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# handout03_stat_default - Outline Failure time Likelihood...

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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

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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
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

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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 .
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handout03_stat_default - Outline Failure time Likelihood...

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