 3,, T, {z it : i =,, n} N(η t, σ 2 ɛ ). {z i : i =,, n} T {β + [( + T )/2]β 2 } σ 2 ɛ + T σ 2 α µ = z = z i, σ τ 2 = SS = n n (z i z ) 2. (3.) t = 2, 3,, T, {z it : i =,, n} N(η t, σ 2 ɛ ), β 2, σ 2 ɛ : [ / T β 2 = (H t X i2 ) 2] T H t X i2 z it, (3.2) σ ɛ 2 T = (z it H t X i2 β2 ) 2. (3.3), 3. β, β 2, σ 2 α σ 2 ɛ β = z + T [ / T β 2, β2 = T 2 t=2 σ α 2 = T (SS σ2 ɛ ) = ( T n σ 2 ɛ = T (z i z ) 2 σ 2 ɛ (z it H t X i2 β2 ) 2, (H t X i2 ) 2] T ), SS = i z ) n (z 2, z = n z i, H t X i2 z it, H t H, H (T ) T, HH = I T, H T = 0.
294 3.2 β, β 2 β, β 2 ; { / [ ]} σ 2 ɛ, (/T ) {[n/(n )] σ 2 τ [/[ ]] σ 2 ɛ } σ 2 ɛ, σ 2 α. :,,, [6] 4..4 β 2 β 2, µ µ, µ = T {β +[( + T )/2]β 2 }, β β. {/[ ]} σ 2 ɛ σ 2 ɛ, [n/(n )] σ 2 τ σ 2 τ, σ 2 τ = σ 2 ɛ +T σ 2 α, (/T ){[n/(n )] σ 2 τ [n(t )/[n(t ) ]] σ 2 ɛ } σ 2 α.?,,. 4., P, ( [4]). 4. Z h E(Z h ) ( E(Z h q / ) = C b b j r ) h { r / j a b q } k k Γ[a k ( + h) + ξ k ] Γ[b j ( + h) + η j ], (4.) C, Z 0 E(Z 0 ) =, C = q / r Γ[b j + η j ] Γ[a k + ξ k ]. (4.2) a k =O(n) (k =, 2,, r) b j =O(n) (j =, 2,, q), ξ k (k =, 2,, r) η j (j =, 2,, q), q r n. r q a k = b j, [ r f = 2 ρ = f [ r ξ k q η j ], 2 (r q) (4.3) a ω 2 = 6ρ 2 { r B 2(ξ k ) a 2 k q B 3[( ρ)a k + ξ k ] B 2 (h) B 3 (h) Bernoulli : ] b B 2(η j ), (4.4) q B 2 (h) = h 2 h + 6, B 3(h) = h 3 3 2 h2 + 2 h. } b 2 j B 3 [( ρ)b j + η j ], (4.5)
: 295 P( 2 ln Z x) = P(χ 2 (f) x) + O(n ), (4.6) P( 2ρ ln Z x) = P(χ 2 (f) x) + O(n 2 ), (4.7) P( 2ρ ln Z x) = P(χ 2 (f) x) + ω 2 [P(χ 2 (f + 4) x) P(χ 2 (f) x)] + O(n 3 ).(4.8) [4] 8.2.4., H 0 : σα 2 = 0 H : σα 2 0., σ 2 τ = σ 2 ɛ + T σ 2 α 3 z i. H 0 : σ 2 τ = σ 2 ɛ H : σ 2 τ σ 2 ɛ, σ 2 α = 0, β, β 2 σ 2 ɛ z = n n : Λ = = β = z + T β 2, T 2 [ / T β 2 = (H t X i2 ) 2] T H t X i2 z it, (4.9) σ ɛ 2 = [ (z i z ) 2 + T (z i H t X i2 β2 ) 2], (4.0) nt z i. 3. max L(β, β 2, σɛ 2 ) H 0 max L(β, β 2, σɛ 2, σα) 2 H 0 H ( [ (z i z ) 2 + T nt t=2 ( ) n/2 ( (z i z ) n 2 ]) nt/2 (z it H t X i2 β2 ) 2 T ). (4.) n(t )/2 (z it H t X i2 β2 ) 2 Λ,. 4, 3, χ 2, 2 ln Λ L χ 2 (), n. P P(χ 2 () 2 ln Λ).
296 P : P = n (z i z ) 2, P 2 = T (z it H t X i2 β2 ) 2, Λ = (nt ) nt/2 ( n n/2 () n(t )/2 P = P /(P + P 2 ), Λ = P ) n/2( P + P 2 P 2 (nt ) nt/2 n n/2 () n(t )/2 (P )n/2 ( P ) n(t )/2. P + P 2 ) n(t )/2. (4.2), P Beta ( (n )/2, [ ]/2 ). Λ h,, E(Λ h ) = r = 2, a = n 2, a 2 = T nt h/2 P. f, ρ ω 2 f = [ 2 ( 2 ) 2 [ 2 ρ = f ω 2 = 6ρ 2 { 2 = 4 6n 2 ρ 2 { T 2 B 3 = 4 6n 2 ρ 2 {( h 3 + + 2 ( h + Γ((nT 2)/2) (T ) n(t )h/2 Γ((n )/2)Γ([ ]/2) Γ([n(h + ) ]/2)Γ([(h + ) ]/2). (4.3) Γ([nT (h + ) 2]/2), ξ = ξ 2 = 2 2, q =, b = nt 2, η =, a B 2( /2) ( ) 2 (2 ) ] =, ] b j B 2 ( ) = T 2 26T + 26, 6T } a 2 k B 3[( ρ)a k + ξ k ] b 2 B 3[( ρ)b j + η j ] [ T 2 32T + 26 [ 5T 2 20T + 26 B 3 2T (T ) [ T 2 38T + 38 ] 2(T ) ] + (T ) 2 h3 2 (T ) 2 h 2 T 2 h 3 T 2 h3 3 )}, (T ) 2 B 3 2T ]} ) 3 ( h 2 + 2 (T ) 2 h2 2 ) T 2 h2 3, h = 5T 2 20T + 26 2T (T ), h 2 = T 2 32T + 26 2T, h 3 = T 2 38T + 38. 2(T )
: 297, O(n ), O(n 2 ) O(n 3 ) P P(χ 2 () 2 ln Λ)+O(n ), P(χ 2 () 2ρ ln Λ)+O(n 2 ), P(χ 2 () 2ρ ln Λ)+ ω 2 [P(χ 2 (5) 2ρ ln Λ) P(χ 2 () 2ρ ln Λ)]+O(n 3 ). O(n 3 ) P., α = 0.05, T = 5, σɛ 2 =, n = 0 n = 30 n = 50 σ 2 α = 0.5 0.28 0.779 0.926 σ 2 α = 0.7 0.595 0.980 0.990 σ 2 α = 0.9 0.607 0.974 0.998,,. : H 0 : β 2 = 0, 5. H 0 : β 2 = 0 H : β 2 0. y ij = β + α i + ɛ ij i =, 2,, n, j =, 2,, T. (5.) : β = n z i, σ ɛ 2 = T σ 2 = n n (z i T β ). : Λ = = = T zit, 2 σ α = T ( σ2 σ ɛ 2 ), max L(β, β 2, σɛ 2 ) H 0 max L(β, β 2, σɛ 2, σα) 2 H 0 H ( ) n/2 ( T ) n(t )/2 (z i z ) 2 zit 2 n nt ( ) n/2 ( (z i z ) n 2 T ) n(t )/2 (z it H t X i2 β2 ) 2 ( T ) n(t )/2 T n(t )/2 zit 2 ( T ). (5.2) n(t )/2 (T ) n(t )/2 (z it H t X i2 β2 ) 2 t=2
298 Q = T zit 2, Q 2 = T, Q σ 2 ɛ χ 2 (). (z it H t X i2 β2 ) 2, Q 2 σɛ 2 χ 2 ( ). Λ, Λ, Q /Q 2,. Q Q 2 = Q Q 2 Q 2 +. (5.3) Q /Q 2, (Q Q 2 )/Q 2,. Q Q 2 σ 2 ɛ χ 2 (), Q 2, F =, F F (, ), P (Q Q 2 )/ Q 2 /[ ]. (5.4) P = P(F (, ) F ). (5.5), n = 0, 30, 50, α = 0.05, T = 5, β =, 2 n = 0 n = 30 n = 50 β 2 = 0. 0.074 0.28 0.202 β 2 = 0.3 0.289 0.72 0.96 β 2 = 0.5 0.692 0.988 0.997,,. 6. Hachemeister (975) [7] 970 7 973 6. 3. [3],, : y ij = β + jβ 2 + α i + ɛ ij i =, 2,, 5, j =, 2,, 2.,,. : H 0 : β 2 = 0 H : β 2 0.
: 299 (5.5) F, P :,. P = P(F (, 54) F ) < 0.00. 3 2 3 4 5 738 364 759 223 456 2 642 408 685 46 499 3 794 597 479 00 609 4 205 444 763 257 74 5 2079 342 674 426 482 6 2234 675 203 532 572 7 2032 470 502 953 606 8 2035 448 622 23 735 9 25 464 828 343 607 0 2262 83 255 243 573 2267 62 2233 762 63 2 257 47 2059 306 690, H 0 : σα 2 = 0 H : σα 2 0. 4 P, P(χ 2 () 2 ln(λ)) < 0.00.,.,. 3 2, 5 3, : ŷ i,3, i =, 2,, 5, 4,, 3. 4 2 3 4 5 3 2257 728 2025 584 82
300. [],, :, 2004. [2] Buhlmann, H. and Gisler, A., A Course in Credibility Theory and Its Application, Springer, 2005. [3] Frees, E.W., Young, V.R. and Luo Y., A longitudinal date analysis interpretation of credibility models, Insurance: Mathematics and Economics, 24(999), 229 248. [4] Anderson, T.W., An Introduction to Multivariate Statistical Analysis (Second Edition), New York: John Wiley, 984. [5],,, :,, 998. [6],, :, 2004. [7] Hachemeister, C.A., Credibility for regression models with application to trend in credibility theory and application, Proceedings of the Berkeley Actuarial Research Conference on Credibility, Academic Press, New York, 975, 29 63. [8] Lo, C.H., Fung, W.K. and Zhu, Z.Y., Generalized estimating equations for variance and covariance parameters in regression credibility models, Insurance: Mathematics and Economics 39(2006), 99 3. [9] Antonio, K. and Beirlant, J., Actuarial statistics with generalized linear mixed models, Insurance: Mathematics and Economics, 40(2007), 58 76. Estimating and Testing Parameter for Regression Credibility Model with Linear Trend Tang Guoqiang (School of Finance and Statistics, East China Normal University, Shanghai, 20024 ) (Department of Mathematics and Physics, Guilin University of Technology, Guilin, 54004 ) In this paper, the parameters of regression credibility model with linear trend are estimated and tested. Orthogonal transformation is used to estimate parameter and unbiased estimate of parameters are obtained. Likelihood ratio test is used to test randomness and linear trend. The better P-value of likelihood ratio test is got and Monte-Carlo simulation is performed. Keywords: Regression credibility model, orthogonal transformation, likelihood ratio test. AMS Subject Classification: 62P05.