13_AS_2_lec_a

Test special cases 442 actuarial statistics module 2

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Unformatted text preview: thesis test Special cases 4/42 Actuarial Statistics – Module 2: Non-parametric methods Censoring and truncation Censoring 1 Introduction 2 Censoring and truncation Censoring Truncation Likelihood for censored and truncated data 3 Estimating the lifetime distribution: non-parametric approach Preliminaries Maximum Likelihood with censoring The likelihood as a function of the discrete hazard function Kaplan-Meier (product limit) estimator Nelson-Aalen estimator 4 Comparing survival functions Introductory example A hypothesis test Special cases 4/42 Actuarial Statistics – Module 2: Non-parametric methods Censoring and truncation Censoring Censoring I Right Censoring (Type I censoring) Event (e.g. failure such as death) is observed only if it occurs prior to some prescribed time CR (right censoring time). The lifetime T is only known if T ≤ CR ; the observation will be CR if T > CR . Examples of right censoring: the mortality investigation ends before all the lives being observed have died life insurance policyholders surrender their policies Right Censoring (Type II censoring) Observations continues until a predetermined number (say r ) of events (failures) have occurred. Data then consists of r smallest lifetimes in a sample of n (order statistics). 4/42 Actuarial Statistics – Module 2: Non-parametric methods Censoring and truncation Censoring Censoring II Left Censoring The event of interest (such as death) has already occurred before the observation starts. So we only know that the lifetime T is less than a left censoring time CL (left censoring time). Interval Censoring The lifetime T is only known to occur within an interval (e.g. actuarial investigations where we only know the calendar year of death) Random censoring Another competing risk can remove the individual from the study (e.g. lapsing of policy in a mortality study of insured lives) lifetime is censored by another random event (not failure) 5/42 Actuarial Statistics – Module 2: Non-parametric methods Censoring and truncation Censoring Censoring III Ci , the time at which the i th observation is censored is a random variable Informative and Non-Informative Censoring Censoring is non-informative if it gives no information about the...
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