with Current Status Data

Transcription

1 Estimation and Testing with Current Status Data Jon A. Wellner University of Washington Estimation and Testing p. 1/4

2 joint work with Moulinath Banerjee, University of Michigan Talk at Université Paul Sabatier, Toulouse III, Laboratoire de Statistique et Probabilités, http: // Estimation and Testing p. 2/4

3 Outline Introduction: current status data Estimation and Testing p. 3/4

4 Outline Introduction: current status data Estimation of F, unconstrained (old, Ayer et al. 1956) Estimation and Testing p. 3/4

5 Outline Introduction: current status data Estimation of F, unconstrained (old, Ayer et al. 1956) Estimation of F, constrained (new) Estimation and Testing p. 3/4

6 Outline Introduction: current status data Estimation of F, unconstrained (old, Ayer et al. 1956) Estimation of F, constrained (new) The likelihood ratio test of H : F (t 0 )=θ 0 Estimation and Testing p. 3/4

7 Outline Introduction: current status data Estimation of F, unconstrained (old, Ayer et al. 1956) Estimation of F, constrained (new) The likelihood ratio test of H : F (t 0 )=θ 0 How big is too big? The limit Gaussian problem Estimation and Testing p. 3/4

8 Outline Introduction: current status data Estimation of F, unconstrained (old, Ayer et al. 1956) Estimation of F, constrained (new) The likelihood ratio test of H : F (t 0 )=θ 0 How big is too big? The limit Gaussian problem Limiting Distribution of the likelihood ratio statistic under H Estimation and Testing p. 3/4

9 Outline Introduction: current status data Estimation of F, unconstrained (old, Ayer et al. 1956) Estimation of F, constrained (new) The likelihood ratio test of H : F (t 0 )=θ 0 How big is too big? The limit Gaussian problem Limiting Distribution of the likelihood ratio statistic under H Confidence intervals for F (t 0 ) Estimation and Testing p. 3/4

10 Outline Introduction: current status data Estimation of F, unconstrained (old, Ayer et al. 1956) Estimation of F, constrained (new) The likelihood ratio test of H : F (t 0 )=θ 0 How big is too big? The limit Gaussian problem Limiting Distribution of the likelihood ratio statistic under H Confidence intervals for F (t 0 ) Further problems Estimation and Testing p. 3/4

11 1. Introduction: current status data X F, Y G, X, Y independent Estimation and Testing p. 4/4

12 1. Introduction: current status data X F, Y G, X, Y independent We observe (Y,1{X Y }) (Y, ) with density p F (y, δ) =F (y) δ (1 F (y)) 1 δ g(y) Estimation and Testing p. 4/4

13 1. Introduction: current status data X F, Y G, X, Y independent We observe (Y,1{X Y }) (Y, ) with density p F (y, δ) =F (y) δ (1 F (y)) 1 δ g(y) Suppose that (Y i, i ) are i.i.d. as (Y, ). Estimation and Testing p. 4/4

14 1. Introduction: current status data X F, Y G, X, Y independent We observe (Y,1{X Y }) (Y, ) with density p F (y, δ) =F (y) δ (1 F (y)) 1 δ g(y) Suppose that (Y i, i ) are i.i.d. as (Y, ). Likelihood: L n (F )= n F (Y i ) i (1 F (Y i )) 1 i i=1 Estimation and Testing p. 4/4

15 Likelihood ratio test of H : F (t 0 )=θ 0 Estimation and Testing p. 5/4

16 Likelihood ratio test of H : F (t 0 )=θ 0 The likelihood ratio statistic: λ n = sup F L n (F ) sup F :F (t0 )=θ 0 L n (F ) = L n( ˆF n ) L n ( ˆF 0 n). Estimation and Testing p. 5/4

17 2. Estimation of F : Nonparametric MLE MLE (Unconstrained) ˆFn (t) =argmax F L n (F ) Estimation and Testing p. 6/4

18 2. Estimation of F : Nonparametric MLE MLE (Unconstrained) ˆFn (t) =argmax F L n (F ) Another description: define Note that G n (t) = 1 n n 1 [Yi t], V n (t) = 1 n i=1 n i 1 [Yi t]. i=1 G n (t) a.s. G(t), V n (t) a.s. t 0 F (y)dg(y) V (t). Estimation and Testing p. 6/4

19 2. Estimation of F : Nonparametric MLE MLE (Unconstrained) ˆFn (t) =argmax F L n (F ) Another description: define Note that G n (t) = 1 n n 1 [Yi t], V n (t) = 1 n i=1 n i 1 [Yi t]. i=1 G n (t) a.s. G(t), V n (t) a.s. t 0 F (y)dg(y) V (t). Thus dv (t) =F (t) dg Estimation and Testing p. 6/4

20 Partial sum diagram: Let Y (1) Y (2) Y (n). The partial sum diagram P = {P i } is given by P i =(G n (Y (i) ), V n (Y (i) )), i =1,...,n. Estimation and Testing p. 7/4

21 Partial sum diagram: Let Y (1) Y (2) Y (n). The partial sum diagram P = {P i } is given by P i =(G n (Y (i) ), V n (Y (i) )), i =1,...,n. The Nonparametric MLE ˆF n of F is: ˆF n (Y (i) )=left derivative of the Greatest Convex Minorant of P at Y (i). Estimation and Testing p. 7/4

22 Partial sum diagram: Let Y (1) Y (2) Y (n). The partial sum diagram P = {P i } is given by P i =(G n (Y (i) ), V n (Y (i) )), i =1,...,n. The Nonparametric MLE ˆF n of F is: ˆF n (Y (i) )=left derivative of the Greatest Greatest Convex Minorant = GCM Convex Minorant of P at Y (i). Estimation and Testing p. 7/4

23 Cumulative Sum Diagram: Example n =5 i (i) Y (i) nv n (Y (i) ) i Estimation and Testing p. 8/4

24 Cumulative Sum Diagram: Example n =5 i (i) Y (i) nv n (Y (i) ) i Estimation and Testing p. 8/4

25 Cumulative Sum Diagram: Example n =5 i (i) Y (i) nv n (Y (i) ) i Estimation and Testing p. 8/4

26 Example continued, n =5 i (i) Y (i) ˆF n 1 2 y Estimation and Testing p. 9/4

27 Example continued, n =5 i (i) Y (i) ˆF n 1 2 y Estimation and Testing p. 9/4

28 Example continued, n =5 i (i) Y (i) ˆF n 1 2 y Estimation and Testing p. 9/4

29 cumvec index Estimation and Testing p. 10/4

30 yaxis xaxis Estimation and Testing p. 11/4

31 3. The constrained MLE ˆF 0 n. Recipe: Break P into P L and P R where P L = {P i : Y (i) t 0 }, P R = {P i : Y (i) >t 0 }. Estimation and Testing p. 12/4

32 3. The constrained MLE ˆF n 0. Recipe: Break P into P L and P R where P L = {P i : Y (i) t 0 }, P R = {P i : Y (i) >t 0 }. Form the GCM s of P L and P R,sayṼL n and ṼR n. Estimation and Testing p. 12/4

33 3. The constrained MLE ˆF n 0. Recipe: Break P into P L and P R where P L = {P i : Y (i) t 0 }, P R = {P i : Y (i) >t 0 }. Form the GCM s of P L and P R,sayṼL n and ṼR n. If the slope of ṼL n exceeds θ 0, replace it by θ 0 ; if the slope of Ṽ R n drops below θ 0, replace it by θ 0. Estimation and Testing p. 12/4

34 3. The constrained MLE ˆF n 0. Recipe: Break P into P L and P R where P L = {P i : Y (i) t 0 }, P R = {P i : Y (i) >t 0 }. Form the GCM s of P L and P R,sayṼL n and ṼR n. If the slope of ṼL n exceeds θ 0, replace it by θ 0 ; if the slope of Ṽ R n drops below θ 0, replace it by θ 0. The resulting (truncated or constrained) slope process yields the constrained MLE ˆF n. 0 Estimation and Testing p. 12/4

35 cumvec cusum diag. Unconstrained Constrained Index Estimation and Testing p. 13/4

36 yaxis xaxis Estimation and Testing p. 14/4

37 4. The likelihood ratio test of H : F (t 0 )=θ 0 Likelihood ratio statistic: λ n = sup F L n (F ) sup F :F (t0 )=θ 0 L n (F ) = L n( ˆF n ) L n ( ˆF 0 n). Estimation and Testing p. 15/4

38 4. The likelihood ratio test of H : F (t 0 )=θ 0 Likelihood ratio statistic: λ n = sup F L n (F ) sup F :F (t0 )=θ 0 L n (F ) = L n( ˆF n ) L n ( ˆF 0 n). How big is too big? Estimation and Testing p. 15/4

39 4. The likelihood ratio test of H : F (t 0 )=θ 0 Likelihood ratio statistic: λ n = sup F L n (F ) sup F :F (t0 )=θ 0 L n (F ) = L n( ˆF n ) L n ( ˆF 0 n). How big is too big? When H : F (t 0 )=θ 0 holds, does 2logλ n d something? Estimation and Testing p. 15/4

40 4. The likelihood ratio test of H : F (t 0 )=θ 0 Likelihood ratio statistic: λ n = sup F L n (F ) sup F :F (t0 )=θ 0 L n (F ) = L n( ˆF n ) L n ( ˆF 0 n). How big is too big? When H : F (t 0 )=θ 0 holds, does 2logλ n d something? Answer: Yes! Banerjee and Wellner (2001) Estimation and Testing p. 15/4

41 5. How big is too big? The limiting Gaussian problem Suppose that we observe {X(t) :t R} where X(t) =F (t)+σw(t) F (t) = t f(s)ds, f monotone non-decreasing, and W is standard (two-sided) Brownian motion. Estimation and Testing p. 16/4

42 5. How big is too big? The limiting Gaussian problem Suppose that we observe {X(t) :t R} where X(t) =F (t)+σw(t) F (t) = t f(s)ds, f monotone non-decreasing, and W is standard (two-sided) Brownian motion. Suppose that we want to estimate the monotone function f. Equivalently dx(t) =f(t)dt + σdw(t). Estimation and Testing p. 16/4

43 5. How big is too big? The limiting Gaussian problem Suppose that we observe {X(t) :t R} where X(t) =F (t)+σw(t) F (t) = t f(s)ds, f monotone non-decreasing, and W is standard (two-sided) Brownian motion. Suppose that we want to estimate the monotone function f. Equivalently dx(t) =f(t)dt + σdw(t). The canonical monotone function is a linear one, and we can change σ to 1 by virtue of scaling arguments so the canonical version of the problem is as follows: dx(t) =2tdt + dw (t), Estimation and Testing p. 16/4

44 estimate 2t when {X(t) : t R}, is observed. Thus X(t) =t 2 + W (t). Unconstrained Estimator : Slope of GCM of X(t). Call this process of slopes of the GCM S. Estimation and Testing p. 17/4

45 estimate 2t when {X(t) : t R}, is observed. Thus X(t) =t 2 + W (t). Unconstrained Estimator : Slope of GCM of X(t). Call this process of slopes of the GCM S. Estimation and Testing p. 17/4

46 W(t) + t^ samplepath unconstr t Estimation and Testing p. 18/4

47 2t g_{1,1} g_{1,1}^0 2t t Estimation and Testing p. 19/4

48 What is the canonical constrained problem? Estimation and Testing p. 20/4

49 What is the canonical constrained problem? Estimate the monontone function f(t) =2t subject to the constraint that f(0) = 0 when {X(t) : t R} is observed. Estimation and Testing p. 20/4

50 What is the constrained estimator? Estimation and Testing p. 21/4

51 What is the constrained estimator? Recipe: Break {X(t) :t R} into X R {X(t) :t 0}. X L {X(t) :t<0} and Form the GCM s of X L and X R say Y L and Y R. If the slope of Y L exceeds 0, replace it by 0; if the slope of Y R drops below 0, replace it by 0. The resulting (truncated or constrained) slope process S 0 is the constrained MLE of f(t) =2t in the Gaussian problem. Estimation and Testing p. 21/4

52 t Estimation and Testing p. 22/4

53 Unconstrained. One-sided Constrained t Estimation and Testing p. 23/4

54 2t g_{1,1} g_{1,1}^0 2t t Estimation and Testing p. 24/4

55 Likelihood ratio test statistic in the Gaussian problem? Suppose {X(t) :t [ c, c]} is given by dx(t) =f(t)dt + dw (t) so X(t) =F (t)+w (t). Estimation and Testing p. 25/4

56 Likelihood ratio test statistic in the Gaussian problem? Suppose {X(t) :t [ c, c]} is given by so dx(t) =f(t)dt + dw (t) X(t) =F (t)+w (t). Radon-Nikodym derivative (drifted process relative to zero drift): ( dp c f =exp fdx 1 c ) f 2 (t)dt. dp 0 c 2 c Estimation and Testing p. 25/4

57 Likelihood ratio test statistic in the Gaussian problem? Suppose {X(t) :t [ c, c]} is given by so dx(t) =f(t)dt + dw (t) X(t) =F (t)+w (t). Radon-Nikodym derivative (drifted process relative to zero drift): ( dp c f =exp fdx 1 c ) f 2 (t)dt. dp 0 c 2 c F(c, K) ={monotone functions f :[ c, c] R, f c K} F 0 (c, K) ={f F(c, K) : f(0) = 0} Estimation and Testing p. 25/4

58 Then 2logλ c = 2log { c = 2 = 2 c ( supf F(c,K) dp f /dp 0 c c sup f F0 (c,k) dp f /dp 0 ˆf c dx 1 2 c c c c ˆf 2 c (t)dt ˆf c,0 dx ( ˆf c ˆf c,0 )dx c c ) c c =2log } ˆf c,0(t)dt 2 ( dp ˆf /dp 0 { ˆf 2 c (t) ˆf 2 c,0(t)}dt. dp ˆf0 /dp 0 ) Estimation and Testing p. 26/4

59 Taking the limit as c with K = K c =5c, this yields 2logλ =2 D ( ˆf ˆf 0 )dx D { ˆf 2 (t) ˆf 2 0 (t)}dt Estimation and Testing p. 27/4

60 Taking the limit as c with K = K c =5c, this yields 2logλ =2 D ( ˆf ˆf 0 )dx D { ˆf 2 (t) ˆf 2 0 (t)}dt From the characterizations of ˆf and ˆf 0 : R (X ˆF )d ˆf =0, R (X ˆF 0 )d ˆf 0 =0. Estimation and Testing p. 27/4

61 Integration by parts: ( ˆf ˆf 0 )dx = ( ˆf ˆf 0 )dx = R D = ˆFdˆf + ˆF 0 d ˆf 0 D D = ˆfdˆF ˆf 0 d ˆF 0 D D = { ˆf 2 (t) ˆf 0 2 (t)}dt. D D Xd( ˆf ˆf 0 ) Estimation and Testing p. 28/4

62 Integration by parts: ( ˆf ˆf 0 )dx = ( ˆf ˆf 0 )dx = R D = ˆFdˆf + ˆF 0 d ˆf 0 D D = ˆfdˆF ˆf 0 d ˆF 0 D D = { ˆf 2 (t) ˆf 0 2 (t)}dt. D D Xd( ˆf ˆf 0 ) Likelihood ratio statistic becomes: 2logλ = { ˆf 2 (t) ˆf 0 2 (t)}dt. D Estimation and Testing p. 28/4

63 6. Limit distribution, LR statistic under H Limit distributions for ˆF n and ˆF 0 n. Set G loc n (t, h) =n 1/3 (G n (t + n 1/3 h) G n (t)) V loc n (t, h) = n 1/3 { n 1/3 (V n (t + n 1/3 h) V n (t)) G loc n (t, h)f (t) }. Estimation and Testing p. 29/4

64 6. Limit distribution, LR statistic under H Limit distributions for ˆF n and ˆF 0 n. Set G loc n (t, h) =n 1/3 (G n (t + n 1/3 h) G n (t)) V loc n (t, h) = n 1/3 { n 1/3 (V n (t + n 1/3 h) V n (t)) G loc n (t, h)f (t) }. Theorem 1. If g(t0 )=G (t 0 ) and f(t 0 )=F (t 0 ) exist, then: A. G loc n (t 0,h) p g(t 0 )h. B. V loc n (t 0,h) aw (h)+bh 2 where a = F (t 0 )(1 F (t 0 ))g(t 0 ), b = f(t 0 )g(t 0 )/2, and W is a two-sided Brownian motion starting from 0. Estimation and Testing p. 29/4

65 Now define Z n (h) =n 1/3 ( ˆF n (t 0 + hn 1/3 ) F (t 0 )), Z 0 n(h) =n 1/3 ( ˆF 0 n(t 0 + hn 1/3 ) F (t 0 )). Estimation and Testing p. 30/4

66 Now define Z n (h) =n 1/3 ( ˆF n (t 0 + hn 1/3 ) F (t 0 )), Z 0 n(h) =n 1/3 ( ˆF 0 n(t 0 + hn 1/3 ) F (t 0 )). Theorem 2. If the hypotheses of Theorem 1 hold with f(t 0 ) > 0, g(t 0 ) > 0, and F (t 0 )=θ 0, then (Z n (h), Z 0 n(h)) (S a,b (h), S 0 a,b (h))/g(t 0) where S a,b and S 0 a,b are the constrained and unconstrained slope processes corresponding to X a,b (h) =aw (h)+bh 2. Estimation and Testing p. 30/4

67 Limit distribution for 2logλ n Estimation and Testing p. 31/4

68 Limit distribution for 2logλ n Theorem 3. (Banerjee and Wellner, 2001). Suppose that F and G have densities f and g which are strictly positive and continuous in a neighborhood in a neighborhood of t 0. Suppose that F (t 0 )=θ 0. Then 2logλ n d 1 g(t 0 )a 2 ((S a,b (z)) 2 (S 0 a,b (z))2 ) dz d = {(S(z)) 2 (S 0 (z)) 2 }dz D, and the distribution of D is universal (free of parameters). Estimation and Testing p. 31/4

69 F(x) Brownian Exponential x Estimation and Testing p. 32/4

70 7. Confidence intervals for F (t 0 ) Wald-type intervals: Z n (0) = n 1/3 ( ˆF n (t 0 ) F (t 0 )) d S a,b (0)/g(t 0 ) d = { } F (t0 )(1 F (t 0 ))f(t 0 ) 1/3 S(0) 2g(t 0 ) C(F, f, g) S(0) where S(0) d =2Z 2argmin(W (h)+h 2 ), S(0) S 1,1 (0). Estimation and Testing p. 33/4

71 7. Confidence intervals for F (t 0 ) Wald-type intervals: Z n (0) = n 1/3 ( ˆF n (t 0 ) F (t 0 )) d S a,b (0)/g(t 0 ) d = { } F (t0 )(1 F (t 0 ))f(t 0 ) 1/3 S(0) 2g(t 0 ) C(F, f, g) S(0) where S(0) d =2Z 2argmin(W (h)+h 2 ), S(0) S 1,1 (0). Wald - interval: ˆF n (t 0 ) ± n 1/3 C( ˆF n, ˆf n, ĝ n ) t α where ˆf n and ĝ n are estimates of f and g (at t 0 ), and t α/2 satisfies P (2Z >t α/2 )=α/2. Estimation and Testing p. 33/4

72 Problem: this involves smoothing to get estimators ˆf n and ĝ n! Estimation and Testing p. 34/4

73 Confidence intervals from the LR test Estimation and Testing p. 35/4

74 Confidence intervals from the LR test Invert the test: {θ :2logλ n (θ) d α }. where P (D d α )=1 α Estimation and Testing p. 35/4

75 Confidence intervals from the LR test Invert the test: {θ :2logλ n (θ) d α }. where P (D d α )=1 α Advantage: no smoothing needed! Estimation and Testing p. 35/4

76 Confidence intervals from the LR test Invert the test: {θ :2logλ n (θ) d α }. where P (D d α )=1 α Advantage: no smoothing needed! Tradeoff: need to compute constrained estimator(s) ˆF 0 n of F and λ n (θ) for many different values of the constraint θ. Estimation and Testing p. 35/4

77 t Estimation and Testing p. 36/4 F_n(t)

78 t Estimation and Testing p. 37/4 F(t)

79 F_n(t) Wald type LRT based t Estimation and Testing p. 38/4

80 F_n(t) Keiding's LRT based t Estimation and Testing p. 39/4

81 8. Further problems Can we find the distribution of D analytically? Estimation and Testing p. 40/4

82 8. Further problems Can we find the distribution of D analytically? Does the same limit D arise as the limit distribution for the likelihood ratio test for a large class of such problems involving monotone functions? (Yes Banerjee (2005)) Estimation and Testing p. 40/4

83 8. Further problems Can we find the distribution of D analytically? Does the same limit D arise as the limit distribution for the likelihood ratio test for a large class of such problems involving monotone functions? (Yes Banerjee (2005)) Power? What is the appropriate contiguity theory? What is the limit distribution of the likelihood ratio statistic under local alternatives? See Banerjee and Wellner (2001), (2005b). Estimation and Testing p. 40/4

84 8. Further problems Can we find the distribution of D analytically? Does the same limit D arise as the limit distribution for the likelihood ratio test for a large class of such problems involving monotone functions? (Yes Banerjee (2005)) Power? What is the appropriate contiguity theory? What is the limit distribution of the likelihood ratio statistic under local alternatives? See Banerjee and Wellner (2001), (2005b). What happens if we constrain at k>1points? (Independence!) Estimation and Testing p. 40/4

85 8. Further problems Can we find the distribution of D analytically? Does the same limit D arise as the limit distribution for the likelihood ratio test for a large class of such problems involving monotone functions? (Yes Banerjee (2005)) Power? What is the appropriate contiguity theory? What is the limit distribution of the likelihood ratio statistic under local alternatives? See Banerjee and Wellner (2001), (2005b). What happens if we constrain at k>1points? (Independence!) Confidence bands for the whole monotone function F? Estimation and Testing p. 40/4

86 8. Further problems Can we find the distribution of D analytically? Does the same limit D arise as the limit distribution for the likelihood ratio test for a large class of such problems involving monotone functions? (Yes Banerjee (2005)) Power? What is the appropriate contiguity theory? What is the limit distribution of the likelihood ratio statistic under local alternatives? See Banerjee and Wellner (2001), (2005b). What happens if we constrain at k>1points? (Independence!) Confidence bands for the whole monotone function F? Confidence intervals (and bands?) for estimating a concave distribution function F? Estimation and Testing p. 40/4