[PPT] - Inference for the difference of two percentile residual life PowerPoint Presentation

SLIDE 1

Introduction The problem A simulation study A real data example References

Inference for the difference of two percentile residual life functions

Alba M. Franco-Pereira

Department of Statistics Universidad Carlos III de Madrid August, 2010

Joint work with Rosa E. Lillo and Juan Romo

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 2

Introduction The problem A simulation study A real data example References

Outline

1 Introduction

Motivation Reliability measures Stochastic orders

2 The problem

Description and previous results Our methodology

3 A simulation study

The sampling models The simulation mechanism The simulation results Conclusions of the simulation study

4 A real data example 5 References

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 3

Introduction The problem A simulation study A real data example References Motivation

Example

Kalbfleisch and Prentice (1982) TYPE OF TREATMENT PATIENTS % CENSORED T1 (Radiotherapy) 100 27% T2 (Radiotherapy + Chemotherapeutic agent) 95 27.37%

T1

T2 500 1000 1500

Figure: Box-and-whisker plots of the survival times of the two groups of patients

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 4

Introduction The problem A simulation study A real data example References Motivation

Example

Interests:

To study the effectiveness of both treatments independently
To compare both types of treatments

Tools:

Reliability measures
Stochastic orderings

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 5

Introduction The problem A simulation study A real data example References Motivation

Example

Interests:

To study the effectiveness of both treatments independently
To compare both types of treatments

Tools:

Reliability measures
Stochastic orderings

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 6

Introduction The problem A simulation study A real data example References Reliability measures

Definitions

Let X be the lifetime of an item or component and let Xt = [X − t|X > t] represents its residual lifetime at time t > 0 Assume that X has an absolutely continuous distribution FX and a density fX

The survival function of X is ¯

FX(t) = P(X > t)

The hazard rate function of X is rX(t) = fX(t)

¯ FX(t)

The mean residual life function of X is mX(t) = E[Xt]
Fix γ ∈ (0, 1). The γ-percentile residual life function of X is the γ-percentile of

Xt; i.e., qX,γ(t) =

F −1

Xt (γ),

t < uX; 0, t ≥ uX,

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 7

Introduction The problem A simulation study A real data example References Reliability measures

Definitions

Let X be the lifetime of an item or component and let Xt = [X − t|X > t] represents its residual lifetime at time t > 0 Assume that X has an absolutely continuous distribution FX and a density fX

The survival function of X is ¯

FX(t) = P(X > t)

The hazard rate function of X is rX(t) = fX(t)

¯ FX(t)

The mean residual life function of X is mX(t) = E[Xt]
Fix γ ∈ (0, 1). The γ-percentile residual life function of X is the γ-percentile of

Xt; i.e., qX,γ(t) =

F −1

Xt (γ),

t < uX; 0, t ≥ uX,

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 8

Introduction The problem A simulation study A real data example References Reliability measures

Motivation

RELIABILITY MEASURES STOCHASTIC ORDERINGS Hazard rate function HR order Survival function ST order Mean residual life function MRL order Percentile residual life function γ-PRL, γ ∈ (0, 1)

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 9

Introduction The problem A simulation study A real data example References Stochastic orders

Definitions

Let X and Y be two absolutely continuous random variables with survival functions ¯ FX and ¯ FY , hazard rate functions rX and rY , and mean residual life functions mX and mY , respectively

X is said to be smaller than Y in the usual stochastic order, denoted by

X ≤st Y , if F X(t) ≤ F Y (t), for all t ∈ R

X is said to be smaller than Y in the hazard rate order, denoted by X ≤hr Y , if

rX(t) ≥ rY (t), for all t ∈ R

X is said to be smaller than Y in the mean residual life order, denoted by

X ≤mrl Y , if mX(t) ≤ mY (t), for all t ∈ R

Fix γ ∈ (0, 1). X is said to be smaller than Y in the γ-percentile residual life
rder, denoted by X ≤γ−rl Y , if

qX,γ(t) ≤ qY,γ(t), for all t ∈ R

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 10

Introduction The problem A simulation study A real data example References Stochastic orders

Definitions

Let X and Y be two absolutely continuous random variables with survival functions ¯ FX and ¯ FY , hazard rate functions rX and rY , and mean residual life functions mX and mY , respectively

X is said to be smaller than Y in the usual stochastic order, denoted by

X ≤st Y , if F X(t) ≤ F Y (t), for all t ∈ R

X is said to be smaller than Y in the hazard rate order, denoted by X ≤hr Y , if

rX(t) ≥ rY (t), for all t ∈ R

X is said to be smaller than Y in the mean residual life order, denoted by

X ≤mrl Y , if mX(t) ≤ mY (t), for all t ∈ R

Fix γ ∈ (0, 1). X is said to be smaller than Y in the γ-percentile residual life
rder, denoted by X ≤γ−rl Y , if

qX,γ(t) ≤ qY,γ(t), for all t ∈ R

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 11

Introduction The problem A simulation study A real data example References Stochastic orders

Example

500 1000 1500 200 400 600 800 1000 500 1000 1500 200 400 600 800 1000

Figure: Comparison of the mrl’s and merl’s of the patients undergoing T1 and T2

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 12

Introduction The problem A simulation study A real data example References

Outline

1 Introduction

Motivation Reliability measures Stochastic orders

2 The problem

Description and previous results Our methodology

3 A simulation study

The sampling models The simulation mechanism The simulation results Conclusions of the simulation study

4 A real data example 5 References

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 13

Introduction The problem A simulation study A real data example References Description and previous results

Description

Given γ, α ∈ (0, 1), X1, X2, . . . Xn and Y1, Y2, . . . Ym how to construct a (1 − α) · 100%-confidence band for qY,γ(t) − qX,γ(t)? The empirical γ-percentile residual life function of X is qX,n,γ(t) = Qn(γ + (1 − γ) ¯ FX,n(t)) − t, t < uX, 0 < γ < 1 where ¯ FX,n is the empirical survival of X and Qn is its sample quantile function: Qn(x) =

Xk

(k−1) n

< x ≤ k

n

(k = 1, . . . , n) X1 x = 0

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 14

Introduction The problem A simulation study A real data example References Description and previous results

Previous results

Cs¨

rgo and Cs¨
rgo (1987)
qX,n,γ(t)

a.s. − → qX,γ(t)

n

1 2 fX(qX,γ(t) + t){qX,γ(t) − qX,n,γ(t)}

d − → N(0, 1) Our methodology is based on

Bootstrap techniques
Statistical depth

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 15

Introduction The problem A simulation study A real data example References Description and previous results

Previous results

Cs¨

rgo and Cs¨
rgo (1987)
qX,n,γ(t)

a.s. − → qX,γ(t)

n

1 2 fX(qX,γ(t) + t){qX,γ(t) − qX,n,γ(t)}

d − → N(0, 1) Our methodology is based on

Bootstrap techniques
Statistical depth

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 16

Introduction The problem A simulation study A real data example References Description and previous results

Statistical depth

STATISTICAL DEPTH FOR MULTIVARIATE DATA Measures the centrality of a d-dimensional observation with respect to a multivariate distribution F or with respect to a set of d-dimensional points Mahalanobis (1936) Tuckey (1975) Oja (1983) Liu (1990) Singh (1991) Koshevoy and Mosler (1997) Fraiman and Meloche (1999) Vardi and Zhang (2000) Zuo (2003)

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 17

Introduction The problem A simulation study A real data example References Description and previous results

Statistical depth

STATISTICAL DEPTH FOR MULTIVARIATE DATA Measures the centrality of a d-dimensional observation with respect to a multivariate distribution F or with respect to a set of d-dimensional points Mahalanobis (1936) Tuckey (1975) Oja (1983) Liu (1990) Singh (1991) Koshevoy and Mosler (1997) Fraiman and Meloche (1999) Vardi and Zhang (2000) Zuo (2003)

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 18

Introduction The problem A simulation study A real data example References Description and previous results

Statistical depth

STATISTICAL DEPTH FOR FUNCTIONAL DATA Measures the centrality of a function with respect to a set of functions Vardi and Zhang (2000) Fraiman and Muniz (2001) L´

pez-Pintado and Romo (2005)

Cuevas, Febrero and Fraiman (2007) Cuesta-Albertos and Nieto-Reyes (2008) L´

pez-Pintado and Romo (2009)

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 19

Introduction The problem A simulation study A real data example References Description and previous results

Statistical depth

STATISTICAL DEPTH FOR FUNCTIONAL DATA Measures the centrality of a function with respect to a set of functions Vardi and Zhang (2000) Fraiman and Muniz (2001) L´

pez-Pintado and Romo (2005)

Cuevas, Febrero and Fraiman (2007) Cuesta-Albertos and Nieto-Reyes (2008) L´

pez-Pintado and Romo (2009)

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 20

Introduction The problem A simulation study A real data example References Description and previous results

The modified band depth

L´

pez-Pintado and Romo (2009) (J = 2)

MBDB,2(x) = B 2 −1

1≤i1<i2≤B

λ(A(x; xi1, xi2)) λ(I) where λ is the Lebesgue measure in I and A(x; xi1, xi2) ≡ {t ∈ I : min

r=i1,i2

xr(t) ≤ x(t) ≤ max

r=i1,i2

xr(t)}

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 21

Introduction The problem A simulation study A real data example References Description and previous results

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4

Figure: Illustration of how to compute the Modified Band Depth J = 2

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 22

Introduction The problem A simulation study A real data example References Our methodology

The algorithm

Let B be the bootstrap size, α ∈ (0, 1) the confidence level, γ ∈ (0, 1) the percentile. X1, X2, . . . Xn and Y1, Y2, . . . Ym

For b = 1, . . . , B;

resample from X1, X2, . . . Xn and Y1, Y2, . . . Ym to obtain X∗b

1 , X∗b 2 , . . . X∗b n

and Y ∗b

1 , Y ∗b 2 , . . . Y ∗b m

For b = 1, . . . , B;

compute q∗b

X,n,γ and q∗b Y,m,γ

For every t ∈ R;

consider q∗

b (t) = q∗b Y,m,γ(t) − q∗b X,n,γ(t), for b = 1, . . . , B

For b = 1, . . . , B;
rder the sample curves q∗

b , from inner to outer using any notion of depth for

curves and take the band given by the (1 − α) · 100% deepest curves

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 23

Introduction The problem A simulation study A real data example References Our methodology

The algorithm

Let B be the bootstrap size, α ∈ (0, 1) the confidence level, γ ∈ (0, 1) the percentile. X1, X2, . . . Xn and Y1, Y2, . . . Ym

For b = 1, . . . , B;

resample from X1, X2, . . . Xn and Y1, Y2, . . . Ym to obtain X∗b

1 , X∗b 2 , . . . X∗b n

and Y ∗b

1 , Y ∗b 2 , . . . Y ∗b m

For b = 1, . . . , B;

compute q∗b

X,n,γ and q∗b Y,m,γ

For every t ∈ R;

consider q∗

b (t) = q∗b Y,m,γ(t) − q∗b X,n,γ(t), for b = 1, . . . , B

For b = 1, . . . , B;
rder the sample curves q∗

b , from inner to outer using any notion of depth for

curves and take the band given by the (1 − α) · 100% deepest curves

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 24

Introduction The problem A simulation study A real data example References Our methodology

The algorithm

X ∼ Pareto(10,10) Y ∼ Pareto(60,10)

20

40 60 80 100 10 20 30 40

10

20 30 40 5 10 15 10 20 30 40 5 10 15 10 20 30 40 5 10 15

Figure: Illustration of the algorithm

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 25

Introduction The problem A simulation study A real data example References

Outline

1 Introduction

Motivation Reliability measures Stochastic orders

2 The problem

Description and previous results Our methodology

3 A simulation study

The sampling models The simulation mechanism The simulation results Conclusions of the simulation study

4 A real data example 5 References

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 26

Introduction The problem A simulation study A real data example References The sampling models

The sampling models

X ∼ Pareto(10,10) Y1 ∼ Pareto(20,10) Y2 ∼ Pareto(40,10) Y3 ∼ Pareto(60,10) Y4 ∼ Pareto(80,10) Y5 ∼ Pareto(100,10) Y6 ∼ Pareto(110,10)

−5 5 10 15 5 10 15 qX, 0.5 qY1, 0.5 qY2, 0.5 qY3, 0.5 qY4, 0.5 qY5, 0.5 qY6, 0.5

X7 ∼ Pareto(10,10) Y7 ∼ Pareto(1,5) X8 ∼ Pareto(20,5) Y8 ∼ Pareto(70,10) X9 ∼ Pareto(160,20) Y9 ∼ Pareto(70,10) X10 ∼ Pareto(10,10) Y10 ∼ Pareto(20,15)

Modified band depth introduced in L´

pez-Pintado and Romo (2009) with J = 2;

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 27

Introduction The problem A simulation study A real data example References The simulation mechanism

The simulation mechanism

Modified band depth introduced in L´
pez-Pintado and Romo (2009) with J = 2
B = 1000, γ = 0.5, 1 − α = 0.90

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 28

Introduction The problem A simulation study A real data example References The simulation results

The simulation results

2 4 6 8 10 20 40 60 5 10 15 20 25 20 40 60

10 20 30 40 20 40 60

20 40 60 80 20 40 60

20 40 60 80 20 40 60 20 40 60 80 20 40 60

Figure: 90%-confidence band for qYi,0.5 − qX,0.5, i = 1, · · · , 6

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 29

Introduction The problem A simulation study A real data example References The simulation results

The simulation results

1 2 3 4 5 6 −20 −10 10 20 5 10 15 20 25 30 −20 −10 10 20 10 20 30 40 −20 −10 10 20 2 4 6 −20 −10 10 20

Figure: 90%-confidence band for qYi,0.5 − qX,0.5, i = 7, · · · , 10

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 30

Introduction The problem A simulation study A real data example References Conclusions of the simulation study

Conclusions of the simulation study

The bands provide us with a criteria of whether two random variables are close or not with respect to a prl order or allow us to compare prl functions in an interval LLB above the x-axis ⇒ the random variables are ordered ULB below the x-axis ⇒ the random variables are ordered LLB below the x-axis and ULB above the x-axis ⇒ we can not say that one variable dominates the other

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 31

Introduction The problem A simulation study A real data example References Conclusions of the simulation study

Conclusions of the simulation study

The bands provide us with a criteria of whether two random variables are close or not with respect to a prl order or allow us to compare prl functions in an interval LLB above the x-axis ⇒ the random variables are ordered ULB below the x-axis ⇒ the random variables are ordered LLB below the x-axis and ULB above the x-axis ⇒ we can not say that one variable dominates the other

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 32

Introduction The problem A simulation study A real data example References Conclusions of the simulation study

Conclusions of the simulation study

The bands provide us with a criteria of whether two random variables are close or not with respect to a prl order or allow us to compare prl functions in an interval LLB above the x-axis ⇒ the random variables are ordered ULB below the x-axis ⇒ the random variables are ordered LLB below the x-axis and ULB above the x-axis ⇒ we can not say that one variable dominates the other

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 33

Introduction The problem A simulation study A real data example References

Outline

1 Introduction

Motivation Reliability measures Stochastic orders

2 The problem

Description and previous results Our methodology

3 A simulation study

The sampling models The simulation mechanism The simulation results Conclusions of the simulation study

4 A real data example 5 References

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 34

Introduction The problem A simulation study A real data example References

Application in medicine

500 1000 1500 −1000 −500 500 1000

Figure: 90%-confidence bands for the difference of the merl of the patients undergoing T1 and T2

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 35

Introduction The problem A simulation study A real data example References

Outline

1 Introduction

Motivation Reliability measures Stochastic orders

2 The problem

Description and previous results Our methodology

3 A simulation study

The sampling models The simulation mechanism The simulation results Conclusions of the simulation study

4 A real data example 5 References

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 36

Introduction The problem A simulation study A real data example References

Cs¨

rg˝
, S. (1987). Estimating percentile residual life under random censorship.

Contributions to stochastics: in honour to the 75th birthday of Walther Eberl, Sr., Springer-Verlag. Cs¨

rg˝
, M. and Cs¨
rg˝
, S. (1987). Estimation of the percentile residual life.

Operations Research 35, 598–606. Cuesta-Albertos, J. and Nieto-Reyes, A. (2008). The random Tukey depth. Computational Statistics and Data Analysis 52, 4979–4988. Cuevas, A., Febrero, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Computational Statistics 22, 481–496. Fraiman, R. and Meloche, J. (1999). Multivariate L-estimation. Test 8, 255–317. Fraiman, R. and Muniz, G. (2001). Trimmed means for functional data. Test 10, 419–440. Franco-Pereira, A. M., Lillo, R. E., Romo, J., and Shaked, M. (2008). Percentile residual life orders. Technical Report, Department of Mathematics, University of Arizona.

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 37

Introduction The problem A simulation study A real data example References

Ghorai, J., Susarla, A., Susarla, V. and van Ryzin, J. (1980). Nonparametric estimation of mean residual life with censored data. In Colloquia Mathematica Societatis Janos Bolyai 32. Nonparametric Statistical Inference (B. V. Gnedenko et al., eds.), North-Holland, Amsterdam, 269-291. Kalbfleisch J. D. and Prentice R. L. (1980). The statistical analysis of failure time data, John Wiley, New York. Koshevoy, G. and Mosler, K. (1997). Zonoid trimming for multivariate

distributions. The Annals of Statistics 25, 1998–2017.

Liu R. (1990). On a notion of data depth based on randomsimplices. The Annals

f Statistics 18, 405–414.

L´

pez-Pintado, S. and Romo, J. (2005). A half-graph depth for functional data.

Working paper 05-16. L´

pez-Pintado, S. and Romo, J. (2009). On the concept of depth for functional
data. Journal of the American Statistical Association 104, 718–734.

Mahalanobis, P. C. (1936). On the generalized distance in statistics. Proceedings

f National Academy of Science of India 12, 49–55.

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 38

Introduction The problem A simulation study A real data example References

Oja, H. (1983). Descriptive statistics for multivariate distributions. Statistics and Probability Letters 1, 327–332. Singh, K. (1991). A notion of majority depth. Unpublished document. Susarla, V. and Van Ryzin, J. (1976). Nonparametric Bayesian estimation of survival curves from incomplete observations. Journal of the American Statistical Association 61, 897–902. Tukey, J. (1975). Mathematics and picturing data. Proceedings of the 1975 International Congress of Mathematics 2, 523–531. Vardi, Y. and Zhang, C. H. (2000). The multivariate L1-median and associated data depth. Proceedings of the National Academy of Science USA 97, 1423–1426. Zuo, Y. (2003). Projection based depth functions and associated medians. The Annals of Statistics 31, 1460–1490.

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010

SLIDE 39

Introduction The problem A simulation study A real data example References

Estimators

Let X1, . . . , Xn be the lifetimes of the patients after a treatment. Only their right censored versions are observed, leading to the information (δ1, Z1), . . . , (δn, Zn), where for i = 1, . . . , n, δi = I{Xi≤Yi} and Zi = Xi ∧ Yi = max{Xi, Yi}, with Yi representing the i-th censoring random variable (I is the indicator function) It is assumed that Y1, . . . , Yn are i.i.d. with G(y) = P(Y > y) > 0 and that G is

continuous. The survival function of X can be estimated by

¯ FX,n(x) = N+(x) + 1 n + 1

n

j=1

2 + N+(Zj) 1 + N+(Zj) I{δj =0,Zj ≤x}, where N+(x) ≡ number of censored and uncensored observations greater than x. Slight variation of the Bayes estimator of Susarla and Van Ryzin (1976)

Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010