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13_AS_2_lec_a - Actuarial Statistics Module 2...

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Actuarial Statistics – Module 2: Non-parametric methods Actuarial Statistics Benjamin Avanzi c University of New South Wales (2013) School of Risk and Actuarial Studies [email protected] Module 2: Non-parametric methods 1/42
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Actuarial Statistics – Module 2: Non-parametric methods Plan 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 2/42
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Actuarial Statistics – Module 2: Non-parametric methods Introduction 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 3/42
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Actuarial Statistics – Module 2: Non-parametric methods Introduction Introduction Given observations (data), the aim is to estimate the distribution of T (remember all F ( t ) , S ( t ) , f ( t ) or μ t provide the same information about the distribution, so any will do) A simple method to estimate S ( t ) would be to observe a (very) large number of newborns and take the survival function as the proportion alive at each age. However, this presents a number of problems The experiment would take an extremely long time to complete Lives under observation may lost to the investigation, for one reason or another, and to exclude these from the analysis might bias the result (censoring) This would be useful only if all cohorts have the same mortality (which is not the case) 3/42
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Actuarial Statistics – Module 2: Non-parametric methods Censoring and truncation 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
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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
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