Specify a convolution of probability distributions

Creates a lazy specification of the convolution of two independent distributions or of an iid sum of a single distribution.

If $X$ and $Y$ are independent, the pairwise form represents

$$Z = X + Y.$$

The iid form represents

$$Z_n = X_1 + \cdots + X_n,$$

where $X_1,\ldots,X_n$ are independent copies of the distribution specified by x.

The function records the convolution model but does not evaluate a density, mass function, distribution function, or numerical convolution.

Usage

distribution_convolution(x, y = NULL, n = NULL, label = NULL)

Arguments

x: A tidyact_distribution_spec object.
y: Optional second tidyact_distribution_spec object. Supply y for a pairwise convolution x + y.
n: Optional nonnegative integer scalar. Supply n for the iid convolution of n copies of x.
label: Optional nonempty character scalar identifying the convolution.

Value

For a genuine convolution, an object with classes "tidyact_convolution_spec" and "tidyact_distribution_spec".

The object follows the common distribution specification schema:

schema_version: Internal schema version.
dist: The identifier "convolution".
params: The lazy convolution definition.
support: Inferred support information.
data: Always NULL.
custom: Always NULL.
label: Optional user label.
metadata: Reserved metadata list.

When n = 1, the original distribution specification is returned instead of wrapping it in a convolution object.

Details

Exactly one of y and n must be supplied.

The pairwise form

distribution_convolution(x, y)

represents the distribution of the sum of two independent random variables.

The iid form

distribution_convolution(x, n = 5)

represents the five-fold convolution of x.

The cases n = 0 and n = 1 follow the usual convolution conventions:

n = 0 represents the degenerate distribution at zero;
n = 1 returns the original distribution specification. If label is supplied, only the label of the returned copy is changed.

The operation is deliberately lazy. It does not choose a computational algorithm, discretize a continuous distribution, truncate an infinite support, or apply FFT or Panjer recursion. Those decisions belong to later computational functions.

Support bounds are propagated algebraically. For example, if x has support bounded below by $a$ and y by $b$, the convolution is bounded below by $a+b$. Infinite upper bounds are preserved.

Support type is inferred conservatively:

count + count remains count;
iid lattice sums remain lattice;
a continuous component convolved with a discrete component is continuous;
any convolution involving a mixed distribution is classified as mixed;
otherwise discrete supports are retained as discrete.

Independence is part of the meaning of a convolution specification and is therefore not stored as a separate flag.

References

Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., and Nesbitt, C. J. (1997). Actuarial Mathematics, 2nd ed. Society of Actuaries.

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
)

y <- severity_spec(
  "lognormal",
  meanlog = 7,
  sdlog = 0.6
)

# Pairwise convolution: X + Y
distribution_convolution(x, y)
#> <tidyact_convolution_spec>
#> Mode: pairwise
#> Components: gamma + lognormal
#> Support: continuous [0, Inf]

# Four iid claim amounts: X1 + ... + X4
distribution_convolution(
  x,
  n = 4
)
#> <tidyact_convolution_spec>
#> Mode: iid
#> Component: gamma
#> Copies: 4
#> Support: continuous [0, Inf]

# The zero-fold convolution is degenerate at zero.
distribution_convolution(
  x,
  n = 0
)
#> <tidyact_convolution_spec>
#> Mode: iid
#> Component: gamma
#> Copies: 0
#> Support: discrete [0, 0]

# The one-fold convolution is the original specification.
identical(
  distribution_convolution(x, n = 1),
  x
)
#> [1] TRUE

# Count distributions can also be convolved.
n_claims <- frequency_spec(
  "poisson",
  lambda = 3
)

distribution_convolution(
  n_claims,
  n = 2
)
#> <tidyact_convolution_spec>
#> Mode: iid
#> Component: poisson
#> Copies: 2
#> Support: count [0, Inf]