13_AS_8_lec_a - Actuarial Statistics Module 8 Method of...

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Actuarial Statistics – Module 8: Method of Graduation Actuarial Statistics Benjamin Avanzi c University of New South Wales (2013) School of Risk and Actuarial Studies [email protected] Module 8: Methods of Graduation 1/72
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Actuarial Statistics – Module 8: Method of Graduation Plan 1 Introduction 2 Testing smoothness 3 Statistical tests Preliminaries Chi-square ( χ 2 ) test Standardised Deviations Test Signs test Cumulative Deviations Test Grouping of Signs Test Serial Correlations Test 4 Methods of graduation Preliminaries Graduation by Parametric Formula Graduation by Reference to a Standard Table Graphical graduation Statistical Tests The Effect of Duplicate policies 2/72
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Actuarial Statistics – Module 8: Method of Graduation Introduction 1 Introduction 2 Testing smoothness 3 Statistical tests Preliminaries Chi-square ( χ 2 ) test Standardised Deviations Test Signs test Cumulative Deviations Test Grouping of Signs Test Serial Correlations Test 4 Methods of graduation Preliminaries Graduation by Parametric Formula Graduation by Reference to a Standard Table Graphical graduation Statistical Tests The Effect of Duplicate policies 3/72
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Actuarial Statistics – Module 8: Method of Graduation Introduction Graduation of estimates I Estimation of rates for each age x to x + 1 produces piecewise constant estimates (crude estimates) Poisson and Markov multiple state models: for each age x = x 1 , x 2 , . . . , x m we have deaths d x waiting time or central exposed to risk E c x estimate b μ x + 1 2 = d x E c x with asymptotic distribution b μ x + 1 2 Normal μ x + 1 2 , μ x + 1 2 E c x or D x Normal E c x μ x + 1 2 , E c x μ x + 1 2 3/72
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Actuarial Statistics – Module 8: Method of Graduation Introduction Graduation of estimates II Binomial model: for each age x = x 1 , x 2 , . . . , x m we have deaths d x initial exposed to risk E x E c x + 1 2 d x estimate ˆ q x = d x E x with asymptotic distribution e q x Normal q x , q x (1 - q x ) E x or D x Normal( E x q x , E x q x (1 - q x )) 4/72
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Actuarial Statistics – Module 8: Method of Graduation Introduction This module The crude estimates { ˆ q x } or { ˆ μ x } are very unlikely to progress smoothly as age varies. Graduation is the process of “smoothing” the crude estimated rates. In this module, we will address three questions: Why do we want smoothed estimates? How to carry out the graduation? how tho decide that a given attempt to graduate the crude estimates is satisfactory? 5/72
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Actuarial Statistics – Module 8: Method of Graduation Introduction Reasons for graduation It is intuitively sensible to think that the quantities q x or μ x should be a smooth functions of age x . There is some evidence from large investigations to support this. A crude estimate at any age x also carries information about the values at adjacent ages. We need to incorporate the information from adjacent ages into the estimate (reduces the sampling errors).
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