13_AS_8_lec_a

Experience formed the basis of a particular standard

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Unformatted text preview: graduation Graduation by Reference to a Standard Table 1 Introduction 2 Testing smoothness 3 Statistical tests Preliminaries Chi-square (χ2 ) 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 58/72 Actuarial Statistics – Module 8: Method of Graduation Methods of graduation Graduation by Reference to a Standard Table Graduation by Reference to a Standard Table Rationale: if the class of lives involved in the graduation is sufficiently similar to the class of lives whose experience formed the basis of a particular standard table, the underlying mortality of both class should be similar (Overall levels of mortality may differ between the two, the overall progression of rates from age to age should be similar). When we do not have much data, a standard table is useful 58/72 Actuarial Statistics – Module 8: Method of Graduation Methods of graduation Graduation by Reference to a Standard Table Graduation by Reference to a Standard Table I Procedure 1 2 Select an appropriate standard table Decide a relationship between standard table and graduated o o s rates q x = f (qx ) or µx = f (µs ) x For example, o s q x = a + bqx o s q x = (a + bx ) qx o qx = (a + bx ) s qx o µx = µs + k x To decide a suitable function f (), we can use some simple plots. For example, s a plot of qx against qx s a plot of ln(− ln (1 − qx )) against ln(− ln (1 − qx )) 59/72 Actuarial Statistics – Module 8: Method of Graduation Methods of graduation Graduation by Reference to a Standard Table Graduation by Reference to a Standard Table II 3 Determine parameter values maximum likelihood least squares: minimize o o x wx (ˆx − µx )2 or µ x wx (ˆx − q x )2 q with the weight wx being the inverse of the estimated variance for µx or qx . ˜ ˜ 4 5 60/72 Determine the graduated rates from the fitted relationship Test the graduation (goodness-of-fit) Actuarial Statistics – Module 8: Method of Graduation Methods of graduation Graduation by Reference to a Standard Table Example Suppose you wish to graduate a set of mortality rates µ by reference to a standard table µs using the relationship o µx = aµs + b . The parameters are to be determined using x weighted least squares. In general, we need to minimise: o...
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