handout03_stat_default

handout03_stat_default - Outline Failure time Likelihood...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Outline Failure time Likelihood inference Counting process Statistical Modeling of Time-to-Default Haipeng Xing Haipeng Xing AMS517, SUNY Stony Brook Statistical Modeling of Time-to-Default Outline Failure time Likelihood inference Counting process Outline 1 Failure-time distributions and models 2 Likelihood inference 3 Couting process and asymptotic theory Haipeng Xing AMS517, SUNY Stony Brook Statistical Modeling of Time-to-Default
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Outline Failure time Likelihood inference Counting process Survival and hazard functions I Let T be the failure time of an individual from a homogeneous population. The survivor function of T is deFned by the probability that T exceeds a value t in its range; that is S ( t )= P ( T > t ) , 0 <t< . Clearly, S ( t ) is a nonincreasing right-continuous function of t with S (0) = 1 and lim t →∞ S ( t )=0 . If T is (absolutely) continuous, the probability density function of T is f ( t - dS ( t ) /dt. Provided that f ( t ) is continuous at t , f ( t ) has the interpretation that, for small h , f ( t ) h P ( t T < t + h S ( t ) - S ( t + h ) Haipeng Xing AMS517, SUNY Stony Brook Statistical Modeling of Time-to-Default Outline Failure time Likelihood inference Counting process Survival and hazard functions II Note that f ( t ) 0 , ± 0 f ( t ) dt =1 , and F ( t ² t 0 f ( u ) du. The hazard function speciFes the instantaneous rate at which failure occurs for items that are surviving at time t , is deFned as λ ( t ) = lim h 0 + P ( t T < t + h | T t ) h . (1) It follows the deFnition of the density function that λ ( t f ( t ) /S ( t - d log S ( t ) dt . (2) Haipeng Xing AMS517, SUNY Stony Brook Statistical Modeling of Time-to-Default
Background image of page 2
Outline Failure time Likelihood inference Counting process Survival and hazard functions III Integrating (2) with respect to t and using F (0) = 1 , we get S ( t ) = exp[ - ± t 0 λ ( u ) du ] = exp[ - Λ ( t )] , (3) where Λ ( t )= ² t 0 λ ( s ) ds is called the cumulative hazard function . The probability density function of T can be obtained by di f erentiating (3) f ( t λ ( t ) · exp[ - Λ ( t )] (4) If T is discrete and takes values at a 1 <a 2 < · · · with associated probability function f ( a i P ( T = a i ) ,i =1 , 2 , · · · Haipeng Xing AMS517, SUNY Stony Brook Statistical Modeling of Time-to-Default Outline Failure time Likelihood inference Counting process Survival and hazard functions IV the survivor function is S ( t ³ j | a j >t f ( a j ) and the hazard function at a i is deFned as λ i = P ( T = a i | T a i f ( a i ) S ( a - i ) , 2 ,... where S ( a - ) = lim t a - S ( t ) . Corresponding to (3) and (4), the survivor function and the probability function are given by S ( t ´ j | a j t (1 - λ j ) (5) Haipeng Xing AMS517, SUNY Stony Brook Statistical Modeling of Time-to-Default
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Outline Failure time Likelihood inference Counting process Survival and hazard functions V and f ( a i )= λ i i - 1 ± j =1 (1 - λ j ) (6) As in the continuous case, the discrete hazard function { λ i ; i =1 , 2 , · · · } uniquely determines the distribution of the failure time variable T .
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 21

handout03_stat_default - Outline Failure time Likelihood...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online