Estimate age-to-age factors via weighted least squares

Internal helper that fits one no-intercept weighted linear model per age-to-age link:

$$C_{i,k+1} = f_k \cdot C_{i,k} + \varepsilon_{i,k}$$

Weights are proportional to \(w_{i,k} / C_{i,k}^{2 - \alpha}\), where \(w_{i,k}\) is either a constant or a column supplied via weights. This corresponds to Mack's variance assumption \(\mathrm{Var}(C_{i,k+1} \mid C_{i,k}) \propto C_{i,k}^{\alpha}\).

When only one observation is available for a link, the factor is computed directly as loss_to / loss_from and standard errors are set to NA.

Near-zero values of f_se and sigma (below tol) are set to zero to avoid numerical noise from essentially perfect fits.

Usage

.lm_ata(x, weights = 1, alpha = 1, na_rm = TRUE, tol = 1e-12)

Arguments

x

An object of class "Link".

weights

Either a length-one numeric scalar (default 1) or a single column name present in the Link data that provides per-row weights.

alpha

Numeric scalar controlling the variance structure. Default is 1.

na_rm

Logical; if TRUE (default), rows with non-finite or non-positive loss_from are dropped before fitting. Note that loss_to = 0 is permitted, as zero cumulative values are valid observations (e.g. no claims yet developed in early development periods).

tol

Non-negative numeric scalar. Values below tol are set to zero. Default is 1e-12.

Value

A data.table with one row per ata link containing f, f_se, sigma, rse, and n_cohorts. rse is defined as \(f\_se / f\) and represents the relative standard error of the WLS-estimated factor. rse is NA when f_se is NA (single observation links) or when f is zero.