Discretize a continuous claim severity distribution

Converts a continuous nonnegative severity specification into an explicit lattice distribution. The transformation is intended for actuarial algorithms that require discrete severities, such as Panjer recursion or FFT-based aggregate-loss methods.

Discretization is always explicit: this function is never called silently by aggregate-loss functions.

Usage

discretize_severity(
  x,
  step,
  to = NULL,
  method = c("unbiased", "upper", "lower", "rounding"),
  tail_prob = 1e-08,
  label = NULL
)

Arguments

x: A continuous tidyact_severity_spec object.
step: Positive finite numeric scalar giving the lattice spacing.
to: Optional positive finite upper grid point. It must be an integer multiple of step. If NULL, the upper point is selected from the (1 - tail_prob) quantile and rounded upward to the next grid point.
method: Discretization method. One of "unbiased", "upper", "lower", or "rounding".
tail_prob: Probability used to select to when to = NULL. Must satisfy 0 < tail_prob < 1.
label: Optional nonempty character scalar. If NULL, the label of x is retained.

Value

A tidyact_severity_spec and tidyact_distribution_spec object with dist = "discrete", lattice support, and discretization information in metadata$discretization.

The metadata records:

method;
step;
to;
whether to was supplied or selected from tail_prob;
the requested tail_prob when applicable;
the actual probability above to;
the explicit tail treatment;
the original distribution identifier; and
the source of the limited mean for the unbiased method.

Details

Let $F$ be the severity CDF, $h$ the lattice step, and $b = \mathrm{to}$. The lattice is

$$0, h, 2h, \ldots, b.$$

The four methods follow standard actuarial discretization conventions.

Upper method

For lattice points below to,

$$ p_x = F(x+h)-F(x). $$

This assigns each interval to its lower lattice endpoint. Probability above to is assigned explicitly to the upper grid point.

Lower method

$$ p_0 = F(0), \qquad p_x = F(x)-F(x-h). $$

This assigns each interval to its upper lattice endpoint. Probability above to is added to the upper grid point.

Rounding method

Probability is allocated according to the midpoint intervals around each lattice point. Thus the first cell extends from zero to step / 2, interior cells have width step, and the last cell contains everything above to - step / 2.

Unbiased method

The unbiased method performs local first-moment matching. If $L(d) = E[\min(X,d)]$, the finite-grid probabilities before the final tail adjustment are

$$ p_0 = 1 - F(0) - \frac{L(h)-L(0)}{h}, $$

$$ p_x = \frac{2L(x)-L(x-h)-L(x+h)}{h}, \qquad 0 < x < b, $$

and

$$ p_b = \frac{L(b)-L(b-h)}{h} - 1 + F(b). $$

The residual tail probability above to is then assigned to to. Consequently, the resulting distribution is a proper probability distribution and its first moment matches $E[\min(X,b)]$, rather than silently discarding probability above the truncation point.

For the exponential, gamma, lognormal, Weibull, and Pareto II families, the first limited moment used by the unbiased method is evaluated analytically. For other continuous specifications with a CDF, including Burr XII, log-logistic, and eligible custom models, the limited mean is obtained by numerical integration of the survival function. The source used is recorded in metadata$discretization$limited_mean_source.

Only continuous severity specifications are accepted in this initial implementation. Finite discrete and empirical severities already have probability masses and should not be silently re-discretized. Support for mixed severities can be added later without changing this public interface.

The returned object uses the existing "discrete" severity operations but has support$type = "lattice". Zero-probability grid points need not be stored explicitly in params$values; the lattice information is retained in support.

References

Klugman, S. A., Panjer, H. H., and Willmot, G. E. (2019). Loss Models: From Data to Decisions, 5th ed. Wiley.

Examples

x <- severity_spec(
  "gamma",
  shape = 2,
  rate = 0.001
)

# Automatic upper grid point from a small tail probability.
dx <- discretize_severity(
  x,
  step = 100
)
dx
#> <tidyact_severity_spec>
#> Distribution: Finite discrete
#> Parameters:
#>   values: 0, 100, 200, 300, ...
#>   probs: 0.001585779, 0.008905010, 0.016251719, 0.022119521, ...
#> Support: lattice [0, 21600]

# Inspect the resulting lattice probabilities.
tibble::tibble(
  amount = dx$params$values,
  probability = dx$params$probs
)
#> # A tibble: 217 × 2
#>    amount probability
#>     <dbl>       <dbl>
#>  1      0     0.00159
#>  2    100     0.00891
#>  3    200     0.0163 
#>  4    300     0.0221 
#>  5    400     0.0267 
#>  6    500     0.0303 
#>  7    600     0.0329 
#>  8    700     0.0347 
#>  9    800     0.0359 
#> 10    900     0.0366 
#> # ℹ 207 more rows

# Use a fixed truncation point and another discretization rule.
discretize_severity(
  x,
  step = 250,
  to = 10000,
  method = "rounding"
)
#> <tidyact_severity_spec>
#> Distribution: Finite discrete
#> Parameters:
#>   values: 0, 250, 500, 750, ...
#>   probs: 0.007190985, 0.047786257, 0.075222937, 0.088183534, ...
#> Support: lattice [0, 10000]

# The discretization choices remain auditable.
dx$metadata$discretization
#> $method
#> [1] "unbiased"
#> 
#> $step
#> [1] 100
#> 
#> $to
#> [1] 21600
#> 
#> $cutoff_source
#> [1] "tail_prob"
#> 
#> $requested_tail_prob
#> [1] 1e-08
#> 
#> $actual_tail_probability
#> [1] 9.404758e-09
#> 
#> $tail_handling
#> [1] "assigned_to_upper_grid_point"
#> 
#> $source_dist
#> [1] "gamma"
#> 
#> $limited_mean_source
#> [1] "analytic"
#>