Using stable names and arithmetic

Named stable and tweedie laws, and arithmetic on stable references

A sum of independent stable laws with a common index \(\alpha\) is a stable law, and so is a scaled or shifted stable law. Using stable describes the four parameters and the two parameterizations, and states the rules for the parameters of a sum, a multiple, and a shift. The class aggregate.stable.Stable computes these combinations in Python, and DecL reaches them through names. A stable statement with a name¹ stores the law in the recipe base, and the reference stable.NAME returns it. References take arithmetic², which is used to combine the stable limits of several units. A tweedie statement takes the same name and reference but no arithmetic, because a sum of independent Tweedie laws is a Tweedie law only when they have the same gamma severity.

stable x0 1.5 0                           # a name, then any form of the statement
stable x1 1.5 0.85
stable Book as "Pareto book" 1000 claims sev 100 * lomax 1.5
tweedie t0 1.5 100 2

3 * stable.x0                             # an anonymous law, not stored
stable.x0 + 2 * stable.x1 + 3
stable z stable.x0 + 2 * stable.x1 + 3    # the same combination, stored as stable.z
3 @ stable.x0                             # the sum of three independent copies

from aggregate import build, __version__
from style import wprint, refuse, figures
figures()
print(__version__)

1.43.1

Named stable and tweedie laws

A named law is a stable or tweedie statement with a name after the keyword, followed by an optional as label and any form of the statement, with optional note, tags, and hints. Printed, it starts with the name, and each form after the colon is valid DecL.

wprint(build('stable x0 1.5 0'))
wprint(build('stable x1 1.5 0.85'))
wprint(build('stable x2 1.5 -0.5'))
wprint(build('stable Book as "Pareto book" 1000 claims sev 100 * lomax 1.5'))
wprint(build('tweedie t0 1.5 100 2'))

x0: stable 1.5 0 1 0 = stable 1.5 0 1 0 s0, mean 0, Pr(X>mean) 0.5, Pr(X>x) ~
0.199471 x^-1.5, Pr(X<-x) ~ 0.199471 x^-1.5
x1: stable 1.5 0.85 1 0 = stable 1.5 0.85 1 -0.85 s0, mean 0, Pr(X>mean)
0.350502, Pr(X>x) ~ 0.369022 x^-1.5, Pr(X<-x) ~ 0.0299207 x^-1.5
x2: stable 1.5 -0.5 1 0 = stable 1.5 -0.5 1 0.5 s0, mean 0, Pr(X>mean) 0.598389,
Pr(X>x) ~ 0.0997356 x^-1.5, Pr(X<-x) ~ 0.299207 x^-1.5
Pareto book (Book): stable 1.5 1 18452.7 200000 = stable 1.5 1 18452.7 181547
s0, mean 200000, Pr(X>mean) 0.333333, Pr(X>x) ~ 1e+06 x^-1.5
t0: tweedie 1.5 100 2 = tweedie 1.5 100 cv 0.447214 = tweedie 10 claims sev
gamma 10 cv 1, Pr(X=0) 4.53999e-05

The recipe base stores numbers whichever form was written: the four S1 parameters for stable, and \(p\), the mean, and the dispersion for tweedie. The program is stored with the entry, so the limit form of Book remains readable while its stored law is four numbers.

book = build.recipe('Book')
wprint(book.program)
print({k: float(book.spec[k]) for k in ('alpha', 'beta', 'scale', 'loc')})

stable Book as "Pareto book" 1000 claims sev 100 * lomax 1.5
{'alpha': 1.5, 'beta': 1.0, 'scale': 18452.701486440274, 'loc': 200000.0}

A later statement refers to a named law as stable.x0 or tweedie.t0, the same dotted form as agg.X and sev.X. Either reference alone returns the stored law, with its name and label. The bare name does too when it is unique across kinds.

print(repr(build('stable.x0')))
print(repr(build('x0')))
print(build('tweedie.t0').dispersion)

Stable(alpha=1.5, beta=0, scale=1, loc=0, name='x0')
Stable(alpha=1.5, beta=0, scale=1, loc=0, name='x0')
2.0

Arithmetic on stable references

Stable references take + and - between laws, +, -, *, and / with a number, unary minus, and @. An expression on its own returns an anonymous Stable and stores nothing. On the numeric side anything a DecL expression accepts works, and a +, -, or * between numbers needs parentheses, as it does everywhere in DecL.

print(repr(build('3 * stable.x0')))
print(repr(build('stable.x1 + stable.x2')))
print(repr(build('stable.x0 + 2 * stable.x1 + 3')))
print(repr(build('(1 + 2) * stable.x0 / 6 - 1')))

Stable(alpha=1.5, beta=0, scale=3, loc=0)
Stable(alpha=1.5, beta=0.175, scale=1.5874010519681994, loc=0)
Stable(alpha=1.5, beta=0.6279767062808198, scale=2.4472608147714756, loc=3)
Stable(alpha=1.5, beta=0, scale=0.5, loc=-1)

A stored combination is written stable z <expression>. A reference resolves when the statement is parsed, so z holds four numbers and redefining x0 afterward leaves z unchanged.

z = build('stable z stable.x0 + 2 * stable.x1 + 3')
wprint(z)
build('stable x0 1.5 0 9')
print(build('stable.z').scale == z.scale)
build('stable x0 1.5 0');

z: stable 1.5 0.627977 2.44726 3 = stable 1.5 0.627977 2.44726 1.46318 s0, mean
3, Pr(X>mean) 0.381008, Pr(X>x) ~ 1.24322 x^-1.5, Pr(X<-x) ~ 0.2841 x^-1.5
True

Independent copies of a stable law

The arithmetic operations act on laws, not on random variables, and each occurrence of a reference is an independent copy. As a result stable.x0 + stable.x0 is different from 2 * stable.x0. At \(\alpha=3/2\) the first has scale \(2^{2/3}=1.5874\) and the second has scale 2, because the scale of a sum of independent copies grows more slowly than the number of copies. Likewise stable.x0 - stable.x0 is a symmetric stable law, not zero.

print(repr(build('stable.x0 + stable.x0')))
print(repr(build('2 * stable.x0')))
print(repr(build('stable.x0 - stable.x0')))

Stable(alpha=1.5, beta=0, scale=1.5874010519681994, loc=0)
Stable(alpha=1.5, beta=0, scale=2, loc=0)
Stable(alpha=1.5, beta=0, scale=1.5874010519681994, loc=0)

The form t @ stable.x is the sum of \(t\) independent copies, with scale \(t^{1/\alpha}\gamma\) and location \(t\delta\). Its characteristic function is \(\varphi^t\), so \(t\) need not be an integer, and the result is the law at time \(t\) of the Lévy process with stable.x at time 1. The operator @ is the same as in 2 @ agg.X, which doubles a claim count. The last line below adds two laws at time \(1/2\), which returns stable.x1.

print(repr(build('2 @ stable.x1')))
print(repr(build('stable.x1 + stable.x1')))
print(repr(build('0.5 @ stable.x1 + 0.5 @ stable.x1')))

Stable(alpha=1.5, beta=0.85, scale=1.5874010519681994, loc=0)
Stable(alpha=1.5, beta=0.85, scale=1.5874010519681994, loc=0)
Stable(alpha=1.5, beta=0.85, scale=1, loc=0)

Limit laws of two units

Two units with Pareto severities and a common tail index combine through their stable limits. The sum has the sum of the means. Its right tail constant is the sum of the two values of \(\lambda\theta^\alpha\), which gives a closed-form check. The difference has both tails, with the larger on the side of the unit with the larger \(\lambda\theta^\alpha\).

build('stable A 1000 claims sev 100 * lomax 1.5')
build('stable B 400 claims sev 250 * lomax 1.5')
total = build('stable.A + stable.B')
wprint(total)
print(f'{total.tail_constant:,.2f} {1000 * 100 ** 1.5 + 400 * 250 ** 1.5:,.2f}')
wprint(build('stable.A - stable.B'))

stable 1.5 1 34721.7 400000 = stable 1.5 1 34721.7 365278 s0, mean 400000,
Pr(X>mean) 0.333333, Pr(X>x) ~ 2.58114e+06 x^-1.5
2,581,138.83 2,581,138.83
stable 1.5 -0.225148 34721.7 0 = stable 1.5 -0.225148 34721.7 7817.53 s0, mean
0, Pr(X>mean) 0.546994, Pr(X>x) ~ 1e+06 x^-1.5, Pr(X<-x) ~ 1.58114e+06 x^-1.5

Stable laws in the recipe library

The recipe library included with aggregate (see Using recipes) holds four stable laws under their classical names: stable.Cauchy (\(\alpha=1\), \(\beta=0\)), stable.Levy (\(\alpha=1/2\), \(\beta=1\)), stable.Landau (\(\alpha=1\), \(\beta=1\)), and stable.Holtsmark (\(\alpha=3/2\), \(\beta=0\)). They are available in any session and to a client that resolves library entries only, such as a spreadsheet, which has no access to a law declared in another cell.

for name in ('Cauchy', 'Levy', 'Landau', 'Holtsmark'):
    wprint(build(f'stable.{name}'))
print(repr(build('stable.Holtsmark + 2 * stable.Holtsmark')))

Cauchy: stable 1 0 1 0 = stable 1 0 1 0 s0, mean undefined, Pr(X>loc) 0.5,
Pr(X>x) ~ 0.31831 x^-1, Pr(X<-x) ~ 0.31831 x^-1
Levy: stable 0.5 1 1 0 = stable 0.5 1 1 1 s0, mean infinite, Pr(X>loc) 1,
Pr(X>x) ~ 0.797885 x^-0.5
Landau: stable 1 1 1 0 = stable 1 1 1 0 s0, mean infinite, Pr(X>x) ~ 0.63662
x^-1
Holtsmark: stable 1.5 0 1 0 = stable 1.5 0 1 0 s0, mean 0, Pr(X>mean) 0.5,
Pr(X>x) ~ 0.199471 x^-1.5, Pr(X<-x) ~ 0.199471 x^-1.5
Stable(alpha=1.5, beta=0, scale=2.4472608147714756, loc=0)

Refused programs

A law must be named to enter arithmetic, because stable 1.5 -0.5 would otherwise have two readings, and DecL has no variables, so 3 * stable 1.5 0 and x0 = stable 1.5 0 are parse errors. A name may contain a minus sign, so a subtraction needs a space before the minus, and the message for a missing name of that shape shows the form with spaces. The laws in a sum must have a common \(\alpha\). Scaling by zero, division by zero, and zero copies are refused. A tweedie reference takes no arithmetic, and a chfagg takes the parameters of a stable law but not a reference to one. A name held under two kinds must be written with its prefix.

build('agg x0 dfreq [1] dsev [1]')
refuse('3 * stable 1.5 0',
       'x0 = stable 1.5 0',
       'stable.x1-stable.x2',
       'stable.x0 + stable.Cauchy',
       '0 * stable.x1',
       'stable.x1 / 0',
       '0 @ stable.x1',
       '2 * tweedie.t0',
       'chfagg C stable.x1',
       'x0')

DecL parse error at line 1, column 5: Unexpected 'stable'.
DecL parse error at line 1, column 1: Unexpected 'x0'.
no recipe named 'x1-stable.x2' of kind 'stable'. A name may contain a minus
sign, so stable.x1-stable.x2 is one name. To subtract, write spaces: stable.x1 -
stable.x2
Stable: only laws with a common alpha add to a stable law, got 1.5 and 1.0
Stable: scaling by zero gives a point mass, not a stable law
Stable: division by zero does not give a stable law
Stable: the number of independent copies in t @ X must be positive and finite,
got 0.0
DecL parse error at line 1, column 5: Unexpected 'tweedie.t0'.
DecL parse error at line 1, column 10: Unexpected 'stable.x1'. Did you mean:
stable?
'x0' is ambiguous across kinds (agg, stable); write agg.x0 or stable.x0, or pass
kind= to choose. Shipped library names are unique across kinds.

Footnotes

1. From aggregate 1.42.0

2. From aggregate 1.43.0