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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
40/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Grouping of Signs Test Grouping of Signs test (Stevens’ test) I
Purpose
Test for overgraduation
Test groups of deviations of the same sign (runs) Compare the number of groups (runs) of deviations of the
same sign with the no. that would be expected if the positive
and negative signs were arranged in random order. (Fewer
runs than expected if overgraduated)
Assume that of the m deviations there are n1 positive and n2
negative
Null hypothesis: the given n1 positive deviations and n2
negative deviations are in random order
Test Statistic
40/72 G = Number of groups of positive zx ’s Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Grouping of Signs Test Grouping of Signs test (Stevens’ test) II
compute probability of G groups given n1 and n2
+1
there are n2t
ways to arrange t positive groups among the
n2 negative deviations
−
there are nt1−11 ways to arrange n1 positive deviations into t
groups
m
there are n1 ways to arrange n1 positive and n2 negative
deviations
probability of exactly t positive groups (hypergeometric
distribution)
n1 −1
t −1 n2 +1
t
m
n1 41/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Grouping of Signs Test Grouping of Signs test (Stevens’ test) III
use onetailed test by ﬁnding the smallest k such that
t =k Pr (G ≤ k ) =
t =1 n1 −1
t −1 n2 +1
t
m
n1 ≥ 0.05 which implies Pr (G ≤ l ) < 0.05 for any l < k .
The critical value of the test can be found on Page 189 of the
Tables
if G ≤ k , we reject the null hypothesis, ie, that the rates are
overgraduated (the rates do not adhere closely enough to the
crude data and may be consistently too high or too low over
certain parts of the table)
42/72 Actuarial Statistics – Module...
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