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
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
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% ● ● ● 1500 ● ● ● ● ● 1000 500 0 T1 T2 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
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
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
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 X t = [ X − t | X > t ] represents its residual lifetime at time t > 0 Assume that X has an absolutely continuous distribution F X and a density f X • The survival function of X is ¯ F X ( t ) = P ( X > t ) • The hazard rate function of X is r X ( t ) = f X ( t ) ¯ F X ( t ) • The mean residual life function of X is m X ( t ) = E [ X t ] • Fix γ ∈ (0 , 1) . The γ -percentile residual life function of X is the γ -percentile of X t ; i.e., � F − 1 X t ( γ ) , t < u X ; q X,γ ( t ) = 0 , t ≥ u X , Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010
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 X t = [ X − t | X > t ] represents its residual lifetime at time t > 0 Assume that X has an absolutely continuous distribution F X and a density f X • The survival function of X is ¯ F X ( t ) = P ( X > t ) • The hazard rate function of X is r X ( t ) = f X ( t ) ¯ F X ( t ) • The mean residual life function of X is m X ( t ) = E [ X t ] • Fix γ ∈ (0 , 1) . The γ -percentile residual life function of X is the γ -percentile of X t ; i.e., � F − 1 X t ( γ ) , t < u X ; q X,γ ( t ) = 0 , t ≥ u X , Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010
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
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 F X and ¯ ¯ F Y , hazard rate functions r X and r Y , and mean residual life functions m X and m Y , 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 r X ( t ) ≥ r Y ( 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 m X ( t ) ≤ m Y ( t ) , for all t ∈ R • Fix γ ∈ (0 , 1) . X is said to be smaller than Y in the γ - percentile residual life order , denoted by X ≤ γ − rl Y , if q X,γ ( t ) ≤ q Y,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
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 F X and ¯ ¯ F Y , hazard rate functions r X and r Y , and mean residual life functions m X and m Y , 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 r X ( t ) ≥ r Y ( 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 m X ( t ) ≤ m Y ( t ) , for all t ∈ R • Fix γ ∈ (0 , 1) . X is said to be smaller than Y in the γ - percentile residual life order , denoted by X ≤ γ − rl Y , if q X,γ ( t ) ≤ q Y,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
Introduction The problem A simulation study A real data example References Stochastic orders Example 1000 1000 800 800 600 600 400 400 200 200 0 0 0 500 1000 1500 0 500 1000 1500 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
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
Introduction The problem A simulation study A real data example References Description and previous results Description Given γ, α ∈ (0 , 1) , X 1 , X 2 , . . . X n and Y 1 , Y 2 , . . . Y m how to construct a (1 − α ) · 100% -confidence band for q Y,γ ( t ) − q X,γ ( t ) ? The empirical γ -percentile residual life function of X is q X,n,γ ( t ) = Q n ( γ + (1 − γ ) ¯ F X,n ( t )) − t, t < u X , 0 < γ < 1 where ¯ F X,n is the empirical survival of X and Q n is its sample quantile function: � ( k − 1) < x ≤ k ( k = 1 , . . . , n ) X k n n Q n ( x ) = X 1 x = 0 Inference for the difference of two percentile residual life functions Alba Mar´ ıa Franco Pereira COMPSTAT, Paris, France 2010
Introduction The problem A simulation study A real data example References Description and previous results Previous results Cs¨ orgo and Cs¨ orgo (1987) • q X,n,γ ( t ) q X,γ ( t ) a.s. − → 1 • n 2 f X ( q X,γ ( t ) + t ) { q X,γ ( t ) − q X,n,γ ( t ) } N (0 , 1) d → − 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
Introduction The problem A simulation study A real data example References Description and previous results Previous results Cs¨ orgo and Cs¨ orgo (1987) • q X,n,γ ( t ) q X,γ ( t ) a.s. − → 1 • n 2 f X ( q X,γ ( t ) + t ) { q X,γ ( t ) − q X,n,γ ( t ) } N (0 , 1) d → − 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
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