Local Linear Estimation with Censored Data

Transcription

1 Introduction Main Result Comparison with Related Works June 6, 2011

2 Introduction Main Result Comparison with Related Works Outline 1 Introduction Self-Consistent Estimators Local Linear Estimation of Regression Function The Buckley-James Estimators 2 Estimation of Regression Function with The localized self-consistency equation 3 Main Result 4 Comparison with Related Works Compare to Fan 1994 Compare to Kim and Truong, 1998

3 Introduction Main Result Comparison with Related Works Outline 1 Introduction Self-Consistent Estimators Local Linear Estimation of Regression Function The Buckley-James Estimators 2 Estimation of Regression Function with The localized self-consistency equation 3 Main Result 4 Comparison with Related Works Compare to Fan 1994 Compare to Kim and Truong, 1998

4 Introduction Main Result Comparison with Related Works Right It is only known that the observation is greater than a certain value. Example Medical follow up study in which an individual is only followed up for a certain length of time.

5 Introduction Main Result Comparison with Related Works Left It is only known that the observation is less than a certain value. Example Medical study of a condition, in which for certain individuals, the onset of the condition is before the inclusion of that individual in the study.

6 Introduction Main Result Comparison with Related Works Doubly Example Leiderman et al. (Nature, 1973). Study on the time needed for an infant to learn to crawl Some already known how to crawl. Some learned to crawl during the study. Others failed to learn by the end of the study.

7 Introduction Main Result Comparison with Related Works Doubly Example (Peer, Van Dijck, et. al,1993). Age-dependent growth rate of primary breast cancer left censored Tumor already developed at the time of the study. uncensored Tumor developed during the study. right censored Tumor had not developed by the end of the study.

8 Introduction Main Result Comparison with Related Works Abstract representation of Right Y Survival time, with DF F Y C Random Right Censoring Variable Observed: (Z, δ) with Z = min{y, C} and δ = 1{Y C} Y i, C i, (Z i, δ i ), i = 1,..., n

9 Introduction Main Result Comparison with Related Works Doubly Y Survival time, with DF F Y U Random right censoring variable V Random left censoring variable observed: (Z, δ), where Z = min{max{y, V }, U} and δ indicates type of censoring uncensored,z = Y, δ = 1 V Y U V U Y Y V U Y i, (U i, V i ), (Z i, δ i ), i = 1,..., n right censored,z = U, δ = 2 left censored,z = V, δ = 3

10 Introduction Main Result Comparison with Related Works Self-Consistent Estimators Outline 1 Introduction Self-Consistent Estimators Local Linear Estimation of Regression Function The Buckley-James Estimators 2 Estimation of Regression Function with The localized self-consistency equation 3 Main Result 4 Comparison with Related Works Compare to Fan 1994 Compare to Kim and Truong, 1998

11 Introduction Main Result Comparison with Related Works Self-Consistent Estimators Estimation of survival functions based on censored data The problem Finding an estimator for S(t) when the data is censored Solution S Y (t) = E ( 1 n n i=1 ) I (Y i > t) = E ( 1 n E ( n i=1 I (Y i > t) (Z i, δ i ) n i=1 )) Ŝ n (t) = 1 n n i=1 E (I (Y i) > t (Z i, δ i ) n i=1 )

12 Introduction Main Result Comparison with Related Works Self-Consistent Estimators The self-consistency equations and self-consistent estimators The self-consistency equations Ŝ n (t) = 1 n n i=1 EŜn(t) (I (Y i) > t (Z i, δ i ) n i=1 ), t (, ) Definition Self-consistent estimators: solutions to the above system of self-consistency equations

13 Introduction Main Result Comparison with Related Works Self-Consistent Estimators The self-consistency equations for right censored data The self-consistency equation t Ŝ n (t) = S (0) Ŝ n (t) (t) Ŝ n (x) ds(1) (s) where S (0) = I (Z i > t) and S (1) = I (Z i > t, δ i = 1). The solution to the above equation is the Kaplan-Meier estimator (product limit estimator).

14 Introduction Main Result Comparison with Related Works Local Linear Estimation of Regression Function Outline 1 Introduction Self-Consistent Estimators Local Linear Estimation of Regression Function The Buckley-James Estimators 2 Estimation of Regression Function with The localized self-consistency equation 3 Main Result 4 Comparison with Related Works Compare to Fan 1994 Compare to Kim and Truong, 1998

15 Introduction Main Result Comparison with Related Works Local Linear Estimation of Regression Function Local Linear Estimation of Regression Function The regression model Y = m (X ) + ε Where m(x) is the regression function. Estimate m(x) from a sample (X 1, Y 1 ),..., (X n, Y n )

16 Introduction Main Result Comparison with Related Works Local Linear Estimation of Regression Function Local Linear Smoother Estimate m(x) with â where â, together with ˆb, minimizes ( ) Σ n j=1 (Y j a b (x X j )) 2 x Xj K h n h n The band width Advantage It is the asymptotic minimax linear smoother

17 Introduction Main Result Comparison with Related Works Local Linear Estimation of Regression Function Adaptive Band Width Selection If the data are ordered according to the X i s then define h n = X i+kn X i kn where k n and n/k n

18 Introduction Main Result Comparison with Related Works The Buckley-James Estimators Outline 1 Introduction Self-Consistent Estimators Local Linear Estimation of Regression Function The Buckley-James Estimators 2 Estimation of Regression Function with The localized self-consistency equation 3 Main Result 4 Comparison with Related Works Compare to Fan 1994 Compare to Kim and Truong, 1998

19 Introduction Main Result Comparison with Related Works The Buckley-James Estimators Linear Regression with The Linear Regression Model Y = β 1 X + ε The Buckley-James estimators can be used when the responses Y i are censored

20 Introduction Main Result Comparison with Related Works The Buckley-James Estimators The Buckley-James estimators ˆβ Initial guess of β. Y i, ˆβ = Y i ˆβX i. Similarly define (U i, ˆβ, V i, ˆβ) and Z i, ˆβ. Z i, ˆβ = min{max{y i, ˆβ, V i, ˆβ}, U i, ˆβ}. ˆF ˆβ Self-consistent estimator based on Z { i, ˆβ Zi Yi, if δ = 1 = (Y (Z = Z i, δ)), if δ 1 EˆF ˆβ ˆβ Least squares estimator based on (Y i, X i ) If ˆβ = ˆβ A Buckley James Estimator for β.

21 Introduction Main Result Comparison with Related Works Estimation of Regression Function with Outline 1 Introduction Self-Consistent Estimators Local Linear Estimation of Regression Function The Buckley-James Estimators 2 Estimation of Regression Function with The localized self-consistency equation 3 Main Result 4 Comparison with Related Works Compare to Fan 1994 Compare to Kim and Truong, 1998

22 Introduction Main Result Comparison with Related Works Estimation of Regression Function with The regression model Y = m (X ) + ε The sample (Z, δ) with Z = min{y, C} and δ = 1{Y C} Y i, C i, (Z i, δ i ), i = 1,..., n

23 Introduction Main Result Comparison with Related Works The localized self-consistency equation Outline 1 Introduction Self-Consistent Estimators Local Linear Estimation of Regression Function The Buckley-James Estimators 2 Estimation of Regression Function with The localized self-consistency equation 3 Main Result 4 Comparison with Related Works Compare to Fan 1994 Compare to Kim and Truong, 1998

24 Introduction Main Result Comparison with Related Works The localized self-consistency equation Define Z i = Z i b (X i x), i = 1,..., n similarly, define Ỹ i and C i Based on these corrected data, construct the following localized self-consistency equation ˆF n (y x) = ( ξ x K h h K h ( ξ x h ) I (z y) d ˆF n,0 (z, ξ) ) I (z y) 1 ˆF n (y x) 1 ˆF n (z x) d ˆF n,2 (z, ξ) Solution to this equation is an estimator for the distribution function of the residuals.

25 Introduction Main Result Comparison with Related Works The localized self-consistency equation Localized Buckley-James type estimators The previous localized self-consistency equation is dependent on the choice of a and b, the local linear coefficient. The resultant localized self-consistent estimator of the distribution function of the residual can be used to recover the censored data. The recovered data can be used to estimate the local linear coefficients a and b. Thus, estimation equations following the idea of Buckley-James estimator can be constructed.

26 Introduction Main Result Comparison with Related Works The localized self-consistency equation Localized Buckley-James type estimators Advantage of using localized Buckley-James type estimator Localized selfconsistent-estimator combined with localized Buckley-James type estimator allows better estimation of the conditional distribution function of the response. Gives better estimation of the transformation for recovering the oringinal data Smaller error in the estimation of the regression function

27 Introduction Main Result Comparison with Related Works Main Result For the estimation equation described previously, under certain regularity conditions Existence and consistency of the solution for the localized self-consistency equation. Existence of the solution â and ˆb Will establish asymptotic property of â as an estimator for m(x).

28 Introduction Main Result Comparison with Related Works Compare to Fan 1994 Outline 1 Introduction Self-Consistent Estimators Local Linear Estimation of Regression Function The Buckley-James Estimators 2 Estimation of Regression Function with The localized self-consistency equation 3 Main Result 4 Comparison with Related Works Compare to Fan 1994 Compare to Kim and Truong, 1998

29 Introduction Main Result Comparison with Related Works Compare to Fan 1994 Compare to Fan 1994 Fan, Jianqing. (1994), Censored Regression: Local Linear Approximations and Their Applications, Journal of the American Statistical Association

30 Introduction Main Result Comparison with Related Works Compare to Kim and Truong, 1998 Outline 1 Introduction Self-Consistent Estimators Local Linear Estimation of Regression Function The Buckley-James Estimators 2 Estimation of Regression Function with The localized self-consistency equation 3 Main Result 4 Comparison with Related Works Compare to Fan 1994 Compare to Kim and Truong, 1998

31 Introduction Main Result Comparison with Related Works Compare to Kim and Truong, 1998 Haesook T. Kim and Young K. Truong, (1998), Nonparametric Regression Estimation with : Local Linear Smoothers and Their Applications, Biometrics