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Actuarial Severity Fitting for Limits and
Deductibles
Greg Sollenberger
What this package assumes
actuarialfitdist is designed around a claim-level
ground-up severity likelihood.
This assumption is important enough to state repeatedly:
loss is the ground-up loss severity.loss is not the insurer payment after deductible.loss is not an excess-of-deductible amount.loss is not an ACV/depreciated payment unless the
analyst has already converted the experience to the intended common
ground-up basis.- The deductible column tells the fitter that otherwise-identical
claims below the deductible are not observed.
- A finite limit tells the fitter that a claim recorded at or above
the limit is right-censored there.
For an exact observation, the row likelihood is proportional to
f(loss) / S(deductible). For a limit-censored observation
it is S(limit) / S(deductible).
If a row has likelihood weight w, its log-likelihood
contribution is w times the ordinary row log-likelihood. An
integer weight of 2 therefore has exactly the same likelihood effect as
repeating that claim row twice.
1. Auto property-damage limit analysis
Suppose an insurer is analyzing property-damage severity. There is no
deductible in this example, but policies have different limits. A claim
that reaches its policy limit is recorded at the limit and is
right-censored.
n <- 1200
true_loss <- rlnorm(n, meanlog = log(9000), sdlog = 0.95)
policy_limit <- sample(c(10000, 25000, 50000, 100000), n, replace = TRUE,
prob = c(0.15, 0.35, 0.35, 0.15))
pd <- data.frame(
loss = pmin(true_loss, policy_limit),
deductible = 0,
limit = policy_limit
)
mean(pd$loss >= pd$limit)
#> [1] 0.1341667
Fit several plausible severity families:
pd_candidates <- fit_severity_candidates(
pd,
loss = "loss",
limit = "limit",
distributions = c("lognormal", "weibull", "gamma", "burr", "loglogistic")
)
pd_candidates$comparison
#> distribution converged logLik AIC BIC parameters n n_censored
#> 1 lognormal TRUE -10843.45 21690.90 21701.08 2 1200 161
#> 2 weibull TRUE -10915.10 21834.20 21844.38 2 1200 161
#> 3 gamma TRUE -10897.90 21799.79 21809.98 2 1200 161
#> 4 burr TRUE -10841.99 21689.99 21705.26 3 1200 161
#> 5 loglogistic TRUE -10843.38 21690.75 21700.93 2 1200 161
#> error elapsed_seconds function_evaluations gradient_evaluations
#> 1 <NA> 0.03 20 6
#> 2 <NA> 0.06 35 11
#> 3 <NA> 0.18 104 26
#> 4 <NA> 0.29 139 39
#> 5 <NA> 0.05 22 6
#> convergence_code
#> 1 0
#> 2 0
#> 3 0
#> 4 0
#> 5 0
The package does not automatically choose one distribution. The
statistical fit can be compared with the business output directly:
pd_factors <- severity_factors(
pd_candidates,
deductible = 0,
limit = c(10000, 25000, 50000, 100000, 250000),
base_deductible = 0,
base_limit = 25000
)
head(pd_factors)
#> distribution model scale_value deductible limit valuation_factor
#> lognormal.1 lognormal single NA 0 10000 1
#> lognormal.2 lognormal single NA 0 25000 1
#> lognormal.3 lognormal single NA 0 50000 1
#> lognormal.4 lognormal single NA 0 100000 1
#> lognormal.5 lognormal single NA 0 250000 1
#> weibull.1 weibull single NA 0 10000 1
#> expected_payment limited_expected_severity unlimited_mean LER
#> lognormal.1 7403.710 7403.710 14275.64 0
#> lognormal.2 11349.075 11349.075 14275.64 0
#> lognormal.3 13230.395 13230.395 14275.64 0
#> lognormal.4 14023.143 14023.143 14275.64 0
#> lognormal.5 14255.733 14255.733 14275.64 0
#> weibull.1 7436.863 7436.863 13367.07 0
#> ILF rating_factor base_deductible base_limit
#> lognormal.1 0.6523624 0.6523624 0 25000
#> lognormal.2 1.0000000 1.0000000 0 25000
#> lognormal.3 1.1657685 1.1657685 0 25000
#> lognormal.4 1.2356199 1.2356199 0 25000
#> lognormal.5 1.2561140 1.2561140 0 25000
#> weibull.1 0.6289921 0.6289921 0 25000
#> base_valuation_factor
#> lognormal.1 1
#> lognormal.2 1
#> lognormal.3 1
#> lognormal.4 1
#> lognormal.5 1
#> weibull.1 1
The factor table is intentionally long-format.
compare_factors() provides a side-by-side view when
desired.
compare_factors(
pd_candidates,
metric = "rating_factor",
deductible = 0,
limit = c(10000, 25000, 50000, 100000),
base_deductible = 0,
base_limit = 25000
)
#> scale_value deductible limit valuation_factor base_deductible
#> lognormal.1 NA 0 10000 1 0
#> base_limit base_valuation_factor value.lognormal value.weibull
#> lognormal.1 25000 1 0.6523624 0.6289921
#> value.gamma value.burr value.loglogistic
#> lognormal.1 0.6394561 0.6637499 0.6652174
2. Auto collision deductible analysis
Now consider collision severity with varying deductibles and no
finite policy limit. Ground-up losses below the deductible never appear
in the claim sample.
n <- 1500
collision_ground_up <- rgamma(n, shape = 2.2, scale = 3200)
collision_ded <- sample(c(250, 500, 1000, 2000), n, replace = TRUE,
prob = c(0.10, 0.35, 0.40, 0.15))
keep <- collision_ground_up > collision_ded
collision <- data.frame(
loss = collision_ground_up[keep],
deductible = collision_ded[keep],
limit = Inf
)
The ground-up amount is retained. We do not subtract
the deductible before fitting.
collision_fit <- fit_severity(
collision,
loss = "loss",
deductible = "deductible",
limit = "limit",
distribution = "gamma"
)
summary(collision_fit)
#> Actuarial severity model summary
#>
#> Model: single
#> Distribution: gamma
#> Observations: 1468 (effective weighted count: 1468)
#> Right-censored: 0
#> LogLik: -14225.1 AIC: 28454.2 BIC: 28464.78
#>
#> estimate std.error
#> scale 3232.648535 140.23679450
#> shape 2.206463 0.09288173
#>
#> The internal scale is the Gamma scale parameter; with common shape, mean severity is proportional to this scale.
Generate deductible factors relative to a $500 deductible:
severity_factors(
collision_fit,
deductible = c(250, 500, 1000, 2000, 5000),
limit = Inf,
base_deductible = 500,
base_limit = Inf
)
#> distribution model scale_value deductible limit valuation_factor
#> 1 gamma single NA 250 Inf 1
#> 2 gamma single NA 500 Inf 1
#> 3 gamma single NA 1000 Inf 1
#> 4 gamma single NA 2000 Inf 1
#> 5 gamma single NA 5000 Inf 1
#> expected_payment limited_expected_severity unlimited_mean LER ILF
#> 1 6882.829 7132.721 7132.721 0.03503457 1
#> 2 6633.681 7132.721 7132.721 0.06996494 1
#> 3 6140.903 7132.721 7132.721 0.13905188 1
#> 4 5197.377 7132.721 7132.721 0.27133320 1
#> 5 2919.909 7132.721 7132.721 0.59063188 1
#> rating_factor base_deductible base_limit base_valuation_factor
#> 1 1.0375581 500 Inf 1
#> 2 1.0000000 500 Inf 1
#> 3 0.9257158 500 Inf 1
#> 4 0.7834831 500 Inf 1
#> 5 0.4401642 500 Inf 1
3. Homeowners deductible analysis with varying Coverage A
For homeowners, claim severity can vary systematically with the
insured amount. The optional scale_by column lets the
fitted severity scale vary linearly with an analyst-supplied numeric
variable.
Here Coverage A is the scaling variable. The same mechanism could
instead use TIV, vehicle value, or a user-created transformation in
another line of business.
n <- 1600
coverage_a <- sample(seq(150000, 900000, by = 25000), n, replace = TRUE)
home_ded <- sample(c(1000, 2500, 5000, 10000), n, replace = TRUE)
shape <- 1.35
scale_coef <- 0.075
latent_scale <- scale_coef * coverage_a
home_ground_up <- rweibull(n, shape = shape, scale = latent_scale)
home_limit <- coverage_a
keep <- home_ground_up > home_ded
home <- data.frame(
loss = pmin(home_ground_up[keep], home_limit[keep]),
deductible = home_ded[keep],
limit = home_limit[keep],
coverage_a = coverage_a[keep]
)
Fit a Weibull model with zero intercept:
home_fit <- fit_severity(
home,
loss = "loss",
deductible = "deductible",
limit = "limit",
distribution = "weibull",
scale_by = "coverage_a",
scale_intercept = FALSE
)
summary(home_fit)
#> Actuarial severity model summary
#>
#> Model: single
#> Distribution: weibull
#> Observations: 1475 (effective weighted count: 1475)
#> Right-censored: 0
#> LogLik: -16679.47 AIC: 33362.93 BIC: 33373.53
#>
#> estimate std.error
#> scale_coef 0.07416593 0.001680669
#> shape 1.33880873 0.033190229
#>
#> The internal scale is the Weibull scale parameter; with common shape, mean severity is proportional to this scale.
When scale_intercept = FALSE, the intercept is exactly
zero, not merely shrunk toward zero or omitted from printed output.
For the Weibull family, the native scale is
scale_coef * coverage_a, so the fitted mean is that
quantity times Gamma(1 + 1 / shape). Other supported
families use parameterizations that preserve the same intended
proportional-mean relationship when their mean is finite.
Factor output can be produced for arbitrary Coverage A values:
severity_factors(
home_fit,
scale_value = c(200000, 400000, 600000, 800000),
deductible = c(1000, 2500, 5000),
limit = c(100000, 250000, 500000),
base_deductible = 1000,
base_limit = 250000
)
#> distribution model scale_value deductible limit valuation_factor
#> 1 weibull single 2e+05 1000 100000 1
#> 2 weibull single 4e+05 1000 100000 1
#> 3 weibull single 6e+05 1000 100000 1
#> 4 weibull single 8e+05 1000 100000 1
#> 5 weibull single 2e+05 2500 100000 1
#> 6 weibull single 4e+05 2500 100000 1
#> 7 weibull single 6e+05 2500 100000 1
#> 8 weibull single 8e+05 2500 100000 1
#> 9 weibull single 2e+05 5000 100000 1
#> 10 weibull single 4e+05 5000 100000 1
#> 11 weibull single 6e+05 5000 100000 1
#> 12 weibull single 8e+05 5000 100000 1
#> 13 weibull single 2e+05 1000 250000 1
#> 14 weibull single 4e+05 1000 250000 1
#> 15 weibull single 6e+05 1000 250000 1
#> 16 weibull single 8e+05 1000 250000 1
#> 17 weibull single 2e+05 2500 250000 1
#> 18 weibull single 4e+05 2500 250000 1
#> 19 weibull single 6e+05 2500 250000 1
#> 20 weibull single 8e+05 2500 250000 1
#> 21 weibull single 2e+05 5000 250000 1
#> 22 weibull single 4e+05 5000 250000 1
#> 23 weibull single 6e+05 5000 250000 1
#> 24 weibull single 8e+05 5000 250000 1
#> 25 weibull single 2e+05 1000 500000 1
#> 26 weibull single 4e+05 1000 500000 1
#> 27 weibull single 6e+05 1000 500000 1
#> 28 weibull single 8e+05 1000 500000 1
#> 29 weibull single 2e+05 2500 500000 1
#> 30 weibull single 4e+05 2500 500000 1
#> 31 weibull single 6e+05 2500 500000 1
#> 32 weibull single 8e+05 2500 500000 1
#> 33 weibull single 2e+05 5000 500000 1
#> 34 weibull single 4e+05 5000 500000 1
#> 35 weibull single 6e+05 5000 500000 1
#> 36 weibull single 8e+05 5000 500000 1
#> expected_payment limited_expected_severity unlimited_mean LER
#> 1 12633.778 13622.32 13622.33 0.07256755
#> 2 26168.293 27157.77 27244.66 0.03653727
#> 3 38690.383 39636.75 40866.99 0.02440477
#> 4 49076.014 49942.21 54489.32 0.01831911
#> 5 11218.044 13622.32 13622.33 0.17649523
#> 6 24710.469 27157.77 27244.66 0.09034749
#> 7 37282.994 39636.75 40866.99 0.06062379
#> 8 47780.975 49942.21 54489.32 0.04559916
#> 9 9085.913 13622.32 13622.33 0.33301285
#> 10 22375.348 27157.77 27244.66 0.17649523
#> 11 34985.114 39636.75 40866.99 0.11959269
#> 12 45644.876 49942.21 54489.32 0.09034749
#> 13 12633.791 13622.33 13622.33 0.07256755
#> 14 26249.214 27244.66 27244.66 0.03653727
#> 15 39868.923 40866.23 40866.99 0.02440477
#> 16 53464.485 54461.65 54489.32 0.01831911
#> 17 11218.054 13622.33 13622.33 0.17649523
#> 18 24783.173 27244.66 27244.66 0.09034749
#> 19 38388.818 40866.23 40866.99 0.06062379
#> 20 51979.490 54461.65 54489.32 0.04559916
#> 21 9085.919 13622.33 13622.33 0.33301285
#> 22 22436.107 27244.66 27244.66 0.17649523
#> 23 35979.022 40866.23 40866.99 0.11959269
#> 24 49543.471 54461.65 54489.32 0.09034749
#> 25 12633.791 13622.33 13622.33 0.07256755
#> 26 26249.214 27244.66 27244.66 0.03653727
#> 27 39869.640 40866.99 40866.99 0.02440477
#> 28 53491.123 54489.32 54489.32 0.01831911
#> 29 11218.054 13622.33 13622.33 0.17649523
#> 30 24783.173 27244.66 27244.66 0.09034749
#> 31 38389.478 40866.99 40866.99 0.06062379
#> 32 52004.651 54489.32 54489.32 0.04559916
#> 33 9085.919 13622.33 13622.33 0.33301285
#> 34 22436.107 27244.66 27244.66 0.17649523
#> 35 35979.596 40866.99 40866.99 0.11959269
#> 36 49566.345 54489.32 54489.32 0.09034749
#> ILF rating_factor base_deductible base_limit base_valuation_factor
#> 1 0.9999990 0.9999990 1000 250000 1
#> 2 0.9969172 0.9969172 1000 250000 1
#> 3 0.9704396 0.9704396 1000 250000 1
#> 4 0.9179180 0.9179180 1000 250000 1
#> 5 0.9999992 0.8879397 1000 250000 1
#> 6 0.9970664 0.9413794 1000 250000 1
#> 7 0.9711941 0.9351392 1000 250000 1
#> 8 0.9192275 0.8936956 1000 250000 1
#> 9 0.9999993 0.7191755 1000 250000 1
#> 10 0.9972919 0.8524197 1000 250000 1
#> 11 0.9723753 0.8775034 1000 250000 1
#> 12 0.9213096 0.8537420 1000 250000 1
#> 13 1.0000000 1.0000000 1000 250000 1
#> 14 1.0000000 1.0000000 1000 250000 1
#> 15 1.0000000 1.0000000 1000 250000 1
#> 16 1.0000000 1.0000000 1000 250000 1
#> 17 1.0000000 0.8879404 1000 250000 1
#> 18 1.0000000 0.9441491 1000 250000 1
#> 19 1.0000000 0.9628757 1000 250000 1
#> 20 1.0000000 0.9722246 1000 250000 1
#> 21 1.0000000 0.7191760 1000 250000 1
#> 22 1.0000000 0.8547344 1000 250000 1
#> 23 1.0000000 0.9024328 1000 250000 1
#> 24 1.0000000 0.9266613 1000 250000 1
#> 25 1.0000000 1.0000000 1000 250000 1
#> 26 1.0000000 1.0000000 1000 250000 1
#> 27 1.0000180 1.0000180 1000 250000 1
#> 28 1.0004982 1.0004982 1000 250000 1
#> 29 1.0000000 0.8879404 1000 250000 1
#> 30 1.0000000 0.9441492 1000 250000 1
#> 31 1.0000172 0.9628923 1000 250000 1
#> 32 1.0004841 0.9726953 1000 250000 1
#> 33 1.0000000 0.7191760 1000 250000 1
#> 34 1.0000000 0.8547344 1000 250000 1
#> 35 1.0000160 0.9024471 1000 250000 1
#> 36 1.0004617 0.9270892 1000 250000 1
Scaling diagnostics
A severity histogram across the entire Coverage A range is not a
comparison to one fitted distribution: each Coverage A value implies a
different fitted severity distribution. fit_plot_data() can
therefore restrict the comparison to a meaningful scale range while
still retaining the exact row-level scale, deductible, and limit values
inside that range.
plot_fit(
home_fit,
type = "density",
scale_range = c(300000, 450000)
)
A separate scaling diagnostic asks whether the assumed relationship
across Coverage A is reasonable.
plot_scaling_fit(home_fit, bins = 10, nsim = 3, seed = 12)
The actual and modeled observed means are directly comparable because
the modeled series reproduces each claim row’s deductible and censoring
structure. The fitted ground-up mean series is also shown to make the
underlying severity-scaling relationship visible.
4. Spliced body-tail fitting
A single family may fit the body of a severity distribution
reasonably while missing the upper tail.
fit_spliced_severity() supports a body family below a
threshold and a separate tail family above it.
The two densities are normalized on their respective sides of the
threshold. A body probability is estimated. Density continuity is
not required.
The threshold may be fixed by the analyst or searched over candidate
values.
n <- 1500
threshold_true <- 30000
is_body <- runif(n) < 0.90
# Draw body values from a lognormal conditional on being below the threshold.
Fb <- plnorm(threshold_true, log(9000), 0.75)
body_draw <- qlnorm(runif(n, 0, Fb), log(9000), 0.75)
# Draw Pareto-II tail values conditional on exceeding the threshold.
shape_tail <- 2.1
scale_tail <- 18000
Ft <- 1 - (1 + threshold_true / scale_tail)^(-shape_tail)
u_tail <- runif(n, Ft, 1)
tail_draw <- scale_tail * ((1 - u_tail)^(-1 / shape_tail) - 1)
splice_ground_up <- ifelse(is_body, body_draw, tail_draw)
splice_ded <- sample(c(0, 1000, 2500), n, replace = TRUE)
splice_lim <- sample(c(50000, 100000, 250000, Inf), n, replace = TRUE)
keep <- splice_ground_up > splice_ded
splice_dat <- data.frame(
loss = pmin(splice_ground_up[keep], splice_lim[keep]),
deductible = splice_ded[keep],
limit = splice_lim[keep]
)
Search for a threshold:
splice_fit <- fit_spliced_severity(
splice_dat,
loss = "loss",
deductible = "deductible",
limit = "limit",
body = "lognormal",
tail = "pareto2",
threshold_method = "search",
threshold_probs = c(0.60, 0.95),
threshold_grid = 4
)
#> Warning: Spliced-fit diagnostic: Extreme tail scale and shape: possible weak
#> identifiability
splice_fit
#> Actuarial severity fit
#> Model: spliced
#> Distribution: lognormal+pareto2
#> Threshold: 50000 (search)
#> Ground-up observations: 1476
#> Right-censored: 23
#> Log-likelihood: -15243.51
#> AIC: 30499.02
#> Converged: yes
#>
#> Coefficients:
#> body_scale body_sdlog tail_scale tail_shape splice_prob
#> 9.400326e+03 8.157510e-01 2.381546e+07 5.053511e+02 9.494472e-01
#>
#> Note: Spliced densities are normalized on each side of the selected threshold; density continuity at the threshold is not imposed.
splice_fit$threshold_search
#> threshold n_body n_exact_tail n_censored logLik AIC BIC
#> 1 12192.85 886 567 23 -15249.63 30509.27 30535.75
#> 2 16508.30 1058 395 23 -15246.43 30502.86 30529.35
#> 3 22961.79 1230 223 23 -15248.76 30507.53 30534.01
#> 4 50000.00 1414 53 9 -15243.51 30497.02 30523.50
#> converged reason tail_warning
#> 1 TRUE <NA> <NA>
#> 2 TRUE <NA> <NA>
#> 3 TRUE <NA> <NA>
#> 4 TRUE <NA> Extreme tail scale and shape: possible weak identifiability
The same factor-generation interface works for the spliced model:
severity_factors(
splice_fit,
deductible = c(0, 1000, 5000),
limit = c(25000, 50000, 100000, 250000),
base_deductible = 0,
base_limit = 50000
)
#> distribution model scale_value deductible limit valuation_factor
#> 1 lognormal+pareto2 spliced NA 0 25000 1
#> 2 lognormal+pareto2 spliced NA 1000 25000 1
#> 3 lognormal+pareto2 spliced NA 5000 25000 1
#> 4 lognormal+pareto2 spliced NA 0 50000 1
#> 5 lognormal+pareto2 spliced NA 1000 50000 1
#> 6 lognormal+pareto2 spliced NA 5000 50000 1
#> 7 lognormal+pareto2 spliced NA 0 100000 1
#> 8 lognormal+pareto2 spliced NA 1000 100000 1
#> 9 lognormal+pareto2 spliced NA 5000 100000 1
#> 10 lognormal+pareto2 spliced NA 0 250000 1
#> 11 lognormal+pareto2 spliced NA 1000 250000 1
#> 12 lognormal+pareto2 spliced NA 5000 250000 1
#> expected_payment limited_expected_severity unlimited_mean LER
#> 1 11814.759 11814.76 16243.18 0.00000000
#> 2 10953.495 11814.76 16243.18 0.06152844
#> 3 7781.570 11814.76 16243.18 0.28611659
#> 4 13851.063 13851.06 16243.18 0.00000000
#> 5 12901.667 13851.06 16243.18 0.06152844
#> 6 9443.465 13851.06 16243.18 0.28611659
#> 7 15410.717 15410.72 16243.18 0.00000000
#> 8 14428.672 15410.72 16243.18 0.06152844
#> 9 10846.576 15410.72 16243.18 0.28611659
#> 10 16207.626 16207.63 16243.18 0.00000000
#> 11 15208.946 16207.63 16243.18 0.06152844
#> 12 11563.720 16207.63 16243.18 0.28611659
#> ILF rating_factor base_deductible base_limit base_valuation_factor
#> 1 0.8529858 0.8529858 0 50000 1
#> 2 0.8489984 0.7908054 0 50000 1
#> 3 0.8240165 0.5618031 0 50000 1
#> 4 1.0000000 1.0000000 0 50000 1
#> 5 1.0000000 0.9314568 0 50000 1
#> 6 1.0000000 0.6817863 0 50000 1
#> 7 1.1126018 1.1126018 0 50000 1
#> 8 1.1183572 1.0417014 0 50000 1
#> 9 1.1485802 0.7830862 0 50000 1
#> 10 1.1701359 1.1701359 0 50000 1
#> 11 1.1788358 1.0980346 0 50000 1
#> 12 1.2245210 0.8348616 0 50000 1
Valuation or depreciation at the payment stage
The fitting layer does not decide whether historical claims are RCV,
ACV, or some other valuation basis. The analyst should first express the
experience on a consistent ground-up basis.
At factor generation, a multiplicative valuation factor can then
represent an ACV-type payment transformation. If the ground-up
replacement-cost severity is X, the package evaluates
payments of the form
min(max(valuation_factor * X - deductible, 0), limit)
For example:
severity_factors(
home_fit,
scale_value = 400000,
deductible = 2500,
limit = 25000,
valuation_factor = c(1.00, 0.80, 0.60, 0.40, 0.20),
base_deductible = 2500,
base_limit = 25000,
base_valuation_factor = 1
)
#> distribution model scale_value deductible limit valuation_factor
#> 1 weibull single 4e+05 2500 25000 1.0
#> 2 weibull single 4e+05 2500 25000 0.8
#> 3 weibull single 4e+05 2500 25000 0.6
#> 4 weibull single 4e+05 2500 25000 0.4
#> 5 weibull single 4e+05 2500 25000 0.2
#> expected_payment limited_expected_severity unlimited_mean LER ILF
#> 1 16897.766 18289.18 27244.66 0.09034749 1
#> 2 14954.852 18289.18 27244.66 0.11232866 1
#> 3 12183.568 18289.18 27244.66 0.14831883 1
#> 4 8238.316 18289.18 27244.66 0.21767889 1
#> 5 3252.768 18289.18 27244.66 0.40285079 1
#> rating_factor base_deductible base_limit base_valuation_factor
#> 1 1.0000000 2500 25000 1
#> 2 0.8850194 2500 25000 1
#> 3 0.7210165 2500 25000 1
#> 4 0.4875388 2500 25000 1
#> 5 0.1924970 2500 25000 1
Weights and fixed parameters
A likelihood weight multiplies the row log-likelihood. Integer
weights therefore behave as frequency weights.
weighted_collision <- collision[1:1000, ]
weighted_collision$w <- sample(1:3, nrow(weighted_collision), replace = TRUE)
weighted_fit <- fit_severity(
weighted_collision,
loss = "loss",
deductible = "deductible",
limit = "limit",
distribution = "gamma",
weights = "w",
fixed = list(shape = 2.2)
)
coef(weighted_fit)
#> scale shape
#> 3223.05 2.20
Hessian and bootstrap uncertainty
vcov() and confint() use a numerical
Hessian of the optimized negative log-likelihood. These are large-sample
Wald approximations; the Hessian is evaluated numerically rather than
supplied as hand-derived second derivatives for every family.
vcov(collision_fit)
#> scale shape
#> scale 19666.35853 -11.885324597
#> shape -11.88532 0.008627015
confint(collision_fit)
#> lower upper
#> scale 2957.789469 3507.507602
#> shape 2.024419 2.388508
A lightweight parametric or case bootstrap is also available.
Parametric bootstrap retains the fitted portfolio’s deductible, limit,
scaling, and weight structure.
boot <- bootstrap_fit(collision_fit, R = 5, type = "parametric", seed = 99)
head(boot$estimates)
boot$success_rate
For a searched spliced fit, refit_threshold = FALSE
reuses the selected threshold in each bootstrap replicate by default.
Set it to TRUE to repeat threshold search, understanding
that this is much more computationally expensive.
Custom ggplot2 and Plotly graphics
The plot-building data are part of the public interface:
d <- fit_plot_data(home_fit, scale_range = c(300000, 450000), nsim = 3, seed = 5)
head(d)
#> row source simulation observed_loss censored deductible limit scale_value
#> 1 1 actual 0 17733.91 FALSE 2500 450000 450000
#> 2 2 actual 0 26398.36 FALSE 10000 350000 350000
#> 3 3 actual 0 48168.44 FALSE 10000 375000 375000
#> 4 4 actual 0 19954.84 FALSE 10000 400000 400000
#> 5 5 actual 0 34167.79 FALSE 1000 400000 400000
#> 6 6 actual 0 15006.80 FALSE 2500 350000 350000
#> weight
#> 1 1
#> 2 1
#> 3 1
#> 4 1
#> 5 1
#> 6 1
This allows analysts to use their own graphics conventions without
reproducing the statistically important observation logic. See
inst/examples/08_plotly_diagnostics.R for Plotly
examples.
Optional terms, anomaly audits, and log-scale plots
The policy terms are optional. A sample with no deductible or
censoring limit can be fitted by naming only the loss column and family.
In particular, Coverage A is not a policy censoring limit merely because
it is an insured value.
set.seed(20261008)
no_terms <- data.frame(loss = rlnorm(250, log(12000), .8))
fit_no_terms <- fit_severity(no_terms, loss = "loss",
distribution = "lognormal", hessian = FALSE)
print(fit_no_terms)
#> Actuarial severity fit
#> Model: single
#> Distribution: lognormal
#> Ground-up observations: 250
#> Right-censored: 0
#> Log-likelihood: -2624.113
#> AIC: 5252.225
#> Converged: yes
#>
#> Coefficients:
#> scale sdlog
#> 1.152856e+04 7.595185e-01
#>
#> Note: The internal scale is exp(meanlog). With scale_by, mean severity is proportional to scale_coef * scale_by + scale_intercept.
plot_fit(fit_no_terms, x_scale = "log10",
xlim = c(1000, 100000))
If a supplied row is inconsistent with the observation mechanism, the
default is to exclude the row after warning; the original positional row
number is reported. This is an audit aid, not a substitute for
understanding how the original ground-up amounts were collected.
problem <- no_terms
problem$deductible <- 0
problem$deductible[3] <- problem$loss[3] + 10
clean_fit <- suppressWarnings(fit_severity(problem, loss = "loss",
deductible = "deductible", distribution = "lognormal", hessian = FALSE))
clean_fit$excluded_rows
#> [1] 3
clean_fit$excluded_details
#> row reason
#> 1 3 loss at/below deductible
For single-fit plots and multi-candidate plots, use
x_scale = "log10" by default; xlim zooms
without discarding data during statistic calculation. The histogram
density is computed in log-severity coordinates when log-scaled.
plot_candidate_fits(pd_candidates, type = "density",
x_scale = "log10", xlim = c(200, 150000))
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.