By NEAS - 8/5/2018 9:13:20 PM
MS Module 24 Least squares bias function practice exam questions
(The attached PDF file has better formatting.)
The mean value and the number of observations in each cell of a 2 × 2 classification table are
Means Column 1 Column 2 Observations Column 1 Column 2
Row 1 71 59 Row 1 11 12
Row 2 36 25 Row 2 14 19
Illustration: The cell in row 1 column 1 has a mean of 71 from a sample of 11 observations.
An actuary is setting class relativities for insurance pricing using a multiplicative model and a least squares bias function with
● a base rate of 10
● a starting relativity for column 1 of 1
● a starting relativity for column 2 of 1.2
Question 24.1: Multiplicative model least squares implied relativity row 1
What is the implied relativity for Row 1, given the starting relativities by column?
Answer 24.1: (71 × 11 × 1.0 + 59 × 12 × 1.2) / (10 × (1.02 × 11 + 1.22 × 12) ) = 5.766
(relativities computed by taking partial derivatives to minimize the sum of the squared errors; see practice problems for the derivation)
Question 24.2: Multiplicative model least squares implied relativity row 2
What is the implied relativity for Row 2, given the starting relativities by column?
Answer 24.2: (36 × 14 × 1.0 + 25 × 19 × 1.2) / (10 × (1.02 × 14 + 1.22 × 19) ) = 2.597
Question 24.3: Multiplicative model least squares implied relativity column 1
What is the implied relativity for Column 1, given the computed relativities by row?
Answer 24.3: (71 × 11 × 5.766 + 36 × 14 × 2.597) / (10 × (5.7662 × 11 + 2.5972 × 14) ) = 1.263
Question 24.4: Multiplicative model least squares implied relativity column 2
What is the implied relativity for Column 2, given the computed relativities by row?
Answer 24.4: (59 × 12 × 5.766 + 25 × 19 × 2.597) / (10 × (5.7662 × 12 + 2.5972 × 19) ) = 1.009
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