(X i,y, i i ), i =1, 2,,n, Y i = min (Y i, C i ), i = I (Y i C i ) C i. Y i = min (C, Xi t θ 0 + i ) E [ρ α (Y t )] t ρ α (y) =(α I (y < 0))y I (A )

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2 Y F Y ( ) α q α q α = if y : F Y (y ) α. α E [ρ α (Y t )] t ρ α (y) =(α I (y < 0))y I (A ) Y 1,Y 2,,Y q α 1 Σ i =1 ρ α (Y i t ) t X = x α q α (x ) Y q α (x )= if y : P (Y y X = x ) α. q α (x ) θ α q α (x ) x q α (x )=x t θ α θ α 1 Σ i =1 ρ α (Y i X t i θ ) θ (X i,y, i i ), i =1, 2,,, Y i = mi (Y i, C i ), i = I (Y i C i ) C i Y i = mi (C, Xi t θ 0 + i ) i C θ 0

3 1 Σ i =1 ρ α (Y i mi (Xi t θ, C )) C i T i (C i, T i ) (X i, Y i ) T i Y i (X i, Ỹ i, i, T i ), i =1, 2,, with Ỹ i T i S (t ) S(t )=P(Y t ) G (t )=P(T t C ) τ =if t : G (t ) > 0 τ =if t > τ : S (t )=0 or G (t )=0 τ τ F τ (y) =P(Y y Y τ ) y < τ

4 a b a > τ b < τ Fˆ a (y ) F a (y) =P(Y y Y a ) y (1) < y (2) < Fˆ a (y )=1 Π i : a y (i ) y 1 d (i) (i) d (i) y (i) (i ) y (i) (i) = Σj =1 I (T j y (i) Ỹ j ) Ŝ a (y ) S a (y) =P(Y y Y a ) Ŝ a (y )=1 Fˆ a (y ) E [h (X, Y )] Y E[h (X, Y ) a Y b ] h 1 Σ Fˆ a (b ) i =1 ii (a Ỹ i b )h (X i,ỹ i ) Ŝ a (Ỹ i ) #(Ỹ i ) #(Ỹ i )= Σj I (T j Ỹ i Ỹ j ) E[h (X, Y ) a Y b ] =1 W i = i I (a Ỹ i b ) Fˆ a (b ) Ŝ a (Ỹ i ) #(Ỹ i ) α q α (x )=if y : P (Y y X = x ) α Y q α (x ) = x t θ α θ α E[ρ α (Y X t θ )] θ E[ρ α (Y X t θ )] E[ρ α (Y X t θ) a Y b] [a, b ] q α c (x) = x t θ α q α c (x )=if y : P(Y y X = x, a Y b ) α q α c (x)

5 q α (x ) q α c (x) q α (x ) q α (x ) q c α (x) E[ρ α (Y X t θ) a Y b] W i θ α q α c (x) = x t θ α Σi =1 ρ α (Ỹ i X t i θ )W i θ = 1 Σ Fˆ a (b ) i =1 ii (a Ỹ i b )ρ α (Ỹ i X t i θ ) Ŝ a (Ỹ i ) #(Ỹ i ) F (τ )=0 F (τ )=1 a Y b E [h (X, Y )] Ŝ (t )= Π i : y (i ) < t Σi =1 ih (X i,ỹ i ) Ŝ (Ỹ i ) #(Ỹ i ) 1 d (i) (i) q α (x ) = x t θ α θ α q α (x ) = x t θ α Σi =1 iρ α (Ỹ i X t i θ ) Ŝ (Ỹ i ) #(Ỹ i ) Ŝ (Ỹ i ) θ i #(Ỹ i ) θ α

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8 Y T (Y, T ) Y T Y 0 = log T 0 = log a b AGE ˆ 0 se ( ˆ 0 ) ˆ 1 se ( ˆ 1 ) a = -4.38, b = age 4 5 age 59 age a = -3.5, b = age 4 5 age 59 age

9 Y X N X, X 2 X X α q α (x ) q α (x )= 1 5 z α z a x z α α

10 true quatiles 10% cesorig itercept slope itercept slope 25% quatile (0.227*) 0.743(0.157) 50% quatile (0.213) 1.004(0.147) =50 75% quatile (0.244) 1.259(0.169) 90% quatile (0.352) 1.468(0.232) least squares 0.023(0.274) 0.988(0.155) 25% quatile (0.149) 0.735(0.107) 50% quatile (0.135) 0.999(0.102) =100 75% quatile (0.147) 1.269(0.119) 90% quatile (0.218) 1.499(0.165) least squares 0.005(0.186) 0.998(0.107) 25% quatile (0.063) 0.731(0.048) 50% quatile (0.059) 1.001(0.046) =500 75% quatile (0.066) 1.270(0.051) 90% quatile (0.088) 1.509(0.067) least squares 0.004(0.082) 0.999(0.047) 30% cesorig 50% cesorig itercept slope itercept slope 25% quatile (0.266) 0.754(0.186) (0.423) 0.785(0.271) 50% quatile (0.254) 1.022(0.189) (0.358) 1.026(0.261) =50 75% quatile 0.172(0.342) 1.263(0.249) 0.292(0.508) 1.170(0.316) 90% quatile 0.477(0.604) 1.383(0.342) 0.749(0.915) 1.177(0.427) least squares 0.025(0.332) 0.981(0.185) 0.083(0.415) 0.934(0.226) 25% quatile (0.164) 0.742(0.120) (0.215) 0.746(0.164) 50% quatile (0.162) 1.007(0.127) (0.218) 1.016(0.187) =100 75% quatile 0.140(0.217) 1.282(0.182) 0.198(0.294) 1.243(0.247) 90% quatile 0.357(0.339) 1.470(0.250) 0.574(0.617) 1.311(0.337) least squares 0.011(0.241) 0.995(0.134) 0.064(0.297) 0.957(0.169) 25% quatile (0.070) 0.734(0.055) (0.082) 0.732(0.067) 50% quatile 0.000(0.070) 0.999(0.056) (0.088) 1.002(0.077) =500 75% quatile 0.137(0.084) 1.270(0.071) 0.141(0.127) 1.269(0.114) 90% quatile 0.269(0.125) 1.508(0.114) 0.329(0.216) 1.472(0.186) least squares 0.004(0.110) 0.997(0.062) 0.025(0.152) 0.983(0.085) * stadard errors

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