Linear on Average
This smoothing option performs a weighted least square regression of the bin characteristic factor on the average characteristic value in each bin. The weight for each bin is the amount of exposure in the bin.
The basic weighted linear regression model is to determine parameters c0 and c1:
In this case, the weight wi for bin 'i' being used is the exposure, modified by the Base Factor and all characteristics other than the one we are considering. (This modified exposure is referred to as xFactor in the SQL Factor Table.) The value of the predictive variable xi for the bin is the average value of the characteristic within the bin. The observed factor yi is the ratio of target/modified exposure. Therefore, wiyi is simply the target amount for the bin.
Since ultimately the factors will be balanced to average 1 across bins, we are concerned only with the slope parameter c1.
Credibility is then applied to arrive at the final slope parameter.
For the Exposure Method, the final slope parameter = c1 * Ztotal, where:
For the t-Statistic Method, the t value of c1 is calculated from the regression above [t = c1/s.e.(c1)], and the critical t value, , is calculated the same way as for the other Credibility calculations. The credibility is calculated using the formula:
Then, the final slope parameter = c1 * Credibility.
The intercept parameter is then adjusted to be =
The revised slope and intercept parameters are finally applied to each bin's average characteristic value to arrive at factors which are then blended and rebalanced.